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Articles 661 - 690 of 2316

Full-Text Articles in Physical Sciences and Mathematics

Egyptian Fractions As Approximators, Olga Kosheleva, Vladik Kreinovich Oct 2020

Egyptian Fractions As Approximators, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In ancient Egypt, fractions were represented as the sum of inverses to natural numbers. Processing fractions in this representation is computationally complicated. Because of this complexity, traditionally, Egyptian fractions used to be considered an early inefficient approach. In our previous papers, we showed, however, that the Egyptian fractions actually provide an optimal solution to problems important for ancient Egypt -- such as the more efficient distribution of food between workers. In these papers, we assumed, for simplicity, that we know the exact amount of food needed for each worker -- and that this value must be maintained with absolute accuracy. …

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Why Number Of Color Difference Works Better In Detecting Melanoma Than Number Of Colors: A Possible Fractal-Based Explanation, Julio Urenda, Olga Kosheleva, Vladik Kreinovich Oct 2020

Why Number Of Color Difference Works Better In Detecting Melanoma Than Number Of Colors: A Possible Fractal-Based Explanation, Julio Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

At present, the best way to detect melanoma based on an image of a skin spot is to count the number of different colors in this image. A recent paper has shown that the detection can improve if instead of the number of colors, we use the difference between numbers of colors computed by using different thresholds. In this paper, we provide a possible fractal-based explanation for this empirical fact.

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How To Separate Absolute And Relative Error Components: Interval Case, Christian Servin, Olga Kosheleva, Vladik Kreinovich Oct 2020

How To Separate Absolute And Relative Error Components: Interval Case, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Usually, measurement errors contain both absolute and relative components. To correctly gauge the amount of measurement error for all possible values of the measured quantity, it is important to separate these two error components. For probabilistic uncertainty, this separation can be obtained by using traditional probabilistic techniques. The problem is that in many practical situations, we do not know the probability distribution, we only know the upper bound on the measurement error. In such situations of interval uncertainty, separation of absolute and relative error components is not easy. In this paper, we propose a technique for such a separation based …

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Coding Overhead Of Mobile Apps, Yoonsik Cheon Oct 2020

Coding Overhead Of Mobile Apps, Yoonsik Cheon

Departmental Technical Reports (CS)

A mobile app runs on small devices such as smartphones and tablets. Perhaps, because of this, there is a common misconception that writing a mobile app is simpler than a desktop application. In this paper, we show that this is indeed a misconception, and it's the other way around. We perform a small experiment to measure the source code sizes of a desktop application and an equivalent mobile app written in the same language. We found that the mobile version is 19% bigger than the desktop version in terms of the source lines of code, and the mobile code is …

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Does Transition To Democracy Lead To Chaos: A Theorem, Olga Kosheleva, Vladik Kreinovich Sep 2020

Does Transition To Democracy Lead To Chaos: A Theorem, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When a country transitions to democracy, at first, many political parties appear. A natural question is whether the number of such parties feasible and reasonable -- or whether this is a complete chaos. In this paper, we formulate a simplified version of this question in precise terms and show that the number of parties will be feasible -- i.e., that transition to democracy does not lead to chaos.

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Rosenzweig, Equality, And Assignment, Olga Kosheleva, Vladik Kreinovich Sep 2020

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 …

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Need For Diversity In Elected Decision-Making Bodies: Economics-Related Analysis, Nguyen Ngoc Thach, Olga Kosheleva, Vladik Kreinovich Aug 2020

Need For Diversity In Elected Decision-Making Bodies: Economics-Related Analysis, Nguyen Ngoc Thach, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

On a qualitative level, everyone understands the need to have diversity in elected decision-making bodies, so that the viewpoint of each group be properly taken into account. However, when only the usual economic criteria are used in this election -- e.g., in the election of company's board -- the resulting bodies often under-represent some groups (e.g., women). A frequent way to remedy this situation is to artificially enforce diversity instead of strictly following purely economic criteria. In this paper, we show the current seeming contradiction between economics and diversity is caused by the imperfection of the use economic models: in …

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Why 3d Fragmentation Usually Leads To Cuboids: A Simple Geometric Explanation, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich Aug 2020

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.

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Why Cutting Trajectories Into Small Pieces Helps To Learn Dynamical Systems Better: A Seemingly Counterintuitive Empirical Result Explained, Olga Kosheleva, Vladik Kreinovich Aug 2020

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.

