Applied Mathematics Commons^™

Recursive Forms For Determinant Of K-Tridiagonal Toeplitz Matrices, Eugene Agyei-Kodie

Open Access Theses & Dissertations

Toeplitz matrices have garnered renewed interest in recent years due to their practical applications in engineering and computational sciences. Additionally, research has shown their connection to other matrices and their significance in matrix theory. For example, one study demonstrated that any matrix can be expressed as the product of Toeplitz matrices \citep{ye2016every}, while another showed that any square matrix is similar to a Toeplitz matrix \citep{mackey1999every}.

Numerous studies have examined various properties of Toeplitz matrices, including ideals of lower triangular Toeplitz matrices \citep{dogan9some}, matrix power computation with band Toeplitz structures \citep{dogan2017matrix}, and norms of Toeplitz matrices. Moreover, the use of …

Volatility Modeling Of Time Series Using Fractal And Self-Similarity Models, William Kubin

Open Access Theses & Dissertations

The study uses various methods to compare financial and geophysical time series scaling parameters and long-term memory behavior. The Cantor Detrended Fluctuation Analysis (CDFA) method is proposed to provide more accurate estimates of Hurst exponents. The CDFA method is applied to real-time series and the results are verified. The study also analyzes the memory behavior of daily Covid-19 cases before and after the announcement of effective vaccines. Low and high-frequency dataâ??s influence on the Hurst Index estimation is investigated, and a new PCDFA method is proposed. The stability of the Dow Jones Industrial Average is analyzed using a multi-scale normalized …

Region Detection & Segmentation Of Nissl-Stained Rat Brain Tissue, Alexandro Arnal

Open Access Theses & Dissertations

People who analyze images of biological tissue rely on the segmentation of structures as a preliminary step. In particular, laboratories studying the rat brain delineate brain regions to position scientific findings on a brain atlas to propose hypotheses about the rat brain and, ultimately, the human brain. Our work intersects with the preliminary step of delineating regions in images of brain tissue via computational methods.

We investigate pixel-wise classification or segmentation of brain regions using ten histological images of brain tissue sections stained for Nissl substance. We present a deep learning approach that uses the fully convolutional neural network, U-Net, …

Decision Making Under Uncertainty With A Special Emphasis On Geosciences And Education, Laxman Bokati

Open Access Theses & Dissertations

In many practical situations, we need to make a decision. In engineering, we need to decideon the best design of a system, and, for existing systems - on the best control strategy. In financial applications, we need to decide what is the best way to invest money. In geosciences, we need to decide whether we should explore a possible mineral deposit - or whether we should perform more experiments and measurements (and what exactly). In some cases, we can compute the exact consequences of each decision - e.g., if we are controlling a satellite. However, in many other cases, we …

A Central Compact Hybrid-Variable Method With Spectral-Like Resolution, Md Mahmudul Hasan

Open Access Theses & Dissertations

Numerical methods for hyperbolic conservation laws have been a driving force for theresearch in scientific computing and simulation science in the past decades, as many physical, biological, and engineering systems are governed by these equations, such as fluid mechanics, tumor growth, and virtual wind tunnel simulations. Despite the existence of many schemes in the literature, people have never stopped searching for more accurate and efficient methods for these problems. Indeed, the increasing complexity of systems in emerging applications demands better resolution of sub-grid scale phenomenon whereas classical methods usually fail to deliver high-fidelity simulation results of such systems within realistic …

How To Extend Interval Arithmetic So That Inverse And Division Are Always Defined, Tahea Hossain, Jonathan Rivera, Yash Sharma, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life data processing situations, we only know the values of the inputs with interval uncertainty. In such situations, it is necessary to take this interval uncertainty into account when processing data. Most existing methods for dealing with interval uncertainty are based on interval arithmetic, i.e., on the formulas that describe the range of possible values of the result of an arithmetic operation when the inputs are known with interval uncertainty. For most arithmetic operations, this range is also an interval, but for division, the range is sometimes a disjoint union of two semi-infinite intervals. It is therefore desirable …

Gene Selection And Classification In High-Throughput Biological Data With Integrated Machine Learning Algorithms And Bioinformatics Approaches, Abhijeet R Patil

