The Experts below are selected from a list of 303 Experts worldwide ranked by ideXlab platform
Christoph Walker - One of the best experts on this subject based on the ideXlab platform.
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Shape derivative of the Dirichlet energy for a transmission problem
Archive for Rational Mechanics and Analysis, 2020Co-Authors: Philippe Laurençot, Christoph WalkerAbstract:For a transmission problem in a truncated two-dimensional cylinder located beneath the graph of a function u, the shape derivative of the Dirichlet energy (with respect to u) is shown to be well-defined and is computed. The main difficulties in this context arise from the weak regularity of the domain and the possible non-Empty Intersection of the graph of u and the transmission interface. The result is applied to establish the existence of a solution to a free boundary transmission problem for an electrostatic actuator.
Xavier Goaoc - One of the best experts on this subject based on the ideXlab platform.
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Helly numbers of acyclic families
Advances in Mathematics, 2014Co-Authors: Éric Colin De Verdière, Grégory Ginot, Xavier GoaocAbstract:The Helly number of a family of sets with Empty Intersection is the size of its largest inclusion-wise minimal sub-family with Empty Intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Γ. Assume that for every sub-family G⊆F the Intersection of the elements of G has at most r connected components, each of which is a Q-homology cell. We show that the Helly number of F is at most r(dΓ+1), where dΓ is the smallest integer j such that every open set of Γ has trivial Q-homology in dimension j and higher. (In particular dRd=d.) This bound is best possible. We also prove a stronger theorem where the Intersection of small sub-families may have more than r connected components, each possibly with nontrivial homology in low dimension. As an application, we obtain several explicit bounds on Helly numbers in geometric transversal theory for which only ad hoc geometric proofs were previously known; in certain cases, the bound we obtain is better than what was previously known. In fact, our proof bounds the Leray number of the nerves of the families under consideration and thus also yields, under similar assumptions, a fractional Helly theorem, a (p,q)-theorem and the existence of small weak ϵ-nets.
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Helly numbers of acyclic families
Advances in Mathematics, 2014Co-Authors: Éric Colin De Verdière, Grégory Ginot, Xavier GoaocAbstract:International audienceThe Helly number of a family of sets with Empty Intersection is the size of its largest inclusion-wise minimal sub-family with Empty Intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Γ. Assume that for every sub-family G⊆F the Intersection of the elements of G has at most r connected components, each of which is a Q-homology cell. We show that the Helly number of F is at most r(dΓ+1), where dΓ is the smallest integer j such that every open set of Γ has trivial Q-homology in dimension j and higher. (In particular dRd=d.) This bound is best possible. We also prove a stronger theorem where the Intersection of small sub-families may have more than r connected components, each possibly with nontrivial homology in low dimension. As an application, we obtain several explicit bounds on Helly numbers in geometric transversal theory for which only ad hoc geometric proofs were previously known; in certain cases, the bound we obtain is better than what was previously known. In fact, our proof bounds the Leray number of the nerves of the families under consideration and thus also yields, under similar assumptions, a fractional Helly theorem, a (p,q)-theorem and the existence of small weak ϵ-nets
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Transversal Helly numbers, pinning theorems and projection of simplicial complexes
2011Co-Authors: Xavier GoaocAbstract:The efficient resolution of various problems in computational geometry, for instance visibility computation or shape approximation, raises new questions in line geometry, a classical area going back to the mid-19th century. This thesis fits into this theme, and studies Helly numbers of certain sets of lines, an index related to certain basis theorems arising in computational geometry and combinatorial optimization. Formally, the Helly number of a family of sets with Empty Intersection is the size of its largest inclusion-wise minimal sub-family with Empty Intersection. For $d\ge 2$ let $h_d$ denote the least integer such that for any family $\{B_1, \ldots, B_n\}$ of pairwise disjoint balls of equal radius in $R^d$, the Helly number of $\{T(B_1), \ldots, T(B_n)\}$ is at most $h_d$, where $T(B_i)$ denotes the set of lines intersecting $B_i$. In 1957, Ludwig Danzer showed that $h_2$ equals $5$ and conjectured that $h_d$ is finite for all $d \ge 2$ and increases with $d$. We establish that $h_d$ is at least $2d-1$ and at most $4d-1$ for any $d \ge 2$, proving the first conjecture and providing evidence in support of the second one. To study Danzer's conjectures, we introduce the pinning number, a local analogue of the Helly number that is related to grasping questions studied in robotics. We further show that pinning numbers can be bounded for sufficiently generic families of polyhedra or ovaloids in $R^3$, two situations where Helly numbers can be arbitrarily large. A theorem of Tverberg asserts that when $\{B_1, \ldots, B_n\}$ are disjoint translates of a convex figure in the plane, the Helly number of $\{T(B_1), \ldots, T(B_n)\}$ is at most $5$. Although quite different, both our and Tverberg's proofs use, in some way, that the Intersection of at least two $T(B_i)$'s has a bounded number of connected components, each contractible. Using considerations on homology of projection of simplicial complexes and posets, we unify the two proofs and show that such topological condition suffice to ensure explicit bounds on Helly numbers.
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Helly numbers of acyclic families
arXiv: Combinatorics, 2011Co-Authors: Éric Colin De Verdière, Grégory Ginot, Xavier GoaocAbstract:The Helly number of a family of sets with Empty Intersection is the size of its largest inclusion-wise minimal sub-family with Empty Intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Gamma. Assume that for every sub-family G of F the Intersection of the elements of G has at most r connected components, each of which is a Q-homology cell. We show that the Helly number of F is at most r(d_Gamma+1), where d_Gamma is the smallest integer j such that every open set of Gamma has trivial Q-homology in dimension j and higher. (In particular d_{R^d} = d). This bound is best possible. We prove, in fact, a stronger theorem where small sub-families may have more than r connected components, each possibly with nontrivial homology in low dimension. As an application, we obtain several explicit bounds on Helly numbers in geometric transversal theory for which only ad hoc geometric proofs were previously known; in certain cases, the bound we obtain is better than what was previously known.
