Physical Sciences and Mathematics Commons^™

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.

Fuzzy Logic Ideas Can Help In Explaining Kahneman And Tversky's Empirical Decision Weights, Joe Lorkowski, Vladik Kreinovich

Departmental Technical Reports (CS)

Analyzing how people actually make decisions, the Nobelist Daniel Kahneman and his co-author Amos Tversky found out that instead of maximizing the expected gain, people maximize a weighted gain, with weights determined by the corresponding probabilities. The corresponding empirical weights can be explained qualitatively, but quantitatively, these weights remains largely unexplained. In this paper, we show that with a surprisingly high accuracy, these weights can be explained by fuzzy logic ideas.

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.

Roadmap For Graduating Students With Expertise In The Analysis And Development Of Secure Cyber-Systems, Ann Q. Gates, Salamah Salamah, Luc Longpre

Departmental Technical Reports (CS)

Modern society is intensely and irreversibly dependent on software systems of extraordinary size and complexity. This includes software systems in domain areas such as defense, energy, communication, transportation, and manufacturing. Due to the rapid expansion and reliance on the global Internet for day-to-day functions of individuals, organizations, governments, and industry around the world, cyber-security has emerged as an essential component of computing curricula. To address regional and national needs, the Computer Science Department has defined a roadmap for educating and preparing students who have expertise in the analysis and development of secure cyber-systems. Toward that vision, the department has set …

From Interval-Valued Probabilities To Interval-Valued Possibilities: Case Studies Of Interval Computation Under Constraints, Luis C. Gutierrez, Martine Ceberio, Vladik Kreinovich, Rebekah L. Gruver, Mariana Peña, Mathew J. Rister, Abraham Saldaña, John Vasquez, Janelle Ybarra, Salem Benferhat

Departmental Technical Reports (CS)

In many engineering situations, we need to make decisions under uncertainty. In some cases, we know the probabilities p_i of different situations i; these probabilities should add up to 1. In other cases, we only have expert estimates of the degree of possibility μ_ii of different situations; in accordance with the possibility theories, the largest of these degrees should be equal to 1.

In practice, we often only know these degrees p_i and μ_ii with uncertainty. Usually, we know the upper bound and the lower bound on each of these values. In other words, …

How To Compare Different Range Estimations: A Symmetry-Based Approach, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

How to compare different range estimators for multivariate functions under uncertainty? To answer this question, we analyze which utility functions can be used for this task. Specifically, we: (1) introduce various invariance assumptions, (2) describe the class of all utility functions which satisfy these assumptions, and (3) show how the resulting utility functions can be used to compare different range estimators.

Fitts's Law: Towards A Geometric Explanation, Olga Kosheleva, Vladik Kreinovich, Octavio Lerma

Departmental Technical Reports (CS)

In designing human-computer interfaces, designers use an empirical Fitts's Law, according to which the average time T of accessing an icon of size w at a distance d from the center of the screen is proportional to the logarithm of the ratio w/d. There exist explanations for this law, but these explanations have gaps. In this paper, we show that these gaps can be explained if we analyze this problem from the geometric viewpoint. Thus, we get a geometric explanation of the Fitts's Law.

Range Estimation Under Constraints Is Computable Unless There Is A Discontinuity, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main problems of interval computations is computing the range of a given function over given intervals. It is known that there is a general algorithm for computing the range of computable functions over computable intervals. However, if we take into account that often in practice, not all possible combinations of the inputs are possible (i.e., that there are constraints), then it becomes impossible to have an algorithm which would always compute this range. In this paper, we explain that the main reason why range estimation under constraints is not always computable is that constraints may introduce discontinuity …

Interleaving Enhances Learning: A Possible Geometric Explanation, Octavio Lerma, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the traditional approach to learning, if we want students to learn how to solve different types of problems, we first teach them how to solve problems of the first type, then how to solve problems of the second type, etc. It turns out that we can speed up learning if we interleave problems of different types. In particular, it has bene empirically shown that interleaving problems of four different types leads to a double speed-up. In this paper, we provide a possible geometric explanation for this empirical fact.

