scholarly journals Plant-pollinator Specialization: Origin and measurement of curvature

2021 ◽  
Author(s):  
Mannfred Masahiro Asada Boehm ◽  
Jill E. Jankowski ◽  
Quentin C.B. Cronk
Keyword(s):  
Differential Geometry ◽  
Comparative Morphology ◽  
Pollination Ecology ◽  
Flowering Plant ◽  
Arc Length ◽  
Curved Structures ◽  
Pollinator Specialization ◽  
Definition Of ◽  
Pollination Systems ◽  
Plant Pollinator

A feature of biodiversity is the abundance of curves displayed by organs and organisms. Curvature is a widespread, convergent trait that has important ecological and evolutionary implications. In pollination ecology, the curvature of flowers and pollinator mouthparts (e.g. hummingbird bills) along the dorsiventral plane has been associated with specialization, competition, and species co-existence. Six methods have historically been used to measure curvature in pollination systems; we provide a solution to this inconsistency by defining curvature using well-established concepts from differential geometry. Intuitively, curvature is the degree to which a line is not straight, but more formally, it is the rate at which the tangent of a curve changes direction with respect to arc length. Here, we establish a protocol wherein a line is fitted against landmarks placed on an image of a curved organ or organism, then curvature is computed at many points along the fitted line and the sum taken. The protocol is demonstrated by studying the development of nectar spur curvature in the flowering plant genus Epimedium (Berberidaceae). By clarifying the definition of curvature, our aim is to make the language of comparative morphology more precise and broadly applicable to capture other curved structures in nature.

scholarly journals The role of individual variation in flowering and pollination in the reproductive success of a crepuscular buzz-pollinated plant

2020 ◽  
Author(s):  
Natalia Costa Soares ◽  
Pietro Kiyoshi Maruyama ◽  
Vanessa Graziele Staggemeier ◽  
Leonor Patrícia Cerdeira Morellato ◽  
Márcio Silva Araújo
Keyword(s):  
Reproductive Success ◽  
Flowering Phenology ◽  
Southeastern Brazil ◽  
Individual Specialization ◽  
Plant Reproductive Success ◽  
Pollinator Specialization ◽  
The Individual ◽  
Pollination Systems ◽  
Plant Pollinator

Abstract Background and Aims Plant individuals within a population differ in their phenology and interactions with pollinators. However, it is still unknown how individual differences affect the reproductive success of plants that have functionally specialized pollination systems. Here, we evaluated whether plant individual specialization in phenology (temporal specialization) and in pollination (pollinator specialization) affect the reproductive success of the crepuscular-bee-pollinated plant Trembleya laniflora (Melastomataceae). Methods We quantified flowering activity (amplitude, duration and overlap), plant–pollinator interactions (number of flowers visited by pollinators) and reproductive success (fruit set) of T. laniflora individuals from three distinct locations in rupestrian grasslands of southeastern Brazil. We estimated the degree of individual temporal specialization in flowering phenology and of individual specialization in plant–pollinator interactions, and tested their relationship with plant reproductive success. Key Results Trembleya laniflora presented overlapping flowering, a temporal generalization and specialized pollinator interactions. Flowering overlap among individuals and populations was higher than expected by chance but did not affect the individual interactions with pollinators and nor their reproductive success. In contrast, higher individual generalization in the interactions with pollinators was related to higher individual reproductive success. Conclusions Our findings suggest that individual generalization in plant–pollinator interaction reduces the potential costs of specialization at the species level, ensuring reproductive success. Altogether, our results highlight the complexity of specialization/generalization of plant–pollinator interactions at distinct levels of organization, from individuals to populations, to species.

Edges of Regression and Limit Normal Point of Conjugate Surfaces1

1998 ◽  
Vol 122 (4) ◽  
pp. 419-425 ◽  
Author(s):  
Ningxin Chen
Keyword(s):  
Differential Geometry ◽  
Contact Boundary ◽  
Boundary Line ◽  
Tangent Point ◽  
Common Tangent ◽  
Numerical Examples ◽  
Envelope Surface ◽  
The Common ◽  
Conjugate Surfaces ◽  
Definition Of

The presented paper utilizes the basic theory of the envelope surface in differential geometry to investigate the undercutting line, the contact boundary line and the limit normal point of conjugate surfaces in gearing. It is proved that (1) the edges of regression of the envelope surfaces are the undercutting line and the contact boundary line in theory of gearing respectively, and (2) the limit normal point is the common tangent point of the two edges of regression of the conjugate surfaces. New equations for the undercutting line, the contact boundary line and the limit normal point of the conjugate surfaces are developed based on the definition of the edges of regression. Numerical examples are taken for illustration of the above-mentioned concepts and equations. [S1050-0472(00)00104-5]

scholarly journals Pseudoinversion of degenerate metrics

2003 ◽  
Vol 2003 (55) ◽  
pp. 3479-3501 ◽  
Author(s):  
C. Atindogbe ◽  
J.-P. Ezin ◽  
Joël Tossa
Keyword(s):  
Differential Geometry ◽  
Large Class ◽  
Smooth Manifold ◽  
Differential Operators ◽  
Type Operator ◽  
Lorentzian Space ◽  
Lightlike Hypersurface ◽  
Definition Of ◽  
Lightlike Hypersurfaces ◽  
Theoretical Results

