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Calculating The Cohomology Of A Lie Algebra Using Maple And The Serre Hochschild Spectral Sequence, Jacob Kullberg 2018 Utah State University

Calculating The Cohomology Of A Lie Algebra Using Maple And The Serre Hochschild Spectral Sequence, Jacob Kullberg

All Graduate Plan B and other Reports

Lie algebra cohomology is an important tool in many branches of mathematics. It is used in the Topology of homogeneous spaces, Deformation theory, and Extension theory. There exists extensive theory for calculating the cohomology of semi simple Lie algebras, but more tools are needed for calculating the cohomology of general Lie algebras. To calculate the cohomology of general Lie algebras, I used the symbolic software program called Maple. I wrote software to calculate the cohomology in several different ways. I wrote several programs to calculate the cohomology directly. This proved to be computationally expensive as the number of differential forms ...


The Influence Of Canalization On The Robustness Of Finite Dynamical Systems, Claus Kadelka 2018 Illinois State University

The Influence Of Canalization On The Robustness Of Finite Dynamical Systems, Claus Kadelka

Annual Symposium on Biomathematics and Ecology: Education and Research

No abstract provided.


Dual Perspectives On Desargues' Theorem, Carl Lienert 2018 Ursinus College

Dual Perspectives On Desargues' Theorem, Carl Lienert

Geometry

No abstract provided.


Galois Groups Of Differential Equations And Representing Algebraic Sets, Eli Amzallag 2018 The Graduate Center, City University of New York

Galois Groups Of Differential Equations And Representing Algebraic Sets, Eli Amzallag

All Dissertations, Theses, and Capstone Projects

The algebraic framework for capturing properties of solution sets of differential equations was formally introduced by Ritt and Kolchin. As a parallel to the classical Galois groups of polynomial equations, they devised the notion of a differential Galois group for a linear differential equation. Just as solvability of a polynomial equation by radicals is linked to the equation’s Galois group, so too is the ability to express the solution to a linear differential equation in "closed form" linked to the equation’s differential Galois group. It is thus useful even outside of mathematics to be able to compute and ...


Webwork Problems For Linear Algebra, Hashim Saber, Beata Hebda 2018 University of North Georgia

Webwork Problems For Linear Algebra, Hashim Saber, Beata Hebda

Mathematics Ancillary Materials

This set of problems for Linear Algebra in the open-source WeBWorK mathematics platform was created under a Round Eleven Mini-Grant for Ancillary Materials Creation. The problems were created for an implementation of the CC-BY Lyrix open textbook A First Course in Linear Algebra. Also included as an additional file are the selected and modified Lyryx Class Notes for the textbook.

Topics covered include:

  • Linear Independence
  • Linear Transformations
  • Matrix of a Transformation
  • Isomorphisms
  • Eigenvalues and Eigenvectors
  • Diagonalization
  • Orthogonality


Constructing Surfaces With (1/(K-2)^2)(1,K-3) Singularities, Liam Patrick Keenan 2018 Lawrence University

Constructing Surfaces With (1/(K-2)^2)(1,K-3) Singularities, Liam Patrick Keenan

Lawrence University Honors Projects

We develop a procedure to construct complex algebraic surfaces which are stable, minimal, and of general type, possessing a T-singularity of the form (1/(k-2)2)(1,k-3).


Mixed Categories Of Sheaves On Toric Varieties, Sean Michael Taylor 2018 Louisiana State University and Agricultural and Mechanical College

Mixed Categories Of Sheaves On Toric Varieties, Sean Michael Taylor

LSU Doctoral Dissertations

In [BGS96], Beilinson, Ginzburg, and Soergel introduced the notion of mixed categories. This idea often underlies many interesting "Koszul dualities." In this paper, we produce a mixed derived category of constructible complexes (in the sense of [BGS96]) for any toric variety associated to a fan. Furthermore, we show that it comes equipped with a t-structure whose heart is a mixed version of the category of perverse sheaves. In chapters 2 and 3, we provide the necessary background. Chapter 2 concerns the categorical preliminaries, while chapter 3 gives the background geometry. This concerns both some basics of toric varieties as well ...


