Mixed Categories Of Sheaves On Toric Varieties, 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 ...

Parametric Polynomials For Small Galois Groups, 2018 Colby College

#### Parametric Polynomials For Small Galois Groups, Claire Huang

*Honors Theses*

Galois theory, named after French mathematician Evariste Galois in 19th-century, is an important part of abstract algebra. It brings together many different branches of mathematics by providing connections among fields, polynomials, and groups.

Specifically, Galois theory allows us to attach a finite field extension with a finite group. We call such a group the Galois group of the finite field extension. A typical way to attain a finite field extension to compute the splitting field of some polynomial. So we can always start with a polynomial and find the finite group associate to the field extension on its splitting field ...

The Boundedness Of The Hardy-Littlewood Maximal Function And The Strong Maximal Function On The Space Bmo, 2018 Claremont Colleges

#### The Boundedness Of The Hardy-Littlewood Maximal Function And The Strong Maximal Function On The Space Bmo, Wenhao Zhang

*CMC Senior Theses*

In this thesis, we present the space BMO, the one-parameter Hardy-Littlewood maximal function, and the two-parameter strong maximal function. We use the John-Nirenberg inequality, the relation between Muckenhoupt weights and BMO, and the Coifman-Rochberg proposition on constructing A_{1} weights with the Hardy- Littlewood maximal function to show the boundedness of the Hardy-Littlewood maximal function on BMO. The analogous statement for the strong maximal function is not yet understood. We begin our exploration of this problem by discussing an equivalence between the boundedness of the strong maximal function on rectangular BMO and the fact that the strong maximal function maps ...

Weighted Inequalities For Dyadic Operators Over Spaces Of Homogeneous Type, 2017 University of New Mexico

#### Weighted Inequalities For Dyadic Operators Over Spaces Of Homogeneous Type, David Edward Weirich

*Mathematics & Statistics ETDs*

A so-called space of homogeneous type is a set equipped with a quasi-metric and a doubling measure. We give a survey of results spanning the last few decades concerning the geometric properties of such spaces, culminating in the description of a system of dyadic cubes in this setting whose properties mirror the more familiar dyadic lattices in R^n . We then use these cubes to prove a result pertaining to weighted inequality theory over such spaces. We develop a general method for extending Bellman function type arguments from the real line to spaces of homogeneous type. Finally, we uses this ...

Thin Blue Seam, 2017 Bard College

#### Thin Blue Seam, Sebastian Anton-Ojeda

*Senior Projects Spring 2017*

This performance began as an exploration of liminal spaces in the context of a post-modern world. The lines between a suburban, consumptive society, fraught with binaries and thresholds while on the fringes of nature, were what interested me the most. However, this line of thought quickly took me to far more ancient places, to a view of nature driven by animism and pervaded by spiritual introspection. In Celtic Ireland, to give an example, every stone and river is witness to a myth. The Celts of the old world as described by anthropologist Marie-Louise Sjoestedt are constantly straddling the line between ...

Compactness Of Isoresonant Potentials, 2017 University of Kentucky

#### Compactness Of Isoresonant Potentials, Robert G. Wolf

*Theses and Dissertations--Mathematics*

Bruning considered sets of isospectral Schrodinger operators with smooth real potentials on a compact manifold of dimension three. He showed the set of potentials associated to an isospectral set is compact in the topology of smooth functions by relating the spectrum to the trace of the heat semi-group. Similarly, we can consider the resonances of Schrodinger operators with real valued potentials on Euclidean space of whose support lies inside a ball of fixed radius that generate the same resonances as some fixed Schrodinger operator, an ``isoresonant" set of potentials. This isoresonant set of potentials is also compact in the topology ...

Explicit Formulae And Trace Formulae, 2016 The Graduate Center, City University of New York

#### Explicit Formulae And Trace Formulae, Tian An Wong

*All Dissertations, Theses, and Capstone Projects*

In this thesis, motivated by an observation of D. Hejhal, we show that the explicit formulae of A. Weil for sums over zeroes of Hecke L-functions, via the Maass-Selberg relation, occur in the continuous spectral terms in the Selberg trace formula over various number fields. In Part I, we discuss the relevant parts of the trace formulae classically and adelically, developing the necessary representation theoretic background. In Part II, we show how show the explicit formulae intervene, using the classical formulation of Weil; then we recast this in terms of Weil distributions and the adelic formulation of Weil. As an ...

