Measuring Solid Angles Beyond Dimension Three

Ribando, Jason M.

doi:10.1007/s00454-006-1253-4

Cited by 56 publications

(43 citation statements)

References 5 publications

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“…Nevertheless, several approaches are available to numerically compute this solid angle [13]. In this paper, we compute the probabilities p(n) for each n by numerically integrating the Gaussian function over the intersection of the four half spaces.…”

Section: Rectangular Latticementioning

confidence: 99%

Statistical topology of perturbed two-dimensional lattices

Leipold¹,

Lazar²,

Brakke³

et al. 2016

J. Stat. Mech.

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The Voronoi cell of any atom in a lattice is identical. If atoms are perturbed from their lattice coordinates, then the topologies of the Voronoi cells of the atoms will change. We consider the distribution of Voronoi cell topologies in two-dimensional perturbed systems. These systems can be thought of as simple models of finite-temperature crystals. We give analytical results for the distribution of Voronoi topologies of points in two-dimensional Bravais lattices under infinitesimal perturbations and present a discussion with numerical results for finite perturbations.

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Section: Rectangular Latticementioning

confidence: 99%

Statistical topology of perturbed two-dimensional lattices

Leipold¹,

Lazar²,

Brakke³

et al. 2016

J. Stat. Mech.

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“…The following result can be found in [20], although under a slightly different notation. The multiple integral in (7) can be computed explicitly only in rare circumstances.…”

Section: Numerical Integration Methodsmentioning

confidence: 81%

“…One also has By plugging all this information in (15), one arrives at the formula (20). On the other hand, ∞ q=0 c q x q converges for any x ∈ R such that the matrix M,…”

Section: Multivariate Power Series Methodsmentioning

confidence: 99%

“…In such a case, To see this, one needs to differentiate the right-hand side of (20) with respect to the variable ψ, and then one has to evaluate the derivative at π/2. Of course, second-order differentiation of (20) leads to an approximation formula that incorporates an additional term in ε 2 .…”

Section: Multivariate Power Series Methodsmentioning

confidence: 99%

“…With such a calibration factor, the ratio (K ) indicates how much room occupies K when compared with a half-space. In contrast to [20], we use a half-space as reference set and not the whole space R n . If one adopts Definition 1.1, then one has the following properties:…”

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confidence: 99%

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Deterministic and stochastic methods for computing volumetric moduli of convex cones

Gourion¹,

Seeger²

2010

Comput. Appl. Math.

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Nested species interactions promote feasibility over stability during the assembly of a pollinator community

Saavedra

Rohr

Olesen

et al. 2016

Ecology and Evolution

140

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The foundational concepts behind the persistence of ecological communities have been based on two ecological properties: dynamical stability and feasibility. The former is typically regarded as the capacity of a community to return to an original equilibrium state after a perturbation in species abundances and is usually linked to the strength of interspecific interactions. The latter is the capacity to sustain positive abundances on all its constituent species and is linked to both interspecific interactions and species demographic characteristics. Over the last 40 years, theoretical research in ecology has emphasized the search for conditions leading to the dynamical stability of ecological communities, while the conditions leading to feasibility have been overlooked. However, thus far, we have no evidence of whether species interactions are more conditioned by the community's need to be stable or feasible. Here, we introduce novel quantitative methods and use empirical data to investigate the consequences of species interactions on the dynamical stability and feasibility of mutualistic communities. First, we demonstrate that the more nested the species interactions in a community are, the lower the mutualistic strength that the community can tolerate without losing dynamical stability. Second, we show that high feasibility in a community can be reached either with high mutualistic strength or with highly nested species interactions. Third, we find that during the assembly process of a seasonal pollinator community located at The Zackenberg Research Station (northeastern Greenland), a high feasibility is reached through the nested species interactions established between newcomer and resident species. Our findings imply that nested mutualistic communities promote feasibility over stability, which may suggest that the former can be key for community persistence.

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Measuring Solid Angles Beyond Dimension Three

Cited by 56 publications

References 5 publications

Statistical topology of perturbed two-dimensional lattices

Statistical topology of perturbed two-dimensional lattices

Deterministic and stochastic methods for computing volumetric moduli of convex cones

Nested species interactions promote feasibility over stability during the assembly of a pollinator community

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