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Articles 151 - 180 of 2396
Full-Text Articles in Mathematics
On The Work Of Cartan And Münzner On Isoparametric Hypersurfaces, Thomas E. Cecil, Patrick J. Ryan
On The Work Of Cartan And Münzner On Isoparametric Hypersurfaces, Thomas E. Cecil, Patrick J. Ryan
Mathematics and Computer Science Department Faculty Scholarship
A hypersurface Mn in a real space form Rn+1, Sn+1, or Hn+1 is isoparametric if it has constant principal curvatures. This paper is a survey of the fundamental work of Cartan and Münzner on the theory of isoparametric hypersurfaces in real space forms, in particular, spheres. This work is contained in four papers of Cartan [3]–[6] published during the period 1938–1940, and two papers of Münzner [47]–[48] that were published in preprint form in the early 1970’s, and as journal articles in 1980–1981. These papers of Cartan and Münzner have been the …
Why Geological Angular Unconformity Is Usually Horizontal: A Geometric Explanation, Julio C. Urenda, Aaron Velasco, Olga Kosheleva, Vladik Kreinovich
Why Geological Angular Unconformity Is Usually Horizontal: A Geometric Explanation, Julio C. Urenda, Aaron Velasco, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In several locations, geologists have observed the presence of two differently oriented rock masses, one horizonal (or almost horizontal) and the other somewhat inclined; this phenomenon is known as angular unconformity. Based on the detailed analysis of geophysical processes, geologists conclude that usually, horizontal rock masses are much newer. This is known as the law of original horizontality. From the fundamental viewpoint, it is desirable to take into account that geophysics is a developing science, its models get modified and adjusted as time progresses. It is therefore desirable to come up with an explanation of this phenomenon that would be …
Discrete Time Series Forecasting Of Hive Weight, In-Hive Temperature, And Hive Entrance Traffic In Non-Invasive Monitoring Of Managed Honey Bee Colonies: Part I, Vladimir A. Kulyukin, Daniel Coster, Aleksey V. Kulyukin, William Meikle, Milagra Weiss
Discrete Time Series Forecasting Of Hive Weight, In-Hive Temperature, And Hive Entrance Traffic In Non-Invasive Monitoring Of Managed Honey Bee Colonies: Part I, Vladimir A. Kulyukin, Daniel Coster, Aleksey V. Kulyukin, William Meikle, Milagra Weiss
Computer Science Faculty and Staff Publications
From June to October, 2022, we recorded the weight, the internal temperature, and the hive entrance video traffic of ten managed honey bee (Apis mellifera) colonies at a research apiary of the Carl Hayden Bee Research Center in Tucson, AZ, USA. The weight and temperature were recorded every five minutes around the clock. The 30 s videos were recorded every five minutes daily from 7:00 to 20:55. We curated the collected data into a dataset of 758,703 records (208,760–weight; 322,570–temperature; 155,373–video). A principal objective of Part I of our investigation was to use the curated dataset to investigate …
Two Is Enough, But Three (Or More) Is Better: In Ai And Beyond, Olga Kosheleva, Vladik Kreinovich, Victor L. Timchenko, Yury P. Kondratenko
Two Is Enough, But Three (Or More) Is Better: In Ai And Beyond, Olga Kosheleva, Vladik Kreinovich, Victor L. Timchenko, Yury P. Kondratenko
Departmental Technical Reports (CS)
At present, the most successful AI technique is deep learning -- the use of neural networks that consist of multiple layers. Interestingly, it is well known that neural networks with two data processing layers are sufficient -- in the sense that they can approximate any function with any given accuracy. Because of this, until reasonably recently, researchers and practitioners used such networks. However, recently it turned out, somewhat unexpectedly, that using three or more data processing layers -- i.e., using what is called deep learning -- makes the neural networks much more efficient. In this paper, on numerous examples from …
Some Studies On Mathematical Morphology In Remotely Sensed Data Analysis, Geetika Barman
Some Studies On Mathematical Morphology In Remotely Sensed Data Analysis, Geetika Barman
Doctoral Theses
