Mathematics Commons^™

A Priori And A Posteriori Error Estimates For The Quad-Curl Eigenvalue Problem, Lixiu Wang, Qian Zhang, Jiguang Sun, Zhimin Zhang

Michigan Tech Publications, Part 1

In this paper, we consider a priori and a posteriori error estimates of the H(curl²)-conforming finite element when solving the quad-curl eigenvalue problem. An a priori estimate of eigenvalues with convergence order 2(s − 1) is obtained if the corresponding eigenvector u ∈ H^{s − 1}(Ω) and ∇ × u ∈ H^s(Ω). For the a posteriori estimate, by analyzing the associated source problem, we obtain lower and upper bounds for the errors of eigenvectors in the energy norm and upper bounds for the errors of eigenvalues. Numerical examples are presented for validation.

The Mathematical Foundation Of The Musical Scales And Overtones, Michaela Dubose-Schmitt

Theses and Dissertations

This thesis addresses the question of mathematical involvement in music, a topic long discussed going all the way back to Plato. It details the mathematical construction of the three main tuning systems (Pythagorean, just intonation, and equal temperament), the methods by which they were built and the mathematics that drives them through the lens of a historical perspective. It also briefly touches on the philosophical aspects of the tuning systems and whether their differences affect listeners. It further details the invention of the Fourier Series and their relation to the sound wave to explain the concept of overtones within the …

The Impact Of Academic Tracking And Mathematics Self-Concept On Mathematics Achievement., Kain M. Schow

Dissertations, Theses, and Projects

ABSTRACT

This study examines the effects of academic tracking, in high school math, on students’ mathematics self-concept (MSC) and how that correlates to students’ mathematics achievement. This study measured students’ MSC through a mathematics self-concept questionnaire and measured mathematics achievement by the students’ latest grade report. Participants included 60 students in grades 10-12 who had been or were currently enrolled in math courses in the researcher’s school district. The data collected will direct the researcher and school administration on the effects of academic tracking on students, allowing for further discussion about continuing tracking in the district.

Sheltered Math Curriculum For Middle School English Learners, Jasmine Ercink

Dissertations, Theses, and Projects

Language barriers have shown a need for differentiation and sheltered instruction in the classroom for English Learners (ELs) to be successful in the United States public school system. This project proposes a mathematics curriculum using SIOP so that both groups of students in the middle school level can increase their proficiency in the mathematics content area as well as experience opportunities for academic and social language development. The purpose of this report is to describe the processes, methods, data, and intent of the mathematics curriculum for these learners. The curriculum acts as an effective intervention to fill gaps in both …

Tiling Rectangles And 2-Deficient Rectangles With L-Pentominoes, Monica Kane

Rose-Hulman Undergraduate Mathematics Journal

We investigate tiling rectangles and 2-deficient rectangles with L-pentominoes. First, we determine exactly when a rectangle can be tiled with L-pentominoes. We then determine locations for pairs of unit squares that can always be removed from an m × n rectangle to produce a tileable 2-deficient rectangle when m ≡ 1 (mod 5), n ≡ 2 (mod 5) and when m ≡ 3 (mod 5), n ≡ 4 (mod 5).

A New Model For Predicting The Drag And Lift Forces Of Turbulent Newtonian Flow On Arbitrarily Shaped Shells On The Seafloor, Carley R. Walker, James V. Lambers, Julian Simeonov

Dissertations

Currently, all forecasts of currents, waves, and seafloor evolution are limited by a lack of fundamental knowledge and the parameterization of small-scale processes at the seafloor-ocean interface. Commonly used Euler-Lagrange models for sediment transport require parameterizations of the drag and lift forces acting on the particles. However, current parameterizations for these forces only work for spherical particles. In this dissertation we propose a new method for predicting the drag and lift forces on arbitrarily shaped objects at arbitrary orientations with respect to the direction of flow that will ultimately provide models for predicting the sediment sorting processes that lead to …

On Isomorphic K-Rational Groups Of Isogenous Elliptic Curves Over Finite Fields, Ben Kuehnert, Geneva Schlafly, Zecheng Yi

