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Articles 1 - 17 of 17
Full-Text Articles in Algebraic Geometry
Acceleration Skinning: Kinematics-Driven Cartoon Effects For Articulated Characters, Niranjan Kalyanasundaram
Acceleration Skinning: Kinematics-Driven 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 well-received 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 …
ℂ-Motivic Modular Forms, Bogdan Gheorghe, Daniel C. Isaksen, Achim Krause, Nicolas Ricka
ℂ-Motivic Modular Forms, Bogdan Gheorghe, Daniel C. Isaksen, Achim Krause, Nicolas Ricka
Mathematics Faculty Research Publications
We construct a topological model for cellular, 2-complete, stable C-motivic homotopy theory that uses no algebro-geometric foundations.We compute the Steenrod algebra in this context, and we construct a “motivic modular forms” spectrum over ℂ.
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 and Computer Science Department Faculty Scholarship
A hypersurface M in Rn or Sn 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 Rn or Sn 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 …
From Mathematics To Medicine: A Practical Primer On Topological Data Analysis (Tda) And The Development Of Related Analytic Tools For The Functional Discovery Of Latent Structure In Fmri Data, Andrew Salch, Adam Regalski, Hassan Abdallah, Raviteja Suryadevara, Michael J. Catanzaro, Vaibhav A. Diwadkar
From Mathematics To Medicine: A Practical Primer On Topological Data Analysis (Tda) And The Development Of Related Analytic Tools For The Functional Discovery Of Latent Structure In Fmri Data, Andrew Salch, Adam Regalski, Hassan Abdallah, Raviteja Suryadevara, Michael J. Catanzaro, Vaibhav A. Diwadkar
Mathematics Faculty Research Publications
fMRI is the preeminent method for collecting signals from the human brain in vivo, for using these signals in the service of functional discovery, and relating these discoveries to anatomical structure. Numerous computational and mathematical techniques have been deployed to extract information from the fMRI signal. Yet, the application of Topological Data Analyses (TDA) remain limited to certain sub-areas such as connectomics (that is, with summarized versions of fMRI data). While connectomics is a natural and important area of application of TDA, applications of TDA in the service of extracting structure from the (non-summarized) fMRI data itself are heretofore nonexistent. …
Elliptic Curves And Their Practical Applications, Henry H. Hayden Iv
Elliptic Curves And Their Practical Applications, Henry H. Hayden Iv
Graduate Theses/Dissertations
Finding rational points that satisfy functions known as elliptic curves induces a finitely-generated abelian group. Such functions are powerful tools that were used to solve Fermat's Last Theorem and are used in cryptography to send private keys over public systems. Elliptic curves are also useful in factoring and determining primality.
Probability Distributions For Elliptic Curves In The Cgl Hash Function, Dhruv Bhatia, Kara Fagerstrom, Max Watson
Probability Distributions For Elliptic Curves In The Cgl Hash Function, Dhruv Bhatia, Kara Fagerstrom, Max Watson
Mathematical Sciences Technical Reports (MSTR)
Hash functions map data of arbitrary length to data of predetermined length. Good hash functions are hard to predict, making them useful in cryptography. We are interested in the elliptic curve CGL hash function, which maps a bitstring to an elliptic curve by traversing an inputdetermined path through an isogeny graph. The nodes of an isogeny graph are elliptic curves, and the edges are special maps betwixt elliptic curves called isogenies. Knowing which hash values are most likely informs us of potential security weaknesses in the hash function. We use stochastic matrices to compute the expected probability distributions of the …
Evaluating The Historical Accuracy Of Blackwork Embroidery With Fractal Analysis, Rhiannon Cire
Evaluating The Historical Accuracy Of Blackwork Embroidery With Fractal Analysis, Rhiannon Cire
Undergraduate Theses and Capstone Projects
The intricate monochromatic embroidery that graced the collars and cuffs of Renaissance nobility and domestic materials from that era has been little studied beyond the historical costuming and crafting communities. This style, known as blackwork, for it was traditionally done in black silk on white linen, exemplifies how complex and visually-appealing designs can arise from repetition of simple forms, often demonstrating the fractal property of self-similarity. Though most blackwork patterns are not true fractals, fractal analysis offers a means of objectively quantifying their complexity and new lens through which to examine this embroidery technique. The purpose of this study was …
On Elliptic Curves, Montana S. Miller
On Elliptic Curves, Montana S. Miller
Graduate Theses/Dissertations
An elliptic curve over the rational numbers is given by the equation y2 = x3+Ax+B. In our thesis, we study elliptic curves. It is known that the set of rational points on the elliptic curve form a finitely generated abelian group induced by the secant-tangent addition law. We present an elementary proof of associativity using Maple. We also present a relatively concise proof of the Mordell-Weil Theorem.
Lecture 00: Opening Remarks: 46th Spring Lecture Series, Tulin Kaman
Lecture 00: Opening Remarks: 46th Spring Lecture Series, Tulin Kaman
Mathematical Sciences Spring Lecture Series
Opening remarks for the 46th Annual Mathematical Sciences Spring Lecture Series at the University of Arkansas, Fayetteville.
