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| 2. |
- Di Rocco, Sandra, et al.
(författare)
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Algebraic C*-actions and the inverse kinematics of a general 6R manipulator
- 2010
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Ingår i: Applied Mathematics and Computation. - : Elsevier BV. - 0096-3003 .- 1873-5649. ; 216:9, s. 2512-2524
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Tidskriftsartikel (refereegranskat)abstract
- Let X be a smooth quadric of dimension 2m in P-C(2m+1) and let Y, Z subset of X be subvarieties both of dimension m which intersect transversely. In this paper we give an algorithm for computing the intersection points of Y boolean AND Z based on a homotopy method. The homotopy is constructed using a C*-action on X whose fixed points are isolated, which induces Bialynicki-Birula decompositions of X into locally closed invariant subsets. As an application we present a new solution to the inverse kinematics problem of a general six-revolute serial-link manipulator. (C) 2010 Elsevier Inc. All rights reserved.
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| 3. |
- Di Rocco, Sandra, et al.
(författare)
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Chern numbers of smooth varieties via homotopy continuation and intersection theory
- 2011
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Ingår i: Journal of symbolic computation. - : Elsevier BV. - 0747-7171 .- 1095-855X. ; 46:1, s. 23-33
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Tidskriftsartikel (refereegranskat)abstract
- Homotopy continuation provides a numerical tool for computing the equivalence of a smooth variety in an intersection product. Intersection theory provides a theoretical tool for relating the equivalence of a smooth variety in an intersection product to the degrees of the Chern classes of the variety. A combination of these tools leads to a numerical method for computing the degrees of Chern classes of smooth projective varieties in Pn. We illustrate the approach through several worked examples.
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| 4. |
- Di Rocco, Sandra, et al.
(författare)
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Computing Geometric Feature Sizes for Algebraic Manifolds
- 2023
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Ingår i: SIAM JOURNAL ON APPLIED ALGEBRA AND GEOMETRY. - : Society for Industrial & Applied Mathematics (SIAM). - 2470-6566. ; 7:4, s. 716-741
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Tidskriftsartikel (refereegranskat)abstract
- We introduce numerical algebraic geometry methods for computing lower bounds on the reach, local feature size, and weak feature size of the real part of an equidimensional and smooth algebraic variety using the variety's defining polynomials as input. For the weak feature size, we also show that nonquadratic complete intersections generically have finitely many geometric bottlenecks, and we describe how to compute the weak feature size directly rather than a lower bound in this case. In all other cases, we describe additional computations that can be used to determine feature size values rather than lower bounds.
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| 5. |
- Di Rocco, Sandra, et al.
(författare)
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Numerical polar calculus and cohomology of line bundles
- 2018
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Ingår i: Advances in Applied Mathematics. - : Academic Press. - 0196-8858 .- 1090-2074. ; 100, s. 148-162
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Tidskriftsartikel (refereegranskat)abstract
- Let L-1,..., L-s be line bundles on a smooth complex variety X subset of P-r and let D-1,..., D-s be divisors on X such that D-i represents L-i. We give a probabilistic algorithm for computing the degree of intersections of polar classes which are in turn used for computing the Euler characteristic of linear combinations of L-1,..., L-s. The input consists of generators for the homogeneous ideals I-X, I-Di subset of C[x(0),..., x(r).] defining X and D-i.
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| 6. |
- Di Rocco, Sandra, et al.
(författare)
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Sampling and Homology via Bottlenecks
- 2022
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Ingår i: Mathematics of Computation. - : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 91:338, s. 2969-2995
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we present an efficient algorithm to produce a provably dense sample of a smooth compact affine variety. The procedure is partly based on computing bottlenecks of the variety. Using geometric information such as the bottlenecks and the local reach we also provide bounds on the density of the sample needed in order to guarantee that the homology of the variety can be recovered from the sample. An implementation of the algorithm is provided together with numerical experiments and a computational comparison to the algorithm by Dufresne et al. [Sampling real algebraic varieties for topological data analysis, arXiv:1802.07716, 2018].
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| 7. |
- Di Rocco, Sandra, et al.
