The Bernoulli Family: Their Massive Contributions To Mathematics And Hostility Toward Each Other, 2014 Chapman University

#### The Bernoulli Family: Their Massive Contributions To Mathematics And Hostility Toward Each Other, Dung (Yom) Bui, Mohamed Allali

*e-Research: A Journal of Undergraduate Work*

Throughout the history of mathematics, there are several individuals with significant contributions. However, if we look at the contribution of a single family in this field, the Bernoulli probably outshines others in terms of both the number of mathematicians it produced and their influence on the development of mathematics. The most outstanding three Bernoulli mathematicians are Jacob I Bernoulli (1654-1705), Johann I Bernoulli (1667-1748), and Daniel Bernoulli (1700-1782), all three of whom were the most influential math experts in the academic community yet very hostile to each other. Their family structure and jealousy toward each other might have fueled their ...

Continuous Dependence Of Solutions Of Equations On Parameters, 2014 Rose-Hulman Institute of Technology

#### Continuous Dependence Of Solutions Of Equations On Parameters, Sean A. Broughton

*Mathematical Sciences Technical Reports (MSTR)*

It is shown under very general conditions that the solutions of equations depend continuously on the coefficients or parameters of the equations. The standard examples are solutions of monic polynomial equations and the eigenvalues of a matrix. However, the proof methods apply to any finite map *T : C^{n} -> C^{n}*.

Tilting Sheaves On Brauer-Severi Schemes And Arithmetic Toric Varieties, 2014 Western University

#### Tilting Sheaves On Brauer-Severi Schemes And Arithmetic Toric Varieties, Youlong Yan

*University of Western Ontario - Electronic Thesis and Dissertation Repository*

The derived category of coherent sheaves on a smooth projective variety is an important object of study in algebraic geometry. One important device relevant for this study is the notion of tilting sheaf.

This thesis is concerned with the existence of tilting sheaves on some smooth projective varieties. The main technique we use in this thesis is Galois descent theory. We first construct tilting bundles on general Brauer-Severi varieties. Our main result shows the existence of tilting bundles on some Brauer-Severi schemes. As an application, we prove that there are tilting bundles on an arithmetic toric variety whose toric variety ...

Composite Dilation Wavelets With High Degrees, 2014 Illinois Wesleyan University

#### Composite Dilation Wavelets With High Degrees, Tian-Xiao He

*Tian-Xiao He*

No abstract provided.

Choosing Between Parametric And Non-Parametric Tests, 2014 Minnesota State University, Mankato

#### Choosing Between Parametric And Non-Parametric Tests, Russ Johnson

*Journal of Undergraduate Research at Minnesota State University, Mankato*

A common question in comparing two sets of measurements is whether to use a parametric testing procedure or a non-parametric procedure. The question is even more important in dealing with smaller samples. Here, using simulation, several parametric and nonparametric tests, such as, t-test, Normal test, Wilcoxon Rank Sum test, van-der Waerden Score test, and Exponential Score test are compared.

On Sign-Solvable Linear Systems And Their Applications In Economics, 2014 Minnesota State University, Mankato

#### On Sign-Solvable Linear Systems And Their Applications In Economics, Eric Hanson

*Journal of Undergraduate Research at Minnesota State University, Mankato*

Sign-solvable linear systems are part of a branch of mathematics called qualitative matrix theory. Qualitative matrix theory is a development of matrix theory based on the sign (¡; 0; +) of the entries of a matrix. Sign-solvable linear systems are useful in analyzing situations in which quantitative data is unknown or had to measure, but qualitative information is known. These situations arise frequently in a variety of disciplines outside of mathematics, including economics and biology. The applications of sign-solvable linear systems in economics are documented and the development of new examples is formalized mathematically. Additionally, recent mathematical developments about sign-solvable linear systems ...

Asymptotic Expansions And Computation Of Generalized Stirling Numbers And Generalized Stirling Functions, 2014 Illinois Wesleyan University

#### Asymptotic Expansions And Computation Of Generalized Stirling Numbers And Generalized Stirling Functions, Tian-Xiao He

*Tian-Xiao He*

Here presented is a unified approach to Stirling numbers and their generalizations as well as generalized Stirling functions by using generalized factorial functions, k-Gamma functions, and generalized divided difference. Previous well-known extensions of Stirling numbers due to Riordan, Carlitz, Howard, Charalambides-Koutras, Gould-Hopper, Hsu-Shiue, Tsylova Todorov, Ahuja-Enneking, and Stirling functions introduced by Butzer and Hauss, Butzer, Kilbas, and Trujilloet and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations and generating functions are discussed. Some asymptotic expansions for the generalized Stirling functions and generalized Stirling numbers are established ...