Infinite-Order Differential Operators Acting On Entire Hyperholomorphic Functions, Daniel Alpay, Fabrizio Colombo, Stefano Pinton, Irene Sabadini, Daniele C. Struppa
Infinite-Order Differential Operators Acting On Entire Hyperholomorphic Functions, Daniel Alpay, Fabrizio Colombo, Stefano Pinton, Irene Sabadini, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
Infinite-order differential operators appear in different fields of mathematics and physics and in the past decade they turned out to be of fundamental importance in the study of the evolution of superoscillations as initial datum for Schrödinger equation. Inspired by the operators arising in quantum mechanics, in this paper, we investigate the continuity of a class of infinite-order differential operators acting on spaces of entire hyperholomorphic functions. We will consider two classes of hyperholomorphic functions, both being natural extensions of holomorphic functions of one complex variable. We show that, even though these two notions of hyperholomorphic functions are quite different …
On Leibniz Cohomology, Jorg Feldvoss, Friedrich Wagemann
On Leibniz Cohomology, Jorg Feldvoss, Friedrich Wagemann
University Faculty and Staff Publications
In this paper we prove the Leibniz analogue of Whitehead's vanishing theorem for the Chevalley-Eilenberg cohomology of Lie algebras. As a consequence, we obtain the second Whitehead lemma for Leibniz algebras. Moreover, we compute the cohomology of several Leibniz algebras with ad joint or irreducible coefficients. Our main tool is a Leibniz analogue of the Hochschild-Serre spectral sequence, which is an extension of the dual of a spectral sequence of Pirashvili for Leibniz homology from symmetric bimodules to arbitrary bimodules.
The Adams Spectral Sequence For Topological Modular Forms, Robert Bruner, John Rognes
The Adams Spectral Sequence For Topological Modular Forms, Robert Bruner, John Rognes
Mathematics Faculty Research Publications
The connective topological modular forms spectrum, 𝑡𝑚𝑓, is in a sense initial among elliptic spectra, and as such is an important link between the homotopy groups of spheres and modular forms. A primary goal of this volume is to give a complete account, with full proofs, of the homotopy of 𝑡𝑚𝑓 and several 𝑡𝑚𝑓-module spectra by means of the classical Adams spectral sequence, thus verifying, correcting, and extending existing approaches. In the process, folklore results are made precise and generalized. Anderson and Brown-Comenetz duality, and the corresponding dualities in homotopy groups, are carefully proved. The volume also includes an account …
Towards Tropical Psi Classes, Jawahar Madan
Towards Tropical Psi Classes, Jawahar Madan
HMC Senior Theses
To help the interested reader get their initial bearings, I present a survey of prerequisite topics for understanding the budding field of tropical Gromov-Witten theory. These include the language and methods of enumerative geometry, an introduction to tropical geometry and its relation to classical geometry, an exposition of toric varieties and their correspondence to polyhedral fans, an intuitive picture of bundles and Euler classes, and finally an introduction to the moduli spaces of n-pointed stable rational curves and their tropical counterparts.
A Tropical Approach To The Brill-Noether Theory Over Hurwitz Spaces, Kaelin Cook-Powell
A Tropical Approach To The Brill-Noether Theory Over Hurwitz Spaces, Kaelin Cook-Powell
Theses and Dissertations--Mathematics
The geometry of a curve can be analyzed in many ways. One way of doing this is to study the set of all divisors on a curve of prescribed rank and degree, known as a Brill-Noether variety. A sequence of results, starting in the 1980s, answered several fundamental questions about these varieties for general curves. However, many of these questions are still unanswered if we restrict to special families of curves. This dissertation has three main goals. First, we examine Brill-Noether varieties for these special families and provide combinatorial descriptions of their irreducible components. Second, we provide a natural generalization …
On The Tropicalization Of Lines Onto Tropical Quadrics, Natasha Crepeau
On The Tropicalization Of Lines Onto Tropical Quadrics, Natasha Crepeau
HMC Senior Theses
Tropical geometry uses the minimum and addition operations to consider tropical versions of the curves, surfaces, and more generally the zero set of polynomials, called varieties, that are the objects of study in classical algebraic geometry. One known result in classical geometry is that smooth quadric surfaces in three-dimensional projective space, $\mathbb{P}^3$, are doubly ruled, and those rulings form a disjoint union of conics in $\mathbb{P}^5$. We wish to see if the same result holds for smooth tropical quadrics. We use the Fundamental Theorem of Tropical Algebraic Geometry to outline an approach to studying how lines lift onto a tropical …
Diagrams In Intra-Configurational Analysis, Marco Panza, Gianluca Longa
Diagrams In Intra-Configurational Analysis, Marco Panza, Gianluca Longa
MPP Published Research
In this paper we would like to attempt to shed some light on the way in which diagrams enter into the practice of ancient Greek geometrical analysis. To this end, we will first distinguish two main forms of this practice, i.e., trans-configurational and intra-configurational. We will then argue that, while in the former diagrams enter in the proof essentially in the same way (mutatis mutandis) they enter in canonical synthetic demonstrations, in the latter, they take part in the analytic argument in a specific way, which has no correlation in other aspects of classical geometry. In intra-configurational analysis, diagrams represent …
Analysis, Constructions And Diagrams In Classical Geometry, Marco Panza
Analysis, Constructions And Diagrams In Classical Geometry, Marco Panza
MPP Published Research
Greek ancient and early modern geometry necessarily uses diagrams. Among other things, these enter geometrical analysis. The paper distinguishes two sorts of geometrical analysis and shows that in one of them, dubbed “intra-confgurational” analysis, some diagrams necessarily enter as outcomes of a purely material gesture, namely not as result of a codifed constructive procedure, but as result of a free-hand drawing.