Browsing by Author "Goldman, Ronald"

Now showing 1 - 7 of 7
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    A Set of Convolution Identities Relating the Blocks of Two Dixon Resultant Matrices
    (1999-06-16) Chionh, Eng-Wee; Goldman, Ronald; Zhang, Ming
    Resultants for bivariate polynomials are often represented by the determinants of very big matrices. Properly grouping the entries of these matrices into blocks is a very effective tool for studying the properties of these resultants. Here we derive a set of convolution identities relating the blocks of two Dixon bivariate resultant representations.
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    Art and Engineering Inspired by Swarm Robotics
    (2017-04-13) Zhou, Yu; Goldman, Ronald
    Swarm robotics has the potential to combine the power of the hive with the sensibility of the individual to solve non-traditional problems in mechanical, industrial, and architectural engineering and to develop exquisite art beyond the ken of most contemporary painters, sculptors, and architects. The goal of this thesis is to apply swarm robotics to the sublime and the quotidian to achieve this synergy between art and engineering. The potential applications of collective behaviors, manipulation, and self-assembly are quite extensive. We will concentrate our research on three topics: fractals, stability analysis, and building an enhanced multi-robot simulator. Self-assembly of swarm robots into fractal shapes can be used both for artistic purposes (fractal sculptures) and in engineering applications (fractal antennas). Stability analysis studies whether distributed swarm algorithms are stable and robust either to sensing or to numerical errors, and tries to provide solutions to avoid unstable robot configurations. Our enhanced multi-robot simulator supports this research by providing real-time simulations with customized parameters, and can become as well a platform for educating a new generation of artists and engineers. The goal of this thesis is to use techniques inspired by swarm robotics to develop a computational framework accessible to and suitable for both artists and engineers. The scope we have in mind for art and engineering is unlimited. Modern museums, stadium roofs, dams, solar power plants, radio telescopes, star networks, fractal sculptures, fractal antennas, fractal floral arrangements, smooth metallic railroad tracks, temporary utilitarian enclosures, permanent modern architectural designs, guard structures, op art, and communication networks can all be built from the bodies of the swarm.
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    Hybrid Dixon Resultants
    (1998-05-13) Chionh, Eng-Wee; Goldman, Ronald; Zhang, Ming
    Dixon [1908] describes three distinct homogeneous determinant representations for the resultant of three bivariate polynomials of bidegree(m,n). These Dixon resultants are the determinants of matrices of orders 6mn, 3mn and 2mn, and the entries of these matrices are respectively homogeneous of degrees 1, 2, and 3 in the coefficients of the original three polynomial equations. Here we mix and match columns from these three Dixon matrices to construct a large assortment of new hybrid determinant representations of orders ranging from 2mn to 6mn for the resultant of three bivariate polynomials of bidegree (m,n).
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    Mu -bases and their applications in geometric modeling
    (2007) Song, Ning; Goldman, Ronald
    This thesis defines the notion of a μ-basis for an arbitrary number of polynomials in one variable. The properties of these μ-bases are derived, and a straightforward algorithm is provided to calculate a μ-basis for any collection of univariate polynomials. Systems where base points are present are also discussed. μ-bases are then applied to solve implicitization, inversion and intersection problems for rational space curves. Next, a natural one to one correspondence is derived between the singular points of rational planar curves and the axial moving lines that follow these curves. This correspondence is applied together with μ-bases to compute and to analyze all the singular points of low degree rational planar curves.
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    Quantum Algebraic Geometry Codes
    (2024-12-06) Han, Zhengyi; Goldman, Ronald
    Quantum error correction is an essential aspect of quantum information theory, providing protection for quantum states against noise and decoherence. This thesis investigates the construction of quantum error correction codes derived from classical algebraic geometry (AG) codes. We present two distinct construction techniques, highlighting the flexibility and self-orthogonality of AG codes, and demonstrate their ability to produce asymptotically good quantum codes. Additionally, we explore strategies to fine-tune the parameters of classical AG codes, ensuring they possess the desired properties for quantum code construction. This work serves as a comprehensive guide to the fundamental concepts and common methodologies underlying quantum algebraic geometry codes.
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    The Block Structure of Three Dixon Resultants and Their Accompanying Transformation Matrices
    (1999-06-16) Chionh, Eng-Wee; Goldman, Ronald; Zhang, Ming
    Dixon [1908] introduces three distinct determinant formulations for the resultant of three bivariate polynomials of bidegree (m,n) . The first technique applies Sylvester's dialytic method to construct the resultant as the determinant of a matrix of order 6mn . The second approach uses Cayley's determinant device to form a more compact representation for the resultant as the determinant of a matrix of order 2mn . The third method employs a combination of Cayley's determinant device with Sylvester's dialytic method to build the resultant as the determinant of a matrix of order 3mn . Here relations between these three resultant formulations are derived and the structure of the transformations between these resultant matrices is investigated. In particular, it is shown that these transformation matrices all have similar, simple, upper triangular, block symmetric structures and the blocks themselves have elegant symmetry properties. Elementary entry formulas for the transformation matrices are also provided. In light of these results, the three Dixon resultant matrices are reexamined and shown to have natural block structures compatible with the block structures of the transformation matrices. These block structures are analyzed here and applied along with the block structures of the transformation matrices to simplify the calculation of the entries of the Dixon resultants of order 2mn and 3mn and to make these calculations more efficient by removing redundant computations.
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    Transformations and Transitions from the Sylvester to the Bezout Resultant
    (1999-06-17) Chionh, Eng-Wee; Goldman, Ronald; Zhang, Ming
    A simple matrix transformation linking the resultant matrices of Sylvester and Bezout is derived. This transformation matrix is then applied to generate an explicit formula for each entry of the Bezout resultant, and this entry formula is used, in turn, to construct an efficient recursive algorithm for computing all the entries of the Bezout matrix. Hybrid resultant matrices consisting of some columns from the Sylvester matrix and some columns from the Bezout matrix provide natural transitions from the Sylvester to the Bezout resultant, and allow as well the Bezout construction to be generalized to two polynomials of different degrees. Such hybrid resultants are derived here, employing again the transformation matrix from the Sylvester to the Bezout resultant.