Physical Sciences and Mathematics Commons^™

Need For Shift-Invariant Fractional Differentiation Explains The Appearance Of Complex Numbers In Physics, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Complex numbers are ubiquitous in physics, they lead to a natural description of different physical processes and to efficient algorithms for solving the corresponding problems. But why this seemingly counterintuitive mathematical construction is so natural here? In this paper, we provide a possible explanation of this phenomenon: namely, we show that complex numbers appear if take into account that some physical system are described by derivatives of fractional order and that a physically meaningful analysis of such derivatives naturally leads to complex numbers.

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.

Why Ancient Egyptians Preferred Some Sum-Of-Inverses Representations Of Fractions?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Ancient Egyptians represented a fraction as a sum of inverses of natural numbers, with the smallest possible number of terms. In our previous paper, we explained that this representation makes sense since it leads to the optimal way of solving a problem frequently mentioned in the Egyptian papyri: dividing bread between workers. However, this does not explain why ancient Egyptians preferred some representations with the same number of terms but not others. For example, to represent 2/3, they used the sum 1/2 + 1/6 but not the sum 1/3 + 1/3 with the same number of terms. In this paper, …

What If You Are Late On Several (Relatively Small) Tasks?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In practice, we sometimes end up in a situation when we are late on several relatively small tasks. We cannot finish them all, so which ones should we do first? We show that in general, this is an NP-complete problem. In the typical situation when all the tasks are of approximately the same importance and requires approximately the same time to finish, we can have an explicit solution to this problem. In a nutshell, the resulting (somewhat counterintuitive) recommendation is to start with things which are not yet late or only a few days late. Actually, this recommendation makes sense: …

Possibility To Algorithmically Check: Yet Another Reason Why Current Definitions Have Been Selected In Elementary Mathematics, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

At first glance, many definitions in mathematics -- especially in elementary mathematics -- seem arbitrary. Why is 1 not considered a prime number? Why is a square considered to be a particular case of a parallelogram -- in some old textbooks, a parallelogram was defined in such a way as to exclude the square. In his 2018 article, Art Duval explained many such definitions by a natural requirement to make the corresponding results (theorems) as simple as possible. However, elementary mathematics is not just about theorems and proofs, it is also about computations. In this paper, we show that from …

Why Quantiles Are A Good Description Of Volatility In Economics: A Pedagogical Explanation, Sean R. Aguilar, Vladik Kreinovich, Uyen Pham

Departmental Technical Reports (CS)

To make investment decisions, we need to know, for each financial instrument, not only its expected return -- but also how the actual return may deviate from its expected value. A numerical measure of such deviations is known as volatility. Originally, volatility was measured by the srabdard deviation from the expected price, but it turned out that this measure does not always adequately describe our perception of volatility. Empirically, it turned out that quantiles are a more adequate description of volatility. In this paper, we provide an explanation of this empirical phenomenon.

Should Fighting Corruption Always Be One Of The Main Pre-Requisites For Economic Help?, Sean R. Aguilar, Vladik Kreinovich

Departmental Technical Reports (CS)

In general, corruption is bad. In many cases, it makes sense to make fighting corruption one of the main pre-requisites for getting financial help: we do not want this money to line the pockets of corrupted officials, we want to help the people. In this paper, we argue, however, that in some cases -- of over-regulated and/or oppressive regimes -- too much emphasis on fighting corruption may be counter-productive: instead of helping people, it may hurt them.

Being Active In Research Makes A Person A Better Teacher And Even Helps When Working For A Company, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

At first glance, it looks like being active in research is not necessarily related to a person's success in being a teacher or being a productive company employee -- moreover, it looks like research distracts from other tasks. Somewhat surprisingly, however, in practice, the best teachers and the best employees are actually the ones who are active in research. In this paper, we provide an explanation for this seemingly counter-intuitive phenomenon.