The Average Measure Of A K-Dimensional Simplex In An N-Cube, John A. Carter 2018 Missouri State University

The Average Measure Of A K-Dimensional Simplex In An N-Cube, John A. Carter

MSU Graduate Theses

Within an n-dimensional unit cube, a number of k-dimensional simplices can be formed whose vertices are the vertices of the n-cube. In this thesis, we analyze the average measure of a k-simplex in the n-cube. We develop exact equations for the average measure when k = 1, 2, and 3. Then we generate data for these cases and conjecture that their averages appear to approach nk/2 times some constant. Using the convergence of Bernstein polynomials and a k-simplex Bernstein generalization, we prove the conjecture is true for the 1-simplex and 2-simplex cases. We then develop a generalized formula for ...


Determinantal Representations Of Elliptic Curves Via Weierstrass Elliptic Functions, Mao-Ting Chien, Hiroshi Nakazato 2018 Soochow University

Determinantal Representations Of Elliptic Curves Via Weierstrass Elliptic Functions, Mao-Ting Chien, Hiroshi Nakazato

Electronic Journal of Linear Algebra

Helton and Vinnikov proved that every hyperbolic ternary form admits a symmetric derminantal representation via Riemann theta functions. In the case the algebraic curve of the hyperbolic ternary form is elliptic, the determinantal representation of the ternary form is formulated by using Weierstrass $\wp$-functions in place of Riemann theta functions. An example of this approach is given.


Dalton State College Apex Calculus, Thomas Gonzalez, Michael Hilgemann, Jason Schmurr 2018 Dalton State College

Dalton State College Apex Calculus, Thomas Gonzalez, Michael Hilgemann, Jason Schmurr

Mathematics Open Textbooks

This text for Analytic Geometry and Calculus I, II, and III is a Dalton State College remix of APEX Calculus 3.0. The text was created through a Round Six ALG Textbook Transformation Grant.

Topics covered in this text include:

  • Limits
  • Derivatives
  • Integration
  • Antidifferentiation
  • Sequences
  • Vectors

Files can also be downloaded on the Dalton State College GitHub:

https://github.com/DaltonStateCollege/calculus-text/blob/master/Calculus.pdf


Analytic Geometry And Calculus I, Ii, & Iii (Dalton), Thomas Gonzalez, Michael Hilgemann, Jason Schmurr 2018 Dalton State College

Analytic Geometry And Calculus I, Ii, & Iii (Dalton), Thomas Gonzalez, Michael Hilgemann, Jason Schmurr

Mathematics Grants Collections

This Grants Collection for Analytic Geometry and Calculus I, II, & III was created under a Round Six ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

  • Linked Syllabus
  • Initial Proposal
  • Final Report


Branching Matrices For The Automorphism Group Lattice Of A Riemann Surface, Sean A. Broughton 2018 Rose-Hulman Institute of Technology

Branching Matrices For The Automorphism Group Lattice Of A Riemann Surface, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

Let S be a Riemann surface and G a large subgroup of Aut(S) (Aut(S) may be unknown). We are particularly interested in regular n-gonal surfaces, i.e., the quotient surface S/G (and hence S/Aut(S)) has genus zero. For various H the ramification information of the branched coverings S/K -> S/H may be captured in a matrix. The ramification information, in particular strong branching, may be then be used in analyzing the structure of Aut(S). The ramification information is conjugation invariant so the matrix's rows and columns may be indexed by conjugacy ...


Schubert Polynomial Multiplication, Sara Amato 2018 Assumption College

Schubert Polynomial Multiplication, Sara Amato

Honors Theses

Schur polynomials are a fundamental object in the field of algebraic combinatorics. The product of two Schur polynomials can be written as a sum of Schur polynomials using non-negative integer coefficients. A simple combinatorial algorithm for generating these coefficients is called the Littlewood-Richardson Rule. Schubert polynomials are generalizations of the Schur polynomials. Schubert polynomials also appear in many contexts, such as in algebraic combinatorics and algebraic geometry. It is known from algebraic geometry that the product of two Schubert polynomials can be written as a sum of Schubert polynomials using non-negative integer coefficients. However, a simple combinatorial algorithm for generating ...


Framed Sheaves On A Quadric Surface, Nguyen Thuc Huy Le 2018 University of Massachusetts Amherst

Framed Sheaves On A Quadric Surface, Nguyen Thuc Huy Le

Doctoral Dissertations

We study framed sheaves on a smooth quadric surface and conjecture that the moduli of such framed sheaves admits a twistor deformation similar to one studied in the paper "Brill-Noether duality for moduli spaces of sheaves on K3 surfaces" by Markman.