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How The Amount Of Cracks And Potholes Grows With Time: Symmetry-Based Explanation Of Empirical Dependencies, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich Aug 2020

How The Amount Of Cracks And Potholes Grows With Time: Symmetry-Based Explanation Of Empirical Dependencies, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Empirical double-exponential formulas are known that describe how the amount of cracks and potholes in a pavement grows with time. In this paper, we show that these formulas can be explained based on natural symmetries (invariances) -- such as invariance with respect to changing the measuring unit or invariance with respect to changing a starting point for measuring time.

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Two Runners In The Time Of Social Distancing, Speedboats In The Gulf Of Finland: How To Best Pass?, Julio Urenda, Olga Kosheleva, Vladik Kreinovich Aug 2020

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.

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The Similarity Between Earth's And Mars's Core-Mantle Boundary Seems To Be Statistically Significant, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich Aug 2020

The Similarity Between Earth's And Mars's Core-Mantle Boundary Seems To Be Statistically Significant, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Latest, most accurate measurements of the depth of the Mars's core-mantle boundary shows that the ratio between this depth and Mars's radius is the same as for the Earth -- and with new measurements, this coincidence has become statistically significance. This coincidence seems to confirm a simple scale-invariant model in which for planets of Earth-Mars type, this depth is proportional to the planet's radius. Of course, we need more observations to confirm this model, but the fact that, for the first time, we got a statistically significant confirmation, is encouraging: it makes us believe that this coincidence is not accidental.

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Under Limited Resources, Lottery-Based Tutoring Is The Most Efficient, Olga Kosheleva, Christian Servin, Vladik Kreinovich Aug 2020

Under Limited Resources, Lottery-Based Tutoring Is The Most Efficient, Olga Kosheleva, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

In the ideal world, every student who needs tutoring should receive intensive one-on-one tutoring. In practical, schools' resources are limited, so the students get only a portion of needed tutoring. It would have been not so bad if, e.g., half-time tutoring would be half as efficient as the intensive one. However, research shows that half-time tutoring is, on average, 15 times less efficient -- and, e.g., for math tutoring 20 times less efficient. To increase the efficiency, we propose to randomly divide the students who need tutoring into equal-size groups, and each year (or each semester) provide intensive tutoring to …

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Gifted And Talented: With Others? Separately? Mathematical Analysis Of The Problem, Olga Kosheleva, Vladik Kreinovich Jul 2020

Gifted And Talented: With Others? Separately? Mathematical Analysis Of The Problem, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Crudely speaking, there are two main suggestions about teaching gifted and talented student: we can move them to a separate class section, or we can mix them with other students. Both options have pluses and minuses. In this paper, we formulate this problem in precise terms, we solve the corresponding mathematical optimization problem, and we come up with a somewhat unexpected optimal solution: mixing, but with an unusual twist.

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Why Quadratic Log-Log Dependence Is Ubiquitous And What Next, Sean R. Aguilar, Vladik Kreinovich, Uyen Pham Jul 2020

Why Quadratic Log-Log Dependence Is Ubiquitous And What Next, Sean R. Aguilar, Vladik Kreinovich, Uyen Pham

Departmental Technical Reports (CS)

In many real-life situations ranging from financial to volcanic data, growth is described either by a power law -- which is linear in log-log scale, or by a quadratic dependence in the log-log scale. In this paper, we use natural scale invariance requirement to explain the ubiquity of such dependencies. We also explain what should be a reasonable choice of the next model, if quadratic turns out to be not too accurate: it turns out that under scale invariance, the next class of models are cubic dependencies in the log-log scale, then fourth order dependencies, etc.

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The Less We Love A Woman, The More She Likes Us: Pushkin's Observation Explained, Olga Kosheleva, Vladik Kreinovich Jul 2020

The Less We Love A Woman, The More She Likes Us: Pushkin's Observation Explained, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Alexander Pushkin, the most famous Russian poet, made this observation in "Eugene Onegin", his novel in verse which is most known to non-Russian readers via Tchaikovsky's opera. This observation may not be an absolute truth -- there are counterexamples -- but the fact that it is still widely cited shows that there is some truth in this statement. In this paper, we recall the usual utility-based explanation for a similar statement, and propose a new explanation, which is even more fundamental -- it is on the biological level.