Yet Another Possible Explanation Of Egyptian Fractions: Motivated By Fairness, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Ancient Egyptians represented fractions as sums of inverses of natural numbers, and they made sure that all these natural numbers are different. The representation as a sum of inverses makes some sense: it is known to lead to an optimal solution to the problem of dividing bread between workers, a problem often described in the Egyptian papyri. However, this does not explain why the corresponding natural numbers should be all different: some representations with the same natural number repeated several times lead to the same smallest number of cuts as the representations that the ancient Egyptians actually used. In this …

Why Min, Max, Opening, And Closing Stock Prices Are Empirically Most Appropriate For Predictions, And Why Their Linear Combination Provides The Best Estimate For Beta, Somsak Chanaim, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

While we have moment-by-moment prices of each stock, we cannot use all this information to predict the future stock prices, we need to combine them into a few characteristics of the daily stock price. Empirically, it turns out that the best characteristics are the lowest daily price, the highest daily price, the opening price, and the closing price. In the paper, we provide a theoretical explanation for this empirical phenomenon. We also explain why empirically, it turns out that the best way to find the stock's beta coefficient is to consider a convex combination of the about four characteristics.

A Natural Formalization Of Changing-One's-Mind Leads To Square Root Of "Not" And To Complex-Valued Fuzzy Logic, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

We show that a natural formalization of the process of changing one's mind leads to such seemingly non-intuitive ideas as square root of "not" and complex-valued fuzzy degrees.

Rosenzweig, Equality, And Assignment, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In his seminal book "The Star of Redemption", the renowned philosopher Franz Rosenzweig illustrated his ideas by the intuitive difference between mathematical statements A=B and B=A. Of course, from the purely mathematical viewpoint, these two statements are always equivalent, so to a person trained in mathematics -- even in simple school mathematics -- this illustration is puzzling. What we show is that from the viewpoint of common folks, there is indeed a subtle difference between how people understand these two equalities. To us, the understanding of this difference helped us better understand Rosenzweig's ideas. But we believe that this difference …

Why 3d Fragmentation Usually Leads To Cuboids: A Simple Geometric Explanation, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It has been empirically observed that the average shape of natural fragmentation results -- such as natural rock fragments -- is a distorted cube (known as cuboid). Recently, a complex explanation was provides for this empirical fact. In this paper, we propose a simple geometry-based physical explanation for the ubiquity of cuboid fragments.

Why Cutting Trajectories Into Small Pieces Helps To Learn Dynamical Systems Better: A Seemingly Counterintuitive Empirical Result Explained, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In general, the more information we use in machine learning, the more accurate predictions we get. However, recently, it was observed that for prediction of the behavior of dynamical systems, the opposite effect happens: when we replace the original trajectories with shorter pieces -- thus ignoring the information about the system's long-term behavior -- the accuracy of machine learning predictions actually increases. In this paper, we provide an explanation for this seemingly counterintuitive result.

Two Runners In The Time Of Social Distancing, Speedboats In The Gulf Of Finland: How To Best Pass?, Julio Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

If two runners follow the same running path, what is the best trajectory for the faster runner to pass the slower one, taking into account that they should always maintain a prescribed social distance? If a speedboat wants to pass a slower ship following a special canal in the Gulf of Finland, what is the best trajectory? In this paper, we provide answers to both questions.

How Mathematics And Computing Can Help Fight The Pandemic: Two Pedagogical Examples, Julio Urenda, Olga Kosheleva, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

With the 2020 pandemic came unexpected mathematical and computational problems. In this paper, we provide two examples of such problems -- examples that we present in simplified pedagogical form. The problems are related to the need for social distancing and to the need for fast testing. We hope that these examples will help students better understand the importance of mathematical models.

