Mathematics Commons^™

Robustness As A Criterion For Selecting A Probability Distribution Under Uncertainty, Songsak Sriboonchitta, Hung T. Nguyen, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

Often, we only have partial knowledge about a probability distribution, and we would like to select a single probability distribution $\rho(x)$ out of all probability distributions which are consistent with the available knowledge. One way to make this selection is to take into account that usually, the values $x$ of the corresponding quantity are also known only with some accuracy. It is therefore desirable to select a distribution which is the most robust -- in the sense the x-inaccuracy leads to the smallest possible inaccuracy in the resulting probabilities. In this paper, we describe the corresponding most robust probability distributions, …

Why Dependence Of Productivity On Group Size Is Log-Normal, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Empirical analysis shows that, on average, the productivity of a group log-normally depends on its size. The current explanations for this empirical fact are based on reasonably complex assumptions about the human behavior. In this paper, we show that the same conclusion can be made in effect, from first principles, without making these complex assumptions.

Why Locating Local Optima Is Sometimes More Complicated Than Locating Global Ones, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In most applications, practitioners are interested in locating global optima. In such applications, local optima that result from some optimization algorithms are an unnecessary side effect. In other words, in such applications, locating global optima is a much more computationally complex problem than locating local optima. In several practical applications, however, local optima themselves are of interest. Somewhat surprisingly, it turned out that in many such applications, locating all local optima is a much more computationally complex problem than locating all global optima. In this paper, we provide a theoretical explanation for this surprising empirical phenomenon.

Voting Aggregation Leads To (Interval) Median, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When we have several results of measuring or estimating the same quantities, it is desirable to aggregate them into a single estimate for the desired quantities. A natural requirement is that if the majority of estimates has some property, then the aggregate estimate should have the same property. It turns out that it is not possible to require this forall possible properties -- but we can require it for bounds, i.e., for properties that the value of the quantity is in between given bounds a and b. In this paper, we prove that if we restrict the …

Why Superellipsoids: A Probability-Based Explanation, Pedro Barragan Olague, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, it turns out that the set of possible values of the deviation vector is (approximately) a super-ellipsoid. In this paper, we provide a theoretical explanation for this empirical fact -- an explanation based on the natural notion of scale-invariance.

On Geometry Of Finsler Causality: For Convex Cones, There Is No Affine-Invariant Linear Order (Similar To Comparing Volumes), Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Some physicists suggest that to more adequately describe the causal structure of space-time, it is necessary to go beyond the usual pseudo-Riemannian causality, to a more general Finsler causality. In this general case, the set of all the events which can be influenced by a given event is, locally, a generic convex cone, and not necessarily a pseudo-Reimannian-style quadratic cone. Since all current observations support pseudo-Riemannian causality, Finsler causality cones should be close to quadratic ones. It is therefore desirable to approximate a general convex cone by a quadratic one. This cane be done if we select a hyperplane, and …

The New Issues In Classification Problems, Md Mahmudul Hasan

Open Access Theses & Dissertations

The data involved with science and engineering getting bigger everyday. To study and organize a big amount of data is difficult without classification. In machine learning, classification is the problem of identifying a given data from a set of categories. There are several classification technique people using to classify a given data. In our work we present a sparse representation technique to perform classification. The popularity of this technique motivates us to use on our collected samples. To find a sparse representation, we used an $l_1$-minimization algorithm which is a convex relaxation algorithm proven very efficient by researchers. The purpose …

Twenty-Seven Element Albertian Semifields, Thomas Joel Hughes

Open Access Theses & Dissertations

In 1958 Abraham Adrian Albert published his findings on the now famous class of semifields known as twisted fields or, as we refer to them in this Thesis, Albertian semifields. Since the publication of his findings, interest in the topic of semifields has grown and, naturally, some progress has been made in the subject, but there is still much to be studied. Moreover, the greater portion of modern efforts have been made to develop a theory of semifields, with less attention paid to more practical considerations. In this Thesis, we will narrow our investigations to a specific subclass of Albert's …

