Related papers: Approximate computations with modular curves
Basic facts and definitions of conformal moduli of rings and quadrilaterals are recalled. Some computational methods are reviewed. For the case of quadrilaterals with polygonal sides, some recent results are given. Some numerical…
In this work, we present an efficient method for computing in the Generalized Jacobian of special singular curves. The efficiency of the operation is due to representation of an element in the Jacobian group by a single polynomial.
Computation of (approximate) polynomials common factors is an important problem in several fields of science, like control theory and signal processing. While the problem has been widely studied for scalar polynomials, the scientific…
We present an algorithm for computation of cell adjacencies for well-based cylindrical algebraic decomposition. Cell adjacency information can be used to compute topological operations e.g. closure, boundary, connected components, and…
The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the curvature error. Parametric polynomial curves of low degree are used and a geometric continuity is…
For affine algebraic plane curves we reduce a calculation of its invariants to calculation of the intersection of kernels of some derivations.
Comparison to traditionally accurate computing, approximate computing focuses on the rapidity of the satisfactory solution, but not the unnecessary accuracy of the solution. Approximate bisimularity is the approximate one corresponding to…
We discuss the use of matrices for providing sequences of rationals that approximate algebraic irrationalities. In particular, we study the regular representation of algebraic extensions, proving that ratios between two entries of the…
This survey provides a practical and algorithmic perspective on Drinfeld modules over $\mathbb F_q[T]$. Starting with the construction of the Carlitz module, we present Drinfeld modules in any rank and some of their arithmetic properties.…
We introduce computable projection operators onto piecewise polynomial spaces, defined via sampling and discrete least-squares polynomial approximations. The resulting mappings exhibit (almost) optimal approximation properties in $L^2$ and…
Many real life problems can be reduced to the solution of a complex exponentials approximation problem which is usually ill posed. Recently a new transform for solving this problem, formulated as a specific moments problem in the plane, has…
We discuss the approximation of real numbers by Fourier coefficients of newforms, following recent work of Alkan, Ford and Zaharescu. The main tools used here, besides the (now proved) Sato-Tate Conjecture, come from metric number theory.
The theory of partition congruences has been a fascinating and difficult subject for over a century now. In attempting to prove a given congruence family, multiple possible complications include the genus of the underlying modular curve,…
These notes survey the theory of (twisted) conformal blocks from an algebro-geometric perspective and have two main goals. The first one is to summarize the construction of conformal blocks from vertex operator algebras, and to describe…
Building on the established methods for superconducting circuit quantization, we present a new theoretical framework for approximate numerical simulation of Josephson quantum circuits. Simulations based on this framework provide access to a…
Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…
We produce algorithms to detect whether a complex affine variety computed and presented numerically by the machinery of numerical algebraic geometry corresponds to an associated component of a polynomial ideal.
Neural networks are trained to learn an approximate mapping from an input domain to a target domain. Incorporating prior knowledge about true mappings is critical to learning a useful approximation. With current architectures, it is…
Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the…
Considered herein is a modified Newton method for the numerical solution of nonlinear equations where the Jacobian is approximated using a complex-step derivative approximation. We show that this method converges for sufficiently small…