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Articles 361 - 390 of 553
Full-Text Articles in Mathematics
Why Ancient Egyptians Preferred Some Sum-Of-Inverses Representations Of Fractions?, Olga Kosheleva, Vladik Kreinovich
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, …
Yet Another Possible Explanation Of Egyptian Fractions: Motivated By Fairness, Olga Kosheleva, Vladik Kreinovich
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 …
Being Active In Research Makes A Person A Better Teacher And Even Helps When Working For A Company, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich
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.
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
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.
Should Fighting Corruption Always Be One Of The Main Pre-Requisites For Economic Help?, Sean R. Aguilar, Vladik Kreinovich
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.
Possibility To Algorithmically Check: Yet Another Reason Why Current Definitions Have Been Selected In Elementary Mathematics, Christian Servin, Olga Kosheleva, Vladik Kreinovich
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 …
What If You Are Late On Several (Relatively Small) Tasks?, Olga Kosheleva, Vladik Kreinovich
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: …
Rosenzweig, Equality, And Assignment, Olga Kosheleva, Vladik Kreinovich
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 Cutting Trajectories Into Small Pieces Helps To Learn Dynamical Systems Better: A Seemingly Counterintuitive Empirical Result Explained, Olga Kosheleva, Vladik Kreinovich
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.
Why 3d Fragmentation Usually Leads To Cuboids: A Simple Geometric Explanation, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich
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.
Two Runners In The Time Of Social Distancing, Speedboats In The Gulf Of Finland: How To Best Pass?, Julio Urenda, Olga Kosheleva, Vladik Kreinovich
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
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
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
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
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
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
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 …
Why Ellipsoids In Mechanical Analysis Of Wood Structures, F. Niklas Schietzold, Julio Urenda, Vladik Kreinovich, Wolfgang Graf, Michael Kaliske
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 …
Can We Preserve Physically Meaningful "Macro" Analyticity Without Requiring Physically Meaningless "Micro" Analyticity?, Olga Kosheleva, Vladik Kreinovich
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 …
Several Years Of Practice May Not Be As Good As Comprehensive Training: Zipf's Law Explains Why, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich
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 …
Why A Classification Based On Linear Approximation To Dynamical Systems Often Works Well In Nonlinear Cases, Julio Urenda, Vladik Kreinovich
Why A Classification Based On Linear Approximation To Dynamical Systems Often Works Well In Nonlinear Cases, Julio Urenda, Vladik Kreinovich
Departmental Technical Reports (CS)
It can be proven that linear dynamical systems exhibit either stable behavior, or unstable behavior, or oscillatory behavior, or transitional behavior. Interesting, the same classification often applies to nonlinear dynamical systems as well. In this paper, we provide a possible explanation for this phenomenon, i.e., we explain why a classification based on linear approximation to dynamical systems often works well in nonlinear cases.
Egyptian Fractions Re-Revisited, Olga Kosheleva, Vladik Kreinovich, Francisco Zapata
Egyptian Fractions Re-Revisited, Olga Kosheleva, Vladik Kreinovich, Francisco Zapata
Departmental Technical Reports (CS)
Ancient Egyptians represented each fraction as a sum of unit fractions, i.e., fractions of the type 1/n. In our previous papers, we explained that this representation makes perfect sense: e.g., it leads to an efficient way of dividing loaves of bread between people. However, one thing remained unclear: why, when representing fractions of the type 2/(2k+1), Egyptians did not use a natural representation 1/(2k+1) + 1/(2k+1), but used a much more complicated representation instead. In this paper, we show that the need for such a complicated representation can be explained if we take into account that instead of cutting a …
Geometric Aspects Of Wound Healing, Julio Urenda, Vladik Kreinovich
Geometric Aspects Of Wound Healing, Julio Urenda, Vladik Kreinovich
Departmental Technical Reports (CS)
In this paper, we show that many aspects of complex biological processes related to wound healing can be explained in terms of the corresponding geometric symmetries.
Decision Theory Explains "Telescoping Effect" -- That Our Time Perception Is Biased, Laxman Bokati, Vladik Kreinovich
Decision Theory Explains "Telescoping Effect" -- That Our Time Perception Is Biased, Laxman Bokati, Vladik Kreinovich
Departmental Technical Reports (CS)
People usually underestimate time passed since distant events, and overestimate time passed since recent events. There are several explanations for this "telescoping effect", but most current explanations utilize specific features of human memory and/or human perception. We show that the telescoping effect can be explained on a much basic level of decision theory, without the need to invoke any specific ways we perceive and process time.