Let(M,g)be a smooth manifoldMendowed with a metricg. A large class of differential operators in differential geometry is intrinsically defined by means of the dual metricg∗on the dual bundleTM∗of 1-forms onM. If the metricgis (semi)-Riemannian, the metricg∗is just the inverse ofg. This paper studies the definition of the above-mentioned geometric differential operators in the case of manifolds endowed with degenerate metrics for whichg∗is not defined. We apply the theoretical results to Laplacian-type operator on a lightlike hypersurface to deduce a Takahashi-like theorem (Takahashi (1966)) for lightlike hypersurfaces in Lorentzian spaceℝ1n+2.

scholarly journals Objets compacts dans les topos

1986 ◽  
Vol 40 (2) ◽  
pp. 203-217 ◽  
Author(s):  
Eduardo J. Dubuc ◽  
Jacques Penon
Keyword(s):  
Differential Geometry ◽  
Topological Space ◽  
Compact Manifolds ◽  
Topological Spaces ◽  
Internal Logic ◽  
Definition Of

AbstractIt is well known that compact topological spaces are those space K for which given any point x0 in any topological space X, and a neighborhood H of the fibre -1 {x0} KXX, then there exists a neighborhood U of x0 such that -1UH. If now is an object in an arbitrary topos, in the internal logic of the topos this property means that, for any A in and B in K, we have (-1AB)=AB. We introduce this formula as a definition of compactness for objects in an arbitrary topos. Then we prove that in the gross topoi of algebraic, analytic, and differential geometry, this property characterizes exactly the complete varieties, the compact (analytic) spaces, and the compact manifolds, respectively.

On generic hypersurfaces in ℝn

1981 ◽  
Vol 90 (3) ◽  
pp. 389-394 ◽  
Author(s):  
J. W. Bruce
Keyword(s):  
Differential Geometry ◽  
Euclidean Space ◽  
Dense Subset ◽  
Precise Definition ◽  
Open Dense Subset ◽  
Curves And Surfaces ◽  
Definition Of

In this paper we consider certain questions concerning the differential geometry of generic hypersurfaces in ℝn. Our results prove, for example, that the curve of rib points of a generic surface in ℝ3 has transverse self-intersections.In (4) Porteous discussed (amongst other things) the generic geometry of curves and surfaces in ℝ3. Subsequently Looijenga ((3) and see also (5)) gave a more precise definition of the term generic and showed that an open dense subset of smooth embeddings of manifolds in Euclidean space were indeed generic.

scholarly journals 60 million years of co-divergence in the fig–wasp symbiosis

2005 ◽  
Vol 272 (1581) ◽  
pp. 2593-2599 ◽  
Author(s):  
Nina Rønsted ◽  
George D Weiblen ◽  
James M Cook ◽  
Nicolas Salamin ◽  
Carlos A Machado ◽  
...  
Keyword(s):  
Phylogenetic Trees ◽  
Time Frame ◽  
Molecular Dating ◽  
Fig Wasp ◽  
Geological Time ◽  
Molecular Phylogenetic ◽  
Pollinator Specialization ◽  
Phylogenetic Hypotheses ◽  
Fossil Data ◽  
Plant Pollinator

Figs ( Ficus ; ca 750 species) and fig wasps (Agaoninae) are obligate mutualists: all figs are pollinated by agaonines that feed exclusively on figs. This extraordinary symbiosis is the most extreme example of specialization in a plant–pollinator interaction and has fuelled much speculation about co-divergence. The hypothesis that pollinator specialization led to the parallel diversification of fig and pollinator lineages (co-divergence) has so far not been tested due to the lack of robust and comprehensive phylogenetic hypotheses for both partners. We produced and combined the most comprehensive molecular phylogenetic trees to date with fossil data to generate independent age estimates for fig and pollinator lineages, using both non-parametric rate smoothing and penalized likelihood dating methods. Molecular dating of ten pairs of interacting lineages provides an unparalleled example of plant–insect co-divergence over a geological time frame spanning at least 60 million years.

scholarly journals Plant-pollinator specialization: Origin and measurement of curvature

2021 ◽  
Author(s):  
Mannfred Boehm ◽  
Jill E Jankowski ◽  
Quentin C.B. Cronk
Keyword(s):  
Pollinator Specialization ◽  
Plant Pollinator

Electromagnetism and differential geometry

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Differential Geometry ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Exterior Derivative ◽  
Exterior Product ◽  
Exterior Calculus ◽  
Definition Of

This chapter begins by examining p-forms and the exterior product, as well as the dual of a p-form. Meanwhile, the exterior derivative is an operator, denoted d, which acts on a p-form to give a (p + 1)-form. It possesses the following defining properties: if f is a 0-form, df(t) = t f (where t is a vector of Eₙ), which coincides with the definition of differential 1-forms. Moreover, d(α‎ + β‎) = dα‎ + dβ‎, where α‎ and β‎ are forms of the same degree. Moreover, the exterior calculus can be used to obtain a compact and elegant formulation of Maxwell’s equations.

Differential geometry

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Differential Geometry ◽  
Vector Fields ◽  
Differential Forms ◽  
Curvilinear Coordinates ◽  
Moving Frames ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Vector Calculus ◽  
Vector Version ◽  
Definition Of

This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant derivative and establishes a relation between the moving frames attached to a trajectory introduced in Chapter 2 and the moving frames of Cartan associated with curvilinear coordinates. It illustrates a differential framework based on formulas drawn from Chapter 2, before discussing cotangent spaces and differential forms. The chapter then turns to the metric tensor, triads, and frame fields as well as vector fields, form fields, and tensor fields. Finally, it performs some vector calculus.

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