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Articles 91 - 114 of 114
Full-Text Articles in Applied Mathematics
Is "No Trade Theorem" Really A Paradox: Analysis Based On Decision Theory, Laxman Bokati, Vladik Kreinovich
Is "No Trade Theorem" Really A Paradox: Analysis Based On Decision Theory, Laxman Bokati, Vladik Kreinovich
Departmental Technical Reports (CS)
One of the challenges in foundations of finance is the so-called "no trade theorem" paradox: if an expert trader wants to sell some stock, that means that this trader believes that this stock will go down; however, the very fact that another expert trader is willing to buy it means that this other expert believes that the stock will go up. The fact that equally good experts have different beliefs should dissuade the first expert from selling -- and thus, trades should be very rare. However, in reality, trades are ubiquitous. In this paper, we show that a detailed application …
How To Determine The Stiffness Of The Pavement's Upper Layer (Base) Based On The Overall Stiffness And The Stiffness Of The Lower Layer (Subgrade), Christian Servin, Vladik Kreinovich
How To Determine The Stiffness Of The Pavement's Upper Layer (Base) Based On The Overall Stiffness And The Stiffness Of The Lower Layer (Subgrade), Christian Servin, Vladik Kreinovich
Departmental Technical Reports (CS)
In road construction, it is important to estimate difficult-measure stiffness of the pavement's upper layer based the easier-to-measure overall stiffness and the stiffness of the lower layer. In situations when the overall stiffness is not yet sufficient, it is also important to estimate how much more we need to add to the upper layer to reach the desired overall stiffness. In this paper, for the cases when a linear approximation is sufficient, we provide analytical formulas for the desired estimations.
Why Compaction Meter Value (Cmv) Is A Good Measure Of Pavement Stiffness: Towards A Possible Theoretical Explanation, Andrzej Pownuk, Pedro Barragan Olague, Vladik Kreinovich
Why Compaction Meter Value (Cmv) Is A Good Measure Of Pavement Stiffness: Towards A Possible Theoretical Explanation, Andrzej Pownuk, Pedro Barragan Olague, Vladik Kreinovich
Departmental Technical Reports (CS)
To measure stiffness of the compacted pavement, practitioners use the Compaction Meter Value (CMV); a ratio between the amplitude for the first harmonic of the compactor's acceleration and the amplitude corresponding to the vibration frequency. Numerous experiments show that CMV is highly correlated with the pavement stiffness, but as of now, there is no convincing theoretical explanation for this correlation. In this paper, we provide a possible theoretical explanation for the empirical correlation. This explanation also explains why, the stiffer the material, the more higher-order harmonics we observe.
Standing On The Shoulders Of The Giants: Why Constructive Mathematics, Probability Theory, Interval Mathematics, And Fuzzy Mathematics Are Important, Vladik Kreinovich
Standing On The Shoulders Of The Giants: Why Constructive Mathematics, Probability Theory, Interval Mathematics, And Fuzzy Mathematics Are Important, Vladik Kreinovich
Departmental Technical Reports (CS)
Recent death of Ray Moore, one of the fathers of interval mathematics, inspired these thoughts on why interval computations -- and several other related areas of study -- are important, and what we can learn from the successes of these areas' founders and promoters.
Why Lattice-Valued Fuzzy Values? A Mathematical Justification, Rujira Ouncharoen, Vladik Kreinovich, Hung T. Nguyen
Why Lattice-Valued Fuzzy Values? A Mathematical Justification, Rujira Ouncharoen, Vladik Kreinovich, Hung T. Nguyen
Departmental Technical Reports (CS)
To take into account that expert's degrees of certainty are not always comparable, researchers have used partially ordered set of degrees instead of the more traditional linearly (totally) ordered interval [0,1]. In most cases, it is assumed that this partially ordered set is a lattice, i.e., every two elements have the greatest lower bound and the least upper bound. In this paper, we prove a theorem explaining why it is reasonable to require that the set of degrees is a lattice.