An A Posteriori Error Estimator For The C0 Interior Penalty Approximations Of Fourth Order Elliptic Boundary Value Problem On Quadrilateral Meshes, Mohammad Arifur Rahman

Open Access Theses & Dissertations

Numerical solutions of fourth order elliptic problems with finite element methods has been the topic of research in computational mechanics for over 50 years. Traditional approaches to solve these problems include using C1 conforming finite element methods which demand the C1 continuity of the underlying shape functions, which is computationally very expensive. In this work, we will present the C0 Interior Penalty Galerkin approximation of the fourth order elliptic problems which relies only on continuous i.e., C0 shape functions, which is much cheaper to implement. The spatial discretization is based on quadrilateral meshes and the underlying C0 shape functions are …

On An Iteration Method For Solving A Class Of Nonlinear Matrix Equations, Mohamed Illafe

Open Access Theses & Dissertations

In this thesis, I looked at the most recent extension of the Banach Contraction Principle [1]. Then I show that a concept of Sorgenfrey space may be used to explain this extension as well as Ran and Reurings fixed point theorem.

Complex Gleason Measures And The Nemytsky Operator, Miguel Angel Valles

Open Access Theses & Dissertations

This Thesis is devoted to generalize previous results on Gleason measures

to complex Gleason measures, and to develop a functional calculus

for complex measures in relation to the Nemytsky operator. Furthermore

we present the interpretation of our results in the field of quantum

mechanics, some concrete examples and further extensions of several

theorems.

Combining Interval And Probabilistic Uncertainty In Engineering Applications, Andrew Martin Pownuk

Open Access Theses & Dissertations

In many practical application, we process measurement results and expert estimates. Measurements and expert estimates are never absolutely accurate, their result are slightly different from the actual (unknown) values of the corresponding quantities. It is therefore desirable to analyze how this measurement and estimation inaccuracy affects the results of data processing. There exist numerous methods for estimating the accuracy of the results of data processing under different models of measurement and estimation inaccuracies: probabilistic, interval, and fuzzy. To be useful in engineering applications, these methods should provide accurate estimate for the resulting uncertainty, should not take too much computation time, …

Secondary Mathematics Teachers' Disposition Toward Challenge And Its Impact On Teaching Practice, Yirah Mariana Valverde

Open Access Theses & Dissertations

This two phase, sequential exploratory study focused on mathematics teachers' disposition toward challenge and its impact in their teaching practice and students' performance. This heavily qualitative study employed a mixed methodology to answer the following guiding research question: how do teachersâ?? disposition towards challenge affect teaching practice and what is the nature of that relationship?

During the initial phase (quantitative), two teachers who were enrolled in a graduate mathematics at a University in the southwestern U.S. were selected for the study (N=5). The goal of the class was to increase the participants' mathematical knowledge for teaching. The purposive selection was …

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 …

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

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

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.

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 …

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 …

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.

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

Departmental Technical Reports (CS)

In statistical analysis, we usually use the observed sample values x₁, ..., x_n to compute the values of several statistics v(x₁, ..., x_n) -- such as sample mean, sample variance, etc. The usual formulas for these statistics implicitly assume that we know the exact values x₁, ..., x_n. In practice, the sample values X₁, ..., X_n come from measurements and are, thus, only approximations to the actual (unknown) values x₁, ..., x_n of the corresponding quantity. Often, the only information that we have …

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.

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 …

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, …

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 …

From Equations To Tri-Quations And Multi-Quations, Mourat Tchoshanov, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In general, an equation A(x₁, ..., x_n)= B(x₁, ..., x_n) corresponds to the situation when we have two quantities A(x₁, ..., x_n) and B(x₁, ..., x_n) which are known to be equal, we know how each of these quantities depends on the unknown parameters x₁, ..., x_n, and we want to find the values of the unknowns x_i from this equality -- and from other similar equalities. In some practical situations, instead of two equal values, we have three …

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.

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.

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, …