Why Some Powers Laws Are Possible And Some Are Not, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich, Olga Kosheleva, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

Many dependencies between quantities are described by power laws, in which y is proportional to x raised to some power a. In some application areas, in different situations, we observe all possible pairs (A,a) of the coefficient of proportionality A and of the exponent a. In other application areas, however, not all combinations (A,a) are possible: once we fix the coefficient A, it uniquely determines the exponent a. In such case, the dependence of a on A is usually described by an empirical logarithmic formula. In this paper, we show that natural scale-invariance ideas lead to a theoretical explanation for …

Absence Of Remotely Triggered Large Earthquakes: A Geometric Explanation, Laxman Bokati, Aaron A. Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that seismic waves from a large earthquake can trigger earthquakes in distant locations. Some of the triggered earthquakes are strong themselves. Interestingly, strong triggered earthquakes only happen within a reasonably small distance (less than 1000 km) from the original earthquake. Even catastrophic earthquakes do not trigger any strong earthquakes beyond this distance. In this paper, we provide a possible geometric explanation for this phenomenon.

Why There Are Only Four Fundamental Forces: A Possible Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that there are exactly four fundamental forces of nature: gravity forces, forces corresponding to weak interactions, electromagnetic forces, and forces corresponding to strong interactions. In this paper, we provide a possible explanation of why there are exactly four fundamental forces: namely, we relate this number with the dimension of physical space-time.

A Recent Result About Random Metrics Explains Why All Of Us Have Similar Learning Potential, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the same class, after the same lesson, the amount of learned material often differs drastically, by a factor of ten. Does this mean that people have that different learning abilities? Not really: experiments show that among different students, learning abilities differ by no more than a factor of two. This fact have been successfully used in designing innovative teaching techniques, techniques that help students realize their full learning potential. In this paper, we deal with a different question: how to explain the above experimental result. It turns out that this result about learning abilities -- which are, due to …

Why Geometric Progression In Selecting The Lasso Parameter: A Theoretical Explanation, William Kubin, Yi Xie, Laxman Bokati, Vladik Kreinovich, Kittawit Autchariyapanitkul

Departmental Technical Reports (CS)

In situations when we know which inputs are relevant, the least squares method is often the best way to solve linear regression problems. However, in many practical situations, we do not know beforehand which inputs are relevant and which are not. In such situations, a 1-parameter modification of the least squares method known as LASSO leads to more adequate results. To use LASSO, we need to determine the value of the LASSO parameter that best fits the given data. In practice, this parameter is determined by trying all the values from some discrete set. It has been empirically shown that …

Several Years Of Practice May Not Be As Good As Comprehensive Training: Zipf's Law Explains Why, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Many professions practice certifications as a way to establish that a person practicing this profession has reached a certain skills level. At first glance, it may sound like several years of practice should help a person pass the corresponding certification test, but in reality, even after several years of practice, most people are not able to pass the test, while after a few weeks of intensive training, most people pass it successfully. This sounds counterintuitive, since the overall number of problems that a person solves during several years of practice is much larger than the number of problems solved during …

Can We Preserve Physically Meaningful "Macro" Analyticity Without Requiring Physically Meaningless "Micro" Analyticity?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Physicists working on quantum field theory actively used "macro" analyticity -- e.g., that an integral of an analytical function over a large closed loop is 0 -- but they agree that "micro" analyticity -- the possibility to expand into Taylor series -- is not physically meaningful on the micro level. Many physicists prefer physical theories with physically meaningful mathematical foundations. So, a natural question is: can we preserve physically meaningful "macro" analyticity without requiring physically meaningless "micro" analyticity? In the 1970s, an attempt to do it was made by using constructive mathematics, in which only objects generated by algorithms are …

Why Ellipsoids In Mechanical Analysis Of Wood Structures, F. Niklas Schietzold, Julio Urenda, Vladik Kreinovich, Wolfgang Graf, Michael Kaliske

Departmental Technical Reports (CS)

Wood is a very mechanically anisotropic material. At each point on the wooden beam, both average values and fluctuations of the local mechanical properties corresponding to a certain direction depend, e.g., on whether this direction is longitudinal, radial or tangential with respect to the grain orientation of the original tree. This anisotropy can be described in geometric terms, if we select a point x and form iso-correlation surfaces -- i.e., surfaces formed by points y with the same level of correlation ρ(x,y) between local changes in the vicinities of the points x and y. Empirical analysis shows that for each …

Classification Of The Subalgebras Of The Algebra Of All 2 By 2 Matrices, Justin Luis Bernal

Open Access Theses & Dissertations

Classification of the subalgebras of the familiar algebra of all $n\times n$ real matrices over the real numbers can get quite unwieldy as all subalgebras are of dimension ranging from $1$ to $n^2$. Classification of the subalgebras of the algebra of all $2\times 2$ real matrices over the real numbers is an interesting first start.