Minimax Portfolio Optimization Under Interval Uncertainty, Meng Yuan, Xu Lin, Junzo Watada, Vladik Kreinovich
Minimax Portfolio Optimization Under Interval Uncertainty, Meng Yuan, Xu Lin, Junzo Watada, Vladik Kreinovich
Departmental Technical Reports (CS)
In the 1950s, Markowitz proposed to combine different investment instruments to design a portfolio that either maximizes the expected return under constraints on volatility (risk) or minimizes the risk under given expected return. Markowitz's formulas are still widely used in financial practice. However, these formulas assume that we know the exact values of expected return and variance for each instrument, and that we know the exact covariance of every two instruments. In practice, we only know these values with some uncertainty. Often, we only know the bounds of these values -- i.e., in other words, we only know the intervals …
Among Several Successful Algorithms, Simpler Ones Usually Work Better: A Possible Explanation Of An Empirical Observation, Vladik Kreinovich, Olga Kosheleva
Among Several Successful Algorithms, Simpler Ones Usually Work Better: A Possible Explanation Of An Empirical Observation, Vladik Kreinovich, Olga Kosheleva
Departmental Technical Reports (CS)
Often, several different algorithms can solve a certain practical problem. Sometimes, algorithms which are successful in solving one problem can solve other problems as well. How can we decide which of the original algorithms is the most promising -- i.e., which is more probable to be able to solve other problem? In many cases, the simplest algorithms turns out to be the most successful. In this paper, we provide a possible explanation for this empirical observation.
Granularity Explains Empirical Factor-Of-Three Relation Between Probabilities Of Pulmonary Embolism In Different Patient Categories, Beverly Rivera, Francisco Zapata, Vladik Kreinovich
Granularity Explains Empirical Factor-Of-Three Relation Between Probabilities Of Pulmonary Embolism In Different Patient Categories, Beverly Rivera, Francisco Zapata, Vladik Kreinovich
Departmental Technical Reports (CS)
Pulmonary embolism is a very dangerous difficult-to-detect medical condition. To diagnose pulmonary embolism, medical practitioners combine indirect signs of this condition into a single score, and then classify patients into low-probability, intermediate-probability, and high-probability categories. Empirical analysis shows that, when we move from each category to the next one, the probability of pulmonary embolism increases by a factor of three. In this paper, we provide a theoretical explanation for this empirical relation between probabilities.
Examining The Consistence Of Futures Margin Levels Using Bivariate Extreme Value Copulas, X. Gong, Hung T. Nguyen, Vladik Kreinovich, Songsak Sriboonchitta
Examining The Consistence Of Futures Margin Levels Using Bivariate Extreme Value Copulas, X. Gong, Hung T. Nguyen, Vladik Kreinovich, Songsak Sriboonchitta
Departmental Technical Reports (CS)
This study examines the consistence of the futures margin levels of different commodities and combinations in the CME group by Extreme Value Copula (EVC). We find that if we ignore the co-movements of the commodities, the margins become consistent with each other, and the margin violation rates hover around 0.5%. However, if we consider the co-movement of the related commodities using EVC, the margin levels are found to be not consistent anymore, especially in the combinations of strongly related commodities which are in the same category. Therefore, we suggest that the CME group should try to harmonize the margins policy …
How To Gauge Unknown Unknowns: A Possible Theoretical Explanation Of The Usual Safety Factor Of 2, Joe Lorkowski, Vladik Kreinovich
How To Gauge Unknown Unknowns: A Possible Theoretical Explanation Of The Usual Safety Factor Of 2, Joe Lorkowski, Vladik Kreinovich
Departmental Technical Reports (CS)
To gauge the accuracy of a measuring instrument, engineers analyze possible factors contributing to the instrument's inaccuracy. In addition to known factors, however, there are usually unknown factors which also contribute to the instrument's inaccuracy. To properly gauge the instrument's accuracy -- and thus, to make sure that we do not compromise our safety by underestimating the inaccuracy -- we need to also take these "unknown unknowns" into account. In practice, this is usually done by multiplying the original estimate for inaccuracy by a "safety" factor of 2. In this paper, we provide a possible theoretical explanation for this empirical …
Formalizing The Informal, Precisiating The Imprecise: How Fuzzy Logic Can Help Mathematicians And Physicists By Formalizing Their Intuitive Ideas, Olga Kosheleva, Renata Reiser, Vladik Kreinovich
Formalizing The Informal, Precisiating The Imprecise: How Fuzzy Logic Can Help Mathematicians And Physicists By Formalizing Their Intuitive Ideas, Olga Kosheleva, Renata Reiser, Vladik Kreinovich
Departmental Technical Reports (CS)
Fuzzy methodology transforms expert ideas -- formulated in terms of words from natural language -- into precise rules and formulas. In this paper, we show that by applying this methodology to intuitive physical and mathematical ideas, we can get known fundamental physical equations and known mathematical techniques for solving these equations. This fact makes us confident that in the future, fuzzy techniques will help physicists and mathematicians to transform their imprecise ideas into new physical equations and new techniques for solving these equations.
