Mathematics Commons^™

Quantitative Justification For The Gravity Model In Economics, Vladik Kreinovich, Songsak Sriboonchitta

Departmental Technical Reports (CS)

The gravity model in economics describes the trade flow between two countries as a function of their Gross Domestic Products (GDPs) and the distance between them. This model is motivated by the qualitative similarity between the desired dependence and the dependence of the gravity force (or potential energy) between the two bodies on their masses and on the distance between them. In this paper, we provide a quantitative justification for this economic formula.

How To Get Beyond Uniform When Applying Maxent To Interval Uncertainty, Songsak Sriboonchitta, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, the Maximum Entropy (MaxEnt) approach leads to reasonable distributions. However, in an important case when all we know is that the value of a random variable is somewhere within the interval, this approach leads to a uniform distribution on this interval -- while our intuition says that we should have a distribution whose probability density tends to 0 when we approach the interval's endpoints. In this paper, we show that in most cases of interval uncertainty, we have additional information, and if we account for this additional information when applying MaxEnt, we get distributions which are …

Efficient Algorithms For Synchroning Localization Sensors Under Interval Uncertainty, Raphael Voges, Bernardo Wagner, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that a practical need for synchronization of localization sensors leads to an interval-uncertainty problem. In principle, this problem can be solved by using the general linear programming algorithms, but this would take a long time -- and this time is not easy to decrease, e.g., by parallelization since linear programming is known to be provably hard to parallelize. To solve the corresponding problem, we propose more efficient and easy-to-parallelize algorithms.

Maximum Entropy As A Feasible Way To Describe Joint Distributions In Expert Systems, Thongchai Dumrongpokaphan, Vladik Kreinovich, Hung T. Nguyen

Departmental Technical Reports (CS)

In expert systems, we elicit the probabilities of different statements from the experts. However, to adequately use the expert system, we also need to know the probabilities of different propositional combinations of the experts' statements -- i.e., we need to know the corresponding joint distribution. The problem is that there are exponentially many such combinations, and it is not practically possible to elicit all their probabilities from the experts. So, we need to estimate this joint distribution based on the available information. For this purpose, many practitioners use heuristic approaches -- e.g., the t-norm approach of fuzzy logic. However, this …

Are Permanent Or Temporary Teams More Efficient: A Possible Explanation Of The Empirical Data, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that in education, stable (long-term) student teams are more effective than temporary (short-term) ones. It turned out that the same phenomenon is true for workers working on a long-term project. However, somewhat surprisingly, for small-scale projects, the opposite is true: teams without any prior collaboration experience are more successful. Moreover, it turns out that if combine in a team members with prior collaboration experience and members without such experience, the efficiency of the team gets even lower. In this paper, we provide a possible explanation for this strange empirical phenomenon.

Maybe The Usual Students' Practice Of Cramming For A Test Makes Sense: A Mathematical Analysis, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

We always teach students that cramming for a test is a bad idea, that they should study at the same speed throughout the semester – but many still cram. We ourselves are not that different: when we prepare papers for a conference, we often “cram” in the last days before the deadline instead of working with a regular speed for the whole time before the conference. The ubiquity of cramming makes us think that maybe it is not necessarily always a bad idea. And indeed, a simple model of a study process shows that an optimal solution often involve some …

No Idea Is A Bad Idea: A Theoretical Explanation, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

Many business publications state that no idea is a bad idea, that even if the idea is, at first glance, not helpful, there are usually some aspects of this idea which are helpful Usually, this statement is based on the experience of the author, and it is given without any theoretical explanation. In this paper, we provide a theoretical explanation for this statement.

Attraction-Repulsion Forces Between Biological Cells: A Theoretical Explanation Of Empirical Formulas, Olga Kosheleva, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

Biological calls attract and repulse each other: if they get too close to each other, they repulse, and if they get too far away from each other, they attract. There are empirical formulas that describe the dependence of the corresponding forces on the distance between the cells. In this paper, we provide a theoretical explanation for these empirical formulas.

How To Teach Implication, Martha Osegueda Escobar, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Logical implication is a somewhat counter-intuitive notion. For students, it is difficult to understand why a false statement implies everything. In this paper, we present a simple pedagogical way to make logical implication more intuitive.

Why Linear Interpolation?, Andrzej Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

Linear interpolation is the computationally simplest of all possible interpolation techniques. Interestingly, it works reasonably well in many practical situations, even in situations when the corresponding computational models are rather complex. In this paper, we explain this empirical fact by showing that linear interpolation is the only interpolation procedure that satisfies several reasonable properties such as consistency and scale-invariance.

