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Dr. Sarah Chen
Stanford · NLP & interpretability
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Fairness + NLP projects overlap with her recent work.
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Ronald N. Goldman is a Professor of Computer Science at Rice University in Houston, Texas. He received his B.S. in Mathematics from the Massachusetts Institute of Technology in 1968 and his M.A. and Ph.D. in Mathematics from Johns Hopkins University in 1973. His current research interests focus on the mathematical representation, manipulation, analysis, and reconstruction of shape using computers. His work includes research in computer aided geometric design, solid modeling, computer graphics, polynomials, and splines. He is particularly interested in algorithms for polynomial and piecewise polynomial curves and surfaces, as well as applications of algebraic and differential geometry to geometric modeling. Recently, he has concentrated on the uses of quaternions and Clifford algebras in computer graphics.
Rocky Mountain Journal of Mathematics · 2025-08-01
articleSenior author
We investigate algebraic properties of the negative degree q-Bernstein bases. Our fundamental tool in this investigation is a recently introduced variant of the blossom, the multirational q-blossom, which provides the dual functionals for the negative degree q-Bernstein basis functions. By applying the dual functional property of the multirational q-blossom, we are readily able to generate several fundamental identities involving the negative degree q-Bernstein bases, including a new variant of Marsden’s identity, a partition of unity property, a reparametrization formula, and a formula for representing monomials. We also show how to use the homogeneous variant of the multirational q-blossom to convert between the q-Taylor bases and the negative degree q-Bernstein bases.
Abstract Narayana numbers are well-known and have many applications in the field of combinatorics. The Narayana numbers form a triangular array, where the sum of the n th row is the n th Catalan number. Here we normalize the Narayana numbers by dividing each entry in the n th row by the n th Catalan number. Now each row of these normalized Narayana numbers defines a discrete probability distribution. We investigate two new properties of these normalized Narayana numbers: instead of summing along the rows, we derive the limit of the sums along the columns and the limit of the sums along the short diagonals.
Abstract We present a general constructive approach to blossoming for arbitrary finite dimensional univariate function spaces by altering the diagonal property of the classical blossom. We apply this approach to blossoming to develop the corresponding theory of Bernstein bases and Bézier curves including de Casteljau type algorithms for evaluation and subdivision as well as a Marsden identity for arbitrary finite dimensional univariate function spaces.
Weutilize formulas for basic hypergeometric series to derive identities and formulas for negative degree q-Bernstein bases, including the Marsden identity, the partition of unity property, the monomial representation formula, the reparametrization formula, and the degree reduction formula. We show that all these identities are just special forms of the q-analogue of Gauss? formula. We also provide a new proof for the q-analogue of Gauss? formula by using the Marsden identity for negative degree q-Bernstein bases together with the identity theorem for analytic functions.
Complex rational maps have been used to construct birational quadratic maps based on two special syzygies of degree one. Similar to complex rational curves, rational curves over generalized complex numbers have also been constructed by substituting the imaginary unit with a new independent quantity. We first establish the relationship between degree one, generalized, complex rational Bézier curves and quadratic rational Bézier curves. Then we provide conditions to determine when a quadratic rational planar map has a generalized complex rational representation. Thus, a rational quadratic planar map can be made birational by suitably choosing the middle Bézier control points and their corresponding weights. In contrast to the edges of complex rational maps of degree one, which are circular arcs, the edges of the planar maps can be generalized to hyperbolic and parabolic arcs by invoking the hyperbolic and parabolic numbers.
