574 research outputs found
On Jacobian group of the -graph
In the present paper we compute the Jacobian group of -graph
The notion of -graph continues the list of
families of -, - and -graphs well-known in the graph theory. In
particular, graph is isomorphic to discrete torus
It this case, the structure of the Jacobian group will be find
explicitly.Comment: arXiv admin note: text overlap with arXiv:2111.0430
Computing the common zeros of two bivariate functions via Bezout resultants
The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and B�ezout matrices with polynomial entries. Using techniques including domain subdivision, B�ezoutian regularization and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (� 1000). We analyze the resultant method and its conditioning by noting that the B�ezout matrices are matrix polynomials. Our robust algorithm is implemented in the roots command in Chebfun2, a software package written in object-oriented MATLAB for computing with bivariate functions
Computing the common zeros of two bivariate functions via BĂ©zout resultants
The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and Bézout matrices with polynomial entries. Using techniques including domain subdivision, Bézoutian regularization, and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (≥ 1000). We analyze the resultant method and its conditioning by noting that the Bézout matrices are matrix polynomials. Two implementations are available: one on the Matlab Central File Exchange and another in the roots command in Chebfun2 that is adapted to suit Chebfun’s methodology
Singularities
Singularity theory is a central part of contemporary mathematics. It is concerned with the local and global structure of maps and spaces that occur in algebraic, analytic or differential geometric context. For its study it uses methods from algebra, topology, algebraic geometry and complex analysis
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