Kekule's Benzene Structure: A Case Study Of Teaching Usefulness Of Symmetry, Olga Kosheleva, Vladik Kreinovich
Kekule's Benzene Structure: A Case Study Of Teaching Usefulness Of Symmetry, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Benzene is one of the basic building blocks of organic molecules. One of the reasons for benzene's ubiquity is its unusual ring structure first discovered by Kekule in 1865. In this paper, we show that a simple symmetry-based analysis can narrow down possible benzene structures to three ring ones, including the Kekule's ring. Thus, Kekule's benzene structure provides a good pedagogical example on which one can explain usefulness of symmetries.
Using Second-Order Probabilities To Make Maximum Entropy Approach To Copulas More Reasonable, Hung T. Nguyen, Vladik Kreinovich, Berlin Wu
Using Second-Order Probabilities To Make Maximum Entropy Approach To Copulas More Reasonable, Hung T. Nguyen, Vladik Kreinovich, Berlin Wu
Departmental Technical Reports (CS)
Copulas are a general way of describing dependence between two or more random variables. When we only have partial information about the dependence, i.e., when several different copulas are consistent with our knowledge, it is often necessary to select one of these copulas. A frequently used method of selecting this copula is the maximum entropy approach, when we select a copula with the largest entropy. However, in some cases, the maximum entropy approach leads to an unreasonable selection -- e.g., even if we know that the two random variables are positively correlated, the maximum entropy approach completely ignores this information. …
Observable Causality Implies Lorentz Group: Alexandrov-Zeeman-Type Theorem For Space-Time Regions, Olga Kosheleva, Vladik Kreinovich
Observable Causality Implies Lorentz Group: Alexandrov-Zeeman-Type Theorem For Space-Time Regions, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
The famous Alexandrov-Zeeman theorem proves that causality implies Lorentz group. The physical meaning of this result is that once we observe which event can causally affect which other events, then, using only this information, we can reconstruct the linear structure of the Minkowski space-time. The original Alexandrov-Zeeman theorem is based on the causality relation between events represented by points in space-time. Knowing such a point means that we know the exact moment of time and the exact location of the corresponding event - and that this event actually occurred at a single moment of time and at a single spatial …
How To Modify Grade Point Average (Gpa) To Make It More Adequate, Joe Lorkowski, Olga Kosheleva, Vladik Kreinovich
How To Modify Grade Point Average (Gpa) To Make It More Adequate, Joe Lorkowski, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
At present, the amounts of knowledge acquired by different graduates of the same program are usually compared by comparing their GPAs. We argue that this is not always the most adequate description: for example, if, after completing all required classes with the highest grade of "excellent" (A), a student takes an additional challenging class and gets a "satisfactory" grade (C), the amount of her knowledge increases, but the GPA goes down. We propose a modification of the GPA which is free of this drawback and is, thus, more adequate for describing the student's knowledge. We also provide a psychological explanation …
In Category Of Sets And Relations, It Is Possible To Describe Functions In Purely Category Terms, Vladik Kreinovich, Martine Ceberio, Quentin Brefort
In Category Of Sets And Relations, It Is Possible To Describe Functions In Purely Category Terms, Vladik Kreinovich, Martine Ceberio, Quentin Brefort
Departmental Technical Reports (CS)
We prove that in the category of sets and relations, it is possible to describe functions in purely category terms.
For Each Mathematical Statement, Only Finitely Many Of Its Generalizations Are Useful: A Formal Proof Of E. Bishop's Idea, Olga Kosheleva, Vladik Kreinovich
For Each Mathematical Statement, Only Finitely Many Of Its Generalizations Are Useful: A Formal Proof Of E. Bishop's Idea, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Generalization is one of the main mathematical activities. Some generalizations turn out to be useful for working mathematics, while many other generalizations have so far been not very useful. E. Bishop believed that most fruitless-so-far generalizations are hopeless, that every mathematical statement has only a few useful generalizations. In this paper, we show that, under a natural definition of the notion of useful generalization, Bishop's belief can be proven -- moreover, it turns out that for each mathematical statement, only finitely many of its generalizations are useful.
Deep Mathematical Results Are The Ones That Connect Seemingly Unrelated Areas: Towards A Formal Proof Of Gian-Carlo Rota's Thesis, Olga Kosheleva, Vladik Kreinovich
Deep Mathematical Results Are The Ones That Connect Seemingly Unrelated Areas: Towards A Formal Proof Of Gian-Carlo Rota's Thesis, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
When is a mathematical result deep? At first glance, the answer to this question is subjective: what is deep for one mathematician may not sound that deep for another. A renowned mathematician Gian-Carlo Rota expressed an opinion that the notion of deepness is more objective that we may think: namely, that a mathematical statement is deep if and only if it connects two seemingly unrelated areas of mathematics. In this paper, we formalize this thesis, and show that in this formalization, Gian Carlo Rota's thesis becomes a provable mathematical result.
