Mathematics Commons^™

An Optimal Control Problem Solution For Chemical Reactor, Dias Nurmagambetov

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we describe one of the solutions of a nonlinear optimal control problem for a chemical reactor. A solution on finding a chemical reactor’s optimal temperature regime for having a maximum concentration of final product is presented. The optimal control has been found by immersion method for boundary value problem with a phase and control restrictions. This method is reducing the original boundary value problem to a special optimal control problem, using the general solution of the Fredholm integral equation of the first kind. With this method's solution had been created a software for the problem calculations. Analysis …

Orthogonality In Terms Of 2-Hh Norm And Bounded Linear Operators In Banach Spaces, Bhuwan P. Ojha, Prakash M. Bajracharya

Applications and Applied Mathematics: An International Journal (AAM)

In the present paper, the generalization of the Carlson orthogonality for functionals to operators in Banach spaces has been studied. We will also investigate various properties related to the Carlsson, Birkhoff-James, and Pythagorean orthogonality for operators. Kikianty and Dragomir (2010) mentioned in their paper by stating that Pythagorean and isosceles orthogonality through the medium of 2 − HH norm satisfies the non-degeneracy, symmetry and continuity properties without mentioning detailed proof. This paper provides the complete proof of these properties as well as the equivalency of additivity and homogeneity of the isosceles orthogonality with the help of 2 − HH norm. …

Why Cauchy Membership Functions: Efficiency, Javier Viana, Stephan Ralescu, Kelly Cohen, Anca Ralescu, Vladik Kreinovich

Departmental Technical Reports (CS)

Fuzzy techniques depend heavily on eliciting meaningful membership functions for the fuzzy sets used. Often such functions are obtained from data. Just as often they are obtained from experts knowledgable of the domain and the problem being addressed. However, there are cases when neither is possible, for example because of insufficient data, or unavailable experts. What functions should one choose and what should guide such choice? This paper argues in favor of using Cauchy membership functions, thus named because their expression is similar to that of the Cauchy distributions. The paper provides a theoretical explanation for this choice.

Many Known Quantum Algorithms Are Optimal: Symmetry-Based Proofs, Vladik Kreinovich, Oscar Galindo, Olga Kosheleva

Departmental Technical Reports (CS)

Many quantum algorithms have been proposed which are drastically more efficient that the best of the non-quantum algorithms for solving the same problems. A natural question is: are these quantum algorithms already optimal -- in some reasonable sense -- or they can be further improved? In this paper, we review recent results showing that many known quantum algorithms are actually optimal. Several of these results are based on appropriate invariances (symmetries).

Why Rectified Linear Neurons: Two Convexity-Related Explanations, Jonatan Contreras, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

At present, the most efficient machine learning technique is deep learning, in which non-linearity is attained by using rectified linear functions s(x)=max(0,x). Empirically, these functions work better than any other nonlinear functions that have been tried. In this paper, we provide a possible theoretical explanation for this empirical fact. This explanation is based on the fact that one of the main applications of neural networks is decision making, when we want to find an optimal solution. We show that the need to adequately deal with situations when the corresponding optimization problem is feasible -- i.e., for which the objective function …

Two Temperature Dual-Phase-Lag Fractional Thermal Investigation Of Heat Flow Inside A Uniform Rod, Vinayak Kulkarni, Gaurav Mittal

Applications and Applied Mathematics: An International Journal (AAM)

A non-classical, coupled, fractionally ordered, dual-phase-lag (DPL) heat conduction model has been presented in the framework of the two-temperature theory in the bounded Cartesian domain. Due to the application of two-temperature theory, the governing heat conduction equation is well-posed and satisfying the required stability criterion prescribed for a DPL model. The mathematical formulation has been applied to a uniform rod of finite length with traction free ends considered in a perfectly thermoelastic homogeneous isotropic medium. The initial end of the rod has been exposed to the convective heat flux and energy dissipated by convection into the surrounding medium through the …

Invariance-Based Approach: General Methods And Pavement Engineering Case Study, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

In many application areas such as pavement engineering, the phenomena are complex, and as a result, we do not have first-principle models describing the corresponding dependencies. Luckily, in many such areas, there is a lot of empirical data and, based on this data, many useful empirical dependencies have been found. The problem is that since many of these dependencies do not have a theoretical explanation, practitioners are often hesitant to use them: there have been many cases when an empirical formula stops being valid when circumstances change. To make the corresponding empirical formulas more reliable, it is therefore desirable to …

Why Dilated Convolutional Neural Networks: A Proof Of Their Optimality, Jonatan Contreras, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the most effective image processing techniques is the use of convolutional neural networks that use convolutional layers. In each such layer, the value of the output at each point is a combination of input data corresponding to several neighboring points. To improve the accuracy, researchers have developed a version of this technique, in which only data from some of the neighboring points is processed. It turns out that the most efficient case -- called dilated convolution -- is when we select the neighboring points whose differences in both coordinates are divisible by some constant l. In this paper, …

How General Is Fuzzy Decision Making?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, users describe their preferences in imprecise (fuzzy) terms. In such situations, fuzzy techniques are a natural way to describe these preferences in precise terms.

