Physical Sciences and Mathematics Commons^™

Why Ideas First Appear In Informal Form? Why It Is Very Difficult To Know Yourself? Fuzzy-Based Explanation, Miroslav Svitek, Vladik Kreinovich

Departmental Technical Reports (CS)

To a lay person reading about history of physics, it may sound as if the progress of physics comes from geniuses whose inspiration leads them to precise equations that -- almost magically -- explain all the data: this is what Newton did with mechanics, this is what Schroedinger did with quantum physics, this is what Einstein did with gravitation. However, a deeper study of history of physics shows that in all these cases, these geniuses did not start from scratch -- they formalized ideas that first appeared in imprecise ("fuzzy") form. In this paper, we explain -- on the qualitative …

Unreachable Statements Are Inevitable In Software Testing: Theoretical Explanation, Francisco Zapata, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

Business gurus recommend that an organization should have, in addition to clearly described realistic goals, also additional aspirational goals -- goals for which we may not have resources and which most probably will not be reached at all. At first glance, adding such a vague goal cannot lead to a drastic change in how the company operates, but surprisingly, for many companies, the mere presence of such aspirational goals boosts the company's performance. In this paper, we show that a simple geometric model of this situation can explain the unexpected success of aspirational goals.

Unexpected Economic Consequence Of Cloud Computing: A Boost To Algorithmic Creativity, Francisco Zapata, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

While theoreticians have been designing more and more efficient algorithms, in the past, practitioners were not very interested in this activity: if a company already owns computers that provide computations in required time, there is nothing to gain by using faster algorithms. We show the situation has drastically changed with the transition to cloud computing: many companies have not yet realized this, but with the transition to cloud computing, any algorithmic speed up leads to immediate financial gain. This also has serious consequences for the whole computing profession: there is a need for professionals better trained in subtle aspects of …

Why Core Curriculum? Why Art And Nature Enhance Creativity? A Mathematical Explanation, Olga Kosheleva, Vladik Kreinovich, Christian Servin

Departmental Technical Reports (CS)

Teaching is not easy. One of the main reasons why it is not easy is that the existing descriptions of the teaching process are not very precise -- and thus, we cannot use the usual optimization techniques, techniques which require a precise model of the corresponding phenomenon. It is therefore desirable to come up with a precise description of the learning process. To come up with such a description, we notice that on the set of all possible states of learning, there is a natural order s ≤ s' meaning that we can bring the student from the state s …

Why Aspirational Goals: Geometric Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Business gurus recommend that an organization should have, in addition to clearly described realistic goals, also additional aspirational goals -- goals for which we may not have resources and which most probably will not be reached at all. At first glance, adding such a vague goal cannot lead to a drastic change in how the company operates, but surprisingly, for many companies, the mere presence of such aspirational goals boosts the company's performance. In this paper, we show that a simple geometric model of this situation can explain the unexpected success of aspirational goals.

A Natural Causality-Motivated Description Of Learning, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Teaching is not easy. One of the main reasons why it is not easy is that the existing descriptions of the teaching process are not very precise -- and thus, we cannot use the usual optimization techniques, techniques which require a precise model of the corresponding phenomenon. It is therefore desirable to come up with a precise description of the learning process. To come up with such a description, we notice that on the set of all possible states of learning, there is a natural order s ≤ s' meaning that we can bring the student from the state s …

Data Processing Under Fuzzy Uncertainty: Towards More Efficient Algorithm, Hung T. Nguyen, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we need to process data under fuzzy uncertainty: we have fuzzy information about the algorithm's input, and we want to find the resulting information about the algorithm's output. It is known that this problem can be reduced to computing the range of the algorithm over alpha-cuts of the input. Since the fuzzy degrees are usually known with accuracy at best 0.1, it is sufficient to repeat this range-computing procedure for 11 values alpha = 0, 0.1, ..., 1.0. However, a straightforward application of this idea requires 11 times longer computation time than each range estimation -- …

Why Pre-Teaching: A Geometric Explanation, Olga Kosheleva, Vladik Kreinovich, Christian Servin

Departmental Technical Reports (CS)

Traditionally, subjects are taught in sequential order: e.g., first, students study algebra, then they use the knowledge of algebra to study the basis ideas of calculus. In this traditional scheme, teachers usually do not explain any calculus ideas before students are ready – since they believe that this would only confuse students. However, lately, empirical evidence has shows that, contrary to this common belief, pre-teaching – when students get a brief introduction to the forthcoming new topic before this topic starts – helps students learn. In this paper, we provide a geometric explanation for this unexpected empirical phenomenon.