Open Access Theses & Dissertations

With the rise of high throughput technologies in biomedical research, large volumes of expression profiling, methylation profiling, and RNA-sequencing data are being generated. These high-dimensional data have large number of features with small number of samples, a characteristic called the "curse of dimensionality." The selection of optimal features, which largely affects the performance of classification algorithms in machine learning models, has led to challenging problems in bioinformatics analyses of such high-dimensional datasets. In this work, I focus on the design of two-stage frameworks of feature selection and classification and their applications in multiple sets of colorectal cancer data. The first …

How To Explain The Relation Between Different Empirical Covid-19 Self-Isolation Periods, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Empirical data implies that, to avoid infecting others, an asymptomatic career of Covid-19 should self-isolate for a period of 10 days, a patient who experiences symptoms for 20 days, and a person who was in contact with a Covid-19 patient should self-isolate for 14 days. In this paper, we use Laplace's Principle of Insufficient Reason to provide a simple explanation for the relation between these three self-isolation periods.

What If We Use Almost-Linear Functions Instead Of Linear Ones As A First Approximation In Interval Computations, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, the only information that we have about measurement errors is the upper bound on their absolute values. In such situations, the only information that we have after the measurement about the actual (unknown) value of the corresponding quantity is that this value belongs to the corresponding interval: e.g., if the measurement result is 1.0, and the upper bound is 0.1, then this interval is [1.0−0.1,1.0+0.1] = [0.9,1.1]. An important practical question is what is the resulting interval uncertainty of indirect measurements, i.e., in other words, how interval uncertainty propagates through data processing. There exist feasible algorithms …

How To Describe Measurement Errors: A Natural Generalization Of The Central Limit Theorem Beyond Normal (And Other Infinitely Divisible) Distributions, Julio Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When precise measurement instruments are designed, designers try their best to decrease the effect of the main factors leading to measurement errors. As a result of this decrease, the remaining measurement error is the joint result of a large number of relatively small independent error components. According to the Central Limit Theorem, under reasonable conditions, when the number of components increases, the resulting distribution tends to Gaussian (normal). Thus, in practice, when the number of components is large, the distribution is close to normal -- and normal distributions are indeed ubiquitous in measurements. However, in some practical situations, the distribution …

Why Significant Wave Height And Rogue Waves Are So Defined: A Possible Explanation, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Data analysis has shown that if we want to describe the wave pattern by a single characteristic, the best characteristic is the average height of the highest one third of the waves; this characteristic is called significant wave height. Once we know the value of this characteristic, a natural next question is: what is the highest wave that we should normally observe -- so that waves higher than this amount would be rare ("rogue"). Empirically, it has been shown that rogue waves are best defined as the ones which are at least twice higher than the significant wave height. In …

Egyptian Fractions As Approximators, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In ancient Egypt, fractions were represented as the sum of inverses to natural numbers. Processing fractions in this representation is computationally complicated. Because of this complexity, traditionally, Egyptian fractions used to be considered an early inefficient approach. In our previous papers, we showed, however, that the Egyptian fractions actually provide an optimal solution to problems important for ancient Egypt -- such as the more efficient distribution of food between workers. In these papers, we assumed, for simplicity, that we know the exact amount of food needed for each worker -- and that this value must be maintained with absolute accuracy. …

How To Separate Absolute And Relative Error Components: Interval Case, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Usually, measurement errors contain both absolute and relative components. To correctly gauge the amount of measurement error for all possible values of the measured quantity, it is important to separate these two error components. For probabilistic uncertainty, this separation can be obtained by using traditional probabilistic techniques. The problem is that in many practical situations, we do not know the probability distribution, we only know the upper bound on the measurement error. In such situations of interval uncertainty, separation of absolute and relative error components is not easy. In this paper, we propose a technique for such a separation based …

Does Transition To Democracy Lead To Chaos: A Theorem, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When a country transitions to democracy, at first, many political parties appear. A natural question is whether the number of such parties feasible and reasonable -- or whether this is a complete chaos. In this paper, we formulate a simplified version of this question in precise terms and show that the number of parties will be feasible -- i.e., that transition to democracy does not lead to chaos.