Derivation Of Gross-Pitaevskii Version Of Nonlinear Schroedinger Equation From Scale Invariance, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that in the usual 3-D space, the Schroedinger equation can be derived from scale-invariance. In view of the fact that, according to modern physics, the actual dimension of proper space may be different from 3, it is desirable to analyze what happens in other spatial dimensions D. It turns out that while for D ≥ 3 we still get only the Schroedinger's equation, for D = 2, we also get the Gross-Pitaevskii version of a nonlinear Schroedinger equation that describes a quantum system of identical bosons, and for D = 1, we also get a new nonlinear …

Towards Decision Making Under General Uncertainty, Andrzej Pownuk, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

There exist techniques for decision making under specific types of uncertainty, such as probabilistic, fuzzy, etc. Each of the corresponding ways of describing uncertainty has its advantages and limitations. As a result, new techniques for describing uncertainty appear all the time. Instead of trying to extend the existing decision making idea to each of these new techniques one by one, we attempt to develop a general approach that would cover all possible uncertainty techniques.

Why Stable Teams Are More Efficient In Education, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that study groups speed up learning. Recent studies have shown that stable study groups are more efficient than shifting-membership groups. In this paper, we provide a theoretical explanation for this empirical observation.

Can We Detect Crisp Sets Based Only On The Subsethood Ordering Of Fuzzy Sets? Fuzzy Sets And/Or Crisp Sets Based On Subsethood Of Interval-Valued Fuzzy Sets?, Christian Servin, Gerardo Muela, Vladik Kreinovich

Departmental Technical Reports (CS)

Fuzzy sets are naturally ordered by the subsethood relation. If we only know which set which fuzzy set is a subset of which -- and have no access to the actual values of the corresponding membership functions -- can we detect which fuzzy sets are crisp? In this paper, we show that this is indeed possible. We also show that if we start with interval-valued fuzzy sets, then we can similarly detect type-1 fuzzy sets and crisp sets.

Why Mixture Of Probability Distributions, Andrzej Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

If we have two random variables ξ₁ and ξ₁, then we can form their mixture if we take ξ₁ with some probability w and ξ₂ with the remaining probability 1 − w. The probability density function (pdf) ρ(x) of the mixture is a convex combination of the pdfs of the original variables: ρ(x) = w * ρ₁(x) +( 1 − w) * ρ₂(x). A natural question is: can we use other functions f(ρ₁, ρ₂) to combine the pdfs, i.e., to produce a new pdf ρ(x) =f(ρ_{1 …}

Experimentally Observed Dark Matter Confinement Clarifies A Discrepancy In Estimating The Universe's Expansion Speed, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is well known the our Universe is expanding. In principle, we can estimate the expansion speed either directly, by observing the current state of the Universe, or indirectly, by analyzing the cosmic background radiation. Surprisingly, these two estimates lead to somewhat different expansion speeds. This discrepancy is a important challenge for cosmologists. Another challenge comes from recent experiments that show that, contrary to the original idea that dark matter and regular (baryonic) matter practically do not interact, dark matter actually "shadows" the normal matter.

In this paper, we show that this "dark matter confinement" can explain the discrepancy between …

A Simple Geometric Explanation Of Occam's Razor, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Occam's razor states that out of possible explanations, plans, and designs, we should select the simplest one. It turns out that in many practical situations, the simplest explanation indeed turns out to be the correct one, the simplest plan is often the most successful, etc. But why this happens is not very clear. In this paper, we provide a simple geometric explanation of Occam's razor.

How To Make Machine Learning Robust Against Adversarial Inputs, Gerardo Muela, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

It has been recently shown that it is possible to "cheat" many machine learning algorithms -- i.e., to perform minor modifications of the inputs that would lead to a wrong classification. This feature can be used by adversaries to avoid spam detection, to create a wrong identification allowing access to classified information, etc. In this paper, we propose a solution to this problem: namely, instead of applying the original machine learning algorithm to the original inputs, we should first perform a random modification of these inputs. Since machine learning algorithms perform well on random data, such a random modification ensures …

Why Product "And"-Operation Is Often Efficient: One More Argument, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is an empirical fact that the algebraic product is one the most efficient "and"-operations in fuzzy logic. In this paper, we provide one of the possible explanations of this empirical phenomenon.

Towards An Algebraic Description Of Set Arithmetic, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To describe the state of the world, we need to describe the values of all physical quantities. In practice, due to inevitable measurement inaccuracy, we do not know the exact values of these quantities, we only know the sets of possible values for these quantities. On the class of such uncertainty-related sets, we can naturally define arithmetic operations that transform, e.g., uncertainty in a and b into uncertainty with which we know the sum a + b.

In many applications, it has been useful to reformulate the problem in purely algebraic terms, i.e., in terms of axioms that the basic …

Why The Presence Of Point-Wise ("Punctate") Calcifications Or Linear Configurations Of Calcifications Makes Breast Cancer More Probable: A Geometric Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When a specialist analyzes a mammogram for signs of possible breast cancer, he or she pays special attention to point-wise and linear-shaped calcifications and point-wise and linear configurations of calcification -- since empirically, such calcifications and combinations of calcifications are indeed most frequently associated with cancer. In this paper, we provide a geometric explanation for this empirical phenomenon.