Rose-Hulman Undergraduate Mathematics Journal

It is well known that two elliptic curves are isogenous if and only if they have same number of rational points. In fact, isogenous curves can even have isomorphic groups of rational points in certain cases. In this paper, we consolidate all the current literature on this relationship and give a extensive classification of the conditions in which this relationship arises. First we prove two ordinary isogenous elliptic curves have isomorphic groups of rational points when they have the same $j$-invariant. Then, we extend this result to certain isogenous supersingular elliptic curves, namely those with equal $j$-invariant of either 0 …

Theoretical Aspects Of Calculation Of Photocurrent In Structures With Periodically Spaced Heterinclusions, R. Аyukhanov, A. Uteniyazov, S. Farmonov, N. Shernazarov

Karakalpak Scientific Journal

The processes of excitation of photoconductivity in structures with heteroinclusions with a band gap greater than the band gap of the base material of the matrix have been studied. Structures are studied when such macroinclusions are located periodically and are located at distances equal to the size of the inclusions. A technique has been developed and analytical expressions have been derived for calculating the photocurrent and specific photoconductivity in such structures.

Injection Depletion Effect In Semiconductors With Pair Impurity Complexes, A. Yu. Leyderman, A. Uteniyazov, R. Turmanova, M. Dauletbayev

Karakalpak Scientific Journal

The possibility of realizing the effect of injection depletion in a ⁺ -structure made on the basis of a semiconductor with paired donor-acceptor complexes is considered. It is shown that the appearance of this effect is possible under conditions of almost complete compensation of the shallow dopant donor impurity by the charge of the acceptor component of the pair complex. A necessary condition is also a strong asymmetry of the coefficients of capture of holes at the acceptor level in comparison with the coefficient of capture of electrons at the donor level.

Nonlinear Photoelasticity Processes Near Resonances At The Frequency Of Electronic Transitions From Deep Levels To The Conduction Zone, R. Аyukhanov, A. Uteniyazov, S. Farmonov, N. Shernazarov

Karakalpak Scientific Journal

The phenomenon of nonlinear photoelasticity, which consists of the occurrence of nonlinear photoelasticity coefficients comparable in magnitude to linear ones near the electronic resonance between the deep impurity levels and the bottom of the conduction zone, is considered theoretically. Analytical expressions for the Fourier decomposition coefficients of the dielectric permittivity of the crystal in the phase of the sound wave, which are proportional to the photoelasticity coefficients, are obtained. A general criterion for the nonlinear photoelasticity effect in the investigated frequency region is found numerically. It is shown that high deformation potentials of deep levels stimulate the appearance of this …

Combining Cardiac Monitoring With Actigraphy Aids Nocturnal Arousal Detection During Ambulatory Sleep Assessment In Insomnia, Lara Rösler, Glenn Van Der Lande, Jeanne Leerssen, Austin G. Vandegriffe, Oti Lakbila-Kamal, Jessica C. Foster-Dingley, Anne C.W. Albers, Eus J.W. Van Someren

Mathematics and Statistics Faculty Research & Creative Works

Study Objectives: The objective assessment of insomnia has remained difficult. Multisensory devices collecting heart rate (HR) and motion are regarded as the future of ambulatory sleep monitoring. Unfortunately, reports on altered average HR or heart rate variability (HRV) during sleep in insomnia are equivocal. Here, we evaluated whether the objective quantification of insomnia improves by assessing state-related changes in cardiac measures. Methods: We recorded electrocardiography, posture, and actigraphy in 33 people without sleep complaints and 158 patients with mild to severe insomnia over 4 d in their home environment. At the microscale, we investigated whether HR changed with proximity to …

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

Departmental Technical Reports (CS)

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

In our analysis, we explore the …

Nessie Notation: A New Tool In Sequential Substitution Systems And Graph Theory For Summarizing Concatenations, Colton Davis