Why 3d Fragmentation Usually Leads To Cuboids: A Simple Geometric Explanation, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It has been empirically observed that the average shape of natural fragmentation results -- such as natural rock fragments -- is a distorted cube (known as cuboid). Recently, a complex explanation was provides for this empirical fact. In this paper, we propose a simple geometry-based physical explanation for the ubiquity of cuboid fragments.

Why Cutting Trajectories Into Small Pieces Helps To Learn Dynamical Systems Better: A Seemingly Counterintuitive Empirical Result Explained, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In general, the more information we use in machine learning, the more accurate predictions we get. However, recently, it was observed that for prediction of the behavior of dynamical systems, the opposite effect happens: when we replace the original trajectories with shorter pieces -- thus ignoring the information about the system's long-term behavior -- the accuracy of machine learning predictions actually increases. In this paper, we provide an explanation for this seemingly counterintuitive result.

How The Amount Of Cracks And Potholes Grows With Time: Symmetry-Based Explanation Of Empirical Dependencies, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Empirical double-exponential formulas are known that describe how the amount of cracks and potholes in a pavement grows with time. In this paper, we show that these formulas can be explained based on natural symmetries (invariances) -- such as invariance with respect to changing the measuring unit or invariance with respect to changing a starting point for measuring time.

Two Runners In The Time Of Social Distancing, Speedboats In The Gulf Of Finland: How To Best Pass?, Julio Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

If two runners follow the same running path, what is the best trajectory for the faster runner to pass the slower one, taking into account that they should always maintain a prescribed social distance? If a speedboat wants to pass a slower ship following a special canal in the Gulf of Finland, what is the best trajectory? In this paper, we provide answers to both questions.

The Similarity Between Earth's And Mars's Core-Mantle Boundary Seems To Be Statistically Significant, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Latest, most accurate measurements of the depth of the Mars's core-mantle boundary shows that the ratio between this depth and Mars's radius is the same as for the Earth -- and with new measurements, this coincidence has become statistically significance. This coincidence seems to confirm a simple scale-invariant model in which for planets of Earth-Mars type, this depth is proportional to the planet's radius. Of course, we need more observations to confirm this model, but the fact that, for the first time, we got a statistically significant confirmation, is encouraging: it makes us believe that this coincidence is not accidental.

Under Limited Resources, Lottery-Based Tutoring Is The Most Efficient, Olga Kosheleva, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

In the ideal world, every student who needs tutoring should receive intensive one-on-one tutoring. In practical, schools' resources are limited, so the students get only a portion of needed tutoring. It would have been not so bad if, e.g., half-time tutoring would be half as efficient as the intensive one. However, research shows that half-time tutoring is, on average, 15 times less efficient -- and, e.g., for math tutoring 20 times less efficient. To increase the efficiency, we propose to randomly divide the students who need tutoring into equal-size groups, and each year (or each semester) provide intensive tutoring to …

Gifted And Talented: With Others? Separately? Mathematical Analysis Of The Problem, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Crudely speaking, there are two main suggestions about teaching gifted and talented student: we can move them to a separate class section, or we can mix them with other students. Both options have pluses and minuses. In this paper, we formulate this problem in precise terms, we solve the corresponding mathematical optimization problem, and we come up with a somewhat unexpected optimal solution: mixing, but with an unusual twist.

Why Quadratic Log-Log Dependence Is Ubiquitous And What Next, Sean R. Aguilar, Vladik Kreinovich, Uyen Pham

Departmental Technical Reports (CS)

In many real-life situations ranging from financial to volcanic data, growth is described either by a power law -- which is linear in log-log scale, or by a quadratic dependence in the log-log scale. In this paper, we use natural scale invariance requirement to explain the ubiquity of such dependencies. We also explain what should be a reasonable choice of the next model, if quadratic turns out to be not too accurate: it turns out that under scale invariance, the next class of models are cubic dependencies in the log-log scale, then fourth order dependencies, etc.