On The Free And G-Saturated Weight Monoids Of Smooth Affine Spherical Varieties For G=Sl(N), 2016 The Graduate Center, City University of New York

#### On The Free And G-Saturated Weight Monoids Of Smooth Affine Spherical Varieties For G=Sl(N), Won Geun Kim

*All Dissertations, Theses, and Capstone Projects*

Let $X$ be an affine algebraic variety over $\mathbb{C}$ equipped with an action of a connected reductive group $G$. The weight monoid $\Gamma(X)$ of $X$ is the set of isomorphism classes of irreducible representations of $G$ that occur in the coordinate ring $\mathbb{C}[X]$ of $X$. Losev has shown that if $X$ is a smooth affine spherical variety, that is, if $X$ is smooth and $\mathbb{C}[X]$ is multiplicity-free as a representation of $G$, then $\Gamma(X)$ determines $X$ up to equivariant automorphism.

Pezzini and Van Steirteghem have recently obtained a combinatorial characterization of the weight ...

Limiting Forms Of Iterated Circular Convolutions Of Planar Polygons, 2016 CUNY New York City College of Technology

#### Limiting Forms Of Iterated Circular Convolutions Of Planar Polygons, Boyan Kostadinov

*Publications and Research*

We consider a complex representation of an arbitrary planar polygon *P* centered at the origin. Let *P(1)* be the normalized polygon obtained from *P* by connecting the midpoints of its sides and normalizing the complex vector of vertex coordinates. We say that *P(1)* is a normalized average of *P. *We identify this averaging process with a special case of a circular convolution. We show that if the convolution is repeated many times, then for a large class of polygons the vertices of the limiting polygon lie either on an ellipse or on a star-shaped polygon. We derive a ...

Extension Theorems On Matrix Weighted Sobolev Spaces, 2016 University of Tennessee, Knoxville

#### Extension Theorems On Matrix Weighted Sobolev Spaces, Christopher Ryan Loga

*Doctoral Dissertations*

Let D a subset of R^{n} [R n] be a domain with Lipschitz boundary and 1 ≤ p < ∞ [1 less than or equal to p less than infinity]. Suppose for each x in R^{n} that W(x) is an m x m [m by m] positive definite matrix which satisfies the matrix A_{p} [A p] condition. For k = 0, 1, 2, 3;... define the matrix weighted, vector valued, Sobolev space [L p k of D,W] with

[the weighted L p k norm of vector valued f over D to the p power equals the sum over all alpha with order less than k of the integral over D of the the pth power ...

Nonlinear Harmonic Modes Of Steel Strings On An Electric Guitar, 2016 Linfield College

#### Nonlinear Harmonic Modes Of Steel Strings On An Electric Guitar, Joel Wenrich

*Senior Theses*

Steel strings used on electric and acoustic guitars are non-ideal oscillators that can produce imperfect intonation. According to theory, this intonation should be a function of the bending stiffness of the string, which is related to the dimensions of length and thickness of the string. To test this theory, solid steel strings of three different linear densities were analyzed using an oscilloscope and a Fast Fourier Transform function. We found that strings exhibited more drastic nonlinear harmonic behavior as their effective length was shortened and as linear density increased.

Unions Of Lebesgue Spaces And A1 Majorants, 2016 Washington University in St. Louis

#### Unions Of Lebesgue Spaces And A1 Majorants, Greg Knese, John E. Mccarthy, Kabe Moen

*Mathematics Faculty Publications*

We study two questions. When does a function belong to the union of Lebesgue spaces, and when does a function have anA1majorant? We provide a systematic study of these questions and show that they are fundamentally related. We show that the union ofLwp(ℝn)spaces withw∈Apis equal to the union of all Banach function spaces for which the Hardy–Littlewood maximal function is bounded on the space itself and its associate space.

