Open Access. Powered by Scholars. Published by Universities.®
- Discipline
-
- Computer Sciences (21)
- Engineering (5)
- Computer Engineering (3)
- Applied Mathematics (1)
- Education (1)
-
- Educational Administration and Supervision (1)
- Electrical and Computer Engineering (1)
- Electrical and Electronics (1)
- Engineering Education (1)
- Higher Education Administration (1)
- Higher Education and Teaching (1)
- Physics (1)
- Science and Mathematics Education (1)
- Statistics and Probability (1)
- Teacher Education and Professional Development (1)
- Keyword
-
- Interval uncertainty (3)
- APOD framework (1)
- APOS theory (1)
- Algorithm simplification (1)
- Bistatic Radar Synchronization (1)
-
- Calculus sequence (1)
- Chaos based frequency modulated (CBFM) signals (1)
- Chaotic Oscillators (1)
- Correlogram and entropy (1)
- Data fitting (1)
- Data processing (1)
- Decision making (1)
- Degree of confidence (1)
- Duality (1)
- Duality conception (1)
- Field extensions (1)
- Fields (1)
- Fuzzy uncertainty (1)
- Gaming the system (1)
- Generalized projective synchronization (1)
- Geophysical time series (1)
- Groups (1)
- High Resolution (1)
- High frequency data (1)
- Indirect measurements (1)
- Infinity concept (1)
- Interval Constraint Solving Techniques (1)
- Knowledge Acquisition (1)
- Knowledge Dispersion (1)
- Knowledge Enhancement (1)
- Publication Type
Articles 1 - 30 of 58
Full-Text Articles in Mathematics
Solving Equations (And Systems Of Equations) Under Uncertainty: How Different Practical Problems Lead To Different Mathematical And Computational Formulations, Vladik Kreinovich
Solving Equations (And Systems Of Equations) Under Uncertainty: How Different Practical Problems Lead To Different Mathematical And Computational Formulations, Vladik Kreinovich
Departmental Technical Reports (CS)
Many practical problems are naturally reduced to solving systems of equations. There are many efficient techniques for solving well-defined systems of equations, i.e., systems in which we know the exact values of all the parameters and coefficients. In practice, we usually know these parameters and coefficients with some uncertainty -- uncertainty usually described by an appropriate granule: interval, fuzzy set, rough set, etc. Many techniques have been developed for solving systems of equations under such granular uncertainty. Sometimes, however, practitioners use previously successful techniques and get inadequate results. In this -- mostly pedagogical -- paper, we explain that to obtain …
Interval Methods For Data Fitting Under Uncertainty: A Probabilistic Treatment, Vladik Kreinovich, Sergey P. Shary
Interval Methods For Data Fitting Under Uncertainty: A Probabilistic Treatment, Vladik Kreinovich, Sergey P. Shary
Departmental Technical Reports (CS)
How to estimate parameters from observations subject to errors and uncertainty? Very often, the measurement errors are random quantities that can be adequately described by the probability theory. When we know that the measurement errors are normally distributed with zero mean, then the (asymptotically optimal) Maximum Likelihood Method leads to the popular least squares estimates. In many situations, however, we do not know the shape of the error distribution, we only know that the measurement errors are located on a certain interval. Then the maximum entropy approach leads to a uniform distribution on this interval, and the Maximum Likelihood Method …
Why It Is Healthy To Regularly Challenge Authority: An Algorithmic Explanation, Olga Kosheleva, Vladik Kreinovich
Why It Is Healthy To Regularly Challenge Authority: An Algorithmic Explanation, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
One way to make group decisions is to select the best decision maker(s) in the group as the authority, and to follow his or her decisions. At first glance, it seems that if the selected authority is indeed the best decision maker, it is beneficial for everyone to obey his or her authority. However, history shows that in many cases, challenges to the authority (even to the authority of the best decision maker) were beneficial to the group. In this paper, we provide an algorithmic explanation for this phenomenon. The main idea behind this explanation is that most practical general …
From Tertullian's Credo Quia Absurdum To Bohr's Crazy Theories: A Rational Explanation Of A Seemingly Irrational Idea, Olga Kosheleva, Vladik Kreinovich
