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Full-Text Articles in Mathematics
Why Seneca Effect?, Sean R. Aguilar, Vladik Kreinovich
Why Seneca Effect?, Sean R. Aguilar, Vladik Kreinovich
Departmental Technical Reports (CS)
Already ancients noticed that decrease is usually faster than growth -- whether we talk about companies or empires. A modern researcher Ugo Bardi confirmed that this phenomenon is still valid today. He called it Seneca effect, after the ancient philosopher Seneca -- one of those who observed this phenomenon. In this paper, we provide a natural explanation for the Seneca effect.
Why Smaller-Size Objects Affect The Flow Much More Than Larger Ones: A Geometric Explanation With Applications Ranging From Volcanoes And Tornadoes To Blood, Fish, And Building Preservation, Laxman Bokati, Vladik Kreinovich
Why Smaller-Size Objects Affect The Flow Much More Than Larger Ones: A Geometric Explanation With Applications Ranging From Volcanoes And Tornadoes To Blood, Fish, And Building Preservation, Laxman Bokati, Vladik Kreinovich
Departmental Technical Reports (CS)
At first glance, the larger the object, the larger should be its effect on the surroundings -- in particular, the larger should be its effect on the surrounding flow. However, in many practical situations, we observe the opposite effect: smaller-size particles affect the flow much more than larger-size particles. This seemingly counterintuitive phenomena has been observed in many situations: lava flow in the volcanoes, air circulation in tornadoes, blood flow in a body, the effect of fish on water circulation in the ocean, and the effect of added particles on seeping water that damages historic buildings. In this paper, we …
Why Exponential Almon Lag Works Well In Econometrics: An Invariance-Based Explanation, Laxman Bokati, Vladik Kreinovich
Why Exponential Almon Lag Works Well In Econometrics: An Invariance-Based Explanation, Laxman Bokati, Vladik Kreinovich
Departmental Technical Reports (CS)
In many econometric situations, we can predict future values of relevant quantities by using an empirical formula known as exponential Almon lag. While this formula is empirically successful, there have been no convincing theoretical explanation for this success. In this paper, we provide such a theoretical explanation based on general invariance ideas.
How Order And Disorder Affect People's Behavior: An Explanation, Sofia Holguin, Vladik Kreinovich
How Order And Disorder Affect People's Behavior: An Explanation, Sofia Holguin, Vladik Kreinovich
Departmental Technical Reports (CS)
Experimental data shows that people placed in orderly rooms donate more to charity and make healthier food choices that people placed in disorderly rooms. On the other hand, people placed in disorderly rooms show more creativity. In this paper, we provide a possible explanation for these empirical phenomena.
How To Make Inflation Optimal And Fair, Sean Aguilar, Vladik Kreinovich
How To Make Inflation Optimal And Fair, Sean Aguilar, Vladik Kreinovich
Departmental Technical Reports (CS)
A reasonably small inflation helps economy as a whole -- by encouraging spending, but it also hurts people by decreasing the value of their savings. It is therefore reasonably to come up with an optimal (and fair) level of inflation, that would stimulate economy without hurting people too much. In this paper, we describe how this can be potentially done.
Why Decision Paralysis, Sean Aguilar, Vladik Kreinovich
Why Decision Paralysis, Sean Aguilar, Vladik Kreinovich
Departmental Technical Reports (CS)
If a person has a small number of good alternatives, this person can usually make a good decision, i.e., select one of the given alternatives. However, when we have a large number of good alternatives, people take much longer to make a decision -- sometimes so long that, as a result, no decision is made. How can we explain this seemingly no-optimal behavior? In this paper, we show that this "decision paralysis" can be naturally explained by using the usual decision making ideas.
Why Five Stages Of Solar Activity, Why Five Stages Of Grief, Why Seven Plus Minus Two: A General Geometric Explanation, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich
Why Five Stages Of Solar Activity, Why Five Stages Of Grief, Why Seven Plus Minus Two: A General Geometric Explanation, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
A recent paper showed that the solar activity cycle has five clear stages, and that taking theses stages into account helps to make accurate predictions of future solar activity. Similar 5-stage models have been effective in many other application area, e.g., in psychology, where a 5-stage model provides an effective description of grief. In this paper, we provide a general geometric explanations of why 5-stage models are often effective. This result also explains other empirical facts, e.g., the seven plus minus two law in psychology and the fact that only five space-time dimensions have found direct physical meaning.