Morphological Operations Applied To Digital Art Restoration, 2014 University of Minnesota Morris Digital Well

#### Morphological Operations Applied To Digital Art Restoration, M. Kirbie Dramdahl

*Scholarly Horizons: University of Minnesota, Morris Undergraduate Journal*

This paper provides an overview of the processes involved in detecting and removing cracks from digitized works of art. Speciﬁc attention is given to the crack detection phase as completed through the use of morphological operations. Mathematical morphology is an area of set theory applicable to image processing, and therefore lends itself eﬀectively to the digital art restoration process.

Castelnuovo–Mumford Regularity And Arithmetic Cohen–Macaulayness Of Complete Bipartite Subspace Arrangements, 2014 Boise State University

#### Castelnuovo–Mumford Regularity And Arithmetic Cohen–Macaulayness Of Complete Bipartite Subspace Arrangements, Zach Teitler, Douglas A. Torrence

*Mathematics Faculty Publications and Presentations*

We give the Castelnuovo–Mumford regularity of arrangements of (n−2)-planes in P^{n} whose incidence graph is a sufficiently large complete bipartite graph, and determine when such arrangements are arithmetically Cohen–Macaulay.

Polynomial Identities On Algebras With Actions, 2014 Western University

#### Polynomial Identities On Algebras With Actions, Chris Plyley

*University of Western Ontario - Electronic Thesis and Dissertation Repository*

When an algebra is endowed with the additional structure of an action or a grading, one can often make striking conclusions about the algebra based on the properties of the structure-induced subspaces. For example, if *A* is an associative *G*-graded algebra such that the homogeneous component *A _{1}* satisfies an identity of degree

*d*, then Bergen and Cohen showed that

*A*is itself a PI-algebra. Bahturin, Giambruno and Riley later used combinatorial methods to show that the degree of the identity satisfied by

*A*is bounded above by a function of

*d*and |

*G|*. Utilizing a similar approach, we ...

Calculation Of The Killing Form Of A Simple Lie Group, 2014 Rose-Hulman Institute of Technology

#### Calculation Of The Killing Form Of A Simple Lie Group, Sean A. Broughton

*Mathematical Sciences Technical Reports (MSTR)*

The Killing form of a simple Lie Algebra is determined from invariants of the extended root diagrams of the Lie algebra.

Active Calculus, 2014 Grand Valley State University

#### Active Calculus, Matthew Boelkins, David Austin, Steven Schlicker

*Open Education Materials*

Active Calculus is different from most existing calculus texts in at least the following ways: the text is free for download by students and instructors in .pdf format; in the electronic format, graphics are in full color and there are live html links to java applets; the text is open source, and interested instructors can gain access to the original source files upon request; the style of the text requires students to be active learners — there are very few worked examples in the text, with there instead being 3-4 activities per section that engage students in connecting ideas, solving problems ...

One-Dimensional Weakly Nonlinear Model Equations For Rossby Waves, 2014 Dublin Institute of Technology

#### One-Dimensional Weakly Nonlinear Model Equations For Rossby Waves, David Henry, Rossen Ivanov

*Articles*

In this study we explore several possibilities for modelling weakly nonlinear Rossby waves in fluid of constant depth, which propagate predominantly in one direction. The model equations obtained include the BBM equation, as well as the integrable KdV and Degasperis-Procesi equations.

Algebraic Properties Of Ext-Modules Over Complete Intersections, 2014 University of Nebraska - Lincoln

#### Algebraic Properties Of Ext-Modules Over Complete Intersections, Jason Hardin

*Dissertations, Theses, and Student Research Papers in Mathematics*

We investigate two algebraic properties of Ext-modules over a complete intersection R of codimension c. Given an R-module M, Ext(M,k) can be viewed as a graded module over a polynomial ring in c variables with an action given by the Eisenbud operators. We provide an upper bound on the degrees of the generators of this graded module in terms of the regularities of two associated coherent sheaves. In the codimension two case, our bound recovers a bound of Avramov and Buchweitz in terms of the Betti numbers of M. We also provide a description of the differential graded ...

