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Articles 241 - 270 of 667
Full-Text Articles in Mathematics
Numerical Study Of Cahn-Hilliard Equations, Oula Khouzam
Numerical Study Of Cahn-Hilliard Equations, Oula Khouzam
Open Access Theses & Dissertations
In this thesis we study the well-known first-order Eyre's convex splitting numerical scheme for solving the Cahn-Hilliard equation and theoretically prove and numerically demonstrate the key properties of the scheme namely: mass conservation, unique solvability and unconditional stability. While the convex splitting scheme has been around for over two decades, explicit proofs for these important properties for the fourth order Cahn-Hillard equation are not directly available in the existing literature. This thesis aims to bridge this gap by providing the complete proofs of the aforementioned key properties of the scheme and numerically demonstrating the performance of the scheme.
Mathematical Modeling Of Potassium Modulated Viral Infection, Zaira Elizabeth Mather
Mathematical Modeling Of Potassium Modulated Viral Infection, Zaira Elizabeth Mather
Open Access Theses & Dissertations
In recent years, there is a growing interest in the investigation of using potassium to treat virus infections. In the region of infection, there is a biological observation of extracel- lular potassium level being typically very low whereas the intracellular potassium levels are much higher. There are numerous biological studies showing that elevated potassium levels in the extracellular membrane tends to block virus infections. A recent effort in this direction is a collaborative research conducted by mathematicians and biologists from the University of Texas at El Paso, New Mexico State University, and the University of New Mexico, where we develop …
Why Rectified Linear Unit Is Efficient In Machine Learning: One More Explanation, Barnabas Bede, Vladik Kreinovich, Uyen Pham
Why Rectified Linear Unit Is Efficient In Machine Learning: One More Explanation, Barnabas Bede, Vladik Kreinovich, Uyen Pham
Departmental Technical Reports (CS)
In many applications, in particular, in econometric application, deep learning techniques are very effective. In this paper, we provide a new explanation for why rectified linear units -- the main units of deep learning -- are so effective. This explanation is similar to the usual explanation of why Gaussian (normal) distributions are ubiquitous -- namely, it is based on an appropriate limit theorem.
Game-Theoretic Approach Explains -- On The Qualitative Level -- The Antigenic Map Of Covid-19 Variants, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
Game-Theoretic Approach Explains -- On The Qualitative Level -- The Antigenic Map Of Covid-19 Variants, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
Departmental Technical Reports (CS)
To effectively defend the population against future variants of Covid-19, it is important to be able to predict how it will evolve. For this purpose, it is necessary to understand the logic behind its evolution so far. At first glance, this evolution looks random and thus, difficult to predict. However, we show that already a simple game-theoretic model can actually explain -- on the qualitative level -- how this virus mutated so far.
Why Constraint Interval Arithmetic Works Well: A Theorem Explains Empirical Success, Barnabas Bede, Marina Tuyako Mizukoshi, Weldon Lodwick, Martine Ceberio, Vladik Kreinovich
Why Constraint Interval Arithmetic Works Well: A Theorem Explains Empirical Success, Barnabas Bede, Marina Tuyako Mizukoshi, Weldon Lodwick, Martine Ceberio, Vladik Kreinovich
Departmental Technical Reports (CS)
Often, we are interested in a quantity that is difficult or impossible to measure directly, e.g., tomorrow's temperature. To estimate this quantity, we measure auxiliary easier-to-measure quantities that are related to the desired ones by a known dependence, and use the known relation to estimate the desired quantity. Measurements are never absolutely accurate, there is always a measurement error, i.e., a non-zero difference between the measurement result and the actual (unknown) value of the corresponding quantity. In many practical situations, the only information that we have about each measurement error is the bound on its absolute value. In such situations, …
When Is Deep Learning Better And When Is Shallow Learning Better: Qualitative Analysis, Salvador Robles Herrera, Martine Ceberio, Vladik Kreinovich
When Is Deep Learning Better And When Is Shallow Learning Better: Qualitative Analysis, Salvador Robles Herrera, Martine Ceberio, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, deep neural networks work better than the traditional "shallow" ones, however, in some cases, the shallow neural networks lead to better results. At present, deciding which type of neural networks will work better is mostly done by trial and error. It is therefore desirable to come up with some criterion of when deep learning is better and when shallow is better. In this paper, we argue that this depends on whether the corresponding situation has natural symmetries: if it does, we expect deep learning to work better, otherwise we expect shallow learning to be more effective. …
Why Deep Neural Networks: Yet Another Explanation, Ricardo Lozano, Ivan Montoya Sanchez, Vladik Kreinovich
