The Experts below are selected from a list of 147 Experts worldwide ranked by ideXlab platform
S. V. Borodachov - One of the best experts on this subject based on the ideXlab platform.
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Low Complexity Methods For Discretizing Manifolds Via Riesz Energy Minimization
Foundations of Computational Mathematics, 2014Co-Authors: S. V. Borodachov, D. P. Hardin, E. B. SaffAbstract:Let $$A$$ A be a compact $$d$$ d -Rectifiable Set embedded in Euclidean space $${\mathbb R}^p, d\le p$$ R p , d ≤ p . For a given continuous distribution $$\sigma (x)$$ σ ( x ) with respect to a $$d$$ d -dimensional Hausdorff measure on $$A$$ A , our earlier results provided a method for generating $$N$$ N -point configurations on $$A$$ A that have an asymptotic distribution $$\sigma (x)$$ σ ( x ) as $$N\rightarrow \infty $$ N → ∞ ; moreover, such configurations are “quasi-uniform” in the sense that the ratio of the covering radius to the separation distance is bounded independently of $$N$$ N . The method is based upon minimizing the energy of $$N$$ N particles constrained to $$A$$ A interacting via a weighted power-law potential $$w(x,y)|x-y|^{-s}$$ w ( x , y ) | x - y | - s , where $$s>d$$ s > d is a fixed parameter and $$w(x,y)=\left( \sigma (x)\sigma (y)\right) ^{-({s}/{2d})}$$ w ( x , y ) = σ ( x ) σ ( y ) - ( s / 2 d ) . Here we show that one can generate points on $$A$$ A with the aforementioned properties keeping in the energy sums only those pairs of points that are located at a distance of at most $$r_N=C_N N^{-1/d}$$ r N = C N N - 1 / d from each other, with $$C_N$$ C N being a positive sequence tending to infinity arbitrarily slowly. To do this, we minimize the energy with respect to a varying truncated weight $$v_N(x,y)=\Phi (|x-y|/r_N)\cdot w(x,y)$$ v N ( x , y ) = Φ ( | x - y | / r N ) · w ( x , y ) , where $$\Phi :(0,\infty )\rightarrow [0,\infty )$$ Φ : ( 0 , ∞ ) → [ 0 , ∞ ) is a bounded function with $$\Phi (t)=0, t\ge 1$$ Φ ( t ) = 0 , t ≥ 1 , and $$\lim _{t\rightarrow 0^+}\Phi (t)=1$$ lim t → 0 + Φ ( t ) = 1 . Under appropriate assumptions, this reduces the complexity of generating $$N$$ N -point “low energy” discretizations to order $$N C_N^d$$ N C N d computations.
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Asymptotics for discrete weighted minimal Riesz energy problems on Rectifiable Sets
Transactions of the American Mathematical Society, 2008Co-Authors: S. V. Borodachov, Douglas P. Hardin, Edward B. SaffAbstract:Given a closed d-Rectifiable Set A embedded in Euclidean space, we investigate minimal weighted Riesz energy points on A; that is, N points constrained to A and interacting via the weighted power law potential V = w(x,y)|x-y| -s ,where s > 0 is a fixed parameter and w is an admissible weight. (In the unweighted case (w ≡ 1) such points for N fixed tend to the solution of the best-packing problem on A as the parameter s →∞.) Our main results concern the asymptotic behavior as N → ∞ of the minimal energies as well as the corresponding equilibrium configurations. Given a distribution ρ(x) with respect to d-dimensional Hausdorff measure on A, our results provide a method for generating N-point configurations on A that are "well-separated" and have asymptotic distribution p(x) as N → ∞.
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Asymptotics for Discrete Weighted Minimal Riesz Energy Problems on Rectifiable Sets
arXiv: Mathematical Physics, 2006Co-Authors: S. V. Borodachov, Douglas P. Hardin, Edward B. SaffAbstract:Given a compact $d$-Rectifiable Set $A$ embedded in Euclidean space and a distribution $\rho(x)$ with respect to $d$-dimensional Hausdorff measure on $A$, we address the following question: how can one generate optimal configurations of $N$ points on $A$ that are "well-separated" and have asymptotic distribution $\rho (x)$ as $N\to \infty$? For this purpose we investigate minimal weighted Riesz energy points, that is, points interacting via the weighted power law potential $V=w(x,y)|x-y|^{-s}$, where $s>0$ is a fixed parameter and $w$ is suitably chosen. In the unweighted case ($w\equiv 1$) such points for $N$ fixed tend to the solution of the best-packing problem on $A$ as the parameter $s\to \infty$.
Tristan Riviere - One of the best experts on this subject based on the ideXlab platform.