The Less We Love A Woman, The More She Likes Us: Pushkin's Observation Explained, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Alexander Pushkin, the most famous Russian poet, made this observation in "Eugene Onegin", his novel in verse which is most known to non-Russian readers via Tchaikovsky's opera. This observation may not be an absolute truth -- there are counterexamples -- but the fact that it is still widely cited shows that there is some truth in this statement. In this paper, we recall the usual utility-based explanation for a similar statement, and propose a new explanation, which is even more fundamental -- it is on the biological level.

Covid-19 Peak Immunity Values Seem To Follow Lognormal Distribution, Julio Urenda, Olga Kosheleva, Vladik Kreinovich, Tonghui Wang

Departmental Technical Reports (CS)

For the current pandemic, an important open problem is immunity: do people who had this disease become immune against further infections? In the immunity study, it is important to know how frequent are different levels of immunity, i.e., what is the probability distribution of the immunity levels. Different people have different rates of immunity dynamics: for some, immunity gets to the level faster, for others the immunity effect is slower. Similarly, in some patients, immunity stays longer, it others, it decreases faster. In view of this, an important characteristic is peak immunity. A recent study provides some statistics on the …

Let Us Use Negative Examples In Regression-Type Problems Too, Jonatan Contreras, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich, Martine Ceberio

Departmental Technical Reports (CS)

In many practical situations, we need to reconstruct the dependence between quantities x and y based on several situations in which we know both x and y values. Such problems are known as regression problems. Usually, this reconstruction is based on positive examples, when we know y -- at least, with some accuracy. However, in addition, we often also know some examples in which we have negative information about y -- e.g., we know that y does not belong to a certain interval. In this paper, we show how such negative examples can be used to make the solution …

How To Decide Which Cracks Should Be Repaired First: Theoretical Explanation Of Empirical Formulas, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Due to stress, cracks appear in constructions: cracks appear in buildings, bridges, pavements, among other structures, cracks appear in pavements, etc. In the long run, cracks need to be repaired. However, our resources are limited, so we need to decide which cracks are more dangerous. For this, we need to be able to predict how different cracks will grow. There are several empirical formulas describing crack growth. In this paper, we show that by using scale invariance, we can provide a theoretical explanation for these empirical formulas.

Euclidean Distance Between Intervals Is The Only Representation-Invariant One, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

An interval can be represented as a point in a plane, e.g., as a point with its endpoints as coordinates. We can thus define distance between intervals as the Euclidean distance between the corresponding points. Alternatively, we can describe an interval by its center and radius, which leads to a different definition of distance. Interestingly, these two definitions lead, in effect, to the same distance -- to be more precise, these two distances differ by a multiplicative constant. In principle, we can have more general distances on the plane. In this paper, we show that only for Euclidean distance, the …

It Is Important To Take All Available Information Into Account When Making A Decision: Case Of The Two Envelopes Problem, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In situations when we know the probabilities of all possible consequences, traditional decision theory recommends selecting the action that maximizes expected utility. In many practical situations, however, we only have partial information about the corresponding probabilities. In this case, for different possible probability distributions, we get different values of expected utility. In general, possible values of expected utility form an interval. One way to approach this situation is to use the optimism-pessimism approach proposed by Nobelist Leo Hurwicz. Another approach is to select one of the possible probability distributions -- e.g., the one that has the largest possible entropy. Both …

Grading Homeworks, Verifying Code: How Thorough Should The Feedback Be?, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the ideal world, we should assign many homeworks and give a thorough feedback for each homework. However, in reality, the instructor's time is limited, so we can either assign few homeworks and give a detailed feed back for all of them, or we can assign many homeworks, but give a less thorough feedback. What is the optimal thoroughness? A similar question can be raised for code verification: what is the optimal amount of feedback that should be provided to each programmer? In this paper, we provide answers to these questions.

How To Detect Future Einsteins: Towards Systems Approach, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Talents are rare. It is therefore important to detect and nurture future talents as early as possible. In many disciplines, this is already being done -- via gifted and talented programs, Olympiads, and other ways to select kids with unusually high achievements. However, the current approach is not perfect: some of the kids are selected simply because they are early bloomers, they do not grow into unusually successful researchers; on the other hand, many of those who later become very successful are not selected since they are late bloomers. To avoid these problems, we propose to use systems approach: to …