Why Gaussian Copulas Are Ubiquitous In Economics: Fuzzy-Related Explanation, Chon Van Le, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life situations, deviations are caused by a large number of independent factors. It is known that in such situations, the distribution of the resulting deviations is close to Gaussian, and thus, that the copulas -- that describe the multi-D distributions as a function of 1-D (marginal) ones -- are also Gaussian. In the past, these conclusions were also applied to economic phenomena, until the 2008 crisis showed that in economics, Gaussian models can lead to disastrous consequences. At present, all economists agree that the economic distributions are not Gaussian -- however, surprisingly, Gaussian copulas still often provide an …

How To Describe Relative Approximation Error? A New Justification For Gustafson's Logarithmic Expression, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

How can we describe relative approximation error? When the value b approximate a value a, the usual description of this error is the ratio |b − a|/|a|. The problem with this approach is that, contrary to our intuition, we get different numbers gauging how well a approximates b and how well b approximates a. To avoid this problem, John Gustafson proposed to use the logarithmic measure |ln(b/a)|. In this paper, we show that this is, in effect, the only regular scale-invariant way to describe the relative approximation error.

Video Or Text? Bullets Or No Bullets? Why Not Both?, Olga Kosheleva, Vladik Kreinovich, Christian Servin

Departmental Technical Reports (CS)

Some students – which are, in terms of pop-psychology – more left-brain – prefer linear exposition, others – more right-brain ones – prefer 2-D images and texts with visual emphasis (e.g., with bullets). At present, instructors try to find a middle grounds between these two audiences, but why not prepare each material in two ways, aimed at both audiences?

Computing The Range Of A Function-Of-Few-Linear-Combinations Under Linear Constraints: A Feasible Algorithm, Salvador Robles, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we need to find the range of a given function under interval uncertainty. For nonlinear functions -- even for quadratic ones -- this problem is, in general, NP-hard; however, feasible algorithms exist for many specific cases. In particular, recently a feasible algorithm was developed for computing the range of the absolute value of a Fourier coefficient under uncertainty. In this paper, we generalize this algorithm to the case when we have a function of a few linear combinations of inputs. The resulting algorithm also handles the case when, in addition to intervals containing each input, we …

Commonsense-Continuous Dynamical Systems -- Stationary States, Prediction, And Reconstruction Of The Past: Fuzzy-Based Analysis, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional analysis of dynamical systems usually assumes that the mapping is continuous -- in precise mathematical sense. However, as many formal definitions, the mathematical definition of continuity does not always adequately capture the commonsense notion of continuity: that small changes in the input should lead to small changes in the output. In this paper, we provide a natural fuzzy-based formalization of this intuitive notion, and analyze how the requirement of commonsense continuity affects the properties of dynamical systems. Specifically, we show that for such systems, the set of fixed points is closed and convex, and that the only such systems …

Why Online Teaching Amplifies The Differences Between Instructors' Success, Olga Kosheleva, Vladik Kreinovich, Christian Servin

Departmental Technical Reports (CS)

Empirical studies show that online teaching amplifies the differences between instructors: more successful instructors become even more successful, while the results of the instructors who were not very successful becomes even worse. There is a simple explanation for why the performance of not-perfect instructors decreases: in online teaching, there is less feedback, so these instructors get an indication that their teaching strategies do not work well even later than usual and thus, have fewer time to correct their teaching. However, the fact that the efficiency of good instructors rises is a mystery. In this paper, we provide a possible explanation …

How To React To Student Evaluations, Olga Kosheleva, Vladik Kreinovich, Christian Servin

Departmental Technical Reports (CS)

If most students comment that the course was too fast, a natural idea is to slow it down. If most students comment that the course was too slow, a natural idea is to speed it up. But what if half the students think the speed was too fast and half that the speed was too slow? A frequent reaction to such a situation is to conclude that the speed was just right and not change the speed the next time, but this may not be the right reaction: under the same speed, half of the students will struggle and may …

How To Deal With Conflict Of Interest Situations When Selecting The Best Submission, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations when we need to select the best submission -- the best paper, the best candidate, etc. -- there are so few experts that we cannot simply dismiss all the experts who have conflict of interest: we do not want them to judge their own submissions, but we would like to take into account their opinions of all other submissions. How can we take these opinions into account? In this paper, we show that a seemingly reasonable idea can actually lead to bias, and we explain how to take these opinions into account without biasing the final …

Why People Tend To Overestimate Joint Probabilities, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that, in general, people overestimate the probabilities of joint events. In this paper, we provide an explanation for this phenomenon -- as explanation based on Laplace Indeterminacy Principle and Maximum Entropy approach.