How To Generate "Nice" Cubic Polynomials -- With Rational Coefficients, Rational Zeros And Rational Extrema: A Fast Algorithm, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich
How To Generate "Nice" Cubic Polynomials -- With Rational Coefficients, Rational Zeros And Rational Extrema: A Fast Algorithm, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Students feel more comfortable with rational numbers than with irrational ones. Thus, when teaching the beginning of calculus, it is desirable to have examples of simple problems for which both zeros and extrema point are rational. Recently, an algorithm was proposed for generating cubic polynomials with this property. However, from the computational viewpoint, the existing algorithm is not the most efficient one: in addition to applying explicit formulas, it also uses trial-and-error exhaustive search. In this paper, we propose a computationally efficient algorithm for generating all such polynomials: namely, an algorithm that uses only explicit formulas.
When Revolutions Happen: Algebraic Explanation, Julio Urenda, Vladik Kreinovich
When Revolutions Happen: Algebraic Explanation, Julio Urenda, Vladik Kreinovich
Departmental Technical Reports (CS)
At first glance, it may seem that revolutions happen when life becomes really intolerable. However, historical analysis shows a different story: that revolutions happen not when life becomes intolerable, but when a reasonably prosperous level of living suddenly worsens. This empirical observation seems to contradict traditional decision theory ideas, according to which, in general, people's happiness monotonically depends on their level of living. A more detailed model of human behavior, however, takes into account not only the current level of living, but also future expectations. In this paper, we show that if we properly take these future expectations into account, …
Use Of Symmetries In Economics: An Overview, Vladik Kreinovich, Olga Kosheleva, Nguyen N. Thach, Nguyen Duc Trung
Use Of Symmetries In Economics: An Overview, Vladik Kreinovich, Olga Kosheleva, Nguyen N. Thach, Nguyen Duc Trung
Departmental Technical Reports (CS)
In this paper, we show that many semi-heuristic econometric formulas can be derived from the natural symmetry requirements. The list of such formulas includes many famous formulas provided by Nobel-prize winners, such as Hurwicz optimism-pessimism criterion for decision making under uncertainty, McFadden's formula for probabilistic decision making, Nash's formula for bargaining solution -- as well as Cobb-Douglas formula for production, gravity model for trade, etc.
Why Bilingualism Helps Autistic Children Function: A Symmetry-Based Explanation, Olga Kosheleva, Vladik Kreinovich
Why Bilingualism Helps Autistic Children Function: A Symmetry-Based Explanation, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
One of the main problems of autistic children is that it is very difficult for them to switch to a different state, to a different activity -- and such switches are often needed. Researchers have recently shown that bilingualism helps autistic children function: namely, it is somewhat easier for bilingual children to switch to a new activity. In this paper, we provide a possible explanation for this empirical phenomenon. Namely, we show that, in general terms, autism means difficulty with breaking symmetries of a state, and we describe how this general reformulation indeed explains the above recently discovered phenomenon.
Why Early Galaxies Were Pickle-Shaped: A Geometric Explanation, Olga Kosheleva, Vladik Kreinovich
Why Early Galaxies Were Pickle-Shaped: A Geometric Explanation, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
The vast majority of currently observed geometric shapes of celestial bodies can be explained by a simple symmetry idea: the initial distribution of matter is invariant with respect to shifts, rotations, and scaling, but this distribution is unstable, so we have spontaneous symmetry breaking. According to statistical physics, among all possible transitions, the most probable are the ones that retain the largest number of symmetries. This explains the currently observed shapes and -- on the qualitative level -- their relative frequency. According to this idea, the most probable first transition is into a planar (pancake) shape, then into …
Bhutan Landscape Anomaly: Possible Effect On Himalayan Economy (In View Of Optimal Description Of Elevation Profiles), Thach N. Nguyen, Laxman Bokati, Aaron A. Velasco, Vladik Kreinovich
Bhutan Landscape Anomaly: Possible Effect On Himalayan Economy (In View Of Optimal Description Of Elevation Profiles), Thach N. Nguyen, Laxman Bokati, Aaron A. Velasco, Vladik Kreinovich
Departmental Technical Reports (CS)
Economies of countries located in seismic zones are strongly effected by this seismicity. If we underestimate the seismic activity, then a reasonably routine earthquake can severely damage the existing structures and thus, lead to huge economic losses. On the other hand, if we overestimate the seismic activity, we waste a lot of resources on unnecessarily fortifying all the buildings -- and this too harms the economies. From this viewpoint, it is desirable to have estimations of regional seismic activities which are as accurate as possible. Current predictions are mostly based on the standard geophysical understanding of earthquakes as being largely …