The application of Mathematical Morphology (MM) techniques has proven to be beneficial in the extraction of shapebased and texture-based features during remote sensing image analysis. The characteristics of these techniques, such as nonlinear adaptability and comprehensive lattice structure, make them useful for contextual spatial feature analysis. Despite the advancements, there are still persistent challenges, including the curse of dimensionality, maintaining spatial correlation, and the adaptability of morphological operators in higher dimensions. The focus of this thesis is to explore the potential of MM-based methods to analyse spatial features in addressing these challenges, specifically in the context of spatialcontextual feature analysis …
Coarse-Gridded Simulation Of The Nonlinear Schrödinger Equation With Machine Learning, Benjamin F. Akers, Kristina O. F. Williams
Coarse-Gridded Simulation Of The Nonlinear Schrödinger Equation With Machine Learning, Benjamin F. Akers, Kristina O. F. Williams
Faculty Publications
A numerical method for evolving the nonlinear Schrödinger equation on a coarse spatial grid is developed. This trains a neural network to generate the optimal stencil weights to discretize the second derivative of solutions to the nonlinear Schrödinger equation. The neural network is embedded in a symmetric matrix to control the scheme’s eigenvalues, ensuring stability. The machine-learned method can outperform both its parent finite difference method and a Fourier spectral method. The trained scheme has the same asymptotic operation cost as its parent finite difference method after training. Unlike traditional methods, the performance depends on how close the initial data …
A Full Description Of All Commutative Associative Polynomial Operations On Probabilities, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
A Full Description Of All Commutative Associative Polynomial Operations On Probabilities, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
Departmental Technical Reports (CS)
When two events are independent, the probability that both events occur is equal to the product p1 * p2 of the probabilities of each of these events. The probability that at least one of these events will occur is equal to p1 + p2 − p1 * p2. In both cases, we have a commutative associative polynomial operation. A natural question is: how can we describe all possible operations of this type? These operations are described in this paper.
Why Kolmolgorov-Arnold Networks (Kan) Work So Well: A Qualitative Explanation, Hung T. Nguyen, Vladik Kreinovich, Olga Kosheleva
Why Kolmolgorov-Arnold Networks (Kan) Work So Well: A Qualitative Explanation, Hung T. Nguyen, Vladik Kreinovich, Olga Kosheleva
Departmental Technical Reports (CS)
In the usual deep neural network, weights are adjusted during training, but the activation function remains the same. Lately, it was experimentally shown that if, instead of using the same activation function always, we train the activation functions as well, we get a much better results -- i.e., for the networks with the same number of parameters, we get a much better accuracy. Such networks are called Kolmogorov-Arnold networks. In this paper, we provide a general explanation of why these new networks work so well.
How To Check Continuity Based On Approximate Measurement Results, Inese Bula, Vladik Kreinovich
How To Check Continuity Based On Approximate Measurement Results, Inese Bula, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, a reasonable conjecture is that, e.g., the dependence of some quantity on the spatial location is continuous, with an appropriate bounds on the difference between the values at nearby points. If we knew the exact values of the corresponding quantity, checking this conjecture would be very straightforward. In reality, however, measurement results are only approximations to the actual values. In this paper, we show how to check continuity based on the approximate measurement results.
Three Applications Of Geometric Reasoning: Why Metastasis Is Mostly Caused By Elongated Cancer Cells? How Body Shape Affects Curiosity? Why Ring Fractures In Ice?, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich
Three Applications Of Geometric Reasoning: Why Metastasis Is Mostly Caused By Elongated Cancer Cells? How Body Shape Affects Curiosity? Why Ring Fractures In Ice?, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In this paper, we describe three applications of geometric reasoning to important practical problems ranging from micro- to macro-level. Specifically, we use geometric reasoning to explain why metastasis is mostly caused by elongated cancer cell, why curiosity in fish is strongly correlated with body shape, and why ring-shaped fractures appear in Antarctica.
