Deterministic Global 3d Fractal Cloud Model For Synthetic Scene Generation, 2024 Air Force Institute of Technology

#### Deterministic Global 3d Fractal Cloud Model For Synthetic Scene Generation, Aaron M. Schinder, Shannon R. Young, Bryan J. Steward, Michael L. Dexter, Andrew Kondrath, Stephen Hinton, Ricardo Davila

*Faculty Publications*

This paper describes the creation of a fast, deterministic, 3D fractal cloud renderer for the AFIT Sensor and Scene Emulation Tool (ASSET). The renderer generates 3D clouds by ray marching through a volume and sampling the level-set of a fractal function. The fractal function is distorted by a displacement map, which is generated using horizontal wind data from a Global Forecast System (GFS) weather file. The vertical windspeed and relative humidity are used to mask the creation of clouds to match realistic large-scale weather patterns over the Earth. Small-scale detail is provided by the fractal functions which are tuned to …

Tasks For Learning Trigonometry, 2024 Utah State University

#### Tasks For Learning Trigonometry, Sydnee Andreasen

*All Graduate Reports and Creative Projects, Fall 2023 to Present*

Many studies have been done using task-based learning within different mathematics courses. Within the field of trigonometry, task-based learning is lacking. The following research aimed to create engaging, mathematically rich tasks that meet the standards for the current trigonometry course at Utah State University and align with the State of Utah Core Standards for 7th through 12th grades. Four lessons were selected and developed based on the alignment of standards, the relevance to the remainder of the trigonometry course, and the relevance to courses beyond trigonometry. The four lessons that were chosen and developed were related to trigonometric ratios, graphing …

Derived Jet Schemes And Arc Spaces, And Arithmetic Arc Space Representability, 2024 University of Arkansas, Fayetteville

#### Derived Jet Schemes And Arc Spaces, And Arithmetic Arc Space Representability, C. Eric Overton-Walker

*Graduate Theses and Dissertations*

Associated to a given scheme X one can define geometric and arithmetic notions of jet schemes and arc spaces. We develop a construction of the geometric jets and arcs in the setting of derived schemes and explore consequences thereof. In particular, we prove an analogous theorem to that of one by Tommaso de Fernex and Roi Docampo concerning the cotangent sheaves of geometric jets and arcs. Our version then produces many subsequent results which allow us to prove stronger versions of results concerning geometric jets and arcs by removing unnecessary hypotheses. Separately, we explore evidence as to why, in contrast …

Geometries Gon Wild, 2024 Bellarmine University

#### Geometries Gon Wild, Naat Ambrosino

*Undergraduate Theses*

A circle is mathematically defined as the collection of points a given distance away from a set point. Thus, the appearance of a circle varies dramatically across different metrics—for example, the taxicab metric (as popularized by Krause and Reynolds) has a circle that is a Euclidean square. As such, metrics can be partially defined by the appearance of their unit circles. This paper focuses on creating and analyzing an infinite set of metrics defined by their circles being regular polygons. Additionally, it provides a method of exactly generating a regular n-gon given a center, included point, and specified orientation.

A Note On Umbilic Points At Infinity, 2024 School of STEM, Munster Technological University, Kerry, Tralee Co., Kerry, Ireland

#### A Note On Umbilic Points At Infinity, Brendan Guilfoyle

*Publications*

In this note a definition of *umbilic point at infinity* is proposed, at least for surfaces that are homogeneous polynomial graphs over a plane in Euclidean 3-space. This is a stronger definition than that of Toponogov in his study of complete convex surfaces, and allows one to distinguish between different umbilic points at infinity. It is proven that all such umbilic points at infinity are isolated, that they occur in pairs and are the zeroes of the projective extension of the third fundamental form, as developed in Guilfoyle and Ortiz-Rodríguez (Math Proc R Ir Acad 123A(2), 63–94, 2023). A geometric …

The Modular Generalized Springer Correspondence For The Symplectic Group, 2024 Louisiana State University

#### The Modular Generalized Springer Correspondence For The Symplectic Group, Joseph Dorta

*LSU Doctoral Dissertations*

The Modular Generalized Springer Correspondence (MGSC), as developed by Achar, Juteau, Henderson, and Riche, stands as a significant extension of the early groundwork laid by Lusztig's Springer Correspondence in characteristic zero which provided crucial insights into the representation theory of finite groups of Lie type. Building upon Lusztig's work, a generalized version of the Springer Correspondence was later formulated to encompass broader contexts.

