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Articles 1 - 30 of 460
Full-Text Articles in Mathematics
Derived Jet Schemes And Arc Spaces, And Arithmetic Arc Space Representability, C. Eric Overton-Walker
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 …
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
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, Sydnee Andreasen
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 …
Geometries Gon Wild, Naat Ambrosino
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, Brendan Guilfoyle
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, Joseph Dorta
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, Lucas Burns, Tatsuya Daniel, Stephon Alexander, Justin Dressel
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, Toby Anderson
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 …
Point Modules And Line Modules Of Certain Quadratic Quantum Projective Spaces, Jose E. Lozano
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, Frank Zimmitti
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, Brandilyn Stigler
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, Eugene Boman, Robert Rogers
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, Bryant Xie
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, Polona Durcik, Vjekoslav Kovač, Mario Stipčić
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 (x0, y0) 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, Jake Kettinger
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 …
Explicit Constructions Of Canonical And Absolute Minimal Degree Lifts Of Twisted Edwards Curves, William Coleman Bitting Iv
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 …
Computational Aspects Of Mixed Characteristic Witt Vectors And Denominators In Canonical Liftings Of Elliptic Curves, Jacob Dennerlein
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 …
Invariants Of 3-Braid And 4-Braid Links, Mark Essa Sukaiti
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 typeW(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, …
Brill--Noether Theory Via K3 Surfaces, Richard Haburcak
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 …
Area Activity, Admin Stem For Success
Area Activity, Admin Stem For Success
STEM for Success Showcase
Lesson plan to teach students about area including an activity plan, activity description, activity video, and additional activity materials
Topological Data Analysis Of Weight Spaces In Convolutional Neural Networks, Adam Wagenknecht
Topological Data Analysis Of Weight Spaces In Convolutional Neural Networks, Adam Wagenknecht
Dissertations
Convolutional Neural Networks (CNNs) have become one of the most commonly used tools for performing image classification. Unfortunately, as with most machine learning algorithms, CNNs suffer from a lack of interpretability. CNNs are trained by using a training data set and a loss function to tune a set of parameters known as the layer weights. This tuning process is based on the classical method of gradient descent, but it relies on a strong stochastic component, which makes the weight behavior during training difficult to understand. However, since CNNs are governed largely by the weights that make up each of the …
Elliptic Curves Over Finite Fields, Christopher S. Calger
Elliptic Curves Over Finite Fields, Christopher S. Calger
Honors Theses
The goal of this thesis is to give an expository report on elliptic curves over finite fields. We begin by giving an overview of the necessary background in algebraic geometry to understand the definition of an elliptic curve. We then explore the general theory of elliptic curves over arbitrary fields, such as the group structure, isogenies, and the endomorphism ring. We then study elliptic curves over finite fields. We focus on the number of Fq-rational solutions, Tate modules, supersingular curves, and applications to elliptic curves over Q. In particular, we approach the topic largely through the use …
Surjectivity Of The Wahl Map On Cubic Graphs, Angela C. Hanson
Surjectivity Of The Wahl Map On Cubic Graphs, Angela C. Hanson
Theses and Dissertations--Mathematics
Much of algebraic geometry is the study of curves. One tool we use to study curves is whether they can be embedded in a K3 surface or not. If the Wahl map is surjective on a curve, that curve cannot be embedded in a K3 surface. Therefore, studying if the Wahl map is surjective for a particular curve gives us more insight into the properties of that curve. We simplify this problem by converting graph curves to dual graphs. Then the information for graphs can be used to study the underlying curves. We will discuss conditions for the Wahl map …
Toric Bundles As Mori Dream Spaces, Courtney George
Toric Bundles As Mori Dream Spaces, Courtney George
Theses and Dissertations--Mathematics
