Web Geometry Research Articles

Preliminary Aerostructural Optimization of a Large Business Jet

An overview of an industrial approach to the aerostructural optimization of a large business jet is presented herein. The optimization methodology is based on the integration of aerodynamic and structural analysis codes that combine computational, analytical, and semi-empirical methods, validated in an aircraft design environment. The aerodynamics subspace is analyzed with a three-dimensional transonic small disturbance code capable of predicting the drag of a complete, trimmed aircraft within engineering accuracy. The design of the wing structure is accomplished using a quasi-analytical method that defines the layout of the ribs and geometry of the spar webs, spar caps, and skin-stringer panels, and predicts the wing flexural properties and weight distribution. In addition, the prediction of operating economics as well as the integrated en route performance is coupled into the scheme by way of fractional change functional transformations. To illustrate the automated design system capabilities, the methodology is applied to the optimization of a large business jet comprising winglets, rear-mounted engines, and a T-tail configuration. The aircraft-level design optimization goal in this instance is to minimize a cost function for a fixed range mission assuming a constant maximum takeoff weight.

Deformation of holomorphic maps onto the blow-up of the projective plane

Let S be the blow-up of the projective plane at d distinct points and ψ : S ′ → S be any surjective holomorphic map from a compact complex manifold S ′ . We will show that all deformations of ψ come from automorphisms of S if d ⩾ 3 . The result is optimal in the sense that it is not true if d ⩽ 2 . The strategy of the proof is to use the infinitesimal automorphisms of the web geometry on S arising from the natural foliations of S induced by the pencils of the lines through the blow-up centers.

Abelian functional equations, planar web geometry and polylogarithms

We study some abelian functional equations (Afe). They are equations in the F i ’s of the form F 1(U 1) + ... + F N (U N ) = 0 where the U i ’s are real rational functions in two variables. First we prove that the local measurable solutions are actually analytic and we characterize their components as solutions of linear differential equations constructed from the U i ’s. Then we propose two methods for solving Afe. Next we apply these methods to the explicit resolution of generalized versions of classical (inhomogeneous) Afe satisfied by low order polylogarithms. Interpreted in the framework of web geometry, these results give us new nonlinearizable maximal rank planar webs. Then we observe that there is a relation between these webs and certain configurations of points in $$\mathbb{C}\mathbb{P}^{2} ,$$ which leads us to define the notion of web associated to a configuration: these webs seem of high rank and could provide numerous new exceptional webs. Finally, we use the preceding results to show that, under weak regularity assumptions, the trilogarithm is the only function which satisfies the Spence–Kummer equation.

Extinction cascades and catastrophe in ancient food webs

A model is developed to explore the potential responses of paleocommunities to disruptions of primary production during times of mass extinction and ecological crisis. Disruptions of primary production are expected to generate bottom-up cascades of secondary extinction, and these are predictable given species richnesses, functional diversity, and trophic link distributions. If, however, consumers are permitted to compensate for the loss of trophic resources by increasing the intensities of their remaining biotic interactions, top-down driven catastrophic increases of secondary extinction emerge from the model. Both bottom-up and top-down effects are themselves controlled by the geometry of the food webs. The general Phanerozoic trends of increasing taxonomic and ecological diversities, as well as the varying strengths of biotic interactions, have led to food webs of increasing complexity. The frequency of catastrophic secondary extinction increases as food web complexity increases, but increased complexity also serves to dampen the magnitude of the secondary extinctions. When intraguild competitive interactions are included in the model, competitively inferior taxa are observed to possess greater probabilities of survival if the guilds are embedded in simple subnetworks of the overall food web. The result is the emergence of postextinction guilds dominated by those inferior taxa. These results are congruent with empirical observations of “disaster taxa” dominance after some mass extinction events, and provide a mechanism for the reorganization of ecosystems that is observed after those events. The model makes the testable prediction that dominance by disaster taxa, however, should be observed only when bottom-up disruptions have caused ecosystems to collapse catastrophically.

Détermination du rang des tissus du plan et autres invariants géométriques

Let W ( d ) be a non singular d-web in the plane with d ⩾ 3 , presented by a first order differential equation of the type F ( x , y , y ′ ) : = a 0 ( x , y ) ⋅ ( y ′ ) d + a 1 ( x , y ) ⋅ ( y ′ ) d − 1 + ⋯ + a d ( x , y ) = 0 , where a i ∈ C { x , y } and let ( E , ∇ ) be the connection associated with F. We show that the trace of its curvature is the sum of the Blaschke curvatures of extracted 3-webs of W ( d ) . Our main result is an explicit determination of the rank of W ( d ) . We also recover some well known results in web geometry. To cite this article: O. Ripoll, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

The effects of neurotoxins on web-geometry and web-building behaviour in Araneus diadematus Cl.

