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Articles 1 - 30 of 553
Full-Text Articles in Mathematics
7 Plus Minus 2 Law Revisited: Alternative Geometric Explanation, Mayan Arithmetic, And Using 9- And 18-Based Numbers In Jewish Tradition, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich
7 Plus Minus 2 Law Revisited: Alternative Geometric Explanation, Mayan Arithmetic, And Using 9- And 18-Based Numbers In Jewish Tradition, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
A recent paper showed that to make sure that the movements in the crowd are not chaotic, the directions of all the motions should deviate from some fixed direction by no more than 13 degrees. We show that this results provides a new geometric explanation for the seven plus minus two law in psychology, according to which we can keep in mind no more than 7 plus minus 2 items. We also show that all this is related to the somewhat mysterious appearance of 9- and 18-based number systems in Jewish and Mayan traditions.
Why Um And U*Log(U) Are The Most Effective Nonlinear Functions In Fuzzy Clustering: Theoretical Explanation Of The Empirical Fact, Olga Kosheleva, Vladik Kreinovich, Yuchi Kanzawa
Why Um And U*Log(U) Are The Most Effective Nonlinear Functions In Fuzzy Clustering: Theoretical Explanation Of The Empirical Fact, Olga Kosheleva, Vladik Kreinovich, Yuchi Kanzawa
Departmental Technical Reports (CS)
In fuzzy clustering, we need to have non-linear functions of the membership degrees. Different nonlinear functions have been tried. Empirical evidence shows that for fuzzy clustering, the most effective nonlinear functions are um and u*log(u). In this paper, we provide a theoretical explanation for this empirical fact.
Egyptian Triangle And Geometry Of Airplane Wings: A Simplified Explanation, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich
Egyptian Triangle And Geometry Of Airplane Wings: A Simplified Explanation, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In historically first planes, wings were orthogonal to the fuselage. However, later it turned out that from the aerodynamic viewpoint, it is most efficient to place the wings at about 37 degrees from this orthogonal direction -- and this is where wings are placed in most modern planes. There exist theoretical explanations for this optimality -- explanations based on solving the equations of aerodynamics. In such situations when only a complex not-very-intuitive explanation exists, it is desirable to come up with a simpler more intuitive explanation. For the wing angles, such an explanation is provided in this paper. Namely, we …
"At Least K Out Of N" Under Fuzzy Uncertainty: Efficient Algorithm For General "And"-Operations, Olga Kosheleva, Vladik Kreinovich, Klaus-Peter Adlassnig
"At Least K Out Of N" Under Fuzzy Uncertainty: Efficient Algorithm For General "And"-Operations, Olga Kosheleva, Vladik Kreinovich, Klaus-Peter Adlassnig
Departmental Technical Reports (CS)
In medicine, many diagnoses are made when, for some value k, at least k of n possible symptoms are present. Many of such symptoms -- such as fever -- are, in reality, fuzzy. For example, it makes no sense that say that 38.0 is fever while 37.9 is not a fever, both are fever to some degree. Once such degrees are given, we need to use them to estimate the degree to which the patient has the corresponding disease. For this problem, the usual fuzzy techniques require exponentially many computational steps -- so it is desirable to have a more …
Why Interval-Valued (And Type-2) Fuzzy Methods Are Often More Effective, Christian Servin, Olga Kosheleva, Vladik Kreinovich
Why Interval-Valued (And Type-2) Fuzzy Methods Are Often More Effective, Christian Servin, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Interval-valued and type-2 fuzzy techniques were designed to provide a more adequate representation of expert knowledge than the traditional (type-1) fuzzy techniques. Somewhat unexpectedly, they also often turn out to be more effective even when there is no expert knowledge at all -- when we are simply using fuzzy rules to fit experimental data. In precise terms, for the same number of parameters, interval-valued and type-2 systems often provide a better fit for the data and/or better quality control than traditional (type-1) fuzzy techniques. In this paper, we provide a theoretical explanation for this surprising phenomenon.
Why Convex Combinations Of Interval Endpoints: Related Explanations For Cases Of Data Processing And Decision Making, Christian Servin, Olga Kosheleva, Vladik Kreinovich
Why Convex Combinations Of Interval Endpoints: Related Explanations For Cases Of Data Processing And Decision Making, Christian Servin, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
There are two cases in which it has been empirically shown that a convex combination of the interval's endpoints works better than any other combination: processing interval data and dealing with situations in which we know both approximate probability and possibility and we need to make a decision. In this paper, we provide an explanation of both phenomena.