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Why Majority Rule Does Not Work In Quantum Computing: A Pedagogical Explanation, Oscar Galindo, Olga Kosheleva, Vladik Kreinovich Jul 2020

Why Majority Rule Does Not Work In Quantum Computing: A Pedagogical Explanation, Oscar Galindo, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To increase the reliability of computations result, a natural idea is to use duplication: we let several computers independently perform the same computations, and then, if their results differ, we select the majority's result. Reliability is an important issue for quantum computing as well, since in quantum physics, all the processes are probabilistic, so there is always a probability that the result will be wrong. It thus seems natural to use the same majority rule for quantum computing as well. However, it is known that for general quantum computing, this scheme does not work. In this paper, we provide a …

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Adversarial Teaching Approach To Cybersecurity: A Mathematical Model Explains Why It Works Well, Christian Servin, Olga Kosheleva, Vladik Kreinovich Jul 2020

Adversarial Teaching Approach To Cybersecurity: A Mathematical Model Explains Why It Works Well, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Teaching cybersecurity means teaching all possible ways how software can be attacked -- and how to fight such attacks. From the usual pedagogical viewpoint, a natural idea seems to be to teach all these ways one by one. Surprisingly, a completely different approach works even better: when the class is divided into sparring mini-teams that try their best to attack each other and defend from each other. In spite of the lack of thoroughness, this approach generates good specialists -- but why? In this paper, by analyzing a simple mathematical model of this situation, we explain why this approach work …

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Two Pens In A Pocket Must Be Different: A Nerd-Oriented Lesson From Statistics, Olga Kosheleva, Vladik Kreinovich Jul 2020

Two Pens In A Pocket Must Be Different: A Nerd-Oriented Lesson From Statistics, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Some people always carry a pen with them, so that if an idea comes to mind, they will always be able to write it down. Pens sometimes run out of ink. So, just in case, people carry two pens. The problem is that often, when one carries two identical pens, they seem to run out of ink at about the same time -- which defeats the whole purpose of carrying two pens. In this paper, we provide a simple statistics-based explanation of this phenomenon, and show that a seemingly natural idea of carrying three pens will not help. The only …

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How To Make Sure That Robot's Behavior Is Human-Like, Vladik Kreinovich, Olga Kosheleva, Laxman Bokati Jul 2020

How To Make Sure That Robot's Behavior Is Human-Like, Vladik Kreinovich, Olga Kosheleva, Laxman Bokati

Departmental Technical Reports (CS)

In many applications -- e.g., in health care -- it is desirable to make robots behave human-like. This means, in particular, that robotic control should not be optimal, it should be similar to human (suboptimal) behavior. People's decisions are based on bounded rationality: since we cannot compute an optimal solution for all possible situations, we divide situations into groups and come up with a solution appropriate for each group. What is optimal here is the division into groups. It is therefore desirable to implement a similar algorithm for robots. To help with such algorithms, we provide techniques that help optimally …

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How To Decide Which Cracks Should Be Repaired First: Theoretical Explanation Of Empirical Formulas, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich Jul 2020

How To Decide Which Cracks Should Be Repaired First: Theoretical Explanation Of Empirical Formulas, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Due to stress, cracks appear in constructions: cracks appear in buildings, bridges, pavements, among other structures, cracks appear in pavements, etc. In the long run, cracks need to be repaired. However, our resources are limited, so we need to decide which cracks are more dangerous. For this, we need to be able to predict how different cracks will grow. There are several empirical formulas describing crack growth. In this paper, we show that by using scale invariance, we can provide a theoretical explanation for these empirical formulas.

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Euclidean Distance Between Intervals Is The Only Representation-Invariant One, Olga Kosheleva, Vladik Kreinovich Jul 2020

Euclidean Distance Between Intervals Is The Only Representation-Invariant One, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

An interval can be represented as a point in a plane, e.g., as a point with its endpoints as coordinates. We can thus define distance between intervals as the Euclidean distance between the corresponding points. Alternatively, we can describe an interval by its center and radius, which leads to a different definition of distance. Interestingly, these two definitions lead, in effect, to the same distance -- to be more precise, these two distances differ by a multiplicative constant. In principle, we can have more general distances on the plane. In this paper, we show that only for Euclidean distance, the …

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It Is Important To Take All Available Information Into Account When Making A Decision: Case Of The Two Envelopes Problem, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich Jul 2020

It Is Important To Take All Available Information Into Account When Making A Decision: Case Of The Two Envelopes Problem, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In situations when we know the probabilities of all possible consequences, traditional decision theory recommends selecting the action that maximizes expected utility. In many practical situations, however, we only have partial information about the corresponding probabilities. In this case, for different possible probability distributions, we get different values of expected utility. In general, possible values of expected utility form an interval. One way to approach this situation is to use the optimism-pessimism approach proposed by Nobelist Leo Hurwicz. Another approach is to select one of the possible probability distributions -- e.g., the one that has the largest possible entropy. Both …

