A5066875022- Advanced Topics in Algebra
- Polynomial and algebraic computation
- Nonlinear Waves and Solitons
- Numerical methods for differential equations
- Advanced Numerical Analysis Techniques
- Algebraic structures and combinatorial models
- graph theory and CDMA systems
- Differential Equations and Numerical Methods
- Matrix Theory and Algorithms
- Advanced Differential Equations and Dynamical Systems
- Homotopy and Cohomology in Algebraic Topology
Florida State University
2023-2024
University of California, Los Angeles
2015
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We present two algorithms for computing what we call the absolute factorization of a difference operator. also give an algorithm solving third order equations in terms second equations, together with applications to OEIS sequences. The latter is similar existing differential [4, 11], except that there additional one symmetric product.
We classify order $3$ linear difference operators over $\mathbb{C}(x)$ that are solvable in terms of lower operators. To prove this result, we introduce the notions restriction, induction, and absolute irreducibility modules, modules irreducible but not absolutely irreducible. also show how restriction induction give coordinate-free formulations for sectioning interlacing sequences.
We present two algorithms for computing what we call the absolute factorization of a difference operator. also give an algorithm to solve third order equations in terms second equations, together with applications OEIS sequences. The latter is similar existing differential equations.
We introduce a cohomology theory that classifies differential objects arise from Picard-Vessiot theory, using the Hopf-Galois descent. To do this, we provide an explicit description of in terms torsors. then use this to give bijective correspondence between and As application, prove universal bound for splitting degree central simple algebras.
We show that the $n\times n$ matrix differential equation $\delta(Y)=AY$ with $n^2$ general coefficients cannot be simplified to an in less than $n$ parameters by using gauge transformations whose are rational functions entries of $A$ and their derivatives. Our proof uses Galois theory a analogue essential dimension. also bound minimum number needed describe some generic Picard-Vessiot extensions.