The process of orb weaving and the resultant orb web constitute a good example of a complex behavioural pattern that is still governed by a relatively simple set of rules. We used the orb spider Araneus diadematus as a model organism to study the effect of the three neurotoxins (scopolamine, amphetamine, and caffeine) on the spider's behaviour. Scopolamine was given at two concentrations, with the lower one showing no effects but the higher one reducing web-building frequency; there also appeared to be a weak effect on web geometry. Amphetamine and caffeine, on the other hand, both resulted in significant changes in both building frequency and web geometry, compared to the controls. Amphetamine webs retained their size but showed an increase in spiral spacing and radius irregularity, as well as a decrease in building efficiency. Caffeine led to a general decrease in size and a slight increase in spiral spacing, as well as radius irregularity. Furthermore, caffeine caused webs to be rounder. Our observations suggest that these neurotoxins disturb different parts of the web-building programme presumably by affecting different actions in the spider's CNS.

On planar web geometry through abelian relations and connections

Web geometry is devoted to the study of families of foliations which are in general position. We restrict ourselves to the local situation, in the neighborhood of the origin in C2, with d > 1 complex analytic foliations of curves in general position. We are interested in the geometry of such configurations, that is, properties of planar d-webs which are invariant with respect to analytic local isomorphisms of C2. The initiators of the subject are W. Blaschke, G. Thomsen and G. Bol

Effect of Water Jet Pressure Profile and Initial Web Geometry on the Physical Properties of Composite Hydroentangled Fabrics

The effect of a water jet pressure profile (WJPP) and initial web geometry on the properties of hydroentangled fabrics is investigated. Different web structures (obtained by parallel, cross, and air-laid formation) are produced and hydroentangled at the same total pressure but with different water jet profiles. Composite structures are also made by combining different web structures followed by hydroentangling. The fabrics are then characterized in terms of key physical properties. Although the total pressure may be constant, the WJPP influences the tensile properties of fabrics, especially those containing air-laid structures. The results also suggest that to produce high tenacity fabrics, the webs should be hydroentangled on both sides at almost the same total pressure. The MD/CD strength ratios indicate that the most isotropic fabrics are air-laid, composite web struc tures combining cross and parallel-laid webs.

Developing a New Processing Route to Manufacture Honeycomb Ceramics with Negative Poisson's Ratio

Materials with negative Poisson's ratios (auxetic materials) have several advantages compared to conventional materials, including increased plane strain fracture resistance and increased shear modulus. A method for the manufacture of honeycomb ceramics with a structure, which yields negative Poisson's ratio is described. Ceramics with cellular or honeycomb structures have applications as substrates in catalytic converters for the emissions from internal combustion engines. In such devices due to the extreme temperature variations, the structure has to provide high thermal shock resistance. It is possible to increase the thermal shock resistance by incorporating specific web geometries. Such structures can absorb or compensate for compressive and tensile stresses generated in the body during thermal expansion and contraction. It is shown that such structures can be produced by the extrusion of ceramic pastes through polymer dies produced by rapid prototyping (selective laser sintering), CAD technology was used to design the die. A ceramic paste based on a zirconia powder was used to make the cellular extruded tube. The negative Poisson's ratio behaviour was confirmed by employing a simple test. (orig.)

Finite-Element Modeling of Beams under Concentrated Loading

A numerical study of aluminum and steel beams subjected to concentrated loading is performed and compared with experimental data from the literature. The modeled test specimen referred to simply supported beams where the concentrated loading is applied either at the midspan or at the end support, and the influence of varying bending moment and beam overhang is investigated. The modeled cross sections cover a wide range of web geometries and flange stiffnesses, and loading through both circular bars and rectangular bars are included. The contact between the beam specimen and the loading bars is modeled with a contact algorithm, and the problem is solved by an explicit code. The correlation between the experimental and numerical results is quite good, especially for the ultimate capacity where the difference between predictions and tests is not prominent when compared to the scatter in the test results. The error in the ultimate capacity from finite-element simulations is within 14% of the measured value, and for the web deformation, simulations can predict the main effects that are obtained from the tests. The results show that small elements are necessary for predicting the correct mode of failure, and the development of the local instability depends on the mass scaling and assumed imperfection field. Furthermore, brick elements seem appropriate for including the effect of hydrostatic pressure, and the adopted fracture model seems useful for predicting the crack development that was observed for some specimens.