Why Most Bright Stars Are Binary But Most Dim Stars Are Single: A Simple Qualitative Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that most visible stars are binary: they have a nearby companion star, and these two stars orbit around each other. Based on this fact, until recently, astronomers believed that, in general, most stars are binary. A few years ago, a surprising paper showed that while most bright stars are indeed binary, most dim stars are single. In this paper, we provide a simple qualitative explanation for this empirical fact.

When Invading, Cancer Cells Do Not Divide: A Geometric (Symmetry-Based) Explanation Of An Empirical Observation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In general, malignant tumors are known to grow fast, cancer cells that form these tumors divide and spread around. Tumors also experience the process of metastasis, when cancer cells invade neighboring organs. A recent experiment has shown that, contrary to the previous assumptions, when cancer cells are invading, they stop dividing. In this paper, we provide a geometric explanation for this empirical phenomenon.

Why Growth Of Cancerous Tumors Is Gompertzian: A Symmetry-Based Explanation, Pedro Barragan Olague, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that the growth of a cancerous tumor is well described by the Gompertz's equation. The existing explanations for this equation rely on specifics of cell dynamics. However, the fact that for many different types of tumors, with different cell dynamics, we observe the same growth pattern, make us believe that there should be a more fundamental explanation for this equation. In this paper, we show that a symmetry-based approach indeed leads to such an explanation: indeed, out of all scale-invariant growth dynamics, the Gompertzian growth is the closest to the linear-approximation exponential growth model.

What Is The Best Way To Add Large Number Of Integers: Number-By-Number As Computers Do Or Lowest-Digits-Than-Next-Digits-Etc As We Humans Do?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When we need to add several integers, computers add them one by one, while we usually add them digit by digit: first, we add all the lowest digits, then we add all next lowest digits, etc. Which way is faster? Should we learn from computers or should we teach computers to add several integers our way?

In this paper, we show that the computer way is faster. This adds one more example to the list of cases when computer-based arithmetic algorithms are much more efficient than the algorithms that we humans normally use.

Yes- And No-Gestures Explained By Symmetry, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In most cultures, "yes" is indicate by a vertical head movement (nod), while "no" is indicated by a left-right movement (shake). In this paper, we show that basic symmetries can explain this cultural phenomenon.

Why 3-D Space? Why 10-D Space? A Possible Simple Geometric Explanation, Vladik Kreinovich

Departmental Technical Reports (CS)

In physics, the number of observed spatial dimensions (three) is usually taken as an empirical fact, without a deep theoretical explanation. In this paper, we provide a possible simple geometric explanation for the 3-D character of the proper space. We also provide a simple geometric explanation for the number of additional spatial dimensions that some physical theories use. Specifically, it is known that for some physical quantities, the 3-D space model with point-wise particles leads to meaningless infinities. To avoid these infinities, physicists have proposed that particles are more adequately described not as 0-D points, but rather as 1-D strings …

How To Predict Nesting Sites And How To Measure Shoreline Erosion: Fuzzy And Probabilistic Techniques For Environment-Related Spatial Data Processing, Stephen Escarzaga, Craig Tweedie, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show how fuzzy and probabilistic techniques can be used in environment-related data processing. Specifically, we will show that these methods help in solving two environment-related problems: how to predict the birds' nesting sites and how to measure shoreline erosion.

How To Estimate Resilient Modulus For Unbound Aggregate Materials: A Theoretical Explanation Of An Empirical Formula, Pedro Barragan Olague, Soheil Nazarian, Vladik Kreinovich, Afshin Gholamy

Departmental Technical Reports (CS)

To ensure the quality of pavement, it is important to make sure that the resilient moduli -- that describe the stiffness of all the pavement layers -- exceed a certain threshold. From the mechanical viewpoint, pavement is a non-linear medium. Several empirical formulas have been proposed to describe this non-linearity. In this paper, we describe a theoretical explanation for the most accurate of these empirical formulas.

How To Make A Solution To A Territorial Dispute More Realistic: Taking Into Account Uncertainty, Emotions, And Step-By-Step Approach, Mahdokhat Afravi, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life situations, it is necessary to divide a disputed territory between several interested parties. The usual way to perform this division is by using Nash's bargaining solution, i.e., by finding a partition that maximizes the product of the participants' utilities. However, this solution is based on several idealized assumptions: that we know the exact values of all the utilities, that division is performed on a purely rational basis, with no emotions involved, and that the entire decision is made once. In practice, we only know the utilities with some uncertainty, emotions are often involved, and the solution is …