Why Deep Neural Networks: Yet Another Explanation, Ricardo Lozano, Ivan Montoya Sanchez, Vladik Kreinovich
Departmental Technical Reports (CS)
One of the main motivations for using artificial neural networks was to speed up computations. From this viewpoint, the ideal configuration is when we have a single nonlinear layer: this configuration is computationally the fastest, and it already has the desired universal approximation property. However, the last decades have shown that for many problems, deep neural networks, with several nonlinear layers, are much more effective. How can we explain this puzzling fact? In this paper, we provide a possible explanation for this phenomena: that the universal approximation property is only true in the idealized setting, when we assume that all …
Why Menzerath's Law?, Julio Urenda, Vladik Kreinovich
Why Menzerath's Law?, Julio Urenda, Vladik Kreinovich
Departmental Technical Reports (CS)
In linguistics, there is a dependence between the length of the sentence and the average length of the word: the longer the sentence, the shorter the words. The corresponding empirical formula is known as the Menzerath's Law. A similar dependence can be observed in many other application areas, e.g., in the analysis of genomes. The fact that the same dependence is observed in many different application domains seems to indicate there should be a general domain-independent explanation for this law. In this paper, we show that indeed, this law can be derived from natural invariance requirements.
How To Select A Representative Sample For A Family Of Functions?, Leobardo Valera, Martine Ceberio, Vladik Kreinovich
How To Select A Representative Sample For A Family Of Functions?, Leobardo Valera, Martine Ceberio, Vladik Kreinovich
Departmental Technical Reports (CS)
Predictions are rarely absolutely accurate. Often, the future values of quantities of interest depend on some parameters that we only know with some uncertainty. To make sure that all possible solutions satisfy desired constraints, it is necessary to generate a representative finite sample, so that if the constraints are satisfied for all the functions from this sample, then we can be sure that these constraints will be satisfied for the actual future behavior as well. At present, such a sample is selected based by Monte-Carlo simulations, but, as we show, such selection may underestimate the danger of violating the constraints. …
Why Hate: Analysis Based On Decision Theory, Olga Kosheleva, Vladik Kreinovich
Why Hate: Analysis Based On Decision Theory, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
At first glance, from the general decision-theory viewpoint, hate (and other negative feelings towards each other) makes no sense, since they decrease the utility (i.e., crudely speaking, level of happiness) of the person who experiences these feelings. Our detailed analysis shows that there are situations when such negative feelings make perfect sense: namely, when you have a large group of people almost all of whom are objectively unhappy. In such situations -- e.g., on the battlefield -- negative feelings help keep their spirits high in spite of the harsh situation. This explanation leads to recommendations on how to decrease the …
How To Solve The Apportionment Paradox, Olga Kosheleva, Vladik Kreinovich
How To Solve The Apportionment Paradox, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
It is known that often, after it is proven that a new statement is equivalent to the original definition, this new statement becomes the accepted new definition of the same notion. In this paper, we provide a natural explanation for this empirical phenomenon.
How To Solve The Apportionment Paradox, Christopher Reyes, Vladik Kreinovich
How To Solve The Apportionment Paradox, Christopher Reyes, Vladik Kreinovich
Departmental Technical Reports (CS)
In the ideal world, the number of seats that each region or each community gets in a representative body should be exactly proportional to the population of this region or community. However, since the number of seats allocated to each region or community is whole, we cannot maintain the exact proportionality. Not only this leads to a somewhat unfair situation, when residents of one region get more votes per person than residents of another one, it also leads to paradoxes -- e.g., sometimes a region that gained the largest number of people loses a number of seats. To avoid this …
A Possible Common Mechanism Behind Skew Normal Distributions In Economics And Hydraulic Fracturing-Induced Seismicity, Laxman Bokati, Aaron Velasco, Vladik Kreinovich, Kittawit Autchariyapanitkul
A Possible Common Mechanism Behind Skew Normal Distributions In Economics And Hydraulic Fracturing-Induced Seismicity, Laxman Bokati, Aaron Velasco, Vladik Kreinovich, Kittawit Autchariyapanitkul
Departmental Technical Reports (CS)
Many economic situations -- and many situations in other application areas -- are well-described by a special asymmetric generalization of normal distributions -- known as skew-normal. However, there is no convincing theoretical explanation for this empirical phenomenon. To be more precise, there are convincing explanations for the ubiquity of normal distributions, but not for the transformation that turns normal into skew-normal. In this paper, we use the analysis of hydraulic fracturing-induced seismicity to show explain the ubiquity of such a transformation.