Why Micro-Size Objects Affect The Flow Much More Than Larger Ones: A Geometric Explanations With Applications Ranging From Volcanoes And Tornadoes To Blood, Fish, And Building Preservation, Laxman Bokati, Vladik Kreinovich
Why Micro-Size Objects Affect The Flow Much More Than Larger Ones: A Geometric Explanations With Applications Ranging From Volcanoes And Tornadoes To Blood, Fish, And Building Preservation, Laxman Bokati, Vladik Kreinovich
Departmental Technical Reports (CS)
At first glance, the larger the object, the larger should be its effect on the surroundings -- in particular, the larger should be its effect on the surrounding flow. However, in many practical situations, we observe the opposite effect: micro-size particles affect the flow much more than larger-size particles. This seemingly counterintuitive phenomena has been observed in many situations: lava flow in the volcanoes, air circulation in tornadoes, blood flow in a body, the effect of fish on water circulation in the ocean, and the effect of added particles on seeping water that damages historic buildings. In this paper, we …
Everyone Is Above Average: Is It Possible? Is It Good?, Vladik Kreinovich, Olga Kosheleva
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 …
How To Detect The Fundamental Frequency: Approach Motivated By Soft Computing And Computational Complexity, Eric Freudenthal, Olga Kosheleva, Vladik Kreinovich
How To Detect The Fundamental Frequency: Approach Motivated By Soft Computing And Computational Complexity, Eric Freudenthal, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Psychologists have shown that most information about the mood and attitude of a speaker is carried by the lowest (fundamental) frequency. Because of this frequency's importance, even when the corresponding Fourier component is weak, the human brain reconstruct this frequency based on higher harmonics. The problems is that many people lack this ability. To help them better understand moods and attitudes in social interaction, it is therefore desirable to come up with devices and algorithms that would reconstruct the fundamental frequency. In this paper, we show that ideas from soft computing and computational complexity can be used for this purpose.
Over-Measurement Paradox: Suspension Of Thermonuclear Research Center And Need To Update Standards, Hector Reyes, Saeid Tizpaz-Niari, Vladik Kreinovich
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 …
Monotonic Bit-Invariant Permutation-Invariant Metrics On The Set Of All Infinite Binary Sequences, Irina Padilla, Vladik Kreinovich
Monotonic Bit-Invariant Permutation-Invariant Metrics On The Set Of All Infinite Binary Sequences, Irina Padilla, Vladik Kreinovich
Departmental Technical Reports (CS)
In a computer, all the information about an object is described by a sequence of 0s and 1s. At any given moment of time, we only have partial information, but as we perform more measurements and observations, we get longer and longer sequence that provides a more and more accurate description of the object. In the limit, we get a perfect description by an infinite binary sequence. If the objects are similar, measurement results are similar, so the resulting binary sequences are similar. Thus, to gauge similarity of two objects, a reasonable idea is to define an appropriate metric on …
Physical Trajectories Are Smooth, With Velocities At Least As Continuous As Brownian Motion, Olga Kosheleva, Vladik Kreinovich
Physical Trajectories Are Smooth, With Velocities At Least As Continuous As Brownian Motion, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
The fact that the kinetic energy of a particle cannot exceed its overall energy implies that the velocity -- i.e. the derivative of the trajectory -- should be bounded. This means, in effect, that all the trajectories are differentiable (smooth). However, at first glance, there seems to be no direct requirement that the velocities continuously depend on time. In this paper, we show that the properties of electromagnetic field necessitate that the velocities are continuous functions of time -- moreover, that they are at least as continuous as the Brownian motion.