Why In Mayan Mathematics, Zero And Infinity Are The Same: A Possible Explanation, Olga Kosheleva, Vladik Kreinovich
Why In Mayan Mathematics, Zero And Infinity Are The Same: A Possible Explanation, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In Mayan mathematics, zero is supposed to be, in some sense, equal to infinity. At first glance, while this statement may have a deep philosophical meaning, it does not seem to make much mathematical sense. In this paper, we show, that this statement may be made mathematically reasonable. Specifically, on a real line, it is often useful to consider both −∞ and +∞ as a single infinity. When we deal with very small and very large numbers, it makes sense to use floating point representation, i.e., in effect, consider logarithms of the original values. In terms of logarithms, the original …
Peak-End Rule: A Utility-Based Explanation, Olga Kosheleva, Martine Ceberio, Vladik Kreinovich
Peak-End Rule: A Utility-Based Explanation, Olga Kosheleva, Martine Ceberio, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, people judge their overall experience by only taking into account the peak and the last levels of pleasantness or unpleasantness. While this peak-end rule is empirically supported by numerous psychological experiments, it seems to contradict our general theoretical ideas about people's preferences. In this paper, we show that, contrary to this impression, the end-peak rule can be justified based on the main ideas of the traditional utility-based decision theory.
Decision Making Under Interval Uncertainty (And Beyond), Vladik Kreinovich
Decision Making Under Interval Uncertainty (And Beyond), Vladik Kreinovich
Departmental Technical Reports (CS)
To make a decision, we must find out the user's preference, and help the user select an alternative which is the best -- according to these preferences. Traditional utility-based decision theory is based on a simplifying assumption that for each two alternatives, a user can always meaningfully decide which of them is preferable. In reality, often, when the alternatives are close, the user is often unable to select one of these alternatives. In this chapter, we show how we can extend the utility-based decision theory to such realistic (interval) cases.
Should Voting Be Mandatory? Democratic Decision Making From The Economic Viewpoint, Olga Kosheleva, Vladik Kreinovich, Boakun Li
Should Voting Be Mandatory? Democratic Decision Making From The Economic Viewpoint, Olga Kosheleva, Vladik Kreinovich, Boakun Li
Departmental Technical Reports (CS)
Many decisions are made by voting. At first glance, the more people participate in the voting process, the more democratic -- and hence, better -- the decision. In this spirit, to encourage everyone's participation, several countries make voting mandatory. But does mandatory voting really make decisions better for the society? In this paper, we show that from the viewpoint of decision making theory, it is better to allow undecided voters not to participate in the voting process. We also show that the voting process would be even better -- for the society as a whole -- if we allow partial …
Ubiquity Of Data And Model Fusion: From Geophysics And Environmental Sciences To Estimating Individual Risk During An Epidemic, Omar Ochoa, Aline Jaimes, Christian Servin, Craig Tweedie, Aaron Velasco, Martine Ceberio, Vladik Kreinovich
Ubiquity Of Data And Model Fusion: From Geophysics And Environmental Sciences To Estimating Individual Risk During An Epidemic, Omar Ochoa, Aline Jaimes, Christian Servin, Craig Tweedie, Aaron Velasco, Martine Ceberio, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, we need to combine the results of measuring a local value of a certain quantity with results of measuring average values of this same quantity. For example, in geosciences, we need to combine the seismic models (which describe density at different locations and depths) with gravity models which describe density averaged over certain regions. Similarly, in estimating the risk of an epidemic to an individual, we need to combine probabilities describe the risk to people of the corresponding age group, to people of the corresponding geographical region, etc. In this paper, we provide general techniques for …
How To Define Relative Approximation Error Of An Interval Estimate: A Proposal, Vladik Kreinovich
How To Define Relative Approximation Error Of An Interval Estimate: A Proposal, Vladik Kreinovich
Departmental Technical Reports (CS)
The traditional definition of a relative approximation error of an estimate X as the ratio |X - x|/|x| does not work when the actual value x is 0. To avoid this problem, we propose a new definition |X - x|/|X|. We show how this definition can be naturally extended to the case when instead of a numerical estimate X, we have an interval estimate [x], i.e., an interval that is guaranteed to contain the actual (unknown) value x.