Mathematics Commons^™

Over-Measurement Paradox: Suspension Of Thermonuclear Research Center And Need To Update Standards, Hector Reyes, Saeid Tizpaz-Niari, Vladik Kreinovich

Departmental Technical Reports (CS)

In general, the more measurements we perform, the more information we gain about the system and thus, the more adequate decisions we will be able to make. However, in situations when we perform measurements to check for safety, the situation is sometimes opposite: the more additional measurements we perform beyond what is required, the worse the decisions will be: namely, the higher the chance that a perfectly safe system will be erroneously classified as unsafe and therefore, unnecessary additional features will be added to the system design. This is not just a theoretical possibility: exactly this phenomenon is one of …

Everyone Is Above Average: Is It Possible? Is It Good?, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

Starting with the 1980s, a popular US satirical radio show described a fictitious town Lake Wobegon where ``all children are above average'' -- parodying the way parents like to talk about their children. This everyone-above-average situation was part of the fiction since, if we interpret the average in the precise mathematical sense, as average over all the town's children, then such a situation is clearly impossible. However, usually, when parents make this claim, they do not mean town-wise average, they mean average over all the kids with whom their child directly interacts. Somewhat surprisingly, it turns out that if we …

Why Shapley Value And Its Variants Are Useful In Machine Learning (And In Other Applications), Laxman Bokati, Olga Kosheleva, Vladik Kreinovich, Nguye Ngoc Thach

Departmental Technical Reports (CS)

Shapley value -- a useful way to allocate gains in cooperative games -- has been very successful in machine learning (and in other applications beyond cooperative games). This success is somewhat puzzling, since the usual derivation of the Shapley value is based on requirements like additivity that are natural in cooperative games and but not ents like additivity and is, thus, applicable in the machine learning case as well.

Development Of The Implementation Of Iot Monitoring System Based On Node-Red Technology, Anvar Kabulov, Inomjon Yarashov, Salamat Mirzataev

Karakalpak Scientific Journal

This article describes how to design and implement a process for storing environmental information in a database using the Internet of Things. The problems that need to be solved with the help of this IoT system are the growing demand for forecasts in the world, the demand of the world market for a new sustainable method of implementing the digitization environment through the Internet of Things. The design was implemented using Arduino, Node-Red and sensors, selected when choosing a component based on the required parameters and sent to the database for monitoring and processing. A study of previous work and …

Using Graph Theoretical Methods And Traceroute To Visually Represent Hidden Networks, Jordan M. Sahs

UNO Student Research and Creative Activity Fair

Within the scope of a Wide Area Network (WAN), a large geographical communication network in which a collection of networking devices communicate data to each other, an example being the spanning communication network, known as the Internet, around continents. Within WANs exists a collection of Routers that transfer network packets to other devices. An issue pertinent to WANs is their immeasurable size and density, as we are not sure of the amount, or the scope, of all the devices that exists within the network. By tracing the routes and transits of data that traverses within the WAN, we can identify …

(R1881) Impatient Customers In Queueing System With Optional Vacation Policies And Power Saving Mode, Poonam Gupta, Rajni Gupta, Sangeeta Malik

Applications and Applied Mathematics: An International Journal (AAM)

In this manuscript, a queueing system with two optional vacation policies, power-saving mode under reneging and retention of reneged customers in both vacations is analyzed. If the server is free, it chooses either of the vacations, classical vacation or working vacation. During vacations, the customers may get impatient due to delays and may leave the system, but they are retained in the system with some convincing mechanisms. On vacation completion, if the system is empty, the server is turned off to facilitate better utilization of the resources. Some of the operating system characteristics are derived using the probability generating functions …

Analysis Of A Quantum Attack On The Blum-Micali Pseudorandom Number Generator, Tingfei Feng

Mathematical Sciences Technical Reports (MSTR)

In 2012, Guedes, Assis, and Lula proposed a quantum attack on a pseudorandom number generator named the Blum-Micali Pseudorandom number generator. They claimed that the quantum attack can outperform classical attacks super-polynomially. However, this paper shows that the quantum attack cannot get the correct seed and provides another corrected algorithm that is in exponential time but still faster than the classical attack. Since the original classical attacks are in exponential time, the Blum-Micali pseudorandom number generator would be still quantum resistant.