Webs, Lenard schemes, and the local geometry of bi-Hamiltonian Toda and Lax structures

We introduce a criterion that a given bi-Hamiltonian structure admits a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bi-Hamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odd-dimensional pair of brackets with constant coefficients. This shows that the open Toda lattice cannot be locally represented as a product of two bi-Hamiltonian structures. Near, a generic point, the bi-Hamiltonian periodic Toda lattice is shown to be isomorphic to a product of two open Toda lattices (one of which is a (trivial) structure of dimension 1). While the above results might be obtained by more traditional methods, we use an approach based on general results on geometry of webs. This demonstrates the possibility of applying a geometric language to problems on bi-Hamiltonian integrable systems; such a possibility may be no less important than the particular results proved in this paper. Based on these geometric approaches, we conjecture that decompositions similar to the decomposition of the periodic Toda lattice exist in local geometry of the Volterra system, the complete Toda lattice, the multidimensional Euler top, and a regular bi-Hamiltonian Lie coalgebra. We also state general conjectures about the geometry of more general finite-dimensional bi-Hamiltonian structures. The class of homogeneous structures is shown to coincide with the class of systems integrable by Lenard scheme. The bi-Hamiltonian structures which admit a non-degenerate Lax structure are shown to be locally isomorphic to the open Toda lattice.

Optimal Area Use in Orb Webs of the Spider Araneus diadematus

We studied the abilities of the garden cross spider Araneus diadematus regarding adaptation of web geometry to spatial constraints. Spiders reacted to a spatial reduction in their building site from a square-shaped frame to a slimmer, rectangular frame (side ratio 1 : 2) by maintaining overall web geometry while reducing the web area covered by the sticky capture spiral. However, when the frames were changed further to a rectangular side ratio of 1 : 3, the spiders changed specific web properties in such a way that a further reduction in the capture spiral area was prevented. Construction of the threads making up the web frame and the auxiliary spiral requires that the spider explores the spatial constraints of its building site. The geometry of both frame and auxiliary spiral threads in turn determine the geometry of the capture threads. Since in very narrow frames the spider adjusted the auxiliary to suit the subsequent capture spiral, we suggest that an initial spatial survey led to the final adaptation of overall web geometry to a web site.

Design Variability in Web Geometry of an Orb-Weaving Spider

We studied the effect of several variables (environmental and physiological) on web geometry in the garden cross spider Araneus diadematus. Variables were: web support, wind, temperature, humidity, and silk supply. All had an effect. The spiders generally attempted to fit their webs to the shape of the supporting frame (standard, small, vertical, or horizontal). Windy conditions (0.5 m s −1) during web construction caused spiders to build smaller and rounder webs, laying down fewer capture spirals while increasing the distances between capture-spiral meshes. Decreasing temperature from 24° to 12°C caused the capture spiral to have fewer and wider spaced meshes, which did not change overall capture area but reduced the length of capture-spiral threads laid down. Subsequent increase of temperature to 24°C restored the number of meshes laid down, but the wider mesh was retained, causing the capture area to be increased over initial control values. Decreased humidity (from 70 to 20% rH) had the effect of reducing web and capture-spiral size, the latter by reducing mesh number while keeping mesh spacing constant. Subsequent increase of humidity to control level (70%) restored web and capture area. However, this was achieved by laying down capture meshes at larger distances, rather than returning to initial mesh numbers. Silk supply also had a strong effect. Webs built in unnaturally rapid succession by the same spider (4 in 24 h when 1 is the norm) became sequentially smaller, had fewer radii, shorter capture spirals, and were wider meshed.

A Test of Habitat Selection at Two Spatial Scales in a Sitand-Wait Predator: A Web Spider in the Namib Desert Dunes

1. Distribution and web characteristics were examined for the dune-burrowing spider Seothyra henscheli in the Namib Desert to determine how these spiders relate to spatially heterogeneous habitat features that influence foraging costs and benefits, growth rate and survival. In particular, we set out to determine the relative importance of (i) habitat selection, (ii) site-dependent growth and survival, and (iii) restricted dispersal, to the occurrence of S. henscheli in different macro- and microhabitats. 2. Spider webs were mapped along transects that crossed dune regions with dense vegetation hummocks, steep dune plinths and bare, gravelly interdune plains. Spiders were more abundant in the hummocks than in the interdunes and plinths. Translocation of spiders from the hummocks to the interdunes and plinths confirmed that spider density was correlated with habitat quality. 3. In a comparison of web sites with dummy sites in terms of factors related to sand stability and food availability, it was found that, except for the avoidance of extreme conditions, web-site characteristics did not differ from dummy sites. Web geometry indicated that good sites were characterized by stable sand and abundant ants. However, many spiders occurred at poor sites when good sites were vacant nearby. 4. It is concluded that S. henscheli do not actively choose web sites, but have restricted dispersal. Site-dependent growth and survival may explain the pattern of local abundance. Site tenacity may be a result of the spiders' inability to predict site quality coupled with high costs of relocation. It is inferred that spiderlings tend to remain near their mother's site, which has a previous record of success. 5. Tolerance of variable conditions of dunes and the ability to sustain populations even at poor sites enable these spiders to occur in extreme desert conditions.

