Fern Or Fractal... Or Both?, 2020 Concordia University St. Paul

#### Fern Or Fractal... Or Both?, Christina Babcock

*Research and Scholarship Symposium Posters*

Fractals are series of self similar sets and can be found in nature. After researching the Barnsley Fern and the iterated function systems using to create the fractal, I was able to apply what I learned to create a fractal shell. This was done using iterated function systems, matrices, random numbers, and Python coding.

Syllabus For Semester Bridge Course: Fundamental Concepts Of Math For Educators: Fundamental Concepts Of Algebra And Geometry & Problem Solving Through Theory And Practice (Math 301a Qbr), 2020 California State University, San Bernardino

#### Syllabus For Semester Bridge Course: Fundamental Concepts Of Math For Educators: Fundamental Concepts Of Algebra And Geometry & Problem Solving Through Theory And Practice (Math 301a Qbr), Lamies Nazzal, Joyce Ahlgren

*Q2S Enhancing Pedagogy*

The Quarter-to-Semester transition at CSUSB brought a number of challenges for many courses or course series. One of those included the math requirement for Liberal Studies series, Math 30x courses. The challenge here is that the 30x series includes four courses, yet the transition to semesters will yield three courses. In the Fall of 2020, the fourth 2-unit course in the series, Math 308 (Problem Solving Through Theory and Practice), will no longer be offered. Instead, it will be embedded into the first three courses. Students beginning the series after Fall 2019, will not have enough time to complete the ...

Compactifications Of Cluster Varieties Associated To Root Systems, 2020 University of Massachusetts Amherst

#### Compactifications Of Cluster Varieties Associated To Root Systems, Feifei Xie

*Doctoral Dissertations*

In this thesis we identify certain cluster varieties with the complement of a union of closures of hypertori in a toric variety. We prove the existence of a compactification $Z$ of the Fock--Goncharov $\mathcal{X}$-cluster variety for a root system $\Phi$ satisfying some conditions, and study the geometric properties of $Z$. We give a relation of the cluster variety to the toric variety for the fan of Weyl chambers and use a modular interpretation of $X(A_n)$ to give another compactification of the $\mathcal{X}$-cluster variety for the root system $A_n$.

Numerical Solution For Solving Two-Points Boundary Value Problems Using Orthogonal Boubaker Polynomials, 2020 University of Technology, Iraq

#### Numerical Solution For Solving Two-Points Boundary Value Problems Using Orthogonal Boubaker Polynomials, Imad Noah Ahmed

*Emirates Journal for Engineering Research*

In this paper, a new technique for solving boundary value problems (BVPs) is introduced. An orthogonal function for Boubaker polynomial was utilizedand by the aid of Galerkin method the BVP was transformed to a system of linear algebraic equations with unknown coefficients, which can be easily solved to find the approximate result. Some numerical examples were added with illustrations, comparing their results with the exact to show the efficiency and the applicability of the method.

Isoperimetric Problems On The Line With Density |X|^P, 2020 Nanjing International School

#### Isoperimetric Problems On The Line With Density |X|^P, Juiyu Huang, Xinkai Qian, Yiheng Pan, Mulei Xu, Lu Yang, Junfei Zhou

*Rose-Hulman Undergraduate Mathematics Journal*

On the line with density |x|^p, we prove that the best single bubble is an interval with endpoint at the origin and that the best double bubble is two adjacent intervals that meet at the origin.

Scrollar Invariants Of Tropical Chains Of Loops, 2020 University of Kentucky

#### Scrollar Invariants Of Tropical Chains Of Loops, Kalila Joelle Sawyer

*Theses and Dissertations--Mathematics*

We define scrollar invariants of tropical curves with a fixed divisor of rank 1. We examine the behavior of scrollar invariants under specialization, and compute these invariants for a much-studied family of tropical curves. Our examples highlight many parallels between the classical and tropical theories, but also point to some substantive distinctions.

Phylogenetic Networks And Functions That Relate Them, 2020 The University of Akron

#### Phylogenetic Networks And Functions That Relate Them, Drew Scalzo

*Williams Honors College, Honors Research Projects*

Phylogenetic Networks are defined to be simple connected graphs with exactly n labeled nodes of degree one, called leaves, and where all other unlabeled nodes have a degree of at least three. These structures assist us with analyzing ancestral history, and its close relative - phylogenetic trees - garner the same visualization, but without the graph being forced to be connected. In this paper, we examine the various characteristics of Phylogenetic Networks and functions that take these networks as inputs, and convert them to more complex or simpler structures. Furthermore, we look at the nature of functions as they relate to the ...