Of course, this description is only an approximation to the ideal decision making that a person would perform if we took time to elicit his/her exact preferences. How accurate is this approximation? When can fuzzy decision making -- potentially -- describe the exact decision making, and when there is a limit to the accuracy of fuzzy approximations?

In this paper, we show that decision making can be precisely described in fuzzy terms …

Green Computing: Three Examples Of How Non-Trivial Mathematical Analysis Can Help, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Environment-related problems are extremely important for mankind, the fate of humanity itself depends on our ability to solve these problems. These problems are complex, we cannot solve them without using powerful computers. Thus, in the environmental research, environment-related computing is one of the main computing-related research directions. Another direction is related to the fact that computing itself can be (and currently is) harmful for the environment. How to make computing more environment-friendly, how to move towards green computing -- this is the second important direction. A third direction is motivated by the very complexity of environmental systems: it is difficult …

Is It Fair That Advanced Workers Get Paid Disproportionally More: Economic Analysis, Olga Kosheleva, Sean R. Aguilar

Departmental Technical Reports (CS)

On the one hand, everyone agrees that economics should be fair, that workers should get equal pay for equal work. Any instance of unfairness causes a strong disagreement. On the other hand, in many companies, advanced workers -- who produce more than others -- get paid dispropotionally more for their work, and this does not seem to cause any negative feelings. In this paper, we analyze this situation from the economic viewpoint. We show that from this viewpoint, additional payments for advanced workers indeed make economic sense, benefit everyone, and thus -- in contrast to the naive literal interpretation of …

Dominating Functions In Graphs, Maria Talanda-Fisher

Dissertations

Domination in graphs has become one of the most popular areas of graph the- ory, no doubt due to its many fascinating problems and applications to modern society, as well as the sheer mathematical beauty of the subject. While this area evidently began with the work by the French mathematician Claude Berge in 1958 and the Norwegian-American mathematician Oystein Ore in 1962, domination did not become an active area of research until 1977 with the appearance of the survey paper by Ernest Cockayne and Stephen Hedetniemi. Since then a large number of variations of domination have surfaced and provided numerous …

On Digital Metric Space Satisfying Certain Rational Inequalities, Krati Shukla

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we have established some new results by extending some existing theorems in the setting of Digital Metric Space. We also proved some results in Digital Metric Space which were established earlier in the context of Complete Metric Space by different authors.

Simplified Intuitionistic Neutrosophic Soft Set And Its Application On Diagnosing Psychological Disorder By Using Similarity Measure, Veerappan Chinnadurai, Albert Bobin

Applications and Applied Mathematics: An International Journal (AAM)

The primary focus of this manuscript comprises three sections. Initially, we introduce the concept of a simplified intuitionistic neutrosophic soft set. We impose an intuitionistic condition between the membership values of truth and falsity such that their sum does not exceed unity. Similarly, for indeterminacy, the membership value is a real number from the closed interval [0, 1]. Hence, the sum of membership values of truth, indeterminacy, and falsity does not exceed two. We present the notion of necessity, possibility, concentration, and dilation operators and establish some of its properties. Second, we define the similarity measure between two simplified intuitionistic …

Characterizing Undergraduate Students’ Proving Processes Around “Stuck Points”, Yaomingxin Lu

Dissertations

Learning to prove mathematical propositions is a cornerstone of mathematics as a discipline (de Villiers, 1990). However, since proving is a different mathematical activity as compared to students’ prior experience, research has also shown that many undergraduate students struggle to learn to prove, including those who major in mathematics (Moore, 1994; Selden, 2012). While the field has generated research that has analyzed the final products of proof (Selden & Selden, 2009) and there are frameworks for analyzing problem-solving processes (e.g., Carlson & Bloom, 2005; Schoenfeld, 1985, 2010), much remains to be known about analyzing undergraduate students’ proving processes. With a …

On The Representation Of Boolean Magmas And Boolean Semilattices, Peter Jipsen, M. Eyad Kurd-Misto, James Wimberley

Mathematics, Physics, and Computer Science Faculty Books and Book Chapters

A magma is an algebra with a binary operation ·, and a Boolean magma is a Boolean algebra with an additional binary operation · that distributes over all finite Boolean joins. We prove that all square-increasing (x ≤ x²) Boolean magmas are embedded in complex algebras of idempotent (x = x²) magmas. This solves a problem in a recent paper [3] by C. Bergman. Similar results are shown to hold for commutative Boolean magmas with an identity element and a unary inverse operation, or with any combination of these properties.