Why Flash Radiotherapy Is Efficient: A Possible Explanation, Julio Urenda, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

Usually, a cancer radiotherapy session lasts between 10 to 20 minutes. Technically, it is possible to transmit the dose faster, but traditionally, medical doctors were reluctant to do it, since they were afraid of negative effects of such a speedy treatment. Recent experiments show, however, that these fears are unfounded; moreover, transmitting the whole radiation dose in a shorter time turns out to be more beneficial for the patients. In this paper, we provide a possible geometric explanation for this empirical phenomenon.

Why Rejuvenation Attempts Often Lead To Cancer And Why Cyclic Rejuvenation Is Better: A Simple Qualitative Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Since the 1960s, biologists have shown that, contrary to the previous belief that ageing is irreversible, many undesirable biological effects of ageing can be reversed. First attempts to perform this reversal on living creatures were not fully successful: while mice achieved some rejuvenation, many of these rejuvenated mice developed cancer. Later experiments showed that these cancers can be avoided if we apply cyclic rejuvenation: a short period of rejuvenation followed by a longer pause. This modified strategy led to recent successes of mice that recovered their age-deteriorated vision and mice that recovered their heart tissue after a heart attack. However, …

Computational Paradox Of Deep Learning: A Qualitative Explanation, Jonatan Contreras, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

In general, the more unknowns in a problem, the more computational efforts is necessary to find all these unknowns. Interestingly, in state-of-the-art machine learning methods like deep learning, computations become easier when we increase the number of unknown parameters way beyond the number of equations. In this paper, we provide a qualitative explanation for this computational paradox.

Efficient Algorithms For Data Processing Under Type-3 (And Higher) Fuzzy Uncertainty, Vladik Kreinovich, Olga Kosheleva, Patricia Melin, Oscar Castillo

Departmental Technical Reports (CS)

It is known that to more adequately describe expert knowledge, it is necessary to go from the traditional (type-1) fuzzy techniques to higher order ones: type-2, probably type-3 and even higher. Until recently, only type-1 and type-2 fuzzy sets were used in practical applications. However, lately, it turned out that type-3 fuzzy sets are also useful in some applications. Because of this practical importance, it is necessary to design efficient algorithms for data processing under such type-3 (and higher order) fuzzy uncertainty. In this paper, we show how we can combine known efficient algorithms for processing type-1 and type-2 uncertainty …

Why Quantiles Are A Good Description Of Volatility In Economics: An Alternative Explanation, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich, Kittawit Autchariyapanitkul

Departmental Technical Reports (CS)

In econometrics, volatility of an investment is usually described by its Value-at-Risk (VaR), i.e., by an appropriate quantile of the corresponding probability distribution. The motivations for selecting VaR are largely empirical: VaR provides a more adequate description of what people intuitively perceive as risk. In this paper, we analyze this situation from the viewpoint of decision theory, and we show that this analysis naturally leads to the Value-at-Risk, i.e., to a quantile.

Interestingly, this analysis also naturally leads to an optimization problem related to quantile regression.

Why Decreased Gaps Between Brain Cells Cause Severe Headaches: A Symmetry-Based Geometric Explanation, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

When we analyze biological tissue under the microscope, cells are directly neighboring each other, with no gaps between them. However, a more detailed analysis shows that in vivo, there are small liquid-filled gaps between the cells, and these gaps are important: e.g., in abnormal situations, when the size of the gaps between brain cells decreases, this leads to severe headaches and other undesired effects. At present, there is no universally accepted explanation for this phenomenon. In this case, we show that the analysis of correpsonding geometric symmetries empirical phenomenon leads to a natural explanation for this effect.

How To Represent Uncertainty Via Qudits: Probability Distributions, Regular, Intuitionistic, And Picture Fuzzy Sets, F-Transforms, Etc., Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

While modern computers are fast, there are still many important practical situations in which we need even faster computations. It turns out that, due to the fact that the speed of all communications is limited by the speed of light, the only way to make computers drastically faster is to drastically decrease the size of computer's components. When we decrease their size to sizes comparable with micro-sizes of individual molecules, it becomes necessary to take into account specific physics of the micro-world -- known as quantum physics. Traditional approach to designing quantum computers -- i.e., computers that take effect of …