In the realm of modular representation theory, Juteau's efforts gave rise to the Modular Springer Correspondence, offering a framework to explore the interplay between algebraic geometry and representation theory in positive characteristic. Achar, Juteau, Henderson, and Riche …

Spacetime Geometry Of Acoustics And Electromagnetism, 2024 Chapman University

#### Spacetime Geometry Of Acoustics And Electromagnetism, Lucas Burns, Tatsuya Daniel, Stephon Alexander, Justin Dressel

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

Both acoustics and electromagnetism represent measurable fields in terms of dynamical potential fields. Electromagnetic force-fields form a spacetime bivector that is represented by a dynamical energy–momentum 4-vector potential field. Acoustic pressure and velocity fields form an energy–momentum density 4-vector field that is represented by a dynamical action scalar potential field. Surprisingly, standard field theory analyses of spin angular momentum based on these traditional potential representations contradict recent experiments, which motivates a careful reassessment of both theories. We analyze extensions of both theories that use the full geometric structure of spacetime to respect essential symmetries enforced by vacuum wave propagation. The …

The Dual Boundary Complex Of The Moduli Space Of Cyclic Compactifications, 2024 Claremont Colleges

#### The Dual Boundary Complex Of The Moduli Space Of Cyclic Compactifications, Toby Anderson

*HMC Senior Theses*

Moduli spaces provide a useful method for studying families of mathematical objects. We study certain moduli spaces of algebraic curves, which are generalizations of familiar lines and conics. This thesis focuses on, Δ^{(r,n)}, the dual boundary complex of the moduli space of genus-zero cyclic curves. This complex is itself a moduli space of graphs and can be investigated with combinatorial methods. Remarkably, the combinatorics of this complex provides insight into the geometry and topology of the original moduli space. In this thesis, we investigate two topologically invariant properties of Δ^{(r,n)}. We compute its Euler characteristic and …

GröBner Bases With An Application To Tame Functions, 2024 University of North Florida

#### GröBner Bases With An Application To Tame Functions, Jessica D. Marconi

*UNF Graduate Theses and Dissertations*

Grobner bases are essential tools in algebraic geometry, used to simplify and solve systems of polynomial equations. These bases revolutionized computational methods in various branches of mathematics after being introduced in 1965 by Bruno Buchberger. This thesis explores the foundational concepts of Grobner bases, including their formation and properties. It also demonstrates their use in solving mathematical problems in algebraic geometry, including the ideal membership problem. As an application, we show how Grobner bases can be used to determine whether a polynomial mapping is tame. This concept is crucial for analyzing the topology near singular points and establishing whether a …

Point Modules And Line Modules Of Certain Quadratic Quantum Projective Spaces, 2024 University of Texas at Arlington

#### Point Modules And Line Modules Of Certain Quadratic Quantum Projective Spaces, Jose E. Lozano

*Mathematics Dissertations*

During the past 36 years, some research in noncommutative algebra has been driven by attempts to classify AS-regular algebras of global dimension four. Such algebras are often considered to be noncommutative analogues of polynomial rings. In the 1980s, Artin, Tate, and Van den Bergh introduced a projective scheme that parametrizes the point modules over a graded algebra generated by elements of degree one. In 2002, Shelton and Vancliff introduced the concept of line scheme, which is a projective scheme that parametrizes line modules.

This dissertation is in two parts. In the first part, we consider a 1-parameter family of quadratic …

Unexpectedness Stratified By Codimension, 2023 University of Nebraska-Lincoln

#### Unexpectedness Stratified By Codimension, Frank Zimmitti

*Department of Mathematics: Dissertations, Theses, and Student Research*

A recent series of papers, starting with the paper of Cook, Harbourne, Migliore, and Nagel on the projective plane in 2018, studies a notion of unexpectedness for finite sets *Z* of points in *N*-dimensional projective space. Say the complete linear system *L* of forms of degree *d* vanishing on *Z* has dimension *t* yet for any general point *P* the linear system of forms vanishing on *Z* with multiplicity *m* at *P* is nonempty. If the dimension of *L* is more than the expected dimension of *t*−*r*, where *r* is *N*+*m*−*1* choose …

Msis-Kadelka: Algebraic Methods For Inferring Discrete Models Of Biological Networks, 2023 Southern Methodist University

#### Msis-Kadelka: Algebraic Methods For Inferring Discrete Models Of Biological Networks, Brandilyn Stigler

*Annual Symposium on Biomathematics and Ecology Education and Research*

No abstract provided.

Differential Calculus: From Practice To Theory, 2023 Pennsylvania State University

#### Differential Calculus: From Practice To Theory, Eugene Boman, Robert Rogers

*Milne Open Textbooks*

*Differential Calculus: From Practice to Theory* covers all of the topics in a typical first course in differential calculus. Initially it focuses on using calculus as a problem solving tool (in conjunction with analytic geometry and trigonometry) by exploiting an informal understanding of differentials (infinitesimals). As much as possible large, interesting, and important historical problems (the motion of falling bodies and trajectories, the shape of hanging chains, the Witch of Agnesi) are used to develop key ideas. Only after skill with the computational tools of calculus has been developed is the question of rigor seriously broached. At that point, the …

Interpolation Problems And The Characterization Of The Hilbert Function, 2023 University of Arkansas, Fayetteville

#### Interpolation Problems And The Characterization Of The Hilbert Function, Bryant Xie

*Mathematical Sciences Undergraduate Honors Theses*

In mathematics, it is often useful to approximate the values of functions that are either too awkward and difficult to evaluate or not readily differentiable or integrable. To approximate its values, we attempt to replace such functions with more well-behaving examples such as polynomials or trigonometric functions. Over the algebraically closed field C, a polynomial passing through r distinct points with multiplicities m1, ..., mr on the affine complex line in one variable is determined by its zeros and the vanishing conditions up to its mi − 1 derivative for each point. A natural question would then be to consider …

