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Articles 1  30 of 417
FullText Articles in Algebraic Geometry
Studying Extended Sets From Young Tableaux, Eric Nofziger
Studying Extended Sets From Young Tableaux, Eric Nofziger
RoseHulman 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 GrahamPrecupRussell, an association has been made between a given rowstrict 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 rowstrict or standard tableau and give bounds on the …
A Cluster Structure On The Coordinate Ring Of Partial Flag Varieties, Fayadh Kadhem
A Cluster Structure On The Coordinate Ring Of Partial Flag Varieties, Fayadh Kadhem
LSU Doctoral Dissertations
The main goal of this dissertation is to show that the (multihomogeneous) coordinate ring of a partial flag variety C[G/P_K^−] contains a cluster algebra for every semisimple complex algebraic group G. We use derivation properties and a canonical lifting map to prove that the cluster algebra structure A of the coordinate ring C[N_K] of a Schubert cell constructed by Goodearl and Yakimov can be lifted, in an explicit way, to a cluster structure \hat{A} living in the coordinate ring of the corresponding partial flag variety. Then we use a minimality condition to prove that the cluster algebra \hat{A} is equal …
Generalization Of BiCanonical Degrees, Joseph Brennan, Laura Ghezzi, Jooyoun Hong, Wolmer Vasconcelos
Generalization Of BiCanonical Degrees, Joseph Brennan, Laura Ghezzi, Jooyoun Hong, Wolmer Vasconcelos
Publications and Research
We discuss invariants of CohenMacaulay local rings that admit a canonical module ω. Attached to each such ring R, when ω is an ideal, there are integers–the type of R, the reduction number of ω–that provide valuable metrics to express the deviation of R from being a Gorenstein ring. In (Ghezzi et al. in JMS 589:506–528, 2017) and (Ghezzi et al. in JMS 571:55–74, 2021) we enlarged this list with the canonical degree and the bicanonical degree. In this work we extend the bicanonical degree to rings where ω is not necessarily an ideal. We also discuss generalizations to rings …
The Local Lifting Problem For Curves With Quaternion Actions, George Mitchell
The Local Lifting Problem For Curves With Quaternion Actions, George Mitchell
Dissertations, Theses, and Capstone Projects
The lifting problem asks whether one can lift Galois covers of curves defined over positive characteristic to Galois covers of curves over characteristic zero. The lifting problem has an equivalent local variant, which asks if a Galois extension of complete discrete valuation rings over positive characteristic, with algebraically closed residue field, can be lifted to characteristic zero. In this dissertation, we content ourselves with the study of the local lifting problem when the prime is 2, and the Galois group of the extension is the group of quaternions. In this case, it is known that certain quaternion extensions cannot be …
Reduction Of LFunctions Of Elliptic Curves Modulo Integers, Félix Baril Boudreau
Reduction Of LFunctions Of Elliptic Curves Modulo Integers, Félix Baril Boudreau
Electronic Thesis and Dissertation Repository
Let $\mathbb{F}_q$ be a finite field of size $q$, where $q$ is a power of a prime $p \geq 5$. Let $C$ be a smooth, proper, and geometrically connected curve over $\mathbb{F}_q$. Consider an elliptic curve $E$ over the function field $K$ of $C$ with nonconstant $j$invariant. One can attach to $E$ its $L$function $L(T,E/K)$, which is a generating function that contains information about the reduction types of $E$ at the different places of $K$. The $L$function of $E/K$ was proven to be a polynomial in $\mathbb{Z}[T]$.
In 1985, Schoof devised an algorithm to compute the zeta function of an …
The Design And Implementation Of A HighPerformance Polynomial System Solver, Alexander Brandt
The Design And Implementation Of A HighPerformance Polynomial System Solver, Alexander Brandt
Electronic Thesis and Dissertation Repository
This thesis examines the algorithmic and practical challenges of solving systems of polynomial equations. We discuss the design and implementation of triangular decomposition to solve polynomials systems exactly by means of symbolic computation.