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connecting rational homotopy type singularities
Acta Mathematica, 2008Co-Authors: Robert Hardt, Tristan RiviereAbstract:Let N be a compact simply connected smooth Riemannian manifold and, for p ∈ {2,3,...}, W 1,p (R p+1, N) be the Sobolev space of measurable maps from R p+1 into N whose gradients are in L p . The restriction of u to almost every p-dimensional sphere S in R p+1 is in W 1,p (S, N) and defines an homotopy class in π p (N) (White 1988). Evaluating a fixed element z of Hom(π p (N), R) on this homotopy class thus gives a real number Φ z,u (S). The main result of the paper is that any W 1,p -weakly convergent limit u of a sequence of smooth maps in C ∞(R p+1, N), Φ z,u has a Rectifiable Poincare dual \( {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} \). Here Γ is a a countable union of C 1 curves in R p+1 with Hausdorff \( {\user1{\mathcal{H}}}^{1} \)-measurable orientation \( {\overrightarrow{\Gamma }} :\Gamma \to S^{p} \) and density function θ: Γ→R. The intersection number between \( {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} \) and S evaluates Φ z,u (S), for almost every p-sphere S. Moreover, we exhibit a non-negative integer n z , depending only on homotopy operation z, such that \( {\int_\Gamma {{\left| \theta \right|}^{{p \mathord{\left/ {\vphantom {p {{\left( {p + n_{z} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {p + n_{z} } \right)}}}} d{\user1{\mathcal{H}}}^{1} < \infty } } \) even though the mass \( {\int_\Gamma {{\left| \theta \right|}d{\user1{\mathcal{H}}}^{1} } } \) may be infinite. We also provide cases of N, p and z for which this rational power p/(p + n z ) is optimal. The construction of this Poincare dual is based on 1-dimensional “bubbling” described by the notion of “scans” which was introduced in Hardt and Riviere (2003). We also describe how to generalize these results to R m for any m ⩾ p + 1, in which case the bubbling is described by an (m–p)-Rectifiable Set with orientation and density function determined by restrictions of the mappings to almost every oriented Euclidean p-sphere.
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Connecting rational homotopy type singularities
Acta Mathematica, 2008Co-Authors: Robert Hardt, Tristan RiviereAbstract:Let N be a compact simply connected smooth Riemannian manifold and, for p ∈ {2,3,...}, W ^1, p ( R ^ p +1, N ) be the Sobolev space of measurable maps from R ^ p +1 into N whose gradients are in L ^ p . The restriction of u to almost every p -dimensional sphere S in R ^ p +1 is in W ^1, p ( S , N ) and defines an homotopy class in π_ p ( N ) (White 1988). Evaluating a fixed element z of Hom(π_ p ( N ), R ) on this homotopy class thus gives a real number Φ_ z , u ( S ). The main result of the paper is that any W ^1, p -weakly convergent limit u of a sequence of smooth maps in C ^∞( R ^ p +1, N ), Φ_ z , u has a Rectifiable Poincaré dual $ {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} $ . Here Γ is a a countable union of C ^1 curves in R ^ p +1 with Hausdorff $ {\user1{\mathcal{H}}}^{1} $ -measurable orientation $ {\overrightarrow{\Gamma }} :\Gamma \to S^{p} $ and density function θ : Γ→ R . The intersection number between $ {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} $ and S evaluates Φ_ z , u ( S ), for almost every p -sphere S . Moreover, we exhibit a non-negative integer n _ z , depending only on homotopy operation z , such that $ {\int_\Gamma {{\left| \theta \right|}^{{p \mathord{\left/ {\vphantom {p {{\left( {p + n_{z} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {p + n_{z} } \right)}}}} d{\user1{\mathcal{H}}}^{1} < \infty } } $ even though the mass $ {\int_\Gamma {{\left| \theta \right|}d{\user1{\mathcal{H}}}^{1} } } $ may be infinite. We also provide cases of N , p and z for which this rational power p /( p + n _ z ) is optimal. The construction of this Poincaré dual is based on 1-dimensional “bubbling” described by the notion of “scans” which was introduced in Hardt and Rivière (2003). We also describe how to generalize these results to R ^ m for any m ⩾ p + 1, in which case the bubbling is described by an ( m – p )-Rectifiable Set with orientation and density function determined by restrictions of the mappings to almost every oriented Euclidean p -sphere.
Edward B. Saff - One of the best experts on this subject based on the ideXlab platform.