Porous structure of thick fiber webs

The bulk properties and stochastic pore geometry of finite-thickness fiber webs are studied using a realistic model for the sedimentation of flexible fibers [K. J. Niskanen and M. J. Alava, Phys. Rev. Lett. 73, 3475 (1994)]. The resulting web structure is controlled by a dimensionless number F=Tfwf/tf, where Tf is fiber flexibility, wf fiber width, and tf fiber thickness. The fiber length (≫wf,tf) is irrelevant. With increasing coverage c̄, a crossover occurs at c̄=c0≈1+2F from a vacancy-controlled two-dimensional (2D) structure to a pore-controlled 3D structure. The 3D structures are isomorphic in that the pore dimensions are exponentially distributed, with the decay rate dependent only on F.

Analysing Spider Web-building Behaviour with Rule-based Simulations and Genetic Algorithms

We analysed the web construction-behaviour of an ideal orb web-building spider using a computer model that constructed artificial webs with a Rule Based Simulation and optimised them with a Genetic Algorithm (GA). The GA simulated adaptation towards an adjustable ecological niche in a population of artificial (cyber) spiders. Web construction behaviour in each cyber spider was controlled by a set of rule parameters (encoded in artificial genes) resulting in different web shapes. We then statistically compared the best cyber webs from our optimisation runs with real webs ofAraneus diadematusbuilt under controlled, experimental conditions in the laboratory. Web characteristics like spiral distances, eccentricities and vertical hub location could to a large degree be accurately simulated with the model. In addition, falsification of working hypotheses using the model pointed to incomplete or wrong assumptions about the behaviour under study which became apparent when web geometry adapted poorly.

New Conservation Laws for the Interaction of Nonlinear Waves

Recently, unexpected conservation laws have been discovered for various nonlinear wave systems. Among these systems is the system of Rossby waves, which describes the global dynamics of the atmosphere and the ocean. It turns out that these conservation laws are intimately related to a geometric theory---the geometry of webs---that originated more than 60 years earlier. This relation helped discover how many conservation laws a nonlinear wave system can have.

The influence of food supply on foraging behaviour in a desert spider.

We tested the alternative hypotheses that foraging effort will increase (energy maximizer model) or decrease (due to increased costs or risks) when food supply increased, using a Namib desert burrowing spider, Seothyra henscheli (Eresidae), which feeds mainly on ants. The web of S. henscheli has a simple geometrical configuration, comprising a horizontal mat on the sand surface, with a variable number of lobes lined with sticky silk. The sticky silk is renewed daily after being covered by wind-blown sand. In a field experiment, we supplemented the spiders' natural prey with one ant on each day that spiders had active webs and determined the response to an increase in prey. We compared the foraging activity and web geometry of prey-supplemented spiders to non-supplemented controls. We compared the same parameters in fooddeprived and supplemented spiders in captivity. The results support the "costs of foraging" hypothesis. Supplemented spiders reduced their foraging activity and web dimensions. They moulted at least once and grew rapidly, more than doubling their mass in 6 weeks. By contrast, food-deprived spiders increased foraging effort by enlarging the diameter of the capture web. We suggest that digestive constraints prevented supplemented spiders from fully utilizing the available prey. By reducing foraging activities on the surface, spiders in a prey-rich habitat can reduce the risk of predation. However, early maturation resulting from a higher growth rate provides no advantage to S. henscheli owing to the fact that the timing of mating and dispersal are fixed by climatic factors (wind and temperature). Instead, large female body size will increase fitness by increasing the investiment in young during the period of extended maternal care.

Unfreezing the behaviour of two orb spiders

Spider's webs reflect the builders behaviour pattern; yet there are aspects of the construction behaviour that cannot be “read” from the geometry of the finished web alone. Using computerised image analysis we developed an automatic surveillance method to track a spider's path during web-building. Thus we collected data on two orb-weaving spiders—the cribellate Uloborus walckenaerius and the ecribellate Araneus diadematus—for web geometry, movement pattern and time allocation. Representatives of these two species built webs of similar geometry but they used different movement patterns both spatially (which we describe qualitatively) and temporally (which we analyse quantitatively). Most importantly, temporal analysis showed that the two spiders differed significantly in some but not all web-building stages; and from this we deduce that Uloborus—unlike Araneus—was constrained by speed of silk production during the construction of its capture but not its auxiliary spiral.

Wave systems with an infinite number of invariants

Can some extra quantities (besides energy and momentum) be conserved in 3-wave (resonance) interactions? The paper describes all 3-wave interactions in which infinitely many independent quantities are conserved. The results are obtained due to the application of web geometry. The correspondence between 