Incremental triangular decomposition solves one equation from the input list of polynomials at a time. Each step may produce several different components (points, curves, surfaces, etc.) of the solution set. Independent components imply that the solving process may proceed on each component concurrently. This socalled componentlevel parallelism is a theoretical and practical challenge characterized by irregular parallelism. Parallelism is not an algorithmic property but rather a geometrical …
Recognizing And Reducing Ambiguity In Mathematics Curriculum And Relations Of ΘFunctions In Genus One And Two: A Geometric Perspective, Shantel Spatig
Recognizing And Reducing Ambiguity In Mathematics Curriculum And Relations Of ΘFunctions In Genus One And Two: A Geometric Perspective, Shantel Spatig
All Graduate Plan B and other Reports
Anxiety and mathematics come hand in hand for many individuals. This is due, in
part, to the fact that the only experience they have with mathematics is what some
mathematics educators refer to as "schoolmath," which uses a somewhat different
language than real mathematics. The language of schoolmath can cause individu
als to have confusion and develop misconceptions related to several mathematical
concepts. One such concept is a fraction. In chapter one of this report, one possible
reason for this is discussed and a possible solution is purposed.
In chapter three of this report, genustwo curves admitting an elliptic involution …
Identifying Trace Affine Linear Sets Using Homotopy Continuation, Julianne Mckay
Identifying Trace Affine Linear Sets Using Homotopy Continuation, Julianne Mckay
All Theses
We investigate how the coefficients of a sparse polynomial system influence the sum, or the trace, of its solutions. We discuss an extension of the classical trace test in numerical algebraic geometry to sparse polynomial systems. Two known methods for identifying a trace affine linear subset of the support of a sparse polynomial system use sparse resultants and polyhedral geometry, respectively. We introduce a new approach which provides more precise classifications of trace affine linear sets than was previously known. For this new approach, we developed software in Macaulay2.
Academic Hats And Ice Cream: Two Optimization Problems, Valery F. Ochkov, Yulia V. Chudova
Academic Hats And Ice Cream: Two Optimization Problems, Valery F. Ochkov, Yulia V. Chudova
Journal of Humanistic Mathematics
This article describes the use of computer software to optimize the design of an academic hat and an ice cream cone!
Bbt Acoustic Alternative Top Bracing Cadd Data SetNorev2022jun28, Bill Hemphill
Bbt Acoustic Alternative Top Bracing Cadd Data SetNorev2022jun28, Bill Hemphill
STEM Guitar Project’s BBT Acoustic Kit
This electronic document file set consists of an overview presentation (PDFformatted) file and companion video (MP4) and CADD files (DWG & DXF) for laser cutting the ETSUdeveloped alternate top bracing designs and marking templates for the STEM Guitar Project’s BBT (OMsized) standard acoustic guitar kit. The three (3) alternative BBT top bracing designs in this release are
(a) a onepiece base for the standard kit's (Martinstyle) bracing,
(b) 277 Ladderstyle bracing, and
(c) an Xbraced fanstyle bracing similar to traditional European or socalled 'classical' acoustic guitars.
The CADD data set for each of the three (3) top bracing designs includes …
Bbt Side Mold Assy, Bill Hemphill
Bbt Side Mold Assy, Bill Hemphill
STEM Guitar Project’s BBT Acoustic Kit
This electronic document file set covers the design and fabrication information of the ETSU Guitar Building Project’s BBT (OMsized) Side Mold Assy for use with the STEM Guitar Project’s standard acoustic guitar kit. The extended 'as built' data set contains an overview file and companion video, the 'parent' CADD drawing, CADD data for laser etching and cutting a drill &/or layout template, CADD drawings in AutoCAD .DWG and .DXF R12 formats of the centerline tool paths for creating the mold assembly pieces on an AXYZ CNC router, and support documentation for CAM applications including router bit specifications, feeds, speed, multipass …
Unomaha Problem Of The Week (20212022 Edition), Brad Horner, Jordan M. Sahs
Unomaha Problem Of The Week (20212022 Edition), Brad Horner, Jordan M. Sahs
UNO Student Research and Creative Activity Fair
The University of Omaha math department's Problem of the Week was taken over in Fall 2019 from faculty by the authors. The structure: each semester (Fall and Spring), three problems are given per week for twelve weeks, with each problem worth ten points  mimicking the structure of arguably the most wellregarded university math competition around, the Putnam Competition, with prizes awarded to topscorers at semester's end. The weekly competition was halted midway through Spring 2020 due to COVID19, but relaunched again in Fall 2021, with massive changes.
