Convergence rates for multivariate smoothing spline functions

AbstractLet Ω be an open bounded subset of Rd, the d-dimensional space, and let ƒ be an unknown function belonging to Hm(Ω), where m is an integer (m > d2). Given the values of ƒ at n scattered data point in Ω known with error, i.e., given zi = ƒ(ti) + εi, i = 1, …, n, where the εi's are i.i.d. random errors, we study the error E[¦ƒ − σλ¦k,Ω2], where ¦·¦k,Ω2 are the Sobolev semi-norms in Hm(Ω) and σλ is the thin plate smoothing spline with parameter λ, i.e., the unique minimizer of λ¦u¦m2 + (1n)∑i = 1n (u(ti) − zi)2. Under the assumption that the boundary of Ω is smooth and the points satisfy a “quasi-uniform” condition, we obtain E[¦ƒ − σλ¦k,Ω2] ⩽ C[λ(m − k)m ¦ƒ¦m,Ω2 + D(nλ(2k + d2m))], k = 0, 1,…, m − 1

Quantum RLCRLC circuits: charge discreteness and resonance

In a recent article, we have advanced a semiclassical theory of quantum circuits with discrete charge and electrical resistance. In this work, we present a few elementary applications of this theory. For the zero resistance, inductive circuit, we obtain the Stark ladder energies in yet another way; and generalize earlier results by Chandia et. al, for the circuit driven by a combination d.c. plus a.c. electromotive force (emf). As a second application, we investigate the effect of electrical resistance, together with charge discreteness, in the current amplitude, and resonance conditions of a general $RLC$ quantum circuit, including nonlinear effects up to third order on the external sinusoidal emf