From Tertullian's Credo Quia Absurdum To Bohr's Crazy Theories: A Rational Explanation Of A Seemingly Irrational Idea, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
At first glance, Tertullian's idea -- that absurdity of a statement makes it more believable -- sounds irrational, maybe appropriate for theology but definitely not for science. However, somewhat surprisingly, a similar idea was successfully used by the Nobelist Niels Bohr in theoretical physics -- an epitome of rationality in science. In this paper, we show that this Tertullian-Bohr idea actually has a simple rational explanation. Specifically, if previous attempts to construct a theory which is consistent with what is perceived as common sense were unsuccessful, this implies that a true theory much contradict common sense -- and thus, the …
Towards Selecting The Best Abstraction For A Patrolling Game, Anjon Basak, Chris Kiekintveld, Vladik Kreinovich
Towards Selecting The Best Abstraction For A Patrolling Game, Anjon Basak, Chris Kiekintveld, Vladik Kreinovich
Departmental Technical Reports (CS)
When the number of possible strategies is large, it is not computationally feasible to compute the optimal strategy for the original game. Instead, we select our strategy based on an approximate approximate description of the original game. The quality of the resulting strategy depends on which approximation we select. In this paper, on an example of a simple game, we show how to find the optimal approximation, the approximation whose use results in the best strategy.
How To Predict Nesting Sites?, Stephen Escarzaga, Craig Tweedie, Vladik Kreinovich
How To Predict Nesting Sites?, Stephen Escarzaga, Craig Tweedie, Vladik Kreinovich
Departmental Technical Reports (CS)
How to predict nesting sites? Usually, all we know is the past nesting sites, and the fact that the birds select a site which is optimal for them (in some reasonable sense), but we do not know the exact objective function describing this optimality. In this paper, we propose a way to make predictions in such a situation.
Simplest Innovations Are, Empirically, The Most Promising: An Explanation, Olga Kosheleva, Vladik Kreinovich
Simplest Innovations Are, Empirically, The Most Promising: An Explanation, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Many examples show that the simplest innovation are the most promising. In this paper, we provide a theoretical explanation for this empirical observation.
From Equations To Tri-Quations And Multi-Quations, Mourat Tchoshanov, Olga Kosheleva, Vladik Kreinovich
From Equations To Tri-Quations And Multi-Quations, Mourat Tchoshanov, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In general, an equation A(x1, ..., xn)= B(x1, ..., xn) corresponds to the situation when we have two quantities A(x1, ..., xn) and B(x1, ..., xn) which are known to be equal, we know how each of these quantities depends on the unknown parameters x1, ..., xn, and we want to find the values of the unknowns xi from this equality -- and from other similar equalities. In some practical situations, instead of two equal values, we have three …
Maximum Entropy Approach Is Not As Arbitrary As It May Seem At First Glance, Olga Kosheleva, Vladik Kreinovich
Maximum Entropy Approach Is Not As Arbitrary As It May Seem At First Glance, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
When we only have partial information about the probability distribution, i.e., when several different probability distributions are consistent with our knowledge, then it makes sense to select a distribution with the largest entropy. In particular, when we only know that the quantity is located within a certain interval -- and we have no information about the probability of different values within this intervals -- then it is reasonable to assume that all these values are equally probable, i.e., that we have a uniform distribution on this interval. The problem with this idea is that if we apply it to the …
Why The Graph Isomorphism Problem Is Easier Than Propositional Satisfiability: A Possible Qualitative Explanation, Vladik Kreinovich, Olga Kosheleva
Why The Graph Isomorphism Problem Is Easier Than Propositional Satisfiability: A Possible Qualitative Explanation, Vladik Kreinovich, Olga Kosheleva
Departmental Technical Reports (CS)