Now there are three difficulty tiers to POW problems, roughly corresponding to …
The ZariskiRiemann Space As A Universal Model For The Birational Geometry Of A Function Field, Giovan Battista Pignatti Morano Di Custoza
The ZariskiRiemann Space As A Universal Model For The Birational Geometry Of A Function Field, Giovan Battista Pignatti Morano Di Custoza
Dissertations, Theses, and Capstone Projects
Given a function field $K$ over an algebraically closed field $k$, we propose to use the ZariskiRiemann space $\ZR (K/k)$ of valuation rings as a universal model that governs the birational geometry of the field extension $K/k$. More specifically, we find an exact correspondence between adhoc collections of open subsets of $\ZR (K/k)$ ordered by quasirefinements and the category of normal models of $K/k$ with morphisms the birational maps. We then introduce suitable Grothendieck topologies and we develop a sheaf theory on $\ZR (K/k)$ which induces, locally at once, the sheaf theory of each normal model. Conversely, given a sheaf …
On Isomorphic KRational Groups Of Isogenous Elliptic Curves Over Finite Fields, Ben Kuehnert, Geneva Schlafly, Zecheng Yi
On Isomorphic KRational Groups Of Isogenous Elliptic Curves Over Finite Fields, Ben Kuehnert, Geneva Schlafly, Zecheng Yi
RoseHulman Undergraduate Mathematics Journal
It is well known that two elliptic curves are isogenous if and only if they have same number of rational points. In fact, isogenous curves can even have isomorphic groups of rational points in certain cases. In this paper, we consolidate all the current literature on this relationship and give a extensive classification of the conditions in which this relationship arises. First we prove two ordinary isogenous elliptic curves have isomorphic groups of rational points when they have the same $j$invariant. Then, we extend this result to certain isogenous supersingular elliptic curves, namely those with equal $j$invariant of either 0 …
The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles
The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles
Electronic Theses, Projects, and Dissertations
This thesis is centered around the construction and analysis of the principal arithmetic surface (3, 5) over Q. By adjoining the two symbols i,j, where i^{2} = 3, j^{2} = 5, such that ij = ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL_{2}(R), which I identify with isometries of the hyperbolic plane. From this quaternion algebra, I produce a large list of matrices and apply them via Mobius transformations to the point (0, 2), which is the center of my Dirichlet domain. This …
Lattice Reduction Algorithms, Juan Ortega
Lattice Reduction Algorithms, Juan Ortega
Electronic Theses, Projects, and Dissertations
The purpose of this thesis is to propose and analyze an algorithm that follows
similar steps of Guassian Lattice Reduction Algorithm in twodimensions and applying
them to threedimensions. We start off by discussing the importance of cryptography in
our day to day lives. Then we dive into some linear algebra and discuss specific topics that
will later help us in understanding lattice reduction algorithms. We discuss two lattice
problems: the shortest vector problem and the closest vector problem. Then we introduce
two types of lattice reduction algorithms: Guassian Lattice Reduction in twodimensions
and the LLL Algortihm. We illustrate how both …
Towards A Generalization Of Fulton's Intersection Multiplicity Algorithm, Ryan Sandford
Towards A Generalization Of Fulton's Intersection Multiplicity Algorithm, Ryan Sandford
Electronic Thesis and Dissertation Repository
In this manuscript we generalize Fulton's bivariate intersection multiplicity algorithm to a partial intersection multiplicity algorithm in the nvariate setting. We extend this generalization of Fulton's algorithm to work at any point, rational or not, using the theory of regular chains. We implement these algorithms in Maple and provide experimental testing. The results indicate the proposed algorithm often outperforms the existing standard basisfree intersection multiplicity algorithm in Maple, typically by one to two orders of magnitude. Moreover, we also provide some examples where the proposed algorithm outperforms intersection multiplicity algorithms which rely on standard bases, indicating the proposed algorithm is …
On The Geometry Of MultiAffine Polynomials, Junquan Xiao
On The Geometry Of MultiAffine Polynomials, Junquan Xiao
Electronic Thesis and Dissertation Repository
This work investigates several geometric properties of the solutions of the multiaffine polynomials. Chapters 1, 2 discuss two different notions of invariant circles. Chapter 3 gives several loci of polynomials of degree three. A locus of a complex polynomial p(z) is a minimal, with respect to inclusion, set that contains at least one point of every solution of the polarization of the polynomial. The study of such objects allows one to improve upon know results about the location of zeros and critical points of complex polynomials, see for example [22] and [24]. A complex polynomial has many loci. It is …
Kissing The Archimedeans, Anthony Webb
Kissing The Archimedeans, Anthony Webb
All NMU Master's Theses