Tessellations: An Artistic And Mathematical Look At The Work Of Maurits Cornelis Escher, 2016 University of Northern Iowa

#### Tessellations: An Artistic And Mathematical Look At The Work Of Maurits Cornelis Escher, Emily E. Bachmeier

*Honors Program Theses*

The purpose of this study was to learn more about the mathematics of tessellations and their artistic potential. Whenever I have seen tessellations, I always admire them. It is baffling how such complicated shapes can be repeated infinitely on the plane. This research aimed to increase the amount of tessellation information and activities available to secondary mathematics teachers by connecting the tessellations of Maurits Cornelis Escher with their underlying mathematics in order to use them for teaching secondary mathematics. The overarching premise of this study was to create something that would also be beneficial in my career as a secondary ...

Filters And Matrix Factorization, 2015 Southern Illinois University Edwardsville

#### Filters And Matrix Factorization, Myung-Sin Song, Palle E. T. Jorgensen

*SIUE Faculty Research, Scholarship, and Creative Activity*

We give a number of explicit matrix-algorithms for analysis/synthesis

in multi-phase filtering; i.e., the operation on discrete-time signals which

allow a separation into frequency-band components, one for each of the

ranges of bands, say N , starting with low-pass, and then corresponding

filtering in the other band-ranges. If there are N bands, the individual

filters will be combined into a single matrix action; so a representation of

the combined operation on all N bands by an N x N matrix, where the

corresponding matrix-entries are periodic functions; or their extensions to

functions of a complex variable. Hence our setting ...

Wave Packet Transform Over Finite Fields, 2015 Faculty of Mathematics, University of Vienna

#### Wave Packet Transform Over Finite Fields, Arash Ghaani Farashahi

*Electronic Journal of Linear Algebra*

In this article we introduce the notion of finite wave packet groups over finite fields as the finite group of dilations, translations, and modulations. Then we will present a unified theoretical linear algebra approach to the theory of wave packet transform (WPT) over finite fields. It is shown that each vector defined over a finite field can be represented as a coherent sum of finite wave packet group elements as well.

Reproducing Kernel Hilbert Space Vs. Frame Estimates, 2015 The University of Iowa

#### Reproducing Kernel Hilbert Space Vs. Frame Estimates, Palle E. T. Jorgensen, Myung-Sin Song

*SIUE Faculty Research, Scholarship, and Creative Activity*

We consider conditions on a given system *F* of vectors in Hilbert space *H*, forming a frame, which turn *H*into a reproducing kernel Hilbert space. It is assumed that the vectors in* F* are functions on some set Ω . We then identify conditions on these functions which automatically give *H* the structure of a reproducing kernel Hilbert space of functions on Ω. We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism. Applications are given to Hilbert spaces associated to families of Gaussian processes.

The Structure And Unitary Representations Of Su(2,1), 2015 Bowdoin College

#### The Structure And Unitary Representations Of Su(2,1), Andrew J. Pryhuber

*Honors Projects*

No abstract provided.

Fast Algorithms For Analyzing Partially Ranked Data, 2014 Harvey Mudd College

#### Fast Algorithms For Analyzing Partially Ranked Data, Matthew Mcdermott

*HMC Senior Theses*

Imagine your local creamery administers a survey asking their patrons to choose their five favorite ice cream flavors. Any data collected by this survey would be an example of *partially ranked data*, as the set of all possible flavors is only ranked into subsets of the chosen flavors and the non-chosen flavors. If the creamery asks you to help analyze this data, what approaches could you take? One approach is to use the natural symmetries of the underlying data space to decompose any data set into smaller parts that can be more easily understood. In this work, I describe how ...

A New Subgroup Chain For The Finite Affine Group, 2014 Harvey Mudd College

#### A New Subgroup Chain For The Finite Affine Group, David Alan Lingenbrink Jr.

*HMC Senior Theses*

The finite affine group is a matrix group whose entries come from a finite field. A natural subgroup consists of those matrices whose entries all come from a subfield instead. In this paper, I will introduce intermediate sub- groups with entries from both the field and a subfield. I will also examine the representations of these intermediate subgroups as well as the branch- ing diagram for the resulting subgroup chain. This will allow us to create a fast Fourier transform for the group that uses asymptotically fewer opera- tions than the brute force algorithm.

On Local Fractional Continuous Wavelet Transform, 2013 D. Baleanu

#### On Local Fractional Continuous Wavelet Transform, Yang Xiaojun

*Xiao-Jun Yang*

We introduce a new wavelet transform within the framework of the local fractional calculus. An illustrative example of local fractional wavelet transform is also presented.