From Machine Learning To Human Learning: What Can Pedagogy Learn From Ai Successes, Victor L. Timchenko, Yury P. Kondratenko, Olga Kosheleva, Vladik Kreinovich
From Machine Learning To Human Learning: What Can Pedagogy Learn From Ai Successes, Victor L. Timchenko, Yury P. Kondratenko, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Many machine learning techniques -- including many techniques behind the current AI-based boom in machine learning -- come from the analysis of successful human learning strategies (and researchers expect that other human learning experiences can lead to even more effective AI-based systems). At this moment, so much experience have been accumulated in AI-based machine learning that it is time to start the analysis in the opposite direction -- to see what can human-based pedagogy learn from AI successes. In this chapter, we provide the first results of such an analysis -- some of which go somewhat against the current pedagogical …
Gurevich's Quizani Dialogs As An Example Of Explainable Mathematics, And How This Is Related To Quantum Space-Time Ideas That Can Speed Up Computations, Olga Kosheleva, Vladik Kreinovich
Gurevich's Quizani Dialogs As An Example Of Explainable Mathematics, And How This Is Related To Quantum Space-Time Ideas That Can Speed Up Computations, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Everyone talks about the need for Explainable AI -- when, to supplement a long difficult-to-understand sequence of computational steps leading to AI's decision, we are looking for a shorter and understandable more-informal explanation for this decision. In this paper, we argue that this need is a particular case of what we call Explainable Mathematics -- when we want to supplement a long sequence of arguments and/or computations with a shorter and understandable more-informal explanation. Important instances of Explainable Mathematics are Yuri Gurevich's Quizani dialogs that help explain complex results from theoretical computer science and physicists' more-informal explanations of complex physical …
Unfortunately, The Universal Predictor Cannot Be Made Constructive, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich
Unfortunately, The Universal Predictor Cannot Be Made Constructive, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
A recent article in the Notices of the American Mathematical Society reminded the mathematics community that, under the Axiom of Choice, it is possible to have a universal predictor: if we input, into this predictor, the values of a function for all moments t < to for some to, then, for almost all to, this predictor correctly predicts the next values of this function on some interval [to, to + ε). This predictor cannot be used for actual predictions: it is based on the Axiom of Choice and is, therefore, not constructive. A natural question is: maybe it is possible to have another universal predictor, which is constructive? In this paper we show that, unfortunately, it is not possible to have a constructive universal predictor. In other words, the above universal predictor result cannot be used for actual predictions.
Best Strategies For Bilingual Education: How Can We Explain Their Success?, Claudia Cabrera, Olga Kosheleva, Christian Servin, Vladik Kreinovich
Best Strategies For Bilingual Education: How Can We Explain Their Success?, Claudia Cabrera, Olga Kosheleva, Christian Servin, Vladik Kreinovich
Departmental Technical Reports (CS)
When designing AI-based tools for education, it is important to take into account the experience of human teachers. In this, it is necessary to distinguish between the education features that are justified by the general features of the corresponding education task -- these features should be taken into account in AI-based learning as well -- and features which are specific for traditional non-AI teaching. In this paper, on the important example of bilingual education, we show that several empirically successful teaching strategies can be explained in the general context -- and thus, should be implemented in AI-based teaching as well.