Since $\2$ is of dimension $4$ then its possible subalgebras are of dimension $1, 2, 3,$ or $4$. The one-dimensional subalgebra and four-dimensional subalgebra need little to no attention. The two-dimensional and three-dimensional subalgebras however turn out to be of significance.

It turns out there …

Mathematical Modeling Of Microemulsification Processes, Numerical Simulations And Applications To Drug Delivery, Ogochukwu Nneka Ifeacho

Open Access Theses & Dissertations

Microemulsion systems are a great pharmaceutical tool for the delivery of formulations containing multiple hydrophilic and hydrophobic ingredients of varying physicochemical properties. These systems are gaining popularity because of its long shelf life, improved drug solubilisation capacity, easy preparation and improvement of bioavailability. Despite the advantages associated with the use of microemulsion systems in pharmaceutical industries, the major challenge impeding their use has been and continues to be the lack of understanding of these systems.

Microemulsions can be mathematically modeled by an initial boundary value problem involving a sixth order nonlinear time dependent equation. In this Thesis, we present a …

Free Semigroups And Identites For A Class Of Monoids, Enrique Salcido

Open Access Theses & Dissertations

The study of words as a mathematical object is a deep and rich field of study. Algebra, Combinatorics, Theoretical Computer Science etc., are major disciplines, which are fully using this study. Combinatorial properties (via Codes, Free Hulls, Infinite Words), and algebraic properties of words are presented in this Thesis. The free semigroup on a set (alphabet) X and finite presentation of semigroups have a central place in the algebraic study of words. The last part of the Thesis is devoted to the study of identities in the alphabet X = {x,y} for a class of monoids. The characterization of such …

Machine Learning Analysis To Characterize Phase Variations In Laser Propagation Through Deep Turbulence, Luis Fernando Rodriguez Sanchez

Open Access Theses & Dissertations

The present Dissertation is focused on the analysis of the atmospheric conditions of a turbulent environmental system and its effects on the diffraction of a laser beam that moves through it. The study is based on the optical communication of two labs placed at the summit of two mountains located in Maui, Hawaii. The emitter system is located at the Mauna Loa mountain and the receiver at the Haleakala. The distance between both mountains is 150 km. The emitter system is at a height of 3.1 km and the receiver at 3.4 km. The maritime environment at the location experiences …

Positivity-Preserving Segregate-Flux Method For Infiltration Dynamics In Tumor Growth Models, Gilbert Danso Acheampong

Open Access Theses & Dissertations

We study the positivity preserving property and an incompressibility condition in a recently proposed tumor growth model as well as its numerical simulations. In this model, the biological process is described by a free-boundary problem of hyperbolic equations that govern the in-tumor motion of cancer cells and the infiltration of immune cells. Particularly, due to an assumption that cells take constant volume (the incompressibility condition), the tumor growth/shrinkage is closely correlated to the magnitude of infiltration of immune cells into the tumor.

Despite the fact that previous simulation results largely reproduced experimental data, there remain unanswered questions that are crucial …

Using Machine Learning On An Imbalanced Cancer Dataset, James Ekow Arthur

Open Access Theses & Dissertations

With an estimated 1.4 million cancer diagnosis worldwide and the increasing death of cancer patients. It is prudent to investigate methods, approaches and smarter ways of predicting and diagnosing of cancer so that a holistic techniques can be used to curb or reduce false predictions , increase exact predictions and also meticulos prognosis information .

Can a feasible technique be developed for the general problem of prognosis and diagnosis of cancer be developed ?

We will show here that this problem of cancer prognosis and diagnosis can be efficiently tackled with the aid of machine learning techniques and the best, …