Shall We Use Logical Approach Or More Traditional Mamdani Approach In Fuzzy Control: Pragmatic Analysis, R. Noah Padilla, Olga Kosheleva, Vladik Kreinovich
Shall We Use Logical Approach Or More Traditional Mamdani Approach In Fuzzy Control: Pragmatic Analysis, R. Noah Padilla, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Fuzzy control methodology transforms the experts' if-then rules into a precise control strategy. From the logical viewpoint, an if-then rule means implication, so it seems reasonable to use fuzzy implication in this transformation. However, this logical approach is not what the first fuzzy controllers used. The traditional fuzzy control approach -- first proposed by Mamdani -- transforms the if-then rules into a statement that only contains and's and or's, and does not use fuzzy implication at all. So, a natural question arises: shall we use logical approach or the traditional approach? In this paper, we analyze this question on the …
Why Optimization Is Faster Than Solving Systems Of Equations: A Qualitative Explanation, Siyu Deng, Bimal K. C, Vladik Kreinovich
Why Optimization Is Faster Than Solving Systems Of Equations: A Qualitative Explanation, Siyu Deng, Bimal K. C, Vladik Kreinovich
Departmental Technical Reports (CS)
Most practical problems lead either to solving a system of equation or to optimization. From the computational viewpoint, both classes of problems can be reduced to each other: optimization can be reduced to finding points at which all partial derivatives are zeros, and solving systems of equations can be reduced to minimizing sums of squares. It is therefore natural to expect that, on average, both classes of problems have the same computational complexity -- i.e., require about the same computation time. However, empirically, optimization problems are much faster to solve. In this paper, we provide a possible explanation for this …
Spiral Arms Around A Star: Geometric Explanation, Juan L. Puebla, Vladik Kreinovich
Spiral Arms Around A Star: Geometric Explanation, Juan L. Puebla, Vladik Kreinovich
Departmental Technical Reports (CS)
Recently, astronomers discovered spiral arms around a star. While their shape is similar to the shape of the spiral arms in the galaxies, however, because of the different scale, galaxy-related physical explanations of galactic spirals cannot be directly applied to explaining star-size spiral arms. In this paper, we show that, in contrast to more specific physical explanation, more general symmetry-based geometric explanations of galactic spiral can explain spiral arms around a star.
Why Self-Esteem Helps To Solve Problems: An Algorithmic Explanation, Oscar Ortiz, Henry Salgado, Olga Kosheleva, Vladik Kreinovich
Why Self-Esteem Helps To Solve Problems: An Algorithmic Explanation, Oscar Ortiz, Henry Salgado, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
It is known that self-esteem helps solve problems. From the algorithmic viewpoint, this seems like a mystery: a boost in self-esteem does not provide us with new algorithms, does not provide us with ability to compute faster -- but somehow, with the same algorithmic tools and the same ability to perform the corresponding computations, students become better problem solvers. In this paper, we provide an algorithmic explanation for this surprising empirical phenomenon.
How To Describe Hypothetic Truly Rare Events (With Probability 0), Luc Longpre, Vladik Kreinovich
How To Describe Hypothetic Truly Rare Events (With Probability 0), Luc Longpre, Vladik Kreinovich
Departmental Technical Reports (CS)
In probability theory, rare events are usually described as events with low probability p, i.e., events for which in N observations, the event happens n(N) ~ p*N times. Physicists and philosophers suggested that there may be events which are even rarer, in which n(N) grows slower than N. However, this idea has not been developed, since it was not clear how to describe it in precise terms. In this paper, we propose a possible precise description of this idea, and we use this description to answer a natural question: when two different functions n(N) lead to the same class of …
One More Physics-Based Explanation For Rectified Linear Neurons, Jonatan Contreras, Martine Ceberio, Vladik Kreinovich
One More Physics-Based Explanation For Rectified Linear Neurons, Jonatan Contreras, Martine Ceberio, Vladik Kreinovich
Departmental Technical Reports (CS)
The main idea behind artificial neural networks is to simulate how data is processed in the data processing devoice that has been optimized by million-years natural selection -- our brain. Such networks are indeed very successful, but interestingly, the most recent successes came when researchers replaces the original biology-motivated sigmoid activation function with a completely different one -- known as rectified linear function. In this paper, we explain that this somewhat unexpected function actually naturally appears in physics-based data processing.