To Which Interdisciplinary Research Collaborations Should We Pay More Attention?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
To Which Interdisciplinary Research Collaborations Should We Pay More Attention?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
Departmental Technical Reports (CS)
Interdisciplinary research is very important in modern science. However, such a research is not easy, it often needs support and help. Resources that can be used for such a support are limited, so we need to decide which of many possible collaborations we should support. In this paper, we provide a natural simple model of collaboration effectiveness. Based on this model, we conclude that we should support collaborations for which the vector product of the participants' knowledge vectors attains the largest values.
Why Decisions Based On The Results Of Worst-Case, Most Realistic, And Best-Case Scenarios Work Well?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich, Chon Van Le
Why Decisions Based On The Results Of Worst-Case, Most Realistic, And Best-Case Scenarios Work Well?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich, Chon Van Le
Departmental Technical Reports (CS)
Often, to make an appropriate decision, people try three scenarios: the worst case, the most realistic case, and the best case. This three-scenarios approach often leads to reasonable decisions. A natural question is: why worst case and best case? These extreme cases mean that all numerous independent random factors work in the same direction: either are all stacked for or are all stacked against. Such stacking of random factors is highly improbable. So, at first glance, it would be more beneficial to use more realistic scenarios than the worst case and the best case. However, empirically, decisions based on the …
Why Green Wavelength Is Closer To Blue Than To Red And How It Is Related To Computations: Information-Based Explanation, Victor L. Timchenko, Yury P. Kondratenko, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
Why Green Wavelength Is Closer To Blue Than To Red And How It Is Related To Computations: Information-Based Explanation, Victor L. Timchenko, Yury P. Kondratenko, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong
Departmental Technical Reports (CS)
In our previous papers, we analyzed the idea of using light signals of three basic color -- red, green, and blue -- to speed up computations, in particular fuzzy-related computations. A natural question is: why red, green, and blue? Why not select some other colors: e.g., from the wavelength viewpoint, green is much closer to blue than to green, so why not select colors whose distribution is more even? In this paper, we show that if we consider this problem from the information viewpoint, then the corresponding equal-information criterion indeed implies that the intermediate wavelength should be closer to the …
Training Neural Networks On Interval Data: Unexpected Results And Their Explanation, Edwin Tomy George, Vladik Kreinovich, Christoph Lauter, Martine Ceberio, Luc Jaulin
Training Neural Networks On Interval Data: Unexpected Results And Their Explanation, Edwin Tomy George, Vladik Kreinovich, Christoph Lauter, Martine Ceberio, Luc Jaulin
Departmental Technical Reports (CS)
In many practically useful numerical computations, training-and-then-using a neural network turned out to be a much faster alternative than running the original computations. When we applied a similar idea to take into account interval uncertainty, we encountered two unexpected results: (1) that while for numerical computations, it is usually better to represent an interval by its midpoint and half-width, for neural networks, it is more efficient to represent an interval by its endpoints, and (2) that while usually, it is better to train a neural network on the whole data processing algorithm, in our problems, it turned out to be …
A Staged Approach Using Machine Learning And Uncertainty Quantification To Predict The Risk Of Hip Fracture, Anjum Shaik, Kristoffer A. Larsen, Nancy E. Lane, Chen Zhao, Kuan Jui Su, Joyce H. Keyak, Qing Tian, Qiuying Sha, Hui Shen, Hong Wen Deng, Weihua Zhou
A Staged Approach Using Machine Learning And Uncertainty Quantification To Predict The Risk Of Hip Fracture, Anjum Shaik, Kristoffer A. Larsen, Nancy E. Lane, Chen Zhao, Kuan Jui Su, Joyce H. Keyak, Qing Tian, Qiuying Sha, Hui Shen, Hong Wen Deng, Weihua Zhou
Michigan Tech Publications
Hip fractures present a significant healthcare challenge, especially within aging populations, where they are often caused by falls. These fractures lead to substantial morbidity and mortality, emphasizing the need for timely surgical intervention. Despite advancements in medical care, hip fractures impose a significant burden on individuals and healthcare systems. This paper focuses on the prediction of hip fracture risk in older and middle-aged adults, where falls and compromised bone quality are predominant factors. The study cohort included 547 patients, with 94 experiencing hip fracture. To assess the risk of hip fracture, clinical variables and clinical variables combined with hip DXA …
