55,539 research outputs found
Approximate computations with modular curves
This article gives an introduction for mathematicians interested in numerical
computations in algebraic geometry and number theory to some recent progress in
algorithmic number theory, emphasising the key role of approximate computations
with modular curves and their Jacobians. These approximations are done in
polynomial time in the dimension and the required number of significant digits.
We explain the main ideas of how the approximations are done, illustrating them
with examples, and we sketch some applications in number theory
Asymptotic Methods for Asset Market Equilibrium Analysis
General equilibrium analysis is difficult when asset markets are incomplete. We make the simplifying assumption that uncertainty is small and use bifurcation methods to compute Taylor series approximations for asset demand and asset market equilibrium. A computer must be used to derive these approximations since they involve large amounts of algebraic manipulation. To illustrate this method, we apply it to analyzing the allocative, price, and welfare effects of introducing a new derivative security. We find that the introduction of any derivative will raise the value of the risky asset relative to bonds.
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