How To Make Quantum Ideas Less Counter-Intuitive: A Simple Analysis Of Measurement Uncertainty Can Help, Olga Kosheleva, Vladik Kreinovich, Louis Ray Lopez
How To Make Quantum Ideas Less Counter-Intuitive: A Simple Analysis Of Measurement Uncertainty Can Help, Olga Kosheleva, Vladik Kreinovich, Louis Ray Lopez
Departmental Technical Reports (CS)
Our intuition about physics is based on macro-scale phenomena, phenomena which are well described by non-quantum physics. As a result, many quantum ideas sound counter-intuitive -- and this slows down students' learning of quantum physics. In this paper, we show that a simple analysis of measurement uncertainty can make many of the quantum ideas much less counter-intuitive and thus, much easier to accept and understand.
Physical Meaning Often Leads To Natural Derivations In Elementary Mathematics: On The Examples Of Solving Quadratic And Cubic Equations, Christian Servin, Olga Kosheleva, Vladik Kreinovich
Physical Meaning Often Leads To Natural Derivations In Elementary Mathematics: On The Examples Of Solving Quadratic And Cubic Equations, Christian Servin, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Usual derivation of many formulas of elementary mathematics -- such as the formulas for solving quadratic equation -- often leave un unfortunate impression that mathematics is a collection of unrelated unnatural trick. In this paper, on the example of formulas for solving quadratic and cubic equations, we show that these derivations can be made much more natural if we take physical meaning into account.
Why Immunodepressive Drugs Often Make People Happier, Joshua Ramos, Dario Vasquez, Ruth Trejo, Vladik Kreinovich
Why Immunodepressive Drugs Often Make People Happier, Joshua Ramos, Dario Vasquez, Ruth Trejo, Vladik Kreinovich
Departmental Technical Reports (CS)
Many immunodepressive drugs have an unusual side effect on the patient's mood: they often make the patient happier. This side hae been observed for many different immunodepressive drugs, with different chemical composition. Thus, it is natural to conclude that there must be some general reason for this empirical phenomenon, a reason not related to the chemical composition of any specific drug -- but rather with their general functionality. In this paper, we provide such an explanation.
Explaining An Empirical Formula For Bioreaction To Similar Stimuli (Covid-19 And Beyond), Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
Explaining An Empirical Formula For Bioreaction To Similar Stimuli (Covid-19 And Beyond), Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
Departmental Technical Reports (CS)
A recent comparative analysis of biological reaction to unchanging vs. rapidly changing stimuli -- such as Covid-19 or flu viruses -- uses an empirical formula describing how the reaction to a similar stimulus depends on the distance between the new and original stimuli. In this paper, we provide a from-first-principles explanation for this empirical formula.
Ordered Weighted Averaging (Owa), Decision Making Under Uncertainty, And Deep Learning: How Is This All Related?, Vladik Kreinovich
Ordered Weighted Averaging (Owa), Decision Making Under Uncertainty, And Deep Learning: How Is This All Related?, Vladik Kreinovich
Departmental Technical Reports (CS)
Among many research areas to which Ron Yager contributed are decision making under uncertainty (in particular, under interval and fuzzy uncertainty) and aggregation -- where he proposed, analyzed, and utilized ordered weighted averaging (OWA). The OWA algorithm itself provides only a specific type of data aggregation. However, it turns out that if we allow several OWA stages, one after another, we obtain a scheme with a universal approximation property -- moreover, a scheme which is perfectly equivalent to modern ReLU-based deep neural networks. In this sense, Ron Yager can be viewed as a (grand)father of ReLU-based deep learning. We also …
Unexpected Economic Consequence Of Cloud Computing: A Boost To Algorithmic Creativity, Francisco Zapata, Eric Smith, Vladik Kreinovich
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 …
Unreachable Statements Are Inevitable In Software Testing: Theoretical Explanation, Francisco Zapata, Eric Smith, Vladik Kreinovich
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.
A Natural Causality-Motivated Description Of Learning, Olga Kosheleva, Vladik Kreinovich
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 …
Why Gaussian Copulas Are Ubiquitous In Economics: Fuzzy-Related Explanation, Chon Van Le, Olga Kosheleva, Vladik Kreinovich
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 …
Video Or Text? Bullets Or No Bullets? Why Not Both?, Olga Kosheleva, Vladik Kreinovich, Christian Servin
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
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 …