Ultrathin-Layer Strain-Based Electronic Devices: From-First-Principles Derivation Of The Corresponding Equation, Julio C. Urenda, Vladik Kreinovich
Ultrathin-Layer Strain-Based Electronic Devices: From-First-Principles Derivation Of The Corresponding Equation, Julio C. Urenda, Vladik Kreinovich
Departmental Technical Reports (CS)
Most information about the world comes from sensors -- and from the results of processing sensor data. In many practical situations -- e.g., in biomedical applications -- it is desirable to make sure that the sensors are as "invisible" as possible, in particular, that they are as small as possible. One way to achieve such small size is to use ultrathin-layer materials such as graphene. It is known that for such materials, strain causes electromagnetic effects -- which can be used to detect small strains. Interestingly, it turned out that the same equation describes the relation between strain and electric …
A Natural Extension Of F-Transform To Triangular And Triangulated Domains Necessitates The Use Of Triangular Membership Functions, Hana Zámečiková, Irina Perfilieva, Olga Kosheleva, Vladik Kreinovich
A Natural Extension Of F-Transform To Triangular And Triangulated Domains Necessitates The Use Of Triangular Membership Functions, Hana Zámečiková, Irina Perfilieva, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations when we process 1-D data, the method of F-transform turned out to be very useful. In this method, we can use either triangular membership functions or more complex ones. Because this method has been so successful in 1-D applications, a natural idea is to extend it to functions defined on 2-D and higher-dimensional domains -- e.g., to images. This method allows natural generalization to rectangular domains, where it indeed turned out to be very effective. A recent paper showed that it can extended to more general domains -- e.g., to triangular domains and to more general …
All We (And Llms) Need Is Fuzzy: An Argument, Olga Kosheleva, Vladik Kreinovich
All We (And Llms) Need Is Fuzzy: An Argument, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Large Language Models (LLMs) like ChatGPT have spectacular successes -- but they also have surprising failures that an average person with common sense could easily avoid. It is therefore desirable to incorporate the imprecise ("fuzzy") common sense into LLMs. A natural question is: to what extent will this help? This way, we may avoid a few simple mistakes, but will it significantly improve the LLMs' performance? What portion of the gap between current LLMs and ideal perfect AI-based agents can be, in principle, covered by using fuzzy techniques? Judging by the fact that few researchers working on LLMs (and on …
How To Share A Success, How To Share A Crisis, And How All This Is Related To Fuzzy, Olga Kosheleva, Vladik Kreinovich
How To Share A Success, How To Share A Crisis, And How All This Is Related To Fuzzy, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, a group of people needs to share a success. What is the fair way to share this success? Nobelist John Nash showed that under reasonable conditions, the group should select the alternative for which the product of utility gains is the largest possible. This solution makes perfect sense from the fuzzy-formalized commonsense viewpoint: it maximizes the degree of confidence that all participants are happy. A natural question is: can we extend this result to a different class of situations, when a group of people needs to share sacrifices caused by a crisis? In this paper, we …
How To Deal With High-Impact Low-Probability Events: Theoretical Explanation Of The Empirically Successful Fuzzy-Like Technique, Juan Ulloa, Aaron Velasco, Olga Kosheleva, Vladik Kreinovich
How To Deal With High-Impact Low-Probability Events: Theoretical Explanation Of The Empirically Successful Fuzzy-Like Technique, Juan Ulloa, Aaron Velasco, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
When making decisions, it is important to take into account high-impact low-probability events. For such events, traditional probability-based approach -- which considers the product of the probability p that this event happens and the probability P that a randomly selected building will be destroyed -- often underestimates risks. Available data has lead to an empirical table that provides a more adequate risk estimate. Most of the entries in this table correspond to the fuzzy-like formula min(p,P). This paper explains this empirical result. Specifically, it explains both the effectiveness of the min formula -- and also explains deviations from this formula.
Why Linear Faults Have Fewer Earthquakes: A Geometric Explanation, Julio C. Urenda, Aaron Velasco, Olga Kosheleva, Vladik Kreinovich
Why Linear Faults Have Fewer Earthquakes: A Geometric Explanation, Julio C. Urenda, Aaron Velasco, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Earthquakes usually occur in the vicinity of fault lines. Until recently, geophysical analysis implied that the fault shape should not strongly affect the frequency of its earthquakes. However, recent statistical analysis has shown that faults whose shape is close to linear experience much fewer earthquakes than faults of more complex shape. Based on this empirical fact, researchers have adjusted the corresponding geophysical models, so the updated models do explain this newly discovered phenomenon. The experience of geophysics shows that the updated model will probably need to be updated again when new data appears. It is therefore desirable to come up …
Fair Economic Division: How To Modify Shapley Value To Take Into Account That Different People Have Different Productivity, Christian Servin, Vladik Kreinovich
Fair Economic Division: How To Modify Shapley Value To Take Into Account That Different People Have Different Productivity, Christian Servin, Vladik Kreinovich
Departmental Technical Reports (CS)
Purpose: When several participants, working together, gained some amount of money, what is the fair way to distribute this amount between them? This is the problem that the future Nobelist Lloyd Shapley was working on when he proposed what is now called the Shapley value -- a division uniquely determined by natural fairness assumptions. However, this solutions is not universal: it assumes that all participants are equal -- in particular, that they have equal productivity. In practice, people have different productivity levels, and these productivity levels can differ a lot: e.g., some software engineers are several times more productive than …
