Applied Mathematics Commons^™

How To Estimate The Stiffness Of The Multi-Layer Road Based On Properties Of Layers: Symmetry-Based Explanation For Odemark's Equation, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich, Olga Kosheleva, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

When we design a road, we would like to check that the current design provides the pavement with sufficient stiffness to withstand traffic loads and climatic conditions. For this purpose, we need to estimate the stiffness of the road based on stiffness and thickness of its different layers. There exists a semi-empirical formula for this estimation. In this paper, we show that this formula can be explained by natural scale-invariance requirements.

How The Proportion Of People Who Agree To Perform A Task Depends On The Stimulus: A Theoretical Explanation Of The Empirical Formula, Laxman Bokati, Vladik Kreinovich, Doan Thanh Ha

Departmental Technical Reports (CS)

For each task, the larger the stimulus, the larger proportion of people agree to perform this task. In many economic situations, it is important to know how much stimulus we need to offer so that a sufficient proportion of the people will agree to perform the needed task. There is an empirical formula describing how this proportion increases as we increase the amount of stimulus. However, this empirical formula lacks a convincing theoretical explanation, as a result of which practitioners are somewhat reluctant to use it. In this paper, we provide a theoretical explanation for this empirical formula, thus making …

Reward For Good Performance Works Better Than Punishment For Mistakes: Economic Explanation, Olga Kosheleva, Julio Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

How should we stimulate people to make them perform better? How should we stimulate students to make them study better? Many experiments have shown that reward for good performance works better than punishment for mistakes. In this paper, we provide a possible theoretical explanation for this empirical fact.

Why Most Empirical Distributions Are Few-Modal, Julio Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In principle, any non-negative function can serve as a probability density function -- provided that it adds up to 1. All kinds of processes are possible, so it seems reasonable to expect that observed probability density functions are random with respect to some appropriate probability measure on the set of all such functions -- and for all such measures, similarly to the simplest case of random walk, almost all functions have infinitely many local maxima and minima. However, in practice, most empirical distributions have only a few local maxima and minima -- often one (unimodal distribution), sometimes two (bimodal), and, …

Economics Of Reciprocity And Temptation, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich, Nguyen Ngoc Thach

Departmental Technical Reports (CS)

Behavioral economics has shown that in many situations, people's behavior differs from what is predicted by simple traditional utility-maximization economic models. It is therefore desirable to be able to accurately describe people's actual behavior. In some cases, the difference from the traditional models is caused by bounded rationality -- our limited ability to process information and to come up with a truly optimal solutions. In such cases, predicting people's behavior is difficult. In other cases, however, people actually optimize -- but the actual expression for utility is more complicated than in the traditional models. In such case, it is, in …

Scale-Invariance Ideas Explain The Empirical Soil-Water Characteristic Curve, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

The prediction of the road's properties under the influence of water infiltration is important for pavement design and management. Traditionally, this prediction heavily relied on expert estimates. In the last decades, complex empirical formulas have been proposed to capture the expert's intuition in estimating the effect of water infiltration on the stiffness of the pavement's payers. Of special importance is the effect of water intrusion on the pavement's foundation -- known as subgrade soil. In this paper, we show that natural scale-invariance ideas lead to a theoretical explanation for an empirical formula describing the dependence between soil suction and water …

Towards A Theoretical Explanation Of How Pavement Condition Index Deteriorates Over Time, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

To predict how the Pavement Condition Index will change over time, practitioners use a complex empirical formula derived in the 1980s. In this paper, we provide a possible theoretical explanation for this formula, an explanation based on general ideas of invariance. In general, the existence of a theoretical explanation makes a formula more reliable; thus, we hope that our explanation will make predictions of road quality more reliable.