Student Research

While doing research looking for ways to categorize causal networks generated by Sequential Substitution Systems, I created a new notation to compactly summarize concatenations of integers or strings of integers, including infinite sequences of these, in the same way that sums, products, and unions of sets can be summarized. Using my method, any sequence of integers or strings of integers with a closed-form iterative pattern can be compactly summarized in just one line of mathematical notation, including graphs generated by Sequential Substitution Systems, many Primitive Pythagorean Triplets, and various Lucas sequences including the Fibonacci sequence and the sequence of square …

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

Departmental Technical Reports (CS)

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

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

Departmental Technical Reports (CS)

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

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

Departmental Technical Reports (CS)

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

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

Departmental Technical Reports (CS)

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

How To Estimate The Present Serviceability Rating Of A Road Segment: Explanation Of An Empirical Formula, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

An accurate estimation of the road quality requires a lot of expertise, and there is not enough experts to provide such estimates for all the road segments. It is therefore desirable to estimate this quality based on easy-to-estimate and easy-to-measure characteristics. Recently, an empirical formula was proposed for such an estimate. In this paper, we provide a theoretical explanation for this empirical formula.

Education In The Era Of Google, Wikipedia, And Deep Learning: Are We Humans Still Needed And If Yes For What?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main purposes of education is to teach skills needed in future life and future jobs. What is important and what is useful changes with time. Before the industrial revolution, routine mechanical work was an important part of human activity – now machines can do it (and do it better). Before printing, copying was an important activity – now copy machines do it. Before computers, humans were needed for computing – now computer do it better. With Wikipedia and Google, there is not much need for scholars being erudite. Even extracting dependencies from data – one of the …

Economy-Related Emotional Attitudes Towards Other People: How Can We Explain Them?, Christopher Reyes, Vladik Kreinovich, Chon Van Le

Departmental Technical Reports (CS)

Research has shown that to properly understand people's economic behavior, it is important to take into account their emotional attitudes towards each other. Behavioral economics shows that different attitudes results in different economy-related behavior. A natural question is: where do these emotional attitudes come from? We show that, in principle, such emotions can be explained by people's objective functions. Specifically, we show it on the example of a person whose main objective is to increase his/her country's GDP: in this case, the corresponding optimization problem leads exactly to natural emotions towards people who contribute a lot or a little towards …

The Primitive Root Problem: A Problem In Bqp, Shixin Wu

Mathematical Sciences Technical Reports (MSTR)

Shor’s algorithm proves that the discrete logarithm problem is in BQP. Based on his algorithm, we prove that the primitive root problem, a problem that verifies if some integer g is a primitive root modulo p where p is the largest prime number smaller than 2n for a given n, which is assumed to be harder than the discrete logarithm problem, is in BQP by using an oracle quantum Turing machine.

Data And Algorithmic Modeling Approaches To Count Data, Andraya Hack

Honors College Theses

Various techniques are used to create predictions based on count data. This type of data takes the form of a non-negative integers such as the number of claims an insurance policy holder may make. These predictions can allow people to prepare for likely outcomes. Thus, it is important to know how accurate the predictions are. Traditional statistical approaches for predicting count data include Poisson regression as well as negative binomial regression. Both methods also have a zero-inflated version that can be used when the data has an overabundance of zeros. Another procedure is to use computer algorithms, also known as …

What About Generic "And"- And "Or"-Operations: From Levels Of Certainty (Philosophical-Physical-Mathematical) To A Natural Interpretation Of Quantum-Like Negative Degrees Of Certainty, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Fuzzy techniques -- techniques designed to convert imprecise human knowledge into precise computer-understandable terms -- have many successful applications. Traditional applications of fuzzy techniques use only important general features of human reasoning and, to make an implement more efficient, ignore subtle details, details which are not important for the corresponding application. But from the more fundamental viewpoint, it is desirable to understand, in all the detail, how people actually reason. In this paper, we use general ideas of fuzzy approach to answer this question. Interestingly -- and somewhat unexpectedly -- the resulting analysis leads to a natural explanation of existence …

How Can The Opposite To A True Theory Be Also True? A Similar Talmudic Discussion Helps Make This Famous Bohr's Statement Logically Consistent, Miroslav Svitek, Vladik Kreinovich

Departmental Technical Reports (CS)

In his famous saying, the Nobelist physicist Niels Bohr claimed that the sign of a deep theory is that while this theory is true, its opposite is also true. While this statement makes heuristic sense, it does not seem to make sense from a logical viewpoint, since, in logic, the opposite to true is false. In this paper, we show how a similar Talmudic discussion can help come up with an interpretation in which Bohr's statement becomes logically consistent.