(författare)
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The Bottleneck Degree of Algebraic Varieties
- 2020
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Ingår i: SIAM Journal on Applied Algebra and Geometry. - : Society for Industrial & Applied Mathematics (SIAM). - 2470-6566. ; 4:1, s. 227-253
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Tidskriftsartikel (refereegranskat)abstract
- A bottleneck of a smooth algebraic variety X subset of C-n is a pair (x, y) of distinct points x, y is an element of X such that the Euclidean normal spaces at x and y contain the line spanned by x and y. The narrowness of bottlenecks is a fundamental complexity measure in the algebraic geometry of data. In this paper we study the number of bottlenecks of affine and projective varieties, which we call the bottleneck degree. The bottleneck degree is a measure of the complexity of computing all bottlenecks of an algebraic variety, using, for example, numerical homotopy methods. We show that the bottleneck degree is a function of classical invariants such as Chern classes and polar classes. We give the formula explicitly in low dimension and provide an algorithm to compute it in the general case.
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| 8. |
- Eklund, David, 1978-
(författare)
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Algebraic C*-actions and homotopy continuation
- 2008
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Let X be a smooth projective variety over C equipped with a C*-action whose fixed points are isolated. Let Y and Z be subvarieties of complementary dimentions in X which intersect properly. In this thesis we present an algorithm for computing the points of intersection between Y and Z based on homotopy continuation and the Bialynicki-Birula decompositions of X into locally closed invariant subsets. As an application we present a new solution to the inverse kinematic problem of a general six-revolute serial-link manipulator.
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| 9. |
- Eklund, David, 1978-
(författare)
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Topics in computation, numerical methods and algebraic geometry
- 2010
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis concerns computation and algebraic geometry. On the computational side we have focused on numerical homotopy methods. These procedures may be used to numerically solve systems of polynomial equations. The thesis contains four papers. In Paper I and Paper II we apply continuation techniques, as well as symbolic algorithms, to formulate methods to compute Chern classes of smooth algebraic varieties. More specifically, in Paper I we give an algorithm to compute the degrees of the Chern classes of smooth projective varieties and in Paper II we extend these ideas to cover also the degrees of intersections of Chern classes. In Paper III we formulate a numerical homotopy to compute the intersection of two complementary dimensional subvarieties of a smooth quadric hypersurface in projective space. If the two subvarieties intersect transversely, then the number of homotopy paths is optimal. As an application we give a new solution to the inverse kinematics problem of a six-revolute serial-link mechanism. Paper IV is a study of curves on certain special quartic surfaces in projective 3-space. The surfaces are invariant under the action of a finite group called the level (2,2) Heisenberg group. In the paper, we determine the Picard group of a very general member of this family of quartics. We have found that the general Heisenberg invariant quartic contains 320 smooth conics and we prove that in the very general case, this collection of conics generates the Picard group.
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| 10. |
- Eklund, Petter, 1989-, et al.
(författare)
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A Complete Design of a Rare Earth Metal-Free Permanent Magnet Generator
- 2014
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Ingår i: Machines. - Basel, Switzerland : Multidisciplinary Digital Publishing Institute. - 2075-1702.
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Tidskriftsartikel (refereegranskat)abstract
- The price of rare-earth metals used in neodymium-iron-boron (NdFeB) permanent magnets (PMs) has fluctuated greatly recently. Replacing the NdFeB PMs with more abundant ferrite PMs will avoid the cost insecurity and insecurity of supply. Ferrite PMs have lower performance than NdFeB PMs and for similar performance more PM material has to be used, requiring more support structure. Flux concentration is also necessary, for example, by a spoke-type rotor. In this paper the rotor of a 12 kW NdFeB PM generator was redesigned to use ferrite PMs, reusing the existing stator and experimental setup. Finite element simulations were used to calculate both electromagnetic and mechanical properties of the design. Focus was on mechanical design and feasibility of construction. The result was a design of a ferrite PM rotor to be used with the old stator with some small changes to the generator support structure. The new generator has the same output power at a slightly lower voltage level. It was concluded that it is possible to use the same stator with either a NdFeB PM rotor or a ferrite PM rotor. A ferrite PM generator might require a larger diameter than a NdFeB generator to generate the same voltage.
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