A Boolean semilattice is …

The “Knapsack Problem” Workbook: An Exploration Of Topics In Computer Science, Steven Cosares

Open Educational Resources

This workbook provides discussions, programming assignments, projects, and class exercises revolving around the “Knapsack Problem” (KP), which is widely a recognized model that is taught within a typical Computer Science curriculum. Throughout these discussions, we use KP to introduce or review topics found in courses covering topics in Discrete Mathematics, Mathematical Programming, Data Structures, Algorithms, Computational Complexity, etc. Because of the broad range of subjects discussed, this workbook and the accompanying spreadsheet files might be used as part of some CS capstone experience. Otherwise, we recommend that individual sections be used, as needed, for exercises relevant to a course in …

On Higher-Order Duality In Nondifferentiable Minimax Fractional Programming, S. Al-Homidan, Vivek Singh, I. Ahmad

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we consider a nondifferentiable minimax fractional programming problem with continuously differentiable functions and formulate two types of higher-order dual models for such optimization problem. Weak, strong and strict converse duality theorems are derived under higher- order generalized invexity.

On Burr Iii-Inverse Weibull Distribution With Covid-19 Applications, Fiaz Ahmad Bhatti, Sedigheh Mirzaei Salehabadi, Gholamhossein G. Hamedani

Mathematical and Statistical Science Faculty Research and Publications

We introduce a flexible lifetime distribution called Burr III-Inverse Weibull (BIII-IW). The new proposed distribution has well-known sub-models. The BIII-IW density function includes exponential, left-skewed, right-skewed and symmetrical shapes. The BIII-IW model’s failure rate can be monotone and non-monotone depending on the parameter values. To show the importance of the BIII-IW distribution, we establish various mathematical properties such as random number generator, ordinary moments, conditional moments, residual life functions, reliability measures and characterizations. We address the maximum likelihood estimates (MLE) for the BIII-IW parameters and estimate the precision of the maximum likelihood estimators via a simulation study. We consider applications …

Rapid Entry Into Masters In Computing Program For Non-Majors, Gary S. Krenz, Thomas Kaczmarek, John C. Moyer

Mathematical and Statistical Science Faculty Research and Publications

The COSMIC: Change Opportunity - Start Masters in Computing graduate curriculum initiative strives to provide a rapid entry pathway to a professional Master of Science (MS) degree for individuals who do not have an undergraduate degree in computing, but who wish to cross over to a career in the computing field. The goal of our curriculum is to minimize the time students spend preparing for graduate study and maximize experiences relevant for work after graduation. The COSMIC curriculum initiative is similar in concept to other post-baccalaureate conversion programs. However, customization of the COSMIC bridge course and curriculum pathway makes it …

Quantization Coefficients For Uniform Distributions On The Boundaries Of Regular Polygons, Joel Hansen, Itzamar Marquez, Mrinal Kanti Roychowdhury, Eduardo Torres

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we give a general formula to determine the quantization coefficients for uniform distributions defined on the boundaries of different regular m-sided polygons inscribed in a circle. The result shows that the quantization coefficient for the uniform distribution on the boundary of a regular m-sided polygon inscribed in a circle is an increasing function of m, and approaches to the quantization coefficient for the uniform distribution on the circle as m tends to infinity.

Modeling The Spread Of Covid-19 Over Varied Contact Networks, Ryan L. Solorzano

Master's Theses

When attempting to mitigate the spread of an epidemic without the use of a vaccine, many measures may be made to dampen the spread of the disease such as physically distancing and wearing masks. The implementation of an effective test and quarantine strategy on a population has the potential to make a large impact on the spread of the disease as well. Testing and quarantining strategies become difficult when a portion of the population are asymptomatic spreaders of the disease. Additionally, a study has shown that randomly testing a portion of a population for asymptomatic individuals makes a small impact …

Energy On Spheres And Discreteness Of Minimizing Measures, Dmitriy Bilyk, Alexey Glazyrin, Ryan Matzke, Josiah Park, Oleksandr Vlasiuk

School of Mathematical & Statistical Sciences Faculty Publications

In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the p-frame energy has empty interior whenever p is not an even integer. A similar effect is also demonstrated for energies with analytic potentials which are not positive definite. In addition, we establish the existence of discrete minimizers for a large class of energies, which includes energies with polynomial potentials.