In The Absence Of Information, The Only Reasonable Negotiation Scheme Is Offering A Certain Percentage Of The Original Request: A Proof, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the case of complete information, a reasonable solution to a negotiation process is Nash's bargaining solution, in which we maximize the product of all agents' utility gains. This is the only solution that does not depend on the order in which we list the agents, and does not change if we use a different scale for describing each agent's utility. In this paper, we apply similar invariance criteria to a situation when practically all information is absent, and all we know is the smallest and largest possible gains. We show that in this situation, the only invariant negotiation strategy …

Why Some Theoretically Possible Representations Of Natural Numbers Were Historically Used And Some Were Not: An Algorithm-Based Explanation, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Historically, people have used many ways to represent natural numbers: from the original "unary" arithmetic, where each number is represented as a sequence of, e.g., cuts (4 is IIII) to modern decimal and binary systems. However, with all this variety, some seemingly reasonable ways of representing natural numbers were never used. For example, it may seem reasonable to represent numbers as products -- e.g., as products of prime numbers -- such a representation was never used in history. So why some theoretically possible representations of natural numbers were historically used and some were not? In this paper, we propose an …

Performance Of Density Estimators In Additive Measurement Error Models Based On Right Censored Data, Hrishabh Khakurel

Mathematics Dissertations - Archive

In the deconvolution problem for right censored data, one is interested in estimating the density of a contaminated variable X when X satisfies Z= X+ E, where E is a measurement error with a known distribution, and the observable variable Z is right-censored. Zhu, Sun, Khakurel, and Wang (2021) applied the Inverse Probability of Censoring Weighted Average method and derived the estimators of the unknown density of X. In this study, we evaluate the performance of the density estimators both in theory and in simulation. We derive the theoretical upper bounds for Mean Squared Error (MSE) of the estimator and …

3-Uniform 4-Path Decompositions Of Complete 3-Uniform Hypergraphs, Rachel Mccann

Mathematical Sciences Undergraduate Honors Theses

The complete 3-uniform hypergraph of order v is denoted as K_v and consists of vertex set V with size v and edge set E, containing all 3-element subsets of V. We consider a 3-uniform hypergraph P₇, a path with vertex set {v₁, v₂, v₃, v₄, v₅, v₆, v₇} and edge set {{v₁, v₂, v₃}, {v₂, v₃, v₄}, {v₄, v₅, v₆}, {v₅, v_{6 …}

Structure Of Number Theoretic Graphs, Lee Trent

Mathematical Sciences Technical Reports (MSTR)

The tools of graph theory can be used to investigate the structure
imposed on the integers by various relations. Here we investigate two
kinds of graphs. The first, a square product graph, takes for its vertices
the integers 1 through n, and draws edges between numbers whose product
is a square. The second, a square product graph, has the same vertex set,
and draws edges between numbers whose sum is a square.
We investigate the structure of these graphs. For square product
graphs, we provide a rather complete characterization of their structure as
a union of disjoint complete graphs. For …

A Necessary And Sufficient Condition For The Asymptotic Normality Of The Quantile Estimator In The Deconvolution Problem, Jeremy R. Valdez

Mathematics Dissertations - Archive

In this study, we examine the estimation of a quantile function when we have n observations coming from the convolution model contaminated by additive measurement errors. Under certain assumptions, a kernel type deconvolution quantile estimator of the unknown quantile function is proposed. Moreover, we discuss the necessary and sufficient condition on the bandwidth in order to investigate the limiting distribution of the deconvolution kernel quantile estimator when the error terms follow either an ordinary smooth or super smooth distribution. A bootstrap approach is used to select the optimal bandwidth to construct approximate distribution free confidence bands for the quantile function …