In this paper the three dimensional kissing problem will be related to the Platonic and Archimedean solids. On each polyhedra presented their vertices will have spheres expanding such that the center of each of these outer spheres are the vertices of the polyhedron, and these outer spheres will continue to expand until they become tangent to each other. The ratio will be found between the radius of each outer sphere, and the radius of an inner sphere such that each inner sphere's center is the circumcenter of the polyhedron, and the inner sphere is tangent to each outer sphere. Every …
Anticanonical Models Of Smoothings Of Cyclic Quotient Singularities, Arie A. Stern Gonzalez
Anticanonical Models Of Smoothings Of Cyclic Quotient Singularities, Arie A. Stern Gonzalez
Doctoral Dissertations
In this thesis we study anticanonical models of smoothings of cyclic quotient singularities. Given a surface cyclic quotient singularity $Q\in Y$, it is an open problem to determine all smoothings of $Y$ that admit an anticanonical model and to compute it. In \cite{HTU}, Hacking, Tevelev and Urz\'ua studied certain irreducible components of the versal deformation space of $Y$, and within these components, they found one parameter smoothings $\Y \to \A^1$ that admit an anticanonical model and proved that they have canonical singularities. Moreover, they compute explicitly the anticanonical models that have terminal singularities using Mori's division algorithm \cite{M02}. We study …
Tropical Geometry Of TVarieties With Applications To Algebraic Statistics, Joseph Cummings
Tropical Geometry Of TVarieties With Applications To Algebraic Statistics, Joseph Cummings
Theses and DissertationsMathematics
Varieties with group action have been of interest to algebraic geometers for centuries. In particular, toric varieties have proven useful both theoretically and in practical applications. A rich theory blending algebraic geometry and polyhedral geometry has been developed for Tvarieties which are natural generalizations of toric varieties. The first results discussed in this dissertation study the relationship between torus actions and the wellpoised property. In particular, I show that the wellpoised property is preserved under a geometric invariant theory quotient by a (quasi)torus. Conversely, I argue that Tvarieties built on a wellpoised base preserve the wellpoised property when the base …
Topics In Moufang Loops, Riley Britten
Topics In Moufang Loops, Riley Britten
Electronic Theses and Dissertations
We will begin by discussing power graphs of Moufang loops. We are able to show that as in groups the directed power graph of a Moufang loop is uniquely determined by the undirected power graph. In the process of proving this result we define the generalized octonion loops, a variety of Moufang loops which behave analogously to the generalized quaternion groups. We proceed to investigate paraF quasigroups, a variety of quasigroups which we show are antilinear over Moufang loops. We briefly depart from the context of Moufang loops to discuss solvability in general loops. We then prove some results on …
Equisingular Approximation Of Analytic Germs, Aftab Yusuf Patel
Equisingular Approximation Of Analytic Germs, Aftab Yusuf Patel
Electronic Thesis and Dissertation Repository
This thesis deals with the problem of approximating germs of real or complex analytic spaces by Nash or algebraic germs. In particular, we investigate the problem of approximating analytic germs in various ways while preserving the HilbertSamuel function, which is of importance in the resolution of singularities. We first show that analytic germs that are complete intersections can be arbitrarily closely approximated by algebraic germs which are complete intersections with the same HilbertSamuel function. We then show that analytic germs whose local rings are CohenMacaulay can be arbitrarily closely approximated by Nash germs whose local rings are Cohen Macaulay and …
Acceleration Skinning: KinematicsDriven Cartoon Effects For Articulated Characters, Niranjan Kalyanasundaram
Acceleration Skinning: KinematicsDriven Cartoon Effects For Articulated Characters, Niranjan Kalyanasundaram
All Theses
Secondary effects are key to adding fluidity and style to animation. This thesis introduces the idea of “Acceleration Skinning” following a recent wellreceived technique, Velocity Skinning, to automatically create secondary motion in character animation by modifying the standard pipeline for skeletal rig skinning. These effects, which animators may refer to as squash and stretch or drag, attempt to create an illusion of inertia. In this thesis, I extend the Velocity Skinning technique to include acceleration for creating a wider gamut of cartoon effects. I explore three new deformers that make use of this Acceleration Skinning framework: followthrough, centripetal stretch, and …
ℂMotivic Modular Forms, Bogdan Gheorghe, Daniel C. Isaksen, Achim Krause, Nicolas Ricka
ℂMotivic Modular Forms, Bogdan Gheorghe, Daniel C. Isaksen, Achim Krause, Nicolas Ricka
Mathematics Faculty Research Publications
We construct a topological model for cellular, 2complete, stable Cmotivic homotopy theory that uses no algebrogeometric foundations.We compute the Steenrod algebra in this context, and we construct a “motivic modular forms” spectrum over ℂ.
