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Creators/Authors contains: "Bishop, Christopher J."

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  1. ; ( , International Mathematics Research Notices)
    We strengthen the classical approximation theorems of Weierstrass, Runge, and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function $f$ on a compact set $K$, the critical points of our approximants may be taken to lie in any given domain containing $K$, and all the critical values in any given neighborhood of the polynomially convex hull of $f(K)$. 
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    Free, publicly-accessible full text available April 29, 2025
  2. ( , Discrete & Computational Geometry)
    We show that any polygon P has an acute triangulation where every angle lies in the interval I=[30, 75] (degrees), except for triangles T that contain a vertex v of P where P has an interior angle theta_v < 30; then T is an isosceles triangle with angles \theta_v and 90 -\theta_v/2. 
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  3. ; ; ( , Mathematische Annalen)
    We show that any dynamics on any planar set S, discrete in some domain D, can be realized by the postcritical dynamics of a function holomorphic in D, up to a small perturbation. A key step in the proof, and a result of independent interest, is that any planar domain D can be equilaterally triangulated with triangles whose diameters tend to 0 at any prescribed rate near the boundary. When D is the whole plane, the dynamical result was proved in "Prescribing the Postsingular Dynamics of Meromorphic Functions", by Bishop and Lazebnik by a different method (QC folding). 
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  4. ( , Discrete & Computational Geometry)
    We show that any PSLG has an acute conforming triangulation with an upper angle bound that is strictly less than 90 degrees and that depends only on the minimal angle occurring in the PSLG. In fact, all angles are inside the interval I_0= [theta_0, 90 -\theta_0/2] for some fixed theta_0>0 independent of the PSLG except for triangles T containing a vertex v where the PSLG has an interior angle theta_v < \theta_0; then T is an isosceles triangle with angles in I_v = [theta_v, 90 -\theta_v/2]. 
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  5. ( , Advances in mathematics)
    This gives an improvment of Peter Jones's traveling salesman theorem that holds for Jordan curves, but not for general sets. His theorem implies that the length of a Jordan arc is bounded by (1+delta)diameter + C(delta) beta-sum, and this paper shows this can be replaced by chord + O(beta-sum), where O(diameter) is replaced by the distance between the endpoints of the curve. This is true in all finite dimensions (with a dimension dependent constant). A corollary of our self-contained argument proves the ususal TST in all dimensions (a result of Okikiolu). An appendix proves a folklore result that several different formulation of Jones's theorem are all equivalent. 
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    Accepted Manuscript Full Text Available
  6. ( , Revista Matemática Iberoamericana)
    This is a companion to the paper "Weil-Petersson curves, conformal energies, beta-numbers, and minimal surfaces". That paper gives various new geometric characterizations of Weil-Petersson in the plane that can be extended to curves in all finite dimensional Euclidean spaces. This paper deals with the 2-dimensional case, giving new proofs of some known characterizations, and giving new results for the conformal weldings of Weil-Petersson curves and a geometric characterization of these curves in terms of Peter Jones's beta-numbers. 
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  7. ( , Publicacions Matemàtiques)
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  8. ; ( , Journal d'Analyse Mathématique)
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  9. ; ( , Communications in Analysis and Geometry)
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  10. ; ( , Mathematische Annalen)
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