Codes, Cryptography, And The Mceliece Cryptosystem, 2020 Liberty University

#### Codes, Cryptography, And The Mceliece Cryptosystem, Bethany Matsick

*Senior Honors Theses*

Over the past several decades, technology has continued to develop at an incredible rate, and the importance of properly securing information has increased significantly. While a variety of encryption schemes currently exist for this purpose, a number of them rely on problems, such as integer factorization, that are not resistant to quantum algorithms. With the reality of quantum computers approaching, it is critical that a quantum-resistant method of protecting information is found. After developing the proper background, we evaluate the potential of the McEliece cryptosystem for use in the post-quantum era by examining families of algebraic geometry codes that allow ...

Heat Kernel Voting With Geometric Invariants, 2020 Minnesota State University, Mankato

#### Heat Kernel Voting With Geometric Invariants, Alexander Harr

*All Theses, Dissertations, and Other Capstone Projects*

Here we provide a method for comparing geometric objects. Two objects of interest are embedded into an infinite dimensional Hilbert space using their Laplacian eigenvalues and eigenfunctions, truncated to a finite dimensional Euclidean space, where correspondences between the objects are searched for and voted on. To simplify correspondence finding, we propose using several geometric invariants to reduce the necessary computations. This method improves on voting methods by identifying isometric regions including shapes of genus greater than 0 and dimension greater than 3, as well as almost retaining isometry.

Algebraic And Geometric Properties Of Hierarchical Models, 2020 University of Kentucky

#### Algebraic And Geometric Properties Of Hierarchical Models, Aida Maraj

*Theses and Dissertations--Mathematics*

In this dissertation filtrations of ideals arising from hierarchical models in statistics related by a group action are are studied. These filtrations lead to ideals in polynomial rings in infinitely many variables, which require innovative tools. Regular languages and finite automata are used to prove and explicitly compute the rationality of some multivariate power series that record important quantitative information about the ideals. Some work regarding Markov bases for non-reducible models is shown, together with advances in the polyhedral geometry of binary hierarchical models.

Introduction To Neutroalgebraic Structures And Antialgebraic Structures (Revisited), 2020 University of New Mexico

#### Introduction To Neutroalgebraic Structures And Antialgebraic Structures (Revisited), Florentin Smarandache

*Mathematics and Statistics Faculty and Staff Publications*

In all classical algebraic structures, the Laws of Compositions on a given set are well-defined. But this is a restrictive case, because there are many more situations in science and in any domain of knowledge when a law of composition defined on a set may be only partially-defined (or partially true) and partially-undefined (or partially false), that we call NeutroDefined, or totally undefined (totally false) that we call AntiDefined. Again, in all classical algebraic structures, the Axioms (Associativity, Commutativity, etc.) defined on a set are totally true, but it is again a restrictive case, because similarly there are numerous situations ...

Albert Forms, Quaternions, Schubert Varieties & Embeddability, 2019 The University of Western Ontario

#### Albert Forms, Quaternions, Schubert Varieties & Embeddability, Jasmin Omanovic

*Electronic Thesis and Dissertation Repository*

The origin of embedding problems can be understood as an effort to find some minimal datum which describes certain algebraic or geometric objects. In the algebraic theory of quadratic forms, Pfister forms are studied for a litany of powerful properties and representations which make them particularly interesting to study in terms of embeddability. A generalization of these properties is captured by the study of central simple algebras carrying involutions, where we may characterize the involution by the existence of particular elements in the algebra. Extending this idea even further, embeddings are just flags in the Grassmannian, meaning that their study ...

Algebraic Methods For Proving Geometric Theorems, 2019 California State University, San Bernardino

#### Algebraic Methods For Proving Geometric Theorems, Lynn Redman

*Electronic Theses, Projects, and Dissertations*

Algebraic geometry is the study of systems of polynomial equations in one or more variables. Thinking of polynomials as functions reveals a close connection between affine varieties, which are geometric structures, and ideals, which are algebraic objects. An affine variety is a collection of tuples that represents the solutions to a system of equations. An ideal is a special subset of a ring and is what provides the tools to prove geometric theorems algebraically. In this thesis, we establish that a variety depends on the ideal generated by its defining equations. The ability to change the basis of an ideal ...