The Neural Ring: Using Algebraic Geometry To Analyze Neural Codes, 2014 University of Nebraska - Lincoln

#### The Neural Ring: Using Algebraic Geometry To Analyze Neural Codes, Nora Youngs

*Dissertations, Theses, and Student Research Papers in Mathematics*

Neurons in the brain represent external stimuli via neural codes. These codes often arise from stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How does the brain do this? To address this question, it is important to determine what stimulus space features can - in principle - be extracted from neural codes. This motivates us to define the neural ring and a related neural ideal, algebraic objects that encode ...

Boundary Value Problems Of Nabla Fractional Difference Equations, 2014 University of Nebraska - Lincoln

#### Boundary Value Problems Of Nabla Fractional Difference Equations, Abigail M. Brackins

*Dissertations, Theses, and Student Research Papers in Mathematics*

In this dissertation we develop the theory of the nabla fractional self-adjoint difference equation,

∇_{a}^{ν}(p∇y)(t)+q(t)y(ρ(t)) = f(t),

where 0 < ν < 1.We begin with an introduction to the nabla fractional calculus. In the second chapter, we show existence and uniqueness of the solution to a fractional self-adjoint initial value problem. We find a variation of constants formula for this fractional initial value problem, and use the variation of constants formula to derive the Green's function for a related boundary value problem. We study the Green's function and its properties in several settings. For a simplified boundary value problem, we show that the Green's function is nonnegative and we find its maximum and the maximum of its integral. For a boundary value problem with generalized boundary conditions, we find the Green's function and show that it is a generalization of the first Green's function. In the third chapter, we use the Contraction Mapping Theorem to prove existence and uniqueness of a positive solution to a forced self-adjoint fractional difference equation with a finite limit. We explore modifications to the forcing term and modifications to the space of functions in which the solution exists, and we provide examples to demonstrate the use of these theorems.

Advisers: Lynn Erbe and Allan Peterson

Light Pollution Research Through Citizen Science, 2014 California Polytechnic State University

#### Light Pollution Research Through Citizen Science, John Kanemoto

*STEM Teacher and Researcher (STAR) Program Posters*

Light pollution (LP) can disrupt and/or degrade the health of all living things, as well as, their environments. The goal of my research at the NOAO was to check the accuracy of the citizen science LP reporting systems entitled: Globe at Night (GaN), Dark Sky Meter (DSM), and Loss of the Night (LoN). On the GaN webpage, the darkness of the night sky (DotNS) is reported by selecting a magnitude chart. Each magnitude chart has a different density/number of stars around a specific constellation. The greater number of stars implies a darker night sky. Within the DSM iPhone ...

Reasoning & Proof In The Hs Common Core, 2014 Boise State University

#### Reasoning & Proof In The Hs Common Core, Laurie O. Cavey

*Laurie O. Cavey*

No abstract provided.

Ghost Number Of Group Algebras, 2014 Western University

#### Ghost Number Of Group Algebras, Gaohong Wang

*University of Western Ontario - Electronic Thesis and Dissertation Repository*

The generating hypothesis for the stable module category of a finite group is the statement that if a map in the thick subcategory generated by the trivial representation induces the zero map in Tate cohomology, then it is stably trivial. It is known that the generating hypothesis fails for most groups. Generalizing work done for p-groups, we define the ghost number of a group algebra, which is a natural number that measures the degree to which the generating hypothesis fails. We describe a close relationship between ghost numbers and Auslander-Reiten triangles, with many results stated for a general projective class ...

Near Oracle Performance And Block Analysis Of Signal Space Greedy Methods, 2014 Claremont Colleges

#### Near Oracle Performance And Block Analysis Of Signal Space Greedy Methods, Raja Giryes, Deanna Needell

*CMC Faculty Publications and Research*

Compressive sampling (CoSa) is a new methodology which demonstrates that sparse signals can be recovered from a small number of linear measurements. Greedy algorithms like CoSaMP have been designed for this recovery, and variants of these methods have been adapted to the case where sparsity is with respect to some arbitrary dictionary rather than an orthonormal basis. In this work we present an analysis of the so-called Signal Space CoSaMP method when the measurements are corrupted with mean-zero white Gaussian noise. We establish near-oracle performance for recovery of signals sparse in some arbitrary dictionary. In addition, we analyze the block ...