New (Simplified) Derivation Of Nash's Bargaining Solution, Nguyen Hoang Phuong, Laxman Bokati, Vladik Kreinovich

Departmental Technical Reports (CS)

According to the Nobelist John Nash, if a group of people wants to selects one of the alternatives in which all of them get a better deal than in a status quo situations, then they should select the alternative that maximizes the product of their utilities. In this paper, we provide a new (simplified) derivation of this result, a derivation which is not only simpler -- it also does not require that the preference relation between different alternatives be linear.

Towards Making Fuzzy Techniques More Adequate For Combining Knowledge Of Several Experts, Nguyen Hoang Phuong, Vladik Kreinovich

Departmental Technical Reports (CS)

In medical and other applications, expert often use rules with several conditions, each of which involve a quantity within the domain of expertise of a different expert. In such situations, to estimate the degree of confidence that all these conditions are satisfied, we need to combine opinions of several experts -- i.e., in fuzzy techniques, combine membership functions corresponding to different experts. In each area of expertise, different experts may have somewhat different membership functions describing the same natural-language ("fuzzy") term like small. It is desirable to present the user with all possible conclusions corresponding to all these membership functions. …

Why Mean, Variance, Moments, Correlation, Skewness Etc. – Invariance-Based Explanations, Olga Kosheleva, Laxman Bokati, Vladik Kreinovich

Departmental Technical Reports (CS)

In principle, we can use many different characteristics of a probability distribution. However, in practice, a few of such characteristics are mostly used: mean, variance, moments, correlation, etc. Why these characteristics and not others? The fact that these characteristics have been successfully used indicates that there must be some reason for their selection. In this paper, we show that the selection of these characteristics can be explained by the fact that these characteristics are invariant with respect to natural transformations -- while other possible characteristics are not invariant.

Strength Of Lime Stabilized Pavement Materials: Possible Theoretical Explanation Of Empirical Dependencies, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

When building a road, it is often necessary to strengthen the underlying soil layer. This strengthening is usually done by adding lime. There are empirical formulas that describe how the resulting strength depends on the amount of added lime. In this paper, we provide a theoretical explanation for these empirical formulas.

Fusion Of Probabilistic Knowledge As Foundation For Sliced-Normal Approach, Michael Beer, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical applications, it turns out to be efficient to use Sliced-Normal multi-D distributions, i.e., distributions for which the logarithm of the probability density function (pdf) is a polynomial -- -- to be more precise, it is a sum of squares of several polynomials. This class is a natural extension of normal distributions, i.e., distributions for which the logarithm of the pdf is a quadratic polynomial.

In this paper, we provide a possible theoretical explanation for this empirical success.

Joule's 19th Century Energy Conservation Meta-Law And The 20th Century Physics (Quantum Mechanics And General Relativity): 21st Century Analysis, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

Joule's Energy Conservation Law was the first "meta-law": a general principle that all physical equations must satisfy. It has led to many important and useful physical discoveries. However, a recent analysis seems to indicate that this meta-law is inconsistent with other principles -- such as the existence of free will. We show that this conclusion about inconsistency is based on a seemingly reasonable -- but simplified -- analysis of the situation. We also show that a more detailed mathematical and physical analysis of the situation reveals that not only Joule's principle remains true -- it is actually strengthened: it is …

How Can We Explain Different Number Systems?, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

At present, we mostly use decimal (base-10) number system, but in the past, many other systems were used: base-20, base-60 -- which is still reflected in how we divide an hour into minutes and a minute into seconds -- and many others. There is a known explanation for the base-60 system: 60 is the smallest number that can be divided by 2, by 3, by 4, by 5, and by 6. Because of this, e.g., half an hour, one-third of an hour, all the way to one-sixth of an hour all correspond to a whole number of minutes. In this …

Deep Learning (Partly) Demystified, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

Successes of deep learning are partly due to appropriate selection of activation function, pooling functions, etc. Most of these choices have been made based on empirical comparison and heuristic ideas. In this paper, we show that many of these choices -- and the surprising success of deep learning in the first place -- can be explained by reasonably simple and natural mathematics.

