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Abstract Designing an algorithm with a singly exponential complexity for computing semialgebraic triangulations of a given semialgebraic set has been a holy grail in algorithmic semialgebraic geometry. More precisely, given a description of a semialgebraic set$$S \subset \mathbb {R}^k$$by a first-order quantifier-free formula in the language of the reals, the goal is to output a simplicial complex$$\Delta $$, whose geometric realization,$$|\Delta |$$, is semialgebraically homeomorphic toS. In this paper, we consider a weaker version of this question. We prove that for any$$\ell \geq 0$$, there exists an algorithm which takes as input a description of a semialgebraic subset$$S \subset \mathbb {R}^k$$given by a quantifier-free first-order formula$$\phi $$in the language of the reals and produces as output a simplicial complex$$\Delta $$, whose geometric realization,$$|\Delta |$$is$$\ell $$-equivalent toS. The complexity of our algorithm is bounded by$$(sd)^{k^{O(\ell )}}$$, wheresis the number of polynomials appearing in the formula$$\phi $$, andda bound on their degrees. For fixed$$\ell $$, this bound issingly exponentialink. In particular, since$$\ell $$-equivalence implies that thehomotopy groupsup to dimension$$\ell $$of$$|\Delta |$$are isomorphic to those ofS, we obtain a reduction (having singly exponential complexity) of the problem of computing the first$$\ell $$homotopy groups ofSto the combinatorial problem of computing the first$$\ell $$homotopy groups of a finite simplicial complex of size bounded by$$(sd)^{k^{O(\ell )}}$$.
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A spatial mutation model with increasing mutation rates
Abstract We consider a spatial model of cancer in which cells are points on thed-dimensional torus$$\mathcal{T}=[0,L]^d$$, and each cell with$$k-1$$mutations acquires akth mutation at rate$$\mu_k$$. We assume that the mutation rates$$\mu_k$$are increasing, and we find the asymptotic waiting time for the first cell to acquirekmutations as the torus volume tends to infinity. This paper generalizes results on waiting for$$k\geq 3$$mutations in Fooet al.(2020), which considered the case in which all of the mutation rates$$\mu_k$$are the same. In addition, we find the limiting distribution of the spatial distances between mutations for certain values of the mutation rates.
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- Award ID(s):
- 1645643
- PAR ID:
- 10548583
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Journal of Applied Probability
- Volume:
- 60
- Issue:
- 4
- ISSN:
- 0021-9002
- Page Range / eLocation ID:
- 1157 to 1180
- Subject(s) / Keyword(s):
- Mutation cancer spatial population model
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jpr.2022.120
