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Algorithms for recognizing knots and 3manifolds
 Chaos, Solitons and Fractals
, 1998
"... Algorithms are of interest to geometric topologists for two reasons. First, they have bearing on the decidability of a problem. Certain topological questions, such as finding a classification of four dimensional manifolds, admit no solution. ..."
Abstract

Cited by 6 (3 self)
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Algorithms are of interest to geometric topologists for two reasons. First, they have bearing on the decidability of a problem. Certain topological questions, such as finding a classification of four dimensional manifolds, admit no solution.
A local deterministic model of quantum spin measurement
 Proc.R.Soc.Lond. A
, 1995
"... The conventional view, that Einstein was wrong to believe that quantum physics is local and deterministic, is challenged. A parametrized model, " Q", for the state vector evolution of spin1/2 particles during measurement is developed. Q draws on recent work on socalled "riddled bas ..."
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Cited by 1 (1 self)
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The conventional view, that Einstein was wrong to believe that quantum physics is local and deterministic, is challenged. A parametrized model, " Q", for the state vector evolution of spin1/2 particles during measurement is developed. Q draws on recent work on socalled "riddled basins " in dynamical systems theory, and is local, deterministic, nonlinear and time asymmetric. Moreover, the evolution of the state vector to one of two chaotic attractors (taken to represent observed spin states) is effectively uncomputable. Motivation for considering this model arises from speculations about the (time asymmetric and uncomputable) nature of quantum gravity, and the (nonlinear) role of gravity in quantum state vector reduction. Although the evolution of Q’s state vector cannot be determined by a numerical algorithm, the probability that initial states in some given region of phase space will evolve to one of these attractors, is itself computable. These probabilities can be made to correspond to observed quantum spin probabilities. In an ensemble sense, the evolution of the state vector to an attractor can be described in by a diffusive random walk process, suggesting that deterministic dynamics may underlie recent attempts to model state vector evolution by stochastic equations. Bell’s theorem and a version of the BellKochenSpecker paradox, as illustrated by Penrose’s "magic dodecahedra", are discussed using quantum entanglement Q as a model of quantum spin measurement. It is shown that in both cases, proving an inconsistency with locality demands the existence of definite truth values to certain counterfactual propositions. In Q, these deterministic propositions are physically uncomputable, and no nonalgorithmic mathematical solution is either known or suspected. Adapting the mathematical formalist approach, the nonexistence of definite truth values to such counterfactual propositions is posited. No inconsistency with experiment is found. As a result, it is claimed that constrained by Bell’s inequality, locality and determinism notwithstanding. Q is not
A Signal Subspace Approach to Multiple Emitter Location and Spectral Estimation
 J. London Math. Soc
, 1980
"... ..."
Algorithmic complexity of finding crosscycles in flag complexes
"... A crosscycle in a flag simplicial complex K is an induced subcomplex that is isomorphic to the boundary of a crosspolytope and that contains a maximal face of K. A crosscycle is the most efficient way to define a nonzero class in the homology of K. For an independence complex of a graph G, a cro ..."
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A crosscycle in a flag simplicial complex K is an induced subcomplex that is isomorphic to the boundary of a crosspolytope and that contains a maximal face of K. A crosscycle is the most efficient way to define a nonzero class in the homology of K. For an independence complex of a graph G, a crosscycle is equivalent to an induced matching containing a maximal independent set of G. We study the complexity of finding crosscycles in independence complexes. We show that in general this problem is NPcomplete when input is a graph whose independence complex we consider. This allows us to study special cases. Unfortunately, not a lot has been done in this direction besides the recent polynomial time algorithm for forests [16]. In this contribution, we focus on the more general class of chordal graphs. This is a natural choice for the problem as the independence complexes of chordal graphs are quite well understood, namely they are wedges of spheres up to homotopy, and any wedge of spheres can be realized as the independence complex of a chordal graph, up to homotopy. As our main result, we present a polynomial time algorithm for detecting a crosscycle in the independence complex of a chordal graph. Our algorithm is based on the geometric intersection representation of chordal graphs and has an efficient implementation. We further prove that for chordal graphs crosscycles detect all of homology of the independence complex. As a corollary, we obtain polynomial time algorithms for such topological properties as contractibility or simpleconnectedness of independence complexes of chordal graphs. These problems are undecidable for general independence complexes. We conclude with a discussion of some related cases and open problems. 1
ALGORITHMIC PROBLEMS IN VARIETIES
, 1994
"... Mankind always sets itself only such problems as it can solve. Karl Marx, The Introduction to "A Critique of Political Economy". ..."
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Mankind always sets itself only such problems as it can solve. Karl Marx, The Introduction to &quot;A Critique of Political Economy&quot;.
Clockwork or Turing U/universe?  Remarks on Causal Determinism and Computability
"... The relevance of the Turing universe as a model for complex physical situations (that is, those showing both computable and incomputable aspects) is discussed. Some wellknown arguments concerning the nature of scientific reality are related to this theoretical context. ..."
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The relevance of the Turing universe as a model for complex physical situations (that is, those showing both computable and incomputable aspects) is discussed. Some wellknown arguments concerning the nature of scientific reality are related to this theoretical context.