What If The Resulting Interval Is Too Wide: From A Heuristic Fuzzy-Technique Idea To A Mathematically Justified Approach, Marc Fina, Vladik Kreinovich
What If The Resulting Interval Is Too Wide: From A Heuristic Fuzzy-Technique Idea To A Mathematically Justified Approach, Marc Fina, Vladik Kreinovich
Departmental Technical Reports (CS)
In engineering designs, we usually need to make sure that the values of some characteristics y do not exceed a certain threshold y0 – e.g., that the stress at each location does not exceed a certain critical value. Usually, we know how each of these characteristics y depends on the design parameters x1, . . . ,xn, i.e., we know the function y= f (x1, . . . ,xn). However, it is not enough to use the nominal values of the design parameters in our analysis, since the actual values are, in general, somewhat different from the nominal values. Often, …
Shapley Value Under Interval Uncertainty And Partial Information, Kittawit Autchariyapanikul, Olga Kosheleva, Vladik Kreinovich
Shapley Value Under Interval Uncertainty And Partial Information, Kittawit Autchariyapanikul, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In the 1950s, the future Nobelist Lloyd Shapley solved the problem of how to fairly divide the common gain. Namely, he showed that some reasonable requirements determine a unique division -- which is now known as the Shapley value. The main limitation of Shapley's solution is that it assumes that for each subgroup of the original group of participants, we know exactly how much this group could gain if it acted by itself, without involving others. In practice, we rarely know these exact values. At best, we know the bounds on each such value -- i.e., in other words, an …
Sociomathematical Norms And Automated Proof Checking In Mathematical Education: Reflections And Experiences, Merlin Carl
Sociomathematical Norms And Automated Proof Checking In Mathematical Education: Reflections And Experiences, Merlin Carl
Journal of Humanistic Mathematics
According to a widely held view, mathematical proofs are essentially (indications of) formal derivations, and thus in principle mechanically checkable (this view is defended, for example, by Azzouni [3]). This should in particular hold for the kind of simple proof exercises typically given to students of mathematics learning to write proofs. If that is so, then automated proof checking should be an attractive option for math education at the undergraduate level. An opposing view would be that mathematical proofs are social objects and that what constitutes a mathematical proof can thus not be separated from the social context in which …
Parallel Multigrid In Time For Chaotic Dynamical Systems, David Alan Vargas
Parallel Multigrid In Time For Chaotic Dynamical Systems, David Alan Vargas
Mathematics & Statistics ETDs
Despite the fact that Parallel-in-Time (PinT) methods are predicted to become necessary to fully utilize next-generation exa- and zettascale machines, there are currently no known practical methods which scale well with the length of the time-domain for chaotic problems, due to exponential dependence of the condition number on the fastest chaotic timescale. I present modifications to the coarse-grid equations along with a novel rediscretization approach which together greatly improve convergence of the multigrid reduction in time (MGRIT) algorithm and allow the first known PinT speedup for a chaotic PDE. The novel Local Shadowing Relaxation (LSR) is presented as an alternative …
If Subsequent Results Are Too Easy To Obtain, The Proof Most Probably Has Errors: Explanation Of The Empirical Observation, Olga Kosheleva, Vladik Kreinovich
If Subsequent Results Are Too Easy To Obtain, The Proof Most Probably Has Errors: Explanation Of The Empirical Observation, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Many modern mathematical proofs are very complex, checking them is difficult; as a result, errors sneak into published proofs, even into proofs published in highly reputable journals. Sometimes, the errors are repairable, but sometimes, it turns out that the supposedly proven result is actually wrong. When the error is not noticed for some time, the published result is used to prove many other results -- and when the error is eventually found, all these new results are invalidated. This happened several times. Since it is not realistic to more thoroughly check all the proofs, and we want to minimize the …
Why Angles Between Galactic Center Filaments And Galactic Plane Follow A Bimodal Distribution: A Symmetry-Based Explanation, Julio C. Urenda, Vladik Kreinovich
Why Angles Between Galactic Center Filaments And Galactic Plane Follow A Bimodal Distribution: A Symmetry-Based Explanation, Julio C. Urenda, Vladik Kreinovich
Departmental Technical Reports (CS)
Recent observations have shown that the angles between the Galaxy Center filaments and the Galactic plane follow a bimodal distribution: a large number of filaments are approximately orthogonal to the Galactic plane, a large number of filaments are approximately parallel to the Galactic plane, and much fewer filaments have other orientations. In this paper, we show this bimodal distribution can be explained by natural geometric symmetries.