How Shapley Value And Its Generalizations Can Help In The Analysis Of Complex Engineering Systems And What Next, Niklas Winnewisser, Michael Beer, Olga Kosheleva, Vladik Kreinovich
How Shapley Value And Its Generalizations Can Help In The Analysis Of Complex Engineering Systems And What Next, Niklas Winnewisser, Michael Beer, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
For a complex engineering system -- such as a city's street network -- it is important to predict how its functionality is decreased when some of these components break down, and, if repairs are needed and repairs budget is limited, which subset of the set of components should be repaired first to maximize the resulting functionality. For systems with a large number of components, the number of possible subsets is astronomical, we cannot try to simulate all these subsets. So, the natural idea is to approximate the actual dependence of functionality on the subset by a simple expression -- linear …
What Is Optimal Granularity When Estimating Reliability Of A Complex Engineering Systems, Niklas Winnewisser, Michael Beer, Olga Kosheleva, Vladik Kreinovich
What Is Optimal Granularity When Estimating Reliability Of A Complex Engineering Systems, Niklas Winnewisser, Michael Beer, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
For complex engineering systems, the usual way to estimate their reliability is to run simulations. If the resulting estimate does not satisfy the desired reliability level, we must replace some components with more reliable and again run simulations. This can take several iterations, so the required computation time often becomes unrealistically long. It is known that it is possible to speed up computations if components belong to a few types, and components of each type are identical. So, a natural idea to deal with the general case is to use the general granularity idea, i.e., to group components with similar …
Is Energy Local? Counterintuitive Non-Locality Of Energy In General Relativity Can Be Naturally Explained On The Newtonian Level, Olga Kosheleva, Vladik Kreinovich
Is Energy Local? Counterintuitive Non-Locality Of Energy In General Relativity Can Be Naturally Explained On The Newtonian Level, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
From the physics viewpoint, energy is the ability to perform work. To estimate how much work we can perform, physicists developed several formalisms. For example, for the fields, once we know the Lagrangian, we can find the energy density and, by integrating it, estimate the overall energy of the field. Usually, this adequately describe how much work this field can perform. However, there is an exception -- gravitational field in General Relativity. The known formalism to compute its energy density leads to 0 -- and by integrating this 0, we get a counterintuitive conclusion that the overall energy of the …
Logarithmic Number System Is Optimal For Ai Computations: Theoretical Explanation Of Empirical Success, Olga Kosheleva, Vladik Kreinovich, Christoph Lauter, Kristalys Ruiz-Rohena
Logarithmic Number System Is Optimal For Ai Computations: Theoretical Explanation Of Empirical Success, Olga Kosheleva, Vladik Kreinovich, Christoph Lauter, Kristalys Ruiz-Rohena
Departmental Technical Reports (CS)
Everyone knows the success story of machine-learning AI. However, the current AI tools are not perfect. We know how to make them better: every time we increase the amount of computations by the order of magnitude, we get a drastic improvement in the performance of the resulting machine learning tools. Training modern AI system requires a tremendous amount of computations -- that already take a lot of time. So, to increase the number of computations, we need to make each computation step faster. One way to do that is to use low-precision arithmetic operations, e.g., with 1 byte per real …
Why Geological Angular Unconformity Is Usually Horizontal: A Geometric Explanation, Julio C. Urenda, Aaron Velasco, Olga Kosheleva, Vladik Kreinovich
Why Geological Angular Unconformity Is Usually Horizontal: A Geometric Explanation, Julio C. Urenda, Aaron Velasco, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In several locations, geologists have observed the presence of two differently oriented rock masses, one horizonal (or almost horizontal) and the other somewhat inclined; this phenomenon is known as angular unconformity. Based on the detailed analysis of geophysical processes, geologists conclude that usually, horizontal rock masses are much newer. This is known as the law of original horizontality. From the fundamental viewpoint, it is desirable to take into account that geophysics is a developing science, its models get modified and adjusted as time progresses. It is therefore desirable to come up with an explanation of this phenomenon that would be …
Two Is Enough, But Three (Or More) Is Better: In Ai And Beyond, Olga Kosheleva, Vladik Kreinovich, Victor L. Timchenko, Yury P. Kondratenko
Two Is Enough, But Three (Or More) Is Better: In Ai And Beyond, Olga Kosheleva, Vladik Kreinovich, Victor L. Timchenko, Yury P. Kondratenko
Departmental Technical Reports (CS)
At present, the most successful AI technique is deep learning -- the use of neural networks that consist of multiple layers. Interestingly, it is well known that neural networks with two data processing layers are sufficient -- in the sense that they can approximate any function with any given accuracy. Because of this, until reasonably recently, researchers and practitioners used such networks. However, recently it turned out, somewhat unexpectedly, that using three or more data processing layers -- i.e., using what is called deep learning -- makes the neural networks much more efficient. In this paper, on numerous examples from …
A Full Description Of All Commutative Associative Polynomial Operations On Probabilities, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
A Full Description Of All Commutative Associative Polynomial Operations On Probabilities, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
Departmental Technical Reports (CS)
When two events are independent, the probability that both events occur is equal to the product p1 * p2 of the probabilities of each of these events. The probability that at least one of these events will occur is equal to p1 + p2 − p1 * p2. In both cases, we have a commutative associative polynomial operation. A natural question is: how can we describe all possible operations of this type? These operations are described in this paper.
