The expressive power of voting polynomials

We consider the problem of approximating a Boolean functionf∶{0,1} ⁿ →{0,1} by the sign of an integer polynomialp of degreek. For us, a polynomialp(x) predicts the value off(x) if, wheneverp(x)≥0,f(x)=1, and wheneverp(x)<0,f(x)=0. A low-degree polynomialp is a good approximator forf if it predictsf at almost all points. Given a positive integerk, and a Boolean functionf, we ask, “how good is the best degreek approximation tof?” We introduce a new lower bound technique which applies to any Boolean function. We show that the lower bound technique yields tight bounds in the casef is parity. Minsky and Papert [10] proved that a perceptron cannot compute parity; our bounds indicate exactly how well a perceptron canapproximate it. As a consequence, we are able to give the first correct proof that, for a random oracleA, PP ^A is properly contained in PSPACE ^A . We are also able to prove the old AC⁰ exponential-size lower bounds in a new way. This allows us to prove the new result that an AC⁰ circuit with one majority gate cannot approximate parity. Our proof depends only on basic properties of integer polynomials.     

Optically monitoring and controlling epitaxial growth

We provide a perspective on current capabilities for optically monitoring and controlling epitaxial growth, and discuss examples taken from recent work at Bellcore.     

A photometric ellipsometer for measuring flux in a general state of polarization

We describe the theory and operation of a rotating analyzer/compensator ellipsometer capable of measuring all four Stokes parameters of generally polarized flux. Applications presented include measurement of gray-body emission, and an experimental study of the effects of internally and externally stray and scattered light and surface roughness on calculated values of the dielectric function.     

Extending scanning ellipsometric spectra into experimentally inaccessible regions

The extrapolation of the dielectric function into experimentally inaccessible regions as a series of finitewidth, constant-amplitude oscillators is discussed and used to investigate accuracy requirements on ellipsometric data intended for this purpose. Up to six oscillators may be determined with current limits of accuracy. These extrapolations also provide information about spectroscopic sum rules. A method of evaluating Kramers-Kronig integrals is presented, which is substantially more accurate than direct integration.