Why Seismicity In Ireland Is Low: A Possible Geometric Explanation, Julio C. Urenda, Aaron Velasco, Vladik Kreinovich
Why Seismicity In Ireland Is Low: A Possible Geometric Explanation, Julio C. Urenda, Aaron Velasco, Vladik Kreinovich
Departmental Technical Reports (CS)
For each geographic location, its seismicity level is usually determined by how close this location is to the boundaries of tectonic plates. However, there is one notable exception: while Ireland and Britain are at approximately the same distance from such boundaries, the seismicity level in Ireland is much lower than in Britain. A recent paper provided a partial explanation for this phenomenon: namely, it turns out that the lithosphere under Ireland is unusually thick, and this can potentially lead to lower seismicity. However, the current explanation of the relation between the lithosphere thickness and seismicity level strongly depends on the …
Is Alaska Negative-Tax Arrangement Fair? Almost: Mathematical Analysis, Chon Van Le, Vladik Kreinovich
Is Alaska Negative-Tax Arrangement Fair? Almost: Mathematical Analysis, Chon Van Le, Vladik Kreinovich
Departmental Technical Reports (CS)
In the State of Alaska there is no state income tax. Instead, there is a negative tex: every year every resident gets some money from the state. At present, every resident -- from the poorest to the richest -- gets the exact same amount of money: in 2024, it is expected to be around $1500. A natural question is: Is this fair? Maybe poor people should get more since their needs are greater? Maybe the rich people should get proportionally more, since fairness means equal added happiness for all, and for rich people, extra $1500 is barely noticeable? There have …
For 2 X N Cases, Proportional Fitting Problem Reduces To A Single Equation, Olga Kosheleva, Vladik Kreinovich
For 2 X N Cases, Proportional Fitting Problem Reduces To A Single Equation, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, for each of two classifications, we know the probabilities that a randomly selected object belong to different categories. For example, we know what proportion of people are below 20 years old, what proportion is between 20 and 30, etc., and we also know what proportion of people earns less than 10K, between 10K and 20K, etc. In such situations, we are often interested in proportion of people who are classified by two classifications into two given categories. For example, we are interested in the proportion of people whose age is between 20 and 30 and whose …
Stochastic Dominance: Cases Of Interval And P-Box Uncertainty, Kittawit Autchariyapanikul, Olga Kosheleva, Vladik Kreinovich
Stochastic Dominance: Cases Of Interval And P-Box Uncertainty, Kittawit Autchariyapanikul, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Traditional decision theory recommendation about making a decision assume that we know both the probabilities of different outcomes of each possible decision, and we know the utility function -- that describes the decision maker's preferences. Sometimes, we can make a recommendation even when we only have partial information about utility. Such cases are known as cases of stochastic dominance. In other cases, in addition to not knowing the utility function, we also only have partial information about the probabilities of different outcomes. For example, we may only known bounds on the outcomes (case of interval uncertainty) or bounds on the …
A Portable Numerical Library For The Calculation Of Multi-Dimensional Integrals, Ioannis Sakiotis
A Portable Numerical Library For The Calculation Of Multi-Dimensional Integrals, Ioannis Sakiotis
Computer Science Theses & Dissertations
Multi-dimensional numerical integration is a prevalent task in physics and other scientific fields, e.g., in the simulation of particle-beam dynamics and Bayesian parameter estimation. Scientific computing applications that simulate complex phenomena may require the solution to numerous multi-variate integrals. However, functions that have features such as sharp peaks or oscillations in high dimensional spaces, can result in an exorbitant number of computations. For many cases, convergence to accurate results in a reasonable amount of time is infeasible with existing numerical libraries. One approach towards making multi-dimensional integration viable is to parallelize existing algorithms. No commonly available algorithms or libraries exist …