CacheFriendly, Modular And Parallel Schemes For Computing Subresultant Chains, Mohammadali Asadi
CacheFriendly, Modular And Parallel Schemes For Computing Subresultant Chains, Mohammadali Asadi
Electronic Thesis and Dissertation Repository
The RegularChains library in Maple offers a collection of commands for solving polynomial systems symbolically with taking advantage of the theory of regular chains. The primary goal of this thesis is algorithmic contributions, in particular, to highperformance computational schemes for subresultant chains and underlying routines to extend that of RegularChains in a C/C++ opensource library.
Subresultants are one of the most fundamental tools in computer algebra. They are at the core of numerous algorithms including, but not limited to, polynomial GCD computations, polynomial system solving, and symbolic integration. When the subresultant chain of two polynomials is involved in a client …
Windows In Algebraic Geometry And Applications To Moduli, Sebastian Torres
Windows In Algebraic Geometry And Applications To Moduli, Sebastian Torres
Doctoral Dissertations
We apply the theory of windows, as developed by HalpernLeistner and by Ballard, Favero and Katzarkov, to study certain moduli spaces and their derived categories. Using quantization and other techniques we show that stable GIT quotients of $(\mathbb{P}^1)^n$ by $PGL_2$ over an algebraically closed field of characteristic zero satisfy a rare property called Bott vanishing, which states that $\Omega^j_Y \otimes L$ has no higher cohomology for every j and every ample line bundle L. Similar techniques are used to reprove the well known fact that toric varieties satisfy Bott vanishing. We also use windows to explore derived categories of moduli …
Equivariant Smoothings Of Cusp Singularities, Angelica Simonetti
Equivariant Smoothings Of Cusp Singularities, Angelica Simonetti
Doctoral Dissertations
Let $p \in X$ be the germ of a cusp singularity and let $\iota$ be an antisymplectic involution, that is an involution free on $X\setminus \{p\}$ and such that there exists a nowhere vanishing holomorphic 2form $\Omega$ on $X\setminus \{p\}$ for which $\iota^*(\Omega)=\Omega$. We prove that a sufficient condiition for such a singularity equipped with an antisymplectic involution to be equivariantly smoothable is the existence of a Looijenga (or anticanonical) pair $(Y,D)$ that admits an involution free on $Y\setminus D$ and that reverses the orientation of $D$.
Using Lie Sphere Geometry To Study Dupin Hypersurfaces In R^N, Thomas E. Cecil
Using Lie Sphere Geometry To Study Dupin Hypersurfaces In R^N, Thomas E. Cecil
Mathematics Department Faculty Scholarship
A hypersurface M in R^{n} or S^{n} is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if each principal curvature has constant multiplicity on M, i.e., the number of distinct principal curvatures is constant on M. The notions of Dupin and proper Dupin hypersurfaces in R^{n }or S^{n} can be generalized to the setting of Lie sphere geometry, and these properties are easily seen to be invariant under Lie sphere transformations. This makes Lie sphere geometry an effective …
Distribution Of The PTorsion Of Jacobian Groups Of Regular Matroids, Sergio R. Zapata Ceballos
Distribution Of The PTorsion Of Jacobian Groups Of Regular Matroids, Sergio R. Zapata Ceballos
Electronic Thesis and Dissertation Repository
Given a regular matroid $M$ and a map $\lambda\colon E(M)\to \N$, we construct a regular matroid $M_\lambda$. Then we study the distribution of the $p$torsion of the Jacobian groups of the family $\{M_\lambda\}_{\lambda\in\N^{E(M)}}$. We approach the problem by parameterizing the Jacobian groups of this family with nontrivial $p$torsion by the $\F_p$rational points of the configuration hypersurface associated to $M$. In this way, we reduce the problem to counting points over finite fields. As a result, we obtain a closed formula for the proportion of these groups with nontrivial $p$torsion as well as some estimates. In addition, we show that the …