Why Kolmolgorov-Arnold Networks (Kan) Work So Well: A Qualitative Explanation, Hung T. Nguyen, Vladik Kreinovich, Olga Kosheleva
Why Kolmolgorov-Arnold Networks (Kan) Work So Well: A Qualitative Explanation, Hung T. Nguyen, Vladik Kreinovich, Olga Kosheleva
Departmental Technical Reports (CS)
In the usual deep neural network, weights are adjusted during training, but the activation function remains the same. Lately, it was experimentally shown that if, instead of using the same activation function always, we train the activation functions as well, we get a much better results -- i.e., for the networks with the same number of parameters, we get a much better accuracy. Such networks are called Kolmogorov-Arnold networks. In this paper, we provide a general explanation of why these new networks work so well.
How To Check Continuity Based On Approximate Measurement Results, Inese Bula, Vladik Kreinovich
How To Check Continuity Based On Approximate Measurement Results, Inese Bula, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, a reasonable conjecture is that, e.g., the dependence of some quantity on the spatial location is continuous, with an appropriate bounds on the difference between the values at nearby points. If we knew the exact values of the corresponding quantity, checking this conjecture would be very straightforward. In reality, however, measurement results are only approximations to the actual values. In this paper, we show how to check continuity based on the approximate measurement results.
Three Applications Of Geometric Reasoning: Why Metastasis Is Mostly Caused By Elongated Cancer Cells? How Body Shape Affects Curiosity? Why Ring Fractures In Ice?, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich
Three Applications Of Geometric Reasoning: Why Metastasis Is Mostly Caused By Elongated Cancer Cells? How Body Shape Affects Curiosity? Why Ring Fractures In Ice?, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In this paper, we describe three applications of geometric reasoning to important practical problems ranging from micro- to macro-level. Specifically, we use geometric reasoning to explain why metastasis is mostly caused by elongated cancer cell, why curiosity in fish is strongly correlated with body shape, and why ring-shaped fractures appear in Antarctica.
To Which Interdisciplinary Research Collaborations Should We Pay More Attention?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
To Which Interdisciplinary Research Collaborations Should We Pay More Attention?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
Departmental Technical Reports (CS)
Interdisciplinary research is very important in modern science. However, such a research is not easy, it often needs support and help. Resources that can be used for such a support are limited, so we need to decide which of many possible collaborations we should support. In this paper, we provide a natural simple model of collaboration effectiveness. Based on this model, we conclude that we should support collaborations for which the vector product of the participants' knowledge vectors attains the largest values.
Why Decisions Based On The Results Of Worst-Case, Most Realistic, And Best-Case Scenarios Work Well?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich, Chon Van Le
Why Decisions Based On The Results Of Worst-Case, Most Realistic, And Best-Case Scenarios Work Well?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich, Chon Van Le
Departmental Technical Reports (CS)
Often, to make an appropriate decision, people try three scenarios: the worst case, the most realistic case, and the best case. This three-scenarios approach often leads to reasonable decisions. A natural question is: why worst case and best case? These extreme cases mean that all numerous independent random factors work in the same direction: either are all stacked for or are all stacked against. Such stacking of random factors is highly improbable. So, at first glance, it would be more beneficial to use more realistic scenarios than the worst case and the best case. However, empirically, decisions based on the …
Why Green Wavelength Is Closer To Blue Than To Red And How It Is Related To Computations: Information-Based Explanation, Victor L. Timchenko, Yury P. Kondratenko, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
Why Green Wavelength Is Closer To Blue Than To Red And How It Is Related To Computations: Information-Based Explanation, Victor L. Timchenko, Yury P. Kondratenko, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
Departmental Technical Reports (CS)
In our previous papers, we analyzed the idea of using light signals of three basic color -- red, green, and blue -- to speed up computations, in particular fuzzy-related computations. A natural question is: why red, green, and blue? Why not select some other colors: e.g., from the wavelength viewpoint, green is much closer to blue than to green, so why not select colors whose distribution is more even? In this paper, we show that if we consider this problem from the information viewpoint, then the corresponding equal-information criterion indeed implies that the intermediate wavelength should be closer to the …