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Best-Packing on Compact Sets
Springer Monographs in Mathematics, 2019Co-Authors: Sergiy V. Borodachov, Douglas P. Hardin, Edward B. SaffAbstract:In this chapter we study the behavior of the leading term as N gets large of the N-point best-packing distance $$\begin{aligned}\delta _N(A)=\sup \limits _{\omega _N\subSet A}\min \limits _{x, y\in \omega _N\atop x\ne y} \left| x-y\right| \end{aligned}$$ (defined earlier in Chapter 3) on a compact \((\mathcal H_d, d)\)-Rectifiable Set A in \(\mathbb R^p\) as well as the weak* limit distribution of point configurations \(\omega _N\) that attain the supremum in (13.0.1).
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Asymptotics for discrete weighted minimal Riesz energy problems on Rectifiable Sets
Transactions of the American Mathematical Society, 2008Co-Authors: S. V. Borodachov, Douglas P. Hardin, Edward B. SaffAbstract:Given a closed d-Rectifiable Set A embedded in Euclidean space, we investigate minimal weighted Riesz energy points on A; that is, N points constrained to A and interacting via the weighted power law potential V = w(x,y)|x-y| -s ,where s > 0 is a fixed parameter and w is an admissible weight. (In the unweighted case (w ≡ 1) such points for N fixed tend to the solution of the best-packing problem on A as the parameter s →∞.) Our main results concern the asymptotic behavior as N → ∞ of the minimal energies as well as the corresponding equilibrium configurations. Given a distribution ρ(x) with respect to d-dimensional Hausdorff measure on A, our results provide a method for generating N-point configurations on A that are "well-separated" and have asymptotic distribution p(x) as N → ∞.
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Asymptotics for Discrete Weighted Minimal Riesz Energy Problems on Rectifiable Sets
arXiv: Mathematical Physics, 2006Co-Authors: S. V. Borodachov, Douglas P. Hardin, Edward B. SaffAbstract:Given a compact $d$-Rectifiable Set $A$ embedded in Euclidean space and a distribution $\rho(x)$ with respect to $d$-dimensional Hausdorff measure on $A$, we address the following question: how can one generate optimal configurations of $N$ points on $A$ that are "well-separated" and have asymptotic distribution $\rho (x)$ as $N\to \infty$? For this purpose we investigate minimal weighted Riesz energy points, that is, points interacting via the weighted power law potential $V=w(x,y)|x-y|^{-s}$, where $s>0$ is a fixed parameter and $w$ is suitably chosen. In the unweighted case ($w\equiv 1$) such points for $N$ fixed tend to the solution of the best-packing problem on $A$ as the parameter $s\to \infty$.
E. B. Saff - One of the best experts on this subject based on the ideXlab platform.
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Low Complexity Methods For Discretizing Manifolds Via Riesz Energy Minimization
Foundations of Computational Mathematics, 2014Co-Authors: S. V. Borodachov, D. P. Hardin, E. B. SaffAbstract:Let $$A$$ A be a compact $$d$$ d -Rectifiable Set embedded in Euclidean space $${\mathbb R}^p, d\le p$$ R p , d ≤ p . For a given continuous distribution $$\sigma (x)$$ σ ( x ) with respect to a $$d$$ d -dimensional Hausdorff measure on $$A$$ A , our earlier results provided a method for generating $$N$$ N -point configurations on $$A$$ A that have an asymptotic distribution $$\sigma (x)$$ σ ( x ) as $$N\rightarrow \infty $$ N → ∞ ; moreover, such configurations are “quasi-uniform” in the sense that the ratio of the covering radius to the separation distance is bounded independently of $$N$$ N . The method is based upon minimizing the energy of $$N$$ N particles constrained to $$A$$ A interacting via a weighted power-law potential $$w(x,y)|x-y|^{-s}$$ w ( x , y ) | x - y | - s , where $$s>d$$ s > d is a fixed parameter and $$w(x,y)=\left( \sigma (x)\sigma (y)\right) ^{-({s}/{2d})}$$ w ( x , y ) = σ ( x ) σ ( y ) - ( s / 2 d ) . Here we show that one can generate points on $$A$$ A with the aforementioned properties keeping in the energy sums only those pairs of points that are located at a distance of at most $$r_N=C_N N^{-1/d}$$ r N = C N N - 1 / d from each other, with $$C_N$$ C N being a positive sequence tending to infinity arbitrarily slowly. To do this, we minimize the energy with respect to a varying truncated weight $$v_N(x,y)=\Phi (|x-y|/r_N)\cdot w(x,y)$$ v N ( x , y ) = Φ ( | x - y | / r N ) · w ( x , y ) , where $$\Phi :(0,\infty )\rightarrow [0,\infty )$$ Φ : ( 0 , ∞ ) → [ 0 , ∞ ) is a bounded function with $$\Phi (t)=0, t\ge 1$$ Φ ( t ) = 0 , t ≥ 1 , and $$\lim _{t\rightarrow 0^+}\Phi (t)=1$$ lim t → 0 + Φ ( t ) = 1 . Under appropriate assumptions, this reduces the complexity of generating $$N$$ N -point “low energy” discretizations to order $$N C_N^d$$ N C N d computations.