Why Would Anyone Invest In A High-Risk Low-Profit Enterprise?, Olga Kosheleva, Vladik Kreinovich
Why Would Anyone Invest In A High-Risk Low-Profit Enterprise?, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Strangely enough, investors invest in high-risk low-profit enterprises as well. At first glance, this seems to contradict common sense and financial basics. However, we show that such investments make perfect sense as long as the related risks are independent from the risks of other investments. Moreover, we show that an optimal investment portfolio should allocate some investment to this enterprise.
Why Time Seems To Pass Slowly For Unpleasant Experiences And Quickly For Pleasant Experiences: An Explanation Based On Decision Theory, Laxman Bokati, Vladik Kreinovich
Why Time Seems To Pass Slowly For Unpleasant Experiences And Quickly For Pleasant Experiences: An Explanation Based On Decision Theory, Laxman Bokati, Vladik Kreinovich
Departmental Technical Reports (CS)
It is known that our perception of time depends on our level of happiness: time seems to pass slower when we have unpleasant experiences and faster if our experiences are pleasant. Several explanations have been proposed for this effect. However, these explanations are based on specific features of human memory and/or human perception, features that, in turn, need explaining. In this paper, we show that this effect can be explained on a much more basic level of decision theory, without utilizing any specific features of human memory or perception.
Why Shapley Value And Its Variants Are Useful In Machine Learning (And In Other Applications), Laxman Bokati, Olga Kosheleva, Vladik Kreinovich, Nguye Ngoc Thach
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.
Why Rectified Power (Repu) Activation Functions Are Efficient In Deep Learning: A Theoretical Explanation, Laxman Bokati, Vladik Kreinovich, Joseph Baca, Natasha Rovelli
Why Rectified Power (Repu) Activation Functions Are Efficient In Deep Learning: A Theoretical Explanation, Laxman Bokati, Vladik Kreinovich, Joseph Baca, Natasha Rovelli
Departmental Technical Reports (CS)
At present, the most efficient machine learning techniques is deep learning, with neurons using Rectified Linear (ReLU) activation function s(z) = max(0,z), in many cases, the use of Rectified Power (RePU) activation functions (s(z))^p -- for some p -- leads to better results. In this paper, we explain these results by proving that RePU functions (or their "leaky" versions) are optimal with respect that all reasonable optimality criteria.
Efficient Algorithms For Data Processing Under Type-3 (And Higher) Fuzzy Uncertainty, Vladik Kreinovich, Olga Kosheleva, Patricia Melin, Oscar Castillo
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 Flash Radiotherapy Is Efficient: A Possible Explanation, Julio Urenda, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
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.
How To Represent Uncertainty Via Qudits: Probability Distributions, Regular, Intuitionistic, And Picture Fuzzy Sets, F-Transforms, Etc., Olga Kosheleva, Vladik Kreinovich
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 Rejuvenation Attempts Often Lead To Cancer And Why Cyclic Rejuvenation Is Better: A Simple Qualitative Explanation, Olga Kosheleva, Vladik Kreinovich
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, …
Hawthorne Effect: An Explanation Based On Decision Theory, Sofia Holguin, Vladik Kreinovich, Nguyen Hoang Phuong
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.
How To Detect (And Analyze) Independent Subsystems Of A Black-Box (Or Grey-Box) System, Saeid Tizpaz-Niari, Olga Kosheleva, Vladik Kreinovich
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.
Computational Paradox Of Deep Learning: A Qualitative Explanation, Jonatan Contreras, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
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.
Why Should Exactly 1/4 Be Returned To The Original Owner: An Economic Explanation Of An Ancient Recommendation, Olga Kosheleva, Vladik Kreinovich
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.
Towards Better Ways To Compute The Overall Grade For A Class, Christian Servin, Olga Kosheleva, Vladik Kreinovich
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 …
Estimating Skewness And Higher Central Moments Of An Interval-Valued Fuzzy Set, Juan Carlos Figueroa-Garcia, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich
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.
Why Convex Combination Is An Effective Crossover Operation In Continuous Optimization: A Theoretical Explanation, Kelly Cohen, Olga Kosheleva, Vladik Kreinovich
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.
Why Quantile Regression Works Well In Economics: A Partial Explanation, Olga Kosheleva, Vassilis G. Kaburlasos, Vladik Kreinovich, Roengchai Tansuchat
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
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.