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Grading Homeworks, Verifying Code: How Thorough Should The Feedback Be?, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich Jul 2020

Grading Homeworks, Verifying Code: How Thorough Should The Feedback Be?, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the ideal world, we should assign many homeworks and give a thorough feedback for each homework. However, in reality, the instructor's time is limited, so we can either assign few homeworks and give a detailed feed back for all of them, or we can assign many homeworks, but give a less thorough feedback. What is the optimal thoroughness? A similar question can be raised for code verification: what is the optimal amount of feedback that should be provided to each programmer? In this paper, we provide answers to these questions.

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Covid-19 Peak Immunity Values Seem To Follow Lognormal Distribution, Julio Urenda, Olga Kosheleva, Vladik Kreinovich, Tonghui Wang Jul 2020

Covid-19 Peak Immunity Values Seem To Follow Lognormal Distribution, Julio Urenda, Olga Kosheleva, Vladik Kreinovich, Tonghui Wang

Departmental Technical Reports (CS)

For the current pandemic, an important open problem is immunity: do people who had this disease become immune against further infections? In the immunity study, it is important to know how frequent are different levels of immunity, i.e., what is the probability distribution of the immunity levels. Different people have different rates of immunity dynamics: for some, immunity gets to the level faster, for others the immunity effect is slower. Similarly, in some patients, immunity stays longer, it others, it decreases faster. In view of this, an important characteristic is peak immunity. A recent study provides some statistics on the …

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Let Us Use Negative Examples In Regression-Type Problems Too, Jonatan Contreras, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich, Martine Ceberio Jul 2020

Let Us Use Negative Examples In Regression-Type Problems Too, Jonatan Contreras, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich, Martine Ceberio

Departmental Technical Reports (CS)

In many practical situations, we need to reconstruct the dependence between quantities x and y based on several situations in which we know both x and y values. Such problems are known as regression problems. Usually, this reconstruction is based on positive examples, when we know y -- at least, with some accuracy. However, in addition, we often also know some examples in which we have negative information about y -- e.g., we know that y does not belong to a certain interval. In this paper, we show how such negative examples can be used to make the solution …

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How Mathematics And Computing Can Help Fight The Pandemic: Two Pedagogical Examples, Julio Urenda, Olga Kosheleva, Martine Ceberio, Vladik Kreinovich Jun 2020

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.

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How To Detect Future Einsteins: Towards Systems Approach, Olga Kosheleva, Vladik Kreinovich Jun 2020

How To Detect Future Einsteins: Towards Systems Approach, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Talents are rare. It is therefore important to detect and nurture future talents as early as possible. In many disciplines, this is already being done -- via gifted and talented programs, Olympiads, and other ways to select kids with unusually high achievements. However, the current approach is not perfect: some of the kids are selected simply because they are early bloomers, they do not grow into unusually successful researchers; on the other hand, many of those who later become very successful are not selected since they are late bloomers. To avoid these problems, we propose to use systems approach: to …

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Are There Traces Of Megacomputing In Our Universe, Olga Kosheleva, Vladik Kreinovich Jun 2020

Are There Traces Of Megacomputing In Our Universe, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

The recent successes of quantum computing encouraged many researchers to search for other unconventional physical phenomena that could potentially speed up computations. Several promising schemes have been proposed that will -- hopefully -- lead to faster computations in the future. Some of these schemes -- similarly to quantum computing -- involve using events from the micro-world, others involve using large-scale phenomena. If some civilization used micro-world for computations, this will be difficult for us to notice, but if they use mega-scale effects, maybe we can notice these phenomena? In this paper, we analyze what possible traces such megacomputing can leave …

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Natural Invariance Explains Empirical Success Of Specific Membership Functions, Hedge Operations, And Negation Operations, Julio Urenda, Orsoly Csiszár, Gábor Csiszár, József Dombi, György Eigner, Vladik Kreinovich Jun 2020

Natural Invariance Explains Empirical Success Of Specific Membership Functions, Hedge Operations, And Negation Operations, Julio Urenda, Orsoly Csiszár, Gábor Csiszár, József Dombi, György Eigner, Vladik Kreinovich

Departmental Technical Reports (CS)

Empirical studies have shown that in many practical problems, out of all symmetric membership functions, special distending functions work best, and out of all hedge operations and negation operations, fractional linear ones work the best. In this paper, we show that these empirical successes can be explained by natural invariance requirements.

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