Approximation by piecewise-regular maps

A real algebraic variety W of dimension m is said to be uniformly rational if each of its points has a Zariski open neighborhood which is biregularly isomorphic to a Zariski open subset of R^m. Let l be any nonnegative integer. We prove that every map of class C^l from a compact subset of a real algebraic variety into a uniformly rational real algebraic variety can be approximated in the C^l topology by piecewise-regular maps of class C^k, where k is an arbitrary integer greater than or equal to l. Next we derive consequences regarding algebraization of topological vector bundles.Comment: 19 pages; Sections 1, 2.3 reorganize

Seshadri constants, Diophantine approximation, and Roth's Theorem for arbitrary varieties

In this paper, we associate an invariant $\alpha_{x}(L)$ to an algebraic point $x$ on an algebraic variety $X$ with an ample line bundle $L$. The invariant $\alpha$ measures how well $x$ can be approximated by rational points on $X$, with respect to the height function associated to $L$. We show that this invariant is closely related to the Seshadri constant $\epsilon_{x}(L)$ measuring local positivity of $L$ at $x$, and in particular that Roth's theorem on $\mathbf{P}^1$ generalizes as an inequality between these two invariants valid for arbitrary projective varieties.Comment: 55 pages, published versio

Steady state analysis of Chua's circuit with RLCG transmission line

In this paper we present a new technique to compute the steady state response of nonlinear autonomous circuits with RLCG transmission lines. Using multipoint Pade approximants, instead of the commonly used expansions around s=0 or s/spl rarr//spl infin/ accurate, low-order lumped equivalent circuits of the characteristic impedance and the exponential propagation function are obtained in an explicit way. Then, with the temporal discretization of the equations that describe the transformed circuit, we obtain a nonlinear algebraic formulation where the unknowns to be determined are the samples of the variables directly in the steady state, along with the oscillation period, the main unknown in autonomous circuits. An efficient scheme to build the Jacobian matrix with exact partial derivatives with respect to the oscillation period and with respect to the samples of the unknowns is obtained. Steady state solutions of the Chua's circuit with RLCG transmission line are computed for selected circuit parameters.Peer ReviewedPostprint (published version