A projective, normal variety is called a Mori dream space when its Cox ring is finitely generated. These spaces are desirable to have, as they behave nicely under the Minimal Model Program, but no complete classification of them yet exists. Some early work identified that all toric varieties are examples of Mori dream spaces, as their Cox rings are polynomial rings. Therefore, a natural next step is to investigate projectivized toric vector bundles. These spaces still carry much of the combinatorial data as toric varieties, but have more variable behavior that means that they aren't as straightforward as Mori dream …
Geometry Of Pipe Dream Complexes, Benjamin Reese
Geometry Of Pipe Dream Complexes, Benjamin Reese
Theses and Dissertations--Mathematics
In this dissertation we study the geometry of pipe dream complexes with the goal of gaining a deeper understanding of Schubert polynomials. Given a pipe dream complex PD(w) for w a permutation in the symmetric group, we show its boundary is Whitney stratified by the set of all pipe dream complexes PD(v) where v > w in the strong Bruhat order. For permutations w in the symmetric group on n elements, we introduce the pipe dream complex poset P(n). The dual of this graded poset naturally corresponds to the poset of strata associated to the Whitney stratification of the boundary of …
On The Cohomology Of Solvable Leibniz Algebras, Jorg Feldvoss, Friedrich Wagemann
On The Cohomology Of Solvable Leibniz Algebras, Jorg Feldvoss, Friedrich Wagemann
University Faculty and Staff Publications
This paper is a sequel to J. Feldvoss and F. Wagemann: On Leibniz cohomology, (2021), where we mainly consider semisimple Leibniz algebras. It turns out that the analogue of the Hochschild-Serre spectral sequence for Leibniz cohomology cannot be applied to many ideals, and therefore this spectral sequence seems not to be applicable for computing the cohomology of non-semi-simple Leibniz algebras. The main idea of the present paper is to use similar tools as developed by Farnsteiner for Hochschild cohomology (see R. Farnsteiner: On the cohomology of associative algebras and Lie algebras (1987) and R. Farnsteiner: On the vanishing of homology …
Explorations In Well-Rounded Lattices, Tanis Nielsen
Explorations In Well-Rounded Lattices, Tanis Nielsen
HMC Senior Theses
Lattices are discrete subgroups of Euclidean spaces. Analogously to vector spaces, they can be described as spans of collections of linearly independent vectors, but with integer (instead of real) coefficients. Lattices have many fascinating geometric properties and numerous applications, and lattice theory is a rich and active field of theoretical work. In this thesis, we present an introduction to the theory of Euclidean lattices, along with an overview of some major unsolved problems, such as sphere packing. We then describe several more specialized topics, including prior work on well-rounded ideal lattices and some preliminary results on the study of planar …
Another Angle On Perspective: Solutions For Fermi Questions, May 2023, John Adam
Another Angle On Perspective: Solutions For Fermi Questions, May 2023, John Adam
Mathematics & Statistics Faculty Publications
No abstract provided.
More Properties Of Optimal Polynomial Approximants In Hardy Spaces, Raymond Cheng, Christopher Felder
More Properties Of Optimal Polynomial Approximants In Hardy Spaces, Raymond Cheng, Christopher Felder
Mathematics & Statistics Faculty Publications
We study optimal polynomial approximants (OPAs) in the classical Hardy spaces on the unit disk, Hp (1 < p < ∞). For fixed f ∈ Hp and n ∈ N, the OPA of degree n associated to f is the polynomial which minimizes the quantity ∥qf −1∥p over all complex polynomials q of degree less than or equal to n. We begin with some examples which illustrate, when p ≠ 2, how the Banach space geometry makes the above minimization problem interesting. We then weave through various results concerning limits and roots of these polynomials, including results which show that OPAs can be witnessed as solutions …
Studying Extended Sets From Young Tableaux, Eric Nofziger
Studying Extended Sets From Young Tableaux, Eric Nofziger
Rose-Hulman Undergraduate Mathematics Journal
Young tableaux are combinatorial objects related to the partitions of an integer and have various applications in representation theory. They are particularly useful in the study of the fibers arising from the Springer resolution. In recent work of Graham-Precup-Russell, an association has been made between a given row-strict tableau and three disjoint subsets of {1,2,...,n}. These subsets are then used in the study of extended Springer fibers, so we call them extended sets. In this project, we use combinatorial techniques to classify which of these extended sets correlate to a valid row-strict or standard tableau and give bounds on the …