How To Fully Represent Expert Information About Imprecise Properties In A Computer System -- Random Sets, Fuzzy Sets, And Beyond: An Overview, Hung T. Nguyen, Vladik Kreinovich

Departmental Technical Reports (CS)

To help computers make better decisions, it is desirable to describe all our knowledge in computer-understandable terms. This is easy for knowledge described in terms on numerical values: we simply store the corresponding numbers in the computer. This is also easy for knowledge about precise (well-defined) properties which are either true or false for each object: we simply store the corresponding "true" and "false" values in the computer. The challenge is how to store information about imprecise properties. In this paper, we overview different ways to fully store the expert information about imprecise properties. We show that in the simplest …

A Simple Geometric Model Provides A Possible Quantitative Explanation Of The Advantages Of Immediate Feedback In Student Learning, Octavio Lerma, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Calculus is a known bottleneck for many students studying science and engineering. Various techniques have been developed to enhance the students' success. A recent study published in the Notices of American Mathematical Society showed that only one factor determines the success of a technique: the presence of immediate feedback. On average, students who receive immediate feedback learn twice faster than students who are taught in a more traditional way, with a serious feedback only once or twice a semester (after a test).

The very fact that immediate feedback is helpful is not surprising: it helps the student clear misconceptions and …

Zipf's Law And 7 Plus Minus 2 Principle Lead To A Possible Explanation Of Daniel's Law, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In 1961, D. R. Daniel observed that the success of a company is usually determined by three to six major factors. This observation has led to many successful management ideas, but they leave one puzzled: why three to six? why not two or seven? In this paper, we provide a possible explanation to this puzzle; namely, we show that these numbers of factors can be derived from Zipf's Law and from the 7 plus minus 2 principle.

Why Injecting Fine Dust Into A Tornado Is More Promising Than Injecting Coarse Dust: A Geometric Explanation, Octavio Lerma, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the promising ways to tame a tornado is to inject dust into it. Somewhat counter-intuitively, injecting coarse dust only makes the tornado stronger, while injecting fine dust can indeed help in the taming. This difference has been explained by a mathematical analysis of the corresponding equations, but (in contrast to the usual physics practice) this mathematical analysis has not yet been accompanied by a simple qualitative physical explanation. We show that such a simple explanation can be obtained if we analyze the problem of taming tornados from the geometric viewpoint.

Diversity Is Beneficial For A Research Group: One More Quantitative Argument, Komsan Suriya, Tatcha Sudtasan, Tonghui Wang, Octavio Lerma, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we propose a natural model describing competition between two research groups of the same average research strength. The analysis of this model enables us to conclude that a more diverse group has an advantage: namely, the more diverse the group, the higher the average quality of its publications.

A Feasible Algorithm For Checking N-Scissors Congruence Of Polyhedra In Rd, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

While in R², every two polygons of the same area are scissors congruent (i.e., they can be both decomposed into the same finite number of pair-wise congruent polygonal pieces), in R³, there are polyhedra P and P' of the same volume which are not scissors-congruent. It is therefore necessary, given two polyhedra, to check whether they are scissors-congruent (and if yes -- to find the corresponding decompositions). It is known that while there are algorithms for performing this checking-and-finding task, no such algorithm can be feasible -- their worst-case computation time grows (at least) exponentially, so …

From Global To Local Constraints: A Constructive Version Of Bloch's Principle, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Generalizing several results from complex analysis, A. Bloch formulated an informal principle -- that for every global implication there is a stronger local implication. This principle has been formalized for complex analysis, but is has been successfully used in other areas as well. In this paper, we propose a new formalization of Bloch's Principle, and we show that in general, the corresponding localized version can be obtained algorithmically.