We study the asymptotic behavior of Pick functions, analytic functions which take the upper half plane to itself. We show that if a two variable Pick function f has real residues to order 2N − 1 at infinity and the imaginary part of the remainder between f and this expansion is of order 2N + 1, then f has real residues to order 2N and directional residues to order 2N + 1. Furthermore, f has rea...

Unlike the Metelli monochrome transparencies, when overlays and their backgrounds have chromatic content, the inferred surface colors may not always be physically realizable, and are in some sense "imaginary." In these cases, the inferred chromatic transmittance or reflectance of the overlay lies outside the RGB spectral boundaries. Using the classical Metelli configuration, we demonstrate this...

Imaginary quadratic fields Q( √ −d), for integers d > 0, are perhaps the simplest number fields afterQ. They are equal parts helpful first example and misleading special case. LikeZ, the Gaussian integersZ[i] (the cased = 1) have unique factorization and a Euclidean algorithm. As d grows, however, these properties eventually fail, first the latter and then the former. The classical Euclidean al...

For all quadratic imaginary number fields F of discriminant d > −5000, we give the conjectural value of the order of Milnor’s group (the tame kernel) K2OF , where OF is the ring of integers of F. Assuming that the order is correct, we determine the structure of the group K2OF and of its subgroup WF (the wild kernel). It turns out that the odd part of the tame kernel is cyclic (with one exceptio...

We find a lower bound on the number of imaginary quadratic extensions of the function field Fq(T ) whose class groups have an element of a fixed order. More precisely, let q ≥ 5 be a power of an odd prime and let g be a fixed positive integer ≥ 3. There are q 1 2+ 1 g ) polynomials D ∈ Fq[T ] with deg(D) ≤ ` such that the class groups of the quadratic extensions Fq(T, √ D) have an element of or...

Early research on imaginary companions viewed the phenomenon negatively. Imaginary companions were often associated with psychopathology and an inability to distinguish between fantasy and reality. The prevailing view on imaginary companions has shifted in recent years with many studies focusing on the positive attributes exhibited by children with imaginary friends. Methodological issues which...

PURPOSE This study was undertaken to test the hypothesis that patients with epilepsy have abnormal imaginary coherence compared with control subjects. METHODS Thirty patients with seizures underwent magnetoencephalography (MEG) recording using a whole cortex MEG system. Conventional equivalent current dipoles (ECDs) and synthetic aperture magnetometry (SAM) were used to analyze MEG data. Neur...

We have prepared films of polymethyl .methacrylate in which fullerenes (a mixture of C,, and C,,) are uniformly embedded. By performing both degenerate four-wave mixing experiments and h-radiance-dependent transmission measurements with nanosecond laser pulses of 608nm wavelength, the third-order nonlinear optical susceptibility of these films is determined to be of the order of 10-l’ esu. This...