Philippe Laurençot - One of the best experts on this subject based on the ideXlab platform.
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Shape derivative of the Dirichlet energy for a transmission problem
Archive for Rational Mechanics and Analysis, 2020Co-Authors: Philippe Laurençot, Christoph WalkerAbstract:For a transmission problem in a truncated two-dimensional cylinder located beneath the graph of a function u, the shape derivative of the Dirichlet energy (with respect to u) is shown to be well-defined and is computed. The main difficulties in this context arise from the weak regularity of the domain and the possible non-Empty Intersection of the graph of u and the transmission interface. The result is applied to establish the existence of a solution to a free boundary transmission problem for an electrostatic actuator.
Ubaid M. Al-saggaf - One of the best experts on this subject based on the ideXlab platform.
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Multiagent Rendezvous With Shortest Distance to Convex Regions With Empty Intersection: Algorithms and Experiments
IEEE transactions on cybernetics, 2018Co-Authors: Peng Lin, Wei Ren, Hao Wang, Ubaid M. Al-saggafAbstract:This paper presents both algorithms and experimental results to solve a distributed rendezvous problem with shortest distance to convex regions. In a multiagent network, each agent is assigned to a certain convex region and has information about only its own region. All these regions might not have an Intersection. Through local interaction with their neighbors, multiple agents collectively rendezvous at an optimal location that is a priori unknown to each agent and has the shortest total squared distance to these regions. First, a distributed time-varying algorithm is introduced, where a corresponding condition is given to guarantee that all agents rendezvous at the optimal location asymptotically for bounded convex regions. Then a distributed tracking algorithm combined with a distributed estimation algorithm is proposed. It is first shown that for general possibly unbounded convex regions, all agents rendezvous in finite time and then collectively slide to the optimal location asymptotically. Then it is shown that for convex regions with certain projection compressibility, all agents collectively rendezvous at the optimal location in finite time, even when the regions are time varying. The algorithms are experimentally implemented on multiple ground robots to illustrate the obtained theoretical results.
Éric Colin De Verdière - One of the best experts on this subject based on the ideXlab platform.
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Helly numbers of acyclic families
Advances in Mathematics, 2014Co-Authors: Éric Colin De Verdière, Grégory Ginot, Xavier GoaocAbstract:The Helly number of a family of sets with Empty Intersection is the size of its largest inclusion-wise minimal sub-family with Empty Intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Γ. Assume that for every sub-family G⊆F the Intersection of the elements of G has at most r connected components, each of which is a Q-homology cell. We show that the Helly number of F is at most r(dΓ+1), where dΓ is the smallest integer j such that every open set of Γ has trivial Q-homology in dimension j and higher. (In particular dRd=d.) This bound is best possible. We also prove a stronger theorem where the Intersection of small sub-families may have more than r connected components, each possibly with nontrivial homology in low dimension. As an application, we obtain several explicit bounds on Helly numbers in geometric transversal theory for which only ad hoc geometric proofs were previously known; in certain cases, the bound we obtain is better than what was previously known. In fact, our proof bounds the Leray number of the nerves of the families under consideration and thus also yields, under similar assumptions, a fractional Helly theorem, a (p,q)-theorem and the existence of small weak ϵ-nets.
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Helly numbers of acyclic families
Advances in Mathematics, 2014Co-Authors: Éric Colin De Verdière, Grégory Ginot, Xavier GoaocAbstract:International audienceThe Helly number of a family of sets with Empty Intersection is the size of its largest inclusion-wise minimal sub-family with Empty Intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Γ. Assume that for every sub-family G⊆F the Intersection of the elements of G has at most r connected components, each of which is a Q-homology cell. We show that the Helly number of F is at most r(dΓ+1), where dΓ is the smallest integer j such that every open set of Γ has trivial Q-homology in dimension j and higher. (In particular dRd=d.) This bound is best possible. We also prove a stronger theorem where the Intersection of small sub-families may have more than r connected components, each possibly with nontrivial homology in low dimension. As an application, we obtain several explicit bounds on Helly numbers in geometric transversal theory for which only ad hoc geometric proofs were previously known; in certain cases, the bound we obtain is better than what was previously known. In fact, our proof bounds the Leray number of the nerves of the families under consideration and thus also yields, under similar assumptions, a fractional Helly theorem, a (p,q)-theorem and the existence of small weak ϵ-nets
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Helly numbers of acyclic families
arXiv: Combinatorics, 2011Co-Authors: Éric Colin De Verdière, Grégory Ginot, Xavier GoaocAbstract:The Helly number of a family of sets with Empty Intersection is the size of its largest inclusion-wise minimal sub-family with Empty Intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Gamma. Assume that for every sub-family G of F the Intersection of the elements of G has at most r connected components, each of which is a Q-homology cell. We show that the Helly number of F is at most r(d_Gamma+1), where d_Gamma is the smallest integer j such that every open set of Gamma has trivial Q-homology in dimension j and higher. (In particular d_{R^d} = d). This bound is best possible. We prove, in fact, a stronger theorem where small sub-families may have more than r connected components, each possibly with nontrivial homology in low dimension. As an application, we obtain several explicit bounds on Helly numbers in geometric transversal theory for which only ad hoc geometric proofs were previously known; in certain cases, the bound we obtain is better than what was previously known.