How To Reconcile Randomness With Physicists' Belief That Every Theory Is Approximate: Informal Knowledge Is Needed, Ricardo Alvarez, Nick Sims, Christian Servin, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that physicists' intuition about randomness is not fully consistent with their belief that every theory is only approximate. We also prove that there is no formal way to reconcile these two intuitions, this reconciliation has to be informal. Thus, there are fundamental reasons why informal knowledge is needed for describing the real world.

How User Ratings Change With Time: Theoretical Explanation Of An Empirical Formula, Julio Urenda, Manuel Hernandez, Natalia Villanueva Rosales, Vladik Kreinovich

Departmental Technical Reports (CS)

In many application areas, it is important to predict the user's reaction to new products. In general, this reaction changes with time. Empirical analysis of this dependence has shown that it can be reasonably accurately described by a power law. In this paper, we provide a theoretical explanation for this empirical formula.

How To Assign Grades To Tasks So As To Maximize Student Efforts, Laxman Bokati, Vyacheslav Kalashnikov, Nataliya Kalashnykova, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In some classes, students want to get a desired passing grade (e.g., C or B) by spending the smallest amount of effort. In such situations, it is reasonable for the instructor to assign the grades for different tasks in such a way that the resulting overall student's effort is the largest possible. In this paper, we show that to achieve this goal, we need to assign, to each task, the number of points proportional to the efforts needed for this task.

Smaller Standard Deviation For Initial Weights Improves Performance Of Classifying Neural Networks: A Theoretical Explanation Of Unexpected Simulation Results, Diego Aguirre, Philip Hassoun, Rafael Lopez, Crystal Serrano, Marcoantonio R. Soto, Andrea Torres, Vladik Kreinovich

Departmental Technical Reports (CS)

Numerical experiments show that for classifying neural networks, it is beneficial to select a smaller deviation for initial weights that what is usually recommended. In this paper, we provide a theoretical explanation for these unexpected simulation results.

Status Quo Bias Actually Helps Decision Makers To Take Nonlinearity Into Account: An Explanation, Griselda Acosta, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main motivations for designing computer models of complex systems is to come up with recommendations on how to best control these systems. Many complex real-life systems are so complicated that it is not computationally possible to use realistic nonlinear models to find the corresponding optimal control. Instead, researchers make recommendations based on simplified -- e.g., linearized -- models. The recommendations based on these simplified models are often not realistic but, interestingly, they can be made more realistic if we "tone them down" -- i.e., consider predictions and recommendations which are close to the current status quo state. …

How To Explain That Changes In Elderlies Depression Level Are Uniformly Distributed, Griselda Acosta, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

Changes in the elderlies depression level result from a large number of small independent factors. Such situations are ubiquitous in applications. In most such cases, due to the Central Limit Theorem, the corresponding distribution is close to Gaussian. For the changes in the elderlies depression level, however, the empirical distribution is far from Gaussian: it is uniform. In this paper, we provide a possible explanation for the emergence of the uniform distribution.

80/20 Rule Partially Explains 7 Plus Minus 2 Law: General System-Based Analysis, Griselda Acosta, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

The 80/20 rule and the 7 plus minus 2 law are examples of difficult to explain empirical facts. According to the 80/20 rule, in each activity, 20% of the people contribute to the 80% of the results. The 7 plus minus 2 law means that we divide objects into 7 plus minus 2 groups -- i.e., into 5 to 9 groups. In this paper, we show that there is a relation between these two facts: namely, we show that, because of the 80/20 rule, the number of classes cannot be smaller than 5. Thus, the 80/20 rule explains the lower …

Towards A Theoretical Explanation Of How Pavement Condition Index Deteriorates Over Time, Edgar Daniel Rodriguez Velasquez, Carlos M. Chang Albitres, Vladik Kreinovich

Departmental Technical Reports (CS)

To predict how the Pavement Condition Index will change over time, practitioners use a complex empirical formula derived in the 1980s. In this paper, we provide a possible theoretical explanation for this formula, an explanation based on general ideas of invariance. In general, the existence of a theoretical explanation makes a formula more reliable; thus, we hope that our explanation will make predictions of road quality more reliable.