A recent result has shown that the graph isomorphism problem can be solved in quasi-polynomial time, while the general belief is that only exponential time algorithms are possible for propositional satisfiability. This is somewhat counter-intuitive, since for propositional satisfiability, we need to look for one of 2n options, while in graph isomorphism, we need to look for one of n! options, and n! is much larger than 2n. Our qualitative explanation for this counter-intuitive fact comes from the fact that, in general, a graph isomorphism problem has a unique solution -- in contrast to propositional satisfiability which, …
Decision Making Under Interval (And More General) Uncertainty: Monetary Vs. Utility Approaches, Vladik Kreinovich
Decision Making Under Interval (And More General) Uncertainty: Monetary Vs. Utility Approaches, Vladik Kreinovich
Departmental Technical Reports (CS)
In many situations, e.g., in financial and economic decision making, the decision results either in a money gain (or loss) and/or in the gain of goods that can be exchanged for money or for other goods. In such situations, interval uncertainty means that we do not know the exact amount of money that we will get for each possible decision, we only know lower and upper bounds on this amount. In this case, a natural idea is to assign a fair price to different alternatives, and then to use these fair prices to select the best alternative. In the talk, …
Waning Influence Of History: Why?, Olga Kosheleva, Vladik Kreinovich
Waning Influence Of History: Why?, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In the past, history played an important role in education: students learned history of science, history of mathematics, etc. In the last decades, the influence of history has waned. In this paper, we provide a natural explanation for this waning.
Science Is Helpful For Engineering Applications: A Theoretical Explanation Of An Empirical Observation, Olga Kosheleva, Vladik Kreinovich
Science Is Helpful For Engineering Applications: A Theoretical Explanation Of An Empirical Observation, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Empirical evidence shows that when engineering design uses scientific analysis, we usually get a much better performance that for the system designed by using a trial-and-error engineering approach. In this paper, we provide a quantitative explanation for this empirical observation.
Why The Range Of A Robust Statistic Under Interval Uncertainty Is Often Easier To Compute, Olga Kosheleva, Vladik Kreinovich
Why The Range Of A Robust Statistic Under Interval Uncertainty Is Often Easier To Compute, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In statistical analysis, we usually use the observed sample values x1, ..., xn to compute the values of several statistics v(x1, ..., xn) -- such as sample mean, sample variance, etc. The usual formulas for these statistics implicitly assume that we know the exact values x1, ..., xn. In practice, the sample values X1, ..., Xn come from measurements and are, thus, only approximations to the actual (unknown) values x1, ..., xn of the corresponding quantity. Often, the only information that we have …
Explaining Boris Pasternak's Observation That Complex Ideas Are Sometimes Easier To Understand, Olga Kosheleva, Vladik Kreinovich
Explaining Boris Pasternak's Observation That Complex Ideas Are Sometimes Easier To Understand, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Probably the most cited lines from the poetry of the Nobel-prize winning Russian writer Boris Pasternak contain the observation that complex ideas are sometimes easier to understand than simpler ones. This is not just a paradoxical poetic statement: many teachers have observed the same seemingly counter-intuitive phenomenon. In this paper, we provide a possible explanation for this phenomenon, by showing that indeed, many easier-to-describe mathematical models lead to more-difficult-to-solve mathematical problems.
Positive Consequences Of Negative Attitude: Game-Theoretic Analysis, Mahdokhat Afravi, Vladik Kreinovich
Positive Consequences Of Negative Attitude: Game-Theoretic Analysis, Mahdokhat Afravi, Vladik Kreinovich
Departmental Technical Reports (CS)
At first glance, the world would be a better place if all the people had positive attitude towards each other. It is known that this is not always the case: sometimes, the excess of positive attitude can lead to negative consequences. In this paper, we show that, vice versa, a reasonable amount of negative attitude can make life better for everyone. What is therefore needed is not the exclusive appearance of positive attitude, but rather a balance -- titled towards moderately positive attitude.