Robert Hardt - One of the best experts on this subject based on the ideXlab platform.
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connecting rational homotopy type singularities
Acta Mathematica, 2008Co-Authors: Robert Hardt, Tristan RiviereAbstract:Let N be a compact simply connected smooth Riemannian manifold and, for p ∈ {2,3,...}, W 1,p (R p+1, N) be the Sobolev space of measurable maps from R p+1 into N whose gradients are in L p . The restriction of u to almost every p-dimensional sphere S in R p+1 is in W 1,p (S, N) and defines an homotopy class in π p (N) (White 1988). Evaluating a fixed element z of Hom(π p (N), R) on this homotopy class thus gives a real number Φ z,u (S). The main result of the paper is that any W 1,p -weakly convergent limit u of a sequence of smooth maps in C ∞(R p+1, N), Φ z,u has a Rectifiable Poincare dual \( {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} \). Here Γ is a a countable union of C 1 curves in R p+1 with Hausdorff \( {\user1{\mathcal{H}}}^{1} \)-measurable orientation \( {\overrightarrow{\Gamma }} :\Gamma \to S^{p} \) and density function θ: Γ→R. The intersection number between \( {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} \) and S evaluates Φ z,u (S), for almost every p-sphere S. Moreover, we exhibit a non-negative integer n z , depending only on homotopy operation z, such that \( {\int_\Gamma {{\left| \theta \right|}^{{p \mathord{\left/ {\vphantom {p {{\left( {p + n_{z} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {p + n_{z} } \right)}}}} d{\user1{\mathcal{H}}}^{1} < \infty } } \) even though the mass \( {\int_\Gamma {{\left| \theta \right|}d{\user1{\mathcal{H}}}^{1} } } \) may be infinite. We also provide cases of N, p and z for which this rational power p/(p + n z ) is optimal. The construction of this Poincare dual is based on 1-dimensional “bubbling” described by the notion of “scans” which was introduced in Hardt and Riviere (2003). We also describe how to generalize these results to R m for any m ⩾ p + 1, in which case the bubbling is described by an (m–p)-Rectifiable Set with orientation and density function determined by restrictions of the mappings to almost every oriented Euclidean p-sphere.
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Connecting rational homotopy type singularities
Acta Mathematica, 2008Co-Authors: Robert Hardt, Tristan RiviereAbstract:Let N be a compact simply connected smooth Riemannian manifold and, for p ∈ {2,3,...}, W ^1, p ( R ^ p +1, N ) be the Sobolev space of measurable maps from R ^ p +1 into N whose gradients are in L ^ p . The restriction of u to almost every p -dimensional sphere S in R ^ p +1 is in W ^1, p ( S , N ) and defines an homotopy class in π_ p ( N ) (White 1988). Evaluating a fixed element z of Hom(π_ p ( N ), R ) on this homotopy class thus gives a real number Φ_ z , u ( S ). The main result of the paper is that any W ^1, p -weakly convergent limit u of a sequence of smooth maps in C ^∞( R ^ p +1, N ), Φ_ z , u has a Rectifiable Poincaré dual $ {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} $ . Here Γ is a a countable union of C ^1 curves in R ^ p +1 with Hausdorff $ {\user1{\mathcal{H}}}^{1} $ -measurable orientation $ {\overrightarrow{\Gamma }} :\Gamma \to S^{p} $ and density function θ : Γ→ R . The intersection number between $ {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} $ and S evaluates Φ_ z , u ( S ), for almost every p -sphere S . Moreover, we exhibit a non-negative integer n _ z , depending only on homotopy operation z , such that $ {\int_\Gamma {{\left| \theta \right|}^{{p \mathord{\left/ {\vphantom {p {{\left( {p + n_{z} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {p + n_{z} } \right)}}}} d{\user1{\mathcal{H}}}^{1} < \infty } } $ even though the mass $ {\int_\Gamma {{\left| \theta \right|}d{\user1{\mathcal{H}}}^{1} } } $ may be infinite. We also provide cases of N , p and z for which this rational power p /( p + n _ z ) is optimal. The construction of this Poincaré dual is based on 1-dimensional “bubbling” described by the notion of “scans” which was introduced in Hardt and Rivière (2003). We also describe how to generalize these results to R ^ m for any m ⩾ p + 1, in which case the bubbling is described by an ( m – p )-Rectifiable Set with orientation and density function determined by restrictions of the mappings to almost every oriented Euclidean p -sphere.
