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- Interval arithmetic (2)
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Articles 1 - 5 of 5
Full-Text Articles in Algebraic Geometry
Parameterized Polynomial Systems: Monodromy, Sparse Polynomials, And Solutions, Julianne Barnhart
Parameterized Polynomial Systems: Monodromy, Sparse Polynomials, And Solutions, Julianne Barnhart
All Dissertations
The lift of a loop in the base space of a branched cover to the cover induces a permutation of points in a fibre. The monodromy group of the branched cover is the permutation group generated by all such permutations. When loops are restricted to a particular subset of the base space, the corresponding permutation group induced by these loops is the restricted monodromy group. Monodromy groups encode structure and symmetries of many enumerative problems. We describe the relationship between the restricted monodromy group and the monodromy group of the original branched cover. Our main result is a local-to-global property: …
Certified Approximation Algorithms Of Algebraic Curves, Michael Byrd Jr.
Certified Approximation Algorithms Of Algebraic Curves, Michael Byrd Jr.
All Dissertations
One of the fundamental problems in mathematics is to determine the set of solutions to a system of equations. In algebraic geometry, the equations studied are polynomials, and the solution set is called an algebraic variety. For single variable polynomials of degree less than five, the roots can be determined exactly using algebraic methods, but for polynomials of degree five or higher, numerical methods are required. When using numerical methods, it is important to know when the computed approximation is indeed a correct solution, which leads to the idea of a certified algorithm. An algorithm is said to be …
Certified Computation Of Julia Sets Via Numerical Methods, Hannah Kaufman
Certified Computation Of Julia Sets Via Numerical Methods, Hannah Kaufman
All Theses
The chaotic and fractal nature of Julia sets makes them difficult to graph. This research aims to provide graphical approximations of Julia sets with known and guaranteed levels of accuracy. We implement three methods to approximate Julia sets with c values chosen from the main cardioid of the Mandelbrot set. Each method utilizes different properties of these Julia sets. The exclusion method makes use of the fact that a Julia set of this type is topologically a circle. Attracting and repelling fixed points are used to find a region on the interior of the Julia set and a region on …
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.
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 …