Essential Dimension Of Parabolic Bundles, 2019 The University of Western Ontario

#### Essential Dimension Of Parabolic Bundles, Dinesh Valluri

*Electronic Thesis and Dissertation Repository*

Essential dimension of a geometric object is roughly the number of algebraically independent parameters needed to define the object. In this thesis we give upper bounds for the essential dimension of parabolic bundles over a non-singular curve X of genus g greater than or equal to 2 using Borne's correspondence between parabolic bundles on a curve and vector bundles on a root stack. This is a generalization of the work of Biswas, Dhillon and Hoffmann on the essential dimension of vector bundles, by following their method for curves and adapting it to root stacks. In this process, we invoke ...

Topology And Dynamics Of Gene Regulatory Networks: A Meta-Analysis, 2019 Iowa State University

#### Topology And Dynamics Of Gene Regulatory Networks: A Meta-Analysis, Claus Kadelka

*Biology and Medicine Through Mathematics Conference*

No abstract provided.

Unifications Of Pythagorean Triple Schema, 2019 East Tennessee State University

#### Unifications Of Pythagorean Triple Schema, Emily Hammes

*Undergraduate Honors Theses*

Euclid’s Method of finding Pythagorean triples is a commonly accepted and applied technique. This study focuses on a myriad of other methods behind finding such Pythagorean triples. Specifically, we discover whether or not other ways of finding triples are special cases of Euclid’s Method.

On The Complexity Of Computing Galois Groups Of Differential Equations, 2019 The Graduate Center, City University of New York

#### On The Complexity Of Computing Galois Groups Of Differential Equations, Mengxiao Sun

*All Dissertations, Theses, and Capstone Projects*

The differential Galois group is an analogue for a linear differential equation of the classical Galois group for a polynomial equation. An important application of the differential Galois group is that a linear differential equation can be solved by integrals, exponentials and algebraic functions if and only if the connected component of its differential Galois group is solvable. Computing the differential Galois groups would help us determine the existence of the solutions expressed in terms of elementary functions (integrals, exponentials and algebraic functions) and understand the algebraic relations among the solutions.

Hrushovski first proposed an algorithm for computing the differential ...

Data Parsing For Optimized Molecular Geometry Calculations, 2019 Stephen F Austin State University

#### Data Parsing For Optimized Molecular Geometry Calculations, Luke Rens

*Undergraduate Research Conference*

The purpose of this project is to optimize and streamline to process of using ADF and ReaxFF. There is no efficient way to effectively add constraints to a compound and run it through ADF, take the ADF output and create a file that can be run through Reaxff, then take that Reaxff output and come to conclusions on it. To streamline this process, scripts were developed using Python to parse information out of data generated by ADF.

Enhanced Koszulity In Galois Cohomology, 2019 The University of Western Ontario

#### Enhanced Koszulity In Galois Cohomology, Marina Palaisti

*Electronic Thesis and Dissertation Repository*

Despite their central role in Galois theory, absolute Galois groups remain rather mysterious; and one of the main problems of modern Galois theory is to characterize which profinite groups are realizable as absolute Galois groups over a prescribed field. Obtaining detailed knowledge of Galois cohomology is an important step to answering this problem. In our work we study various forms of enhanced Koszulity for quadratic algebras. Each has its own importance, but the common ground is that they all imply Koszulity. Applying this to Galois cohomology, we prove that, in all known cases of finitely generated pro-$p$-groups, Galois ...

Dense Geometry Of Music And Visual Arts: Vanishing Points, Continuous Tonnetz, And Theremin Performance, 2019 Independent researcher, Palermo, Italy

#### Dense Geometry Of Music And Visual Arts: Vanishing Points, Continuous Tonnetz, And Theremin Performance, Maria Mannone, Irene Iaccarino, Rosanna Iembo

*The STEAM Journal*

The dualism between continuous and discrete is relevant in music theory as well as in performance practice of musical instruments. Geometry has been used since longtime to represent relationships between notes and chords in tonal system. Moreover, in the field of mathematics itself, it has been shown that the continuity of real numbers can arise from geometrical observations and reasoning. Here, we consider a geometrical approach to generalize representations used in music theory introducing continuous pitch. Such a theoretical framework can be applied to instrument playing where continuous pitch can be naturally performed. Geometry and visual representations of concepts of ...