On The Importance Of Duality And Multi-Ality In Mathematics Education, Mourat Tchoshanov, Olga Kosheleva, Vladik Kreinovich
On The Importance Of Duality And Multi-Ality In Mathematics Education, Mourat Tchoshanov, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
For each mathematical object, there are usually several different equivalent representations: for example, a spatial object can be represented either in geometric terms, or by a function that describes its shape. The need for several representations comes from the fact that each of these representations is useful in solving some problems for which the use of other representations is less helpful. Thus, the more representations a student knows, the more capable this student is of solving mathematical problems. In this paper, we propose a general formal description of the corresponding notion of duality (and, more generally, "multi-ality"), and we explain …
How To Divide A Territory: An Argument In Favor Of Private Property, Mahdokhat Afravi, Vladik Kreinovich
How To Divide A Territory: An Argument In Favor Of Private Property, Mahdokhat Afravi, Vladik Kreinovich
Departmental Technical Reports (CS)
No abstract provided.
A Possible Utility-Based Explanation Of Deaton's Paradox (And Habits Of Mind), Hung T. Nguyen, Songsak Sriboonchitta, Olga Kosheleva, Vladik Kreinovich
A Possible Utility-Based Explanation Of Deaton's Paradox (And Habits Of Mind), Hung T. Nguyen, Songsak Sriboonchitta, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
No abstract provided.
Oscillating Exam Averages And Their Control-Theory Explanation, Olga Kosheleva, Vladik Kreinovich
Oscillating Exam Averages And Their Control-Theory Explanation, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
When a student misses one of the exams, his overall grade for the class is often interpolated based on his available grades. This would have been a fair procedure if the grades for different tests were equally distributed. In practice, often, the average grades for different tests are oscillating. As a result, the usual interpolation techniques may inadvertently bias the student grade for the class. In this paper, we explain this oscillation, and analyze how to avoid the corresponding bias.
How To Take Into Account Student's Degree Of Confidence When Grading Exams, Olga Kosheleva, Joe Lorkowski, Viannette Felix, Vladik Kreinovich
How To Take Into Account Student's Degree Of Confidence When Grading Exams, Olga Kosheleva, Joe Lorkowski, Viannette Felix, Vladik Kreinovich
Departmental Technical Reports (CS)
When grading exams, it is important to take into account how confident the student is in the answer. If the answer is correct, then it is better �- and thus, deserves a better grade -- if the student is absolutely confident in this correct answer. On the other hand, if the answer is wrong, then, the more confident the student is in this wrong answer, the worse. The grading scheme should be such that provides an incentive for the students to report their true degree of confidence. In this paper, we explain how to design such a grading scheme.
How To Compute Von Neumann-Morgenstern Solutions, Martha Osegueda Escobar, Vladik Kreinovich
How To Compute Von Neumann-Morgenstern Solutions, Martha Osegueda Escobar, Vladik Kreinovich
Departmental Technical Reports (CS)
No abstract provided.
How To Make Sure That Everyone Works Towards A Common Goal: Towards Optimal Incentives, Christian Servin, Vladik Kreinovich
How To Make Sure That Everyone Works Towards A Common Goal: Towards Optimal Incentives, Christian Servin, Vladik Kreinovich
Departmental Technical Reports (CS)
No abstract provided.
How To Explain The Empirical Success Of Generalized Trigonometric Functions In Processing Discontinuous Signals, Pedro Barragan Olague, Vladik Kreinovich
How To Explain The Empirical Success Of Generalized Trigonometric Functions In Processing Discontinuous Signals, Pedro Barragan Olague, Vladik Kreinovich
Departmental Technical Reports (CS)
Trigonometric functions form the basis of Fourier analysis - one of the main signal processing tools. However, while they are very efficient in describing smooth signals, they do not work well for signals that contain discontinuities - such as signals describing phase transitions, earthquakes, etc. It turns out that empirically, one of the most efficient ways of describing and processing such signals is to use a certain generalization of trigonometric functions. In this paper, we provide a theoretical explanation of why this particular generalization is the most empirically efficient one.