Why Rarity Score Is A Good Evaluation Of A Non-Fungible Token, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the new forms of investment is investing in so-called non-fungible tokens -- unique software objects associated with different real-life objects like songs, painting, photos, videos, characters in computer games, etc. Since these tokens are a form of financial investment, investors would like to estimate the fair price of such tokens. For tokens corresponding to objects that have their own price -- such as a song or a painting -- a reasonable estimate is proportional to the price of the corresponding object. However, for tokens corresponding to computer game characters, we cannot estimate their price this way. Based on …

Towards Better Ways To Compute The Overall Grade For A Class, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional way to compute the overall grade for the class is to use the weighted sum of the grades for all the assignments and exams, including the final exam. In terms of encouraging students to study hard throughout the semester, this grading scheme is better than an alternative scheme, in which all that matters is the grade on the final exam: in contrast to this alternative scheme, in the weighted-sum approach, students are penalized if they did not do well in the beginning of the semester. In practice, however, instructors sometimes deviate from the weighted-sum scheme: indeed, if the weighted …

How To Combine Expert Estimates? How To Estimate Probability In The Intersection Of Two Populations?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we consider two different practical problems that turned to be mathematically similar: (1) how to combine two expert-provided probabilities of some event and (2) how to estimate the frequency of a certain phenomenon (e.g., illness) in an intersection of two populations if we know the frequencies in each of these populations. In both cases, we use the maximum entropy approach to come up with a solution.

Why Quantile Regression Works Well In Economics: A Partial Explanation, Olga Kosheleva, Vassilis G. Kaburlasos, Vladik Kreinovich, Roengchai Tansuchat

Departmental Technical Reports (CS)

To get a better picture of the future behavior of different economics-related quantities, we need to be able to predict not only their mean values, but also their distribution. For example, it is desirable not only to predict future average income, but also to predict the future distribution of income. One of the convenient ways to describe a probability distribution is by using alpha-quantiles such as medians (corresponding to alpha = 0.5), quartiles (corresponding to alpha = 0.25 and alpha = 0.75), etc. In principle, an alpha-quantile of the desired future quantity can depend on beta-quantiles of current distributions corresponding …

Invariance-Based Approach Explains Empirical Formulas From Pavement Engineering To Deep Learning, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many application areas, there are effective empirical formulas that need explanation. In this paper, we focus on two such challenges: neural networks, where a so-called softplus activation function is known to be very efficient, and pavement engineering, where there are empirical formulas describing the dependence of the pavement strength on the properties of the underlying soil. We show that similar scale-invariance ideas can explain both types of formulas -- and, in the case of pavement engineering, invariance ideas can lead to a new formula that combines the advantages of several known ones.

How To Detect (And Analyze) Independent Subsystems Of A Black-Box (Or Grey-Box) System, Saeid Tizpaz-Niari, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Often, we deal with black-box or grey-box systems where we can observe the overall system's behavior, but we do not have access to the system's internal structure. In many such situations, the system actually consists of two (or more) independent components: a) how can we detect this based on observed system's behavior? b) what can we learn about the independent subsystems based on the observation of the system as a whole? In this paper, we provide (partial) answers to these questions.

Hawthorne Effect: An Explanation Based On Decision Theory, Sofia Holguin, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

It is known that people feel better (and even work better) if someone pays attention to them; this is known as the Hawthorne effect. At first glance, it sounds counter-intuitive: this attention does not bring you any material benefits, so why would you feel better? If you are sick and someone gives you medicine, this will make you feel better, but if someone just pays attention, why does that make you feel better? In this paper, we use the general ideas of decision theory to explain this seemingly counterintuitive phenomenon.

Why Should Exactly 1/4 Be Returned To The Original Owner: An Economic Explanation Of An Ancient Recommendation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

What if someone bought a property in good faith, not realizing that this property was unjustly confiscated from the previous owner? In such situations, if the new owner decided to sell this property, Talmud recommended that a fair way is to return 1/4 of the selling price to the original owner. However, it does not provide any explanation of why exactly 1/4 -- and not any other portion -- is to be returned. In this paper, we provide an economic explanation for this recommendation, an explanation that fits well with other ancient recommendations about debts.

Why Convex Combination Is An Effective Crossover Operation In Continuous Optimization: A Theoretical Explanation, Kelly Cohen, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When evolutionary computation techniques are used to solve continuous optimization problems, usually, convex combination is used as a crossover operation. Empirically, this crossover operation works well, but this success is, from the foundational viewpoint, a challenge: why this crossover operation works well is not clear. In this paper, we provide a theoretical explanation for this empirical success.

