25,294 research outputs found
Complete Subdivision Algorithms, II: Isotopic Meshing of Singular Algebraic Curves
Given a real valued function f(X,Y), a box region B_0 in R^2 and a positive
epsilon, we want to compute an epsilon-isotopic polygonal approximation to the
restriction of the curve S=f^{-1}(0)={p in R^2: f(p)=0} to B_0. We focus on
subdivision algorithms because of their adaptive complexity and ease of
implementation. Plantinga and Vegter gave a numerical subdivision algorithm
that is exact when the curve S is bounded and non-singular. They used a
computational model that relied only on function evaluation and interval
arithmetic. We generalize their algorithm to any bounded (but possibly
non-simply connected) region that does not contain singularities of S. With
this generalization as a subroutine, we provide a method to detect isolated
algebraic singularities and their branching degree. This appears to be the
first complete purely numerical method to compute isotopic approximations of
algebraic curves with isolated singularities
Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity
Previous work on applications of Abstract Differential Geometry (ADG) to
discrete Lorentzian quantum gravity is brought to its categorical climax by
organizing the curved finitary spacetime sheaves of quantum causal sets
involved therein, on which a finitary (:locally finite), singularity-free,
background manifold independent and geometrically prequantized version of the
gravitational vacuum Einstein field equations were seen to hold, into a topos
structure. This topos is seen to be a finitary instance of both an elementary
and a Grothendieck topos, generalizing in a differential geometric setting, as
befits ADG, Sorkin's finitary substitutes of continuous spacetime topologies.
The paper closes with a thorough discussion of four future routes we could take
in order to further develop our topos-theoretic perspective on ADG-gravity
along certain categorical trends in current quantum gravity research.Comment: 49 pages, latest updated version (errata corrected, references
polished) Submitted to the International Journal of Theoretical Physic
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