A Strong-Type Furstenberg–Sárközy Theorem For Sets Of Positive Measure, 2023 Chapman University

#### A Strong-Type Furstenberg–Sárközy Theorem For Sets Of Positive Measure, Polona Durcik, Vjekoslav Kovač, Mario Stipčić

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

For every β ∈ (0,∞), β ≠ 1, we prove that a positive measure subset A of the unit square contains a point (x_{0}, y_{0}) such that A nontrivially intersects curves y − y0 = a(x −x0)^{β} for a whole interval I ⊆ (0,∞) of parameters a ∈ I . A classical Nikodym set counterexample prevents one to take β = 1, which is the case of straight lines. Moreover, for a planar set A of positive density, we show that the interval I can be arbitrarily large on the logarithmic scale. These results can …

On The Superabundance Of Singular Varieties In Positive Characteristic, 2023 University of Nebraska-Lincoln

#### On The Superabundance Of Singular Varieties In Positive Characteristic, Jake Kettinger

*Department of Mathematics: Dissertations, Theses, and Student Research*

The geproci property is a recent development in the world of geometry. We call a set of points Z\subseq\P_k^3 an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a and b. Examples known as grids have been known since 2011. Previously, the study of the geproci property has taken place within the characteristic 0 setting; prior to the work in this thesis, a procedure has been known for creating an (a,b)-geproci half-grid for 4\leq a\leq b, but it was not …

Invariants Of 3-Braid And 4-Braid Links, 2023 United Arab Emirates University

#### Invariants Of 3-Braid And 4-Braid Links, Mark Essa Sukaiti

*Theses*

In this study, we established a connection between the Chebyshev polynomial of the first kind and the Jones polynomial of generalized weaving knots of type *W*(3,n,m).

Through our analysis, we demonstrated that the coefficients of the Jones polynomial of weaving knots are essentially the Whitney numbers of Lucas lattices which allowed us to find an explicit formula for the Alexander polynomial of weaving knots of type*W*(3,n).

In addition to confirming Fox’s trapezoidal conjecture, we also discussed the zeroes of the Alexander Polynomial of weaving knots of type *W*(3,n) as they relate to Hoste’s conjecture. In addition, …

Computational Aspects Of Mixed Characteristic Witt Vectors And Denominators In Canonical Liftings Of Elliptic Curves, 2023 University of Tennessee, Knoxville

#### Computational Aspects Of Mixed Characteristic Witt Vectors And Denominators In Canonical Liftings Of Elliptic Curves, Jacob Dennerlein

*Doctoral Dissertations*

Given an ordinary elliptic curve *E* over a field 𝕜 of characteristic *p*, there is an elliptic curve **E** over the Witt vectors **W**(𝕜) for which we can lift the Frobenius morphism, called the canonical lifting of *E*. The Weierstrass coefficients and the elliptic Teichmüller lift of **E** are given by rational functions over 𝔽_*p* that depend only on the coefficients and points of *E*. Finotti studied the properties of these rational functions over fields of characteristic *p* ≥ 5. We investigate the same properties for fields of characteristic 2 and 3, make progress on …

Explicit Constructions Of Canonical And Absolute Minimal Degree Lifts Of Twisted Edwards Curves, 2023 University of Tennessee, Knoxville

#### Explicit Constructions Of Canonical And Absolute Minimal Degree Lifts Of Twisted Edwards Curves, William Coleman Bitting Iv

*Doctoral Dissertations*

Twisted Edwards Curves are a representation of Elliptic Curves given by the solutions of bx^2 + y^2 = 1 + ax^2y^2. Due to their simple and unified formulas for adding distinct points and doubling, Twisted Edwards Curves have found extensive applications in fields such as cryptography. In this thesis, we study the Canonical Liftings of Twisted Edwards Curves and the associated lift of points Elliptic Teichmu ̈ller Lift. The coordinate functions of the latter are proved to be polynomials, and their degrees and derivatives are computed. Moreover, an algorithm is described for explicit computations, and some properties of the general …

Brill--Noether Theory Via K3 Surfaces, 2023 Dartmouth College

#### Brill--Noether Theory Via K3 Surfaces, Richard Haburcak

*Dartmouth College Ph.D Dissertations*

Brill--Noether theory studies the different projective embeddings that an algebraic curve admits. For a curve with a given projective embedding, we study the question of what other projective embeddings the curve can admit. Our techniques use curves on K3 surfaces. Lazarsfeld's proof of the Gieseker--Petri theorem solidified the role of K3 surfaces in the Brill--Noether theory of curves. In this thesis, we further the study of the Brill--Noether theory of curves on K3 surfaces.

We prove results concerning lifting line bundles from curves to K3 surfaces. Via an analysis of the stability of Lazarsfeld--Mukai bundles, we deduce a bounded version …