Brad Osgood

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The modern quantum geometry of strings is concerned to a large extent with the set of all surfaces, with varying metrics on those surfaces, and with determinants of associated Laplacians. Our aim in this paper is to study this determinant as a function of the metric on a given surface and in particular its extreme values when the metric is suitably(More)
A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass-Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian defined for conformal metrics and on a(More)
—We study the problem of interpolating all values of a discrete signal f of length N when d < N values are known, especially in the case when the Fourier transform of the signal is zero outside some prescribed index set J ; these comprise the (generalized) bandlimited spaces B J. The sampling pattern for f is specified by an index set I, and is said to be a(More)
Gehring and Pommerenke have shown that if the Schwarzian derivative Sf of an analytic function f in the unit disk D satisfies ISf(z)] ~_ 2(1-Iz[2)-2, then f(D) is a Jordan domain except when f(D) is the image under a M6bius transformation of an infinite parallel strip. The condition ISf(z)l <_ 2(1-lzl2)-2 is the classical sufficient condition for univalence(More)
Any isospectral family of two-dimensional Euclidean domains is shown to be compact in the C(infinity) topology. Previously Melrose, using heat invariants, was able to establish the C(infinity) compactness of the curvature of the boundary curves. The additional ingredient used in this paper to obtain the compactness of the domains is the behavior of the(More)
—We study the problem of finding unitary submatrices of the discrete Fourier transform matrix. This problem is related to a diverse set of questions on idempotents on Z N , tiling Z N , difference graphs and maximal cliques. Each of these is related to the problem of interpolating a discrete bandlimited signal using an orthogonal basis.
Let f be a continuous, increasing function of R into itself. Let k f (x, h) = f(x + h) − f(x) f(x) − f(x − h) , x ∈ R, h > 0. Observe that if g = af + b, a, b ∈ R, then k g = k f. One can show conversely that if k g = k f , then g = f up to a real affine transformation. The quantity k f is called the quasisymmetry quotient of f. A function is(More)