How To Modify Data Processing Algorithms So That They Detect Only Dependencies Which Make Sense To Domain Experts, Geovany Ramirez, Craig Tweedie, Jason Carlsson, Vladik Kreinovich
How To Modify Data Processing Algorithms So That They Detect Only Dependencies Which Make Sense To Domain Experts, Geovany Ramirez, Craig Tweedie, Jason Carlsson, Vladik Kreinovich
Departmental Technical Reports (CS)
No abstract provided.
Paradox Of Choice: A Possible Explanation, Vladik Kreinovich, Olga Kosheleva
Paradox Of Choice: A Possible Explanation, Vladik Kreinovich, Olga Kosheleva
Departmental Technical Reports (CS)
At first glance, we would expect that the more choices we have, the happier we will be. Experiments show, however, then when the number of choices increases, customers become less happy. In this paper, we provide a possible explanation for this paradox.
Combining Interval And Probabilistic Uncertainty: What Is Computable?, Vladik Kreinovich, Andrzej Pownuk, Olga Kosheleva
Combining Interval And Probabilistic Uncertainty: What Is Computable?, Vladik Kreinovich, Andrzej Pownuk, Olga Kosheleva
Departmental Technical Reports (CS)
In many practical problems, we need to process measurement results. For example, we need such data processing to predict future values of physical quantities. In these computations, it is important to take into account that measurement results are never absolutely exact, that there is always measurement uncertainty, because of which the measurement results are, in general, somewhat different from the actual (unknown) values of the corresponding quantities. In some cases, all we know about measurement uncertainty is an upper bound; in this case, we have an interval uncertainty, meaning that all we know about the actual value is that is …
Occam's Razor Explains Matthew Effect, Olga Kosheleva, Vladik Kreinovich
Occam's Razor Explains Matthew Effect, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Sociologists of science noticed that the results of many collaborative projects and discoveries are often attributed only to their most famous collaborators, even when the contributions of these famous collaborators were minimal. This phenomenon is known as the Matthew effect, after a famous citation from the Gospel of Matthew. In this article, we show that Occam's razor provides a possible explanation for the Matthew effect.
Constructive Mathematics Is Seemingly Simple But There Are Still Open Problems: Kreisel's Observation Explained, Olga Kosheleva, Vladik Kreinovich
Constructive Mathematics Is Seemingly Simple But There Are Still Open Problems: Kreisel's Observation Explained, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In his correspondence with Grigory Mints, the famous logician Georg Kreisel noticed that many results of constructive mathematics seem easier-to-prove than the corresponding classical (non-constructive) results -- although he noted that these results are still far from being simple and the corresponding open problems are challenging. In this paper, we provide a possible explanation for this empirical observation.
Conditional Dimension In Metric Spaces: A Natural Metric-Space Counterpart Of Kolmogorov-Complexity-Based Mutual Dimension, Vladik Kreinovich, Luc Longpre, Olga Kosheleva
Conditional Dimension In Metric Spaces: A Natural Metric-Space Counterpart Of Kolmogorov-Complexity-Based Mutual Dimension, Vladik Kreinovich, Luc Longpre, Olga Kosheleva
Departmental Technical Reports (CS)
It is known that dimension of a set in a metric space can be characterized in information-related terms -- in particular, in terms of Kolmogorov complexity of different points from this set. The notion of Kolmogorov complexity K(x) -- the shortest length of a program that generates a sequence x -- can be naturally generalized to conditionalKolmogorov complexity K(x:y) -- the shortest length of a program that generates x by using y as an input. It is therefore reasonable to use conditional Kolmogorov complexity to formulate a conditional analogue of dimension. Such a generalization has indeed been proposed, under …