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In this paper we consider the problem of bounding the Betti numbers , b i (S), of a semi-algebraic set S ⊂ R k defined by polynomial inequalities b i (S) ≤ 1 2 + (k − s) + 1 2 · min{s+1,k−i} X j=0 2 j " s + 1 j " " k j − 1 " ≤ 3 2 · " 6ek s « s + k. This improves the bound of k O(s) proved by Barvinok in [2]. This improvement is made possible by a new… (More)

We describe an algorithm for computing the zero-th and the first Betti numbers of the union of n simply connected compact semi-algebraic sets in R k , where each such set is defined by a constant number of polynomials of constant degrees. The complexity of the algorithm is O(n 3). We also describe an implementation of this algorithm in the particular case… (More)

- Michael Kettner
- 2006

In this paper, we present a new algorithm for computing in a very efficient way the real intersection of three quadric surfaces. Our approach is based on the cylindrical decomposition ([4]) and the TOP algorithm ([10]) for analyzing the topology of a planar curve. We preform a symbolic pre-processing that allows us later to execute all numerical… (More)

We prove a nearly optimal bound on the number of stable homo-topy types occurring in a k-parameter semi-algebraic family of sets in R , each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in. More precisely, we prove the following. Let R be a real closed field and let k be the projection on the last k… (More)

- Michael Kettner, Saugata Basu, Laureano González-Vega, John Etnyre, Mohammad Ghomi, Eric Carlen +4 others
- 2007

For my Mum and the memory of my Dad iii ACKNOWLEDGEMENTS The writing of this thesis has been one of the most significant academic challenges I have had to face. Without the support, patience and guidance of the following people and institutes, this study would not have been completed. It is to them that I owe my deepest gratitude. In the first place I would… (More)

- MICHAEL KETTNER
- 2007

In this paper, we present a new algorithm for computing in a very efficient way the real intersection of three cubic surfaces. Our approach is based on the cylindrical decomposition ([8]) and the TOP algorithm ([10]) for analyzing the topology of a planar curve. We perform a symbolic pre-processing that allows us later to execute all numerical computations… (More)

BACKGROUND
An ambulance equipped with a computed tomography (CT) scanner, point-of-care laboratory, and telemedicine capabilities (Mobile Stroke Unit [MSU]) has been shown to enable delivery of thrombolysis to stroke patients at the emergency site, thereby significantly decreasing time to treatment. However, the MSU frequently assesses patients with… (More)

Degenerative alterations of the spine occur in an individual-specific manner with increasing age. This is not only dependent on external factors, such as hard physical labor over many years but can also be genetically influenced as demonstrated in recent studies. The spinal cord is well-protected within the spinal canal but can be impaired by degenerative… (More)

In this thesis, we consider semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials. Semi-algebraic sets of $R^k$ are defined as the smallest family of sets in $R^k$ that contains the algebraic sets as well as the sets defined by polynomial inequalities, and which is also closed under the boolean operations (complementation, finite… (More)

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