Enhancing Tumor Classification Through Machine Learning Algorithms For Breast Cancer Diagnosis, Lawrence Agbota, Edmund F. Agyemang, Priscilla Kissi-Appiah, Lateef Moshood, Akua Osei- Nkwantabisa, Vincent Agbenyeavu, Abraham Nsiah, Augustina Adjei
Enhancing Tumor Classification Through Machine Learning Algorithms For Breast Cancer Diagnosis, Lawrence Agbota, Edmund F. Agyemang, Priscilla Kissi-Appiah, Lateef Moshood, Akua Osei- Nkwantabisa, Vincent Agbenyeavu, Abraham Nsiah, Augustina Adjei
School of Mathematical & Statistical Sciences Faculty Publications
In cancer diagnosis, machine learning helps improve cancer detection by providing doctors with a second perspective and allowing for faster and more accurate determination and decisions. Numerous studies have used both classic machine learning approaches and deep learning to address cancer classification. In this study, we examine the efficacy of five commonly used machine learning algorithms; both traditional and deep learning models namely, Logistic Regression, Support Vector Machines (SVM), Random Forest (RF), Decision Tree and Deep Neural Networks (DNN). We analyze their ability to properly classify tumors as Benign or Malignant using the Wisconsin breast cancer dataset (WBCD). Random Forest …
Mixed Uncertainty Analysis On Pumping By Peristaltic Hearts Using Dempster-Shafer Theory, Yanyan He, Nicholas A. Battista, Lindsay D. Waldrop
Mixed Uncertainty Analysis On Pumping By Peristaltic Hearts Using Dempster-Shafer Theory, Yanyan He, Nicholas A. Battista, Lindsay D. Waldrop
Biology, Chemistry, and Environmental Sciences Faculty Articles and Research
In this paper, we introduce the numerical strategy for mixed uncertainty propagation based on probability and Dempster–Shafer theories, and apply it to the computational model of peristalsis in a heart-pumping system. Specifically, the stochastic uncertainty in the system is represented with random variables while epistemic uncertainty is represented using non-probabilistic uncertain variables with belief functions. The mixed uncertainty is propagated through the system, resulting in the uncertainty in the chosen quantities of interest (QoI, such as flow volume, cost of transport and work). With the introduced numerical method, the uncertainty in the statistics of QoIs will be represented using belief …
How To Propagate Uncertainty Via Ai Algorithms, Olga Kosheleva, Vladik Kreinovich
How To Propagate Uncertainty Via Ai Algorithms, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Any data processing starts with measurement results. Measurement results are never absolutely accurate. Because of this measurement uncertainty, the results of processing measurement results are, in general, somewhat different from what we would have obtained if we knew the exact values of the measured quantities. To make a decision based on the result of data processing, we need to know how accurate is this result, i.e., we need to propagate the measurement uncertainty through the data processing algorithm. There are many techniques for uncertainty propagation. Usually, they involve applying the same data processing algorithm several times to appropriately modified data. …
For Discrete-Time Linear Dynamical Systems Under Interval Uncertainty, Predicting Two Moments Ahead Is Np-Hard, Luc Jaulin, Olga Kosheleva, Vladik Kreinovich
For Discrete-Time Linear Dynamical Systems Under Interval Uncertainty, Predicting Two Moments Ahead Is Np-Hard, Luc Jaulin, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In the first approximation, when changes are small, most real-world systems are described by linear dynamical equations. If we know the initial state of the system, and we know its dynamics, then we can, in principle, predict the system's state many moments ahead. In practice, however, we usually know both the initial state and the coefficients of the system's dynamics with some uncertainty. Frequently, we encounter interval uncertainty, when for each parameter, we only know its range, but we have no information about the probability of different values from this range. In such situations, we want to know the range …