What Is A Natural Probability Distribution On The Class Of All Continuous Functions: Maximum Entropy Approach Leads To Wiener Measure, Vladik Kreinovich, Saeid Tizpaz-Niari

Departmental Technical Reports (CS)

While many data processing techniques assume that we know the probability distributions, in practice, we often only have a partial information about these probabilities -- so that several different distributions are consistent with our knowledge. Thus, to apply these data processing techniques, we need to select one of the possible probability distributions. There is a reasonable approach for such selection -- the Maximum Entropy approach. This approach selects a uniform distribution if all we know is that the random variable if located in an interval; it selects a normal distribution if all we know is the mean and the variance. …

Search Under Uncertainty Should Be Randomized: A Lesson From The 2021 Nobel Prize In Medicine, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life situations, we know that one of several objects has the desired property, but we do not know which one. To find the desired object, we need to test these objects one by one. In situations when we have no additional information, there is no reason to prefer any testing order and thus, a usual recommendation is to test them in any order. This is usually interpreted as ordering the objects in the increasing value of some seemingly unrelated quantity. A possible drawback of this approach is that it may turn out that the selected quantity is correlated …

Why Physical Power Laws Usually Have Rational Exponents, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Many physical dependencies are described by power laws y=A*xa, for some exponent a. This makes perfect sense: in many cases, there are no preferred measuring units for the corresponding quantities, so the form of the dependence should not change if we simply replace the original unit with a different one. It is known that such invariance implies a power law. Interestingly, not all exponents are possible in physical dependencies: in most cases, we have power laws with rational exponents. In this paper, we explain the ubiquity of rational exponents by taking into account that in many case, there is also …

Can Physics Attain Its Goals: Extending D'Agostino's Analysis To 21st Century And Beyond, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In his 2000 seminal book, Silvo D'Agostino provided the detailed overview of the history of ideas underlying 19th and 20th century physics. Now that we are two decades into the 21st century, a natural question is: how can we extend his analysis to the 21st century physics -- and, if possible, beyond, to try to predict how physics will change? To perform this analysis, we go beyond an analysis of what happened and focus more on why para-digm changes happened in the history of physics. To better understand these paradigm changes, we analyze now only what were the main ideas …

Why Sine Membership Functions, Sofia Holguin, Javier Viaña, Kelly Cohen, Anca Ralescu, Vladik Kreinovich

Departmental Technical Reports (CS)

In applications of fuzzy techniques to several practical problems -- in particular, to the problem of predicting passenger flows in the airports -- the most efficient membership function is a sine function; to be precise, a portion of a sine function between the two zeros. In this paper, we provide a theoretical explanation for this empirical success.

Need To Combine Interval And Probabilistic Uncertainty: What Needs To Be Computed, What Can Be Computed, What Can Be Feasibly Computed, And How Physics Can Help, Julio Urenda, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

In many practical situations, the quantity of interest is difficult to measure directly. In such situations, to estimate this quantity, we measure easier-to-measure quantities which are related to the desired one by a known relation, and we use the results of these measurement to estimate the desired quantity. How accurate is this estimate?

Traditional engineering approach assumes that we know the probability distributions of measurement errors; however, in practice, we often only have partial information about these distributions. In some cases, we only know the upper bounds on the measurement errors; in such cases, the only thing we know about …

Macrocausality Implies Lorenz Group: A Physics-Related Comment On Guts's Results, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that, in the space-time of Special Relativity, causality implies Lorenz group, i.e., if we know which events can causally influence each other, then, based on this information, we can uniquely reconstruct the affine structure of space-time. When the two events are very close, quantum effects, with their probabilistic nature, make it difficult to detect causality. So, the following question naturally arises: can we uniquely reconstruct the affine structure if we only know causality for events which are sufficiently far away from each other? Several positive answers to this question were provided in a recent paper by Alexander …

Towards Optimal Techniques Intermediate Between Interval And Affine, Affine And Taylor, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In data processing, it is important to gauge how input uncertainty affects the results of data processing. Several techniques have been proposed for this gauging, from interval to affine to Taylor techniques. Some of these techniques result in more accurate estimates but require longer computation time, others' results are less accurate but can be obtained faster. Sometimes, we do not have enough time to use more accurate (but more time-consuming) techniques, but we have more time than needed for less accurate ones. In such cases, it is desirable to come up with intermediate techniques that would utilize the available additional …

Discrete Causality Implies Lorenz Group: Case Of 2-D Space-Times, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that for Minkowski space-times of dimension larger than 2, any causality-preserving transformation is linear. It is also known that in a 2-D space-time, there are many nonlinear causality-preserving transformations. In this paper, we show that for 2-D space-times, if we restrict ourselves to discrete space-times, then linearity is retained: only linear transformation preserve causality.

How To Elicit Complex-Valued Fuzzy Degrees, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the traditional fuzzy logic, an expert's degree of certainty in a statement is described by a single number from the interval [0,1]. However, there are situations when a single number is not sufficient: e.g., a situation when we know nothing and a situation in which we have a lot of arguments for a given statement and an equal number of arguments against it are both described by the same number 0.5. Several techniques have been proposed to distinguish between such situations. The most widely used is interval-valued technique, where we allow the expert to describe his/her degree of certainty …