Centroidal Voronoi Tessellations With Few Generator Points, Kirill Shakhnovskiy 2018 Bard College

Centroidal Voronoi Tessellations With Few Generator Points, Kirill Shakhnovskiy

Senior Projects Spring 2018

A Voronoi tessellation with $n$ generator points is the partitioning of a bounded region in $\rr^2$ into polygons such that every point in a given polygon is closer to its generator point than to any other generator point. A centroidal Voronoi tessellation (CVT) is a Voronoi tessellation where each polygon’s generator point is also its center of mass. In this project I will demonstrate what kinds of CVTs can exists within specific parameters, such as a square or rectangular region, and a set number generator points. I will also prove that the examples I present are the only ...


A Journey To The Adic World, Fayadh Kadhem 2018 Georgia Southern University

A Journey To The Adic World, Fayadh Kadhem

Electronic Theses and Dissertations

The first idea of this research was to study a topic that is related to both Algebra and Topology and explore a tool that connects them together. That was the entrance for me to the “adic world”. What was needed were some important concepts from Algebra and Topology, and so they are treated in the first two chapters.

The reader is assumed to be familiar with Abstract Algebra and Topology, especially with Ring theory and basics of Point-set Topology.

The thesis consists of a motivation and four chapters, the third and the fourth being the main ones. In the third ...


Ideal Containments Under Flat Extensions And Interpolation On Linear Systems In P2, Solomon Akesseh 2017 University of Nebraska-Lincoln

Ideal Containments Under Flat Extensions And Interpolation On Linear Systems In P2, Solomon Akesseh

Dissertations, Theses, and Student Research Papers in Mathematics

Fat points and their ideals have stimulated a lot of research but this dissertation concerns itself with aspects of only two of them, broadly categorized here as, the ideal containments and polynomial interpolation problems.

Ein-Lazarsfeld-Smith and Hochster-Huneke cumulatively showed that for all ideals I in k[Pn], I(mn) ⊆ Im for all m ∈ N. Over the projective plane, we obtain I(4)< ⊆ I2. Huneke asked whether it was the case that I(3) ⊆ I2. Dumnicki, Szemberg and Tutaj-Gasinska show that if I is the saturated homogeneous radical ideal of the 12 points of the Hesse configuration, then ...


Descartes Comes Out Of The Closet, Nora E. Culik 2017 Vassar College

Descartes Comes Out Of The Closet, Nora E. Culik

Journal of Humanistic Mathematics

While “Descartes Comes Out of the Closet” is ostensibly about a young woman’s journey to Paris, the descriptive detail borrows language and images from Cartesian coordinate geometry, dualistic philosophy, neuroanatomy (the pineal), and projections of three dimensions onto planes. This mathematical universe is counterpointed in the natural language of the suppressed love story that locates the real in the human. Thus, at the heart of the story is the tension between competing notions of mathematics, i.e., as either an independent realm apart from history or as a culturally produced and historical set of practices. Of course, the central ...


College Algebra, Trigonometry, And Precalculus (Clayton), Chaogui Zhang, Scott Bailey, Billie May, Jelinda Spotorno, Kara Mullen 2017 Clayton State University

College Algebra, Trigonometry, And Precalculus (Clayton), Chaogui Zhang, Scott Bailey, Billie May, Jelinda Spotorno, Kara Mullen

Mathematics Grants Collections

This Grants Collection for College Algebra, Trigonometry, and Precalculus was created under a Round Five ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

  • Linked Syllabus
  • Initial Proposal
  • Final Report


Counting Rational Points, Integral Points, Fields, And Hypersurfaces, Joseph Gunther 2017 The Graduate Center, City University of New York

Counting Rational Points, Integral Points, Fields, And Hypersurfaces, Joseph Gunther

All Dissertations, Theses, and Capstone Projects

This thesis comes in four parts, which can be read independently of each other.

In the first chapter, we prove a generalization of Poonen's finite field Bertini theorem, and use this to show that the obvious obstruction to embedding a curve in some smooth surface is the only obstruction over perfect fields, extending a result of Altman and Kleiman. We also prove a conjecture of Vakil and Wood on the asymptotic probability of hypersurface sections having a prescribed number of singularities.

In the second chapter, for a fixed base curve over a finite field of characteristic at least 5 ...


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