N-Fold Matrix Factorizations, Eric Hopkins

Department of Mathematics: Dissertations, Theses, and Student Research

The study of matrix factorizations began when they were introduced by Eisenbud; they have since been an important topic in commutative algebra. Results by Eisenbud, Buchweitz, and Yoshino relate matrix factorizations to maximal Cohen-Macaulay modules over hypersurface rings. There are many important properties of the category of matrix factorizations, as well as tensor product and hom constructions. More recently, Backelin, Herzog, Sanders, and Ulrich used a generalization of matrix factorizations -- so called N-fold matrix factorizations -- to construct Ulrich modules over arbitrary hypersurface rings. In this dissertation we build up the theory of N-fold matrix factorizations, proving analogues of …

Chudnovsky's Conjecture And The Stable Harbourne-Huneke Containment, Sankhaneel Bisui, Eloísa Grifo, Huy Tài Hà, Thái Thành Nguyên

Department of Mathematics: Faculty Publications

We investigate containment statements between symbolic and ordinary powers and bounds on the Waldschmidt constant of defining ideals of points in projective spaces. We establish the stable Harbourne conjecture for the defining ideal of a general set of points. We also prove Chudnovsky’s Conjecture and the stable version of the Harbourne–Huneke containment conjectures for a general set of sufficiently many points.

Demailly's Conjecture And The Containment Problem, Sankhaneel Bisui, Eloisa Grifo, Huy Tài Hà, Thái Thành Nguyên

Department of Mathematics: Faculty Publications

We investigate Demailly’s Conjecture for a general set of sufficiently many points. Demailly’s Conjecture generalizes Chudnovsky’s Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective space. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly’s bound, and prove that a general version of that containment holds for generic determinantal ideals and defining ideals of star configurations.

Hexagon Tilings Of The Plane That Are Not Edge-To-Edge, Dirk Frettlöh, Alexey Glazyrin, Z. Lángi

School of Mathematical & Statistical Sciences Faculty Publications

An irregular vertex in a tiling by polygons is a vertex of one tile and belongs to the interior of an edge of another tile. In this paper we show that for any integer k≥3, there exists a normal tiling of the Euclidean plane by convex hexagons of unit area with exactly k irregular vertices. Using the same approach we show that there are normal edge-to-edge tilings of the plane by hexagons of unit area and exactly k many n-gons (n>6) of unit area. A result of Akopyan yields an upper bound for k depending on the maximal diameter …

Hierarchical Hyperbolicity Of Graph Products And Graph Braid Groups, Daniel James Solomon Berlyne

Dissertations, Theses, and Capstone Projects

This thesis comprises three original contributions by the author concerning hierarchical hyperbolicity, a coarse geometric tool developed by Behrstock, Hagen, and Sisto to provide a common framework for studying aspects of non-positive curvature in a wide variety of groups and spaces.

We show that any graph product of finitely generated groups is hierarchically hyperbolic relative to its vertex groups. We apply this to answer two questions of Genevois about the electrification of a graph product of finite groups. We also answer two questions of Behrstock, Hagen, and Sisto: we show that the syllable metric on a graph product forms a …

Asymmetric Multivariate Archimedean Copula Models And Semi-Competing Risks Data Analysis, Ziyan Guo

Dissertations

Many multivariate models have been proposed and developed to model high dimensional data when the dimension of a data set is greater than 2 (d ≥ 3). The existing multivariate models often force the “exchangeable” structure for part or the whole model, are not very flexible which tends to be of limited use in practice. There is a demand for developing and studying multivariate models with any pre-specified bivariate margins.

Suppose there exists such a class of flexible models with any pre-specified bivariate margins. Given a multivariate data, what is the distribution function and how to easily estimate the parameters …

Mechanisms Of Oscillations And Polyglot Entrainment In Neuronal And Circadian Models, Emel Khan

Dissertations

Entrainment is a type of synchronization in which the period of an endogenous oscillator matches the period of an external forcing signal and a stable phase relationship is maintained between them. Entrainment patterns are described in terms of the number of input oscillations (N) that are phase-locked to a number of output oscillations (M), referred to as N:M patterns. Arnold tongue diagrams are used to depict the regions of N:M entrainment patterns in the input period-amplitude parameter space. Although the entrainment of self-sustained oscillators by periodic forcing are well investigated is a well-studied problem, entrainment of damped oscillators has been …