Estimating Skewness And Higher Central Moments Of An Interval-Valued Fuzzy Set, Juan Carlos Figueroa-Garcia, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A known relation between membership functions and probability density functions allows us to naturally extend statistical characteristics like central moments to the fuzzy case. In case of interval-valued fuzzy sets, we have several possible membership functions consistent with our knowledge. For different membership functions, in general, we have different values of the central moments. It is therefore desirable to compute, in the interval-valued fuzzy case, the range of possible values for each such moment. In this paper, we provide efficient algorithms for this computation.

Implementation Of A Least Squares Method To A Navier-Stokes Solver, Jada P. Lytch, Taylor Boatwright, Ja'nya Breeden

Rose-Hulman Undergraduate Mathematics Journal

The Navier-Stokes equations are used to model fluid flow. Examples include fluid structure interactions in the heart, climate and weather modeling, and flow simulations in computer gaming and entertainment. The equations date back to the 1800s, but research and development of numerical approximation algorithms continues to be an active area. To numerically solve the Navier-Stokes equations we implement a least squares finite element algorithm based on work by Roland Glowinski and colleagues. We use the deal.II academic library , the C++ language, and the Linux operating system to implement the solver. We investigate convergence rates and apply the least squares …

From Historically First "Unary" Numbers, Through Egyptian Fractions, Roman Numerals, Leibniz's Binary Numbers And Kepler's Fractions To Modern Ideas Such As Calkin-Wilf Tree: A Unified Approach To Representing Natural Numbers And Fractions, Olga Kosheleva, Vladik Kreinovich, Christian Servin

Departmental Technical Reports (CS)

In elementary mathematics classes, students are often overwhelmed by different representations of numbers and corresponding operations: usual fractions, decimal representations, binary numbers, etc. What often helps is when students learn the history of these representations, see the limitations of seemingly reasonable representations like Roman numerals, and how other representations overcame these limitations. Still, history was developed somewhat randomly, so the historical sequence is still somewhat chaotic. We believe that providing a unified approach for all these representations would help describe their sequence in a more logical way and thus, help the students even more.

In our analysis, we explore the …

Fair Bankruptcy Solutions Under Interval Uncertainty, Uyen Pham, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

If the overall amount of the company's assets is smaller than its total debts, then a fair solution is to give, to each creditor, the amount proportional to the corresponding debt, e.g., 10 center for each dollar or 50 cents for each dollar. But what if the debt amounts are not known exactly, and for some creditors, we only know the lower and upper bounds on the actual debt amount? What division will be fair in such a situation? In this paper, we show that the only fair solution is to make payments proportional to an appropriate convex combination of …

How Probable Is A Revolution? A Natural Relu-Like Formula That Fits The Historical Data, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In his recent book "Principles for Dealing with the Changing World Order", Ray Dalio considered many historical crisis situations, and came up with several data points showing how the probability of a revolution or a civil war depends on the number of economic red flags. In this paper, we provide a simple empirical formula that is consistent with these data points.

Shape Of An Egg: Towards A Natural Simple Universal Formula, Sofia Holguin, Vladik Kreinovich

Departmental Technical Reports (CS)

Eggs of different bird species have different shapes. There exists formulas for describing the usual egg shapes -- e.g., the shapes of chicken eggs. However, some egg shapes are more complex. A recent paper proposed a general formula for describing all possible egg shapes; however this formula is purely empirical, it does not have any theoretical foundations. In this paper, we use the theoretical analysis of the problem to provide an alternative -- theoretically justified -- general formula. Interestingly, the new general formula is easier to compute than the previously proposed one.

Freedom Of Will, Non-Uniqueness Of Cauchy Problem, Fractal Processes, Renormalization, Phase Transitions, And Stealth Aircraft, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

We all know that we can make different decisions, decisions that change -- at least locally -- the state of the world. This is what is known as freedom of will. On the other hand, according to physics, the future state of the world is uniquely pre-determined by its current state, so there is no room for freedom of will. How can we resolve this contradiction? In this paper, we analyze this problem, and we show that many physical phenomena can help resolve this contradiction: fractal character of equations, renormalization, phase transitions, etc. Usually, these phenomena are viewed as somewhat …