Geometric Explanation For An Empirical Formula Describing Our Galaxy's Warping, Julio Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the first approximation, the shape of our Galaxy -- as well as the shape of many other celestial bodies -- can be naturally explained by geometric symmetries and the corresponding invariances. As a result, we get the familiar shape of a planar spiral. A recent more detailed analysis of our Galaxy's shape has shown that the Galaxy somewhat deviates from this ideal shape: namely, it is not perfectly planar, it is somewhat warped in the third dimension. In this paper, we show that the empirical formula for this warping can also be explained by geometric symmetries and invariance.

Why Filtering Out Higher Harmonics Makes It Easier To Carry A Tune, Griselda Acosta, Eric Freudenthal, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent patent shows that filtering out higher harmonics helps people sing in-tune. In this paper, we use the general signal processing ideas to explain this empirical phenomenon. We also show that filtering out higher harmonics is the optimal way of increasing the signal-to-noise ratio -- and thus, of making it easier for people to recognize when they are signing out of tune.

Epicycles Are Almost As Good As Trigonometric Series: General System-Based Analysis, Griselda Acosta, Eric Smith, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To adequately describe the planets' motion, ancient astronomers used epicycles, when a planet makes a circular motion not around the Earth, but around a special auxiliary point which, in turn, performs a circular motion around the Earth -- or around a second auxiliary point which, in turns, rotates around the Earth, etc. Standard textbooks malign this approach by calling it bad science, but in reality, this is, in effect, trigonometric (Fourier) series -- an extremely useful tool in science and engineering. It should be mentioned, however, that the epicycles are almost as good as trigonometric series -- in the sense …

Common Sense Addition Explained By Hurwicz Optimism-Pessimism Criterion, Bibek Aryal, Laxman Bokati, Karla Godinez, Shammir Ibarra, Heyi Liu, Bofei Wang, Vladik Kreinovich

Departmental Technical Reports (CS)

If we place a can of coke that weigh 0.35 kg into a car that weighs 1 ton = 1000 kg, what will be the resulting weight of the car? Mathematics says 1000.35 kg, but common sense says 1 ton. In this paper, we show that this common sense answer can be explained by the Hurwicz optimism-pessimism criterion of decision making under interval uncertainty.

Why Area Under The Curve In Hypothesis Testing?, Griselda Acosta, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

To compare two different hypothesis testing techniques, researchers use the following heuristic idea: for each technique, they form a curve describing how the probabilities of type I and type II errors are related for this technique, and then compare areas under the resulting curves. In this paper, we provide a justification for this heuristic idea.

If Space-Time Is Discrete, We May Be Able To Solve Np-Hard Problems In Polynomial Time, Ricardo Alvarez, Nick Sims, Christian Servin, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional physics assumes that space and time are continuous. However, this reasonable model leads to some serious problems. One the approaches that physicists follow to solve these problems is to assume that the space-time is actually discrete. In this paper, we analyze possible computational consequences of this discreteness. It turns out that in a discrete space-time, we may be able to solve NP-hard problems in polynomial time.

A Natural Explanation For The Minimum Entropy Production Principle, Griselda Acosta, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

It is well known that, according to the second law of thermodynamics, the entropy of a closed system increases (or at least stays the same). In many situations, this increase is the smallest possible. The corresponding minimum entropy production principle was first formulated and explained by a future Nobelist Ilya Prigogine. Since then, many possible explanations of this principle appeared, but all of them are very technical, based on complex analysis of differential equations describing the system's dynamics. Since this phenomenon is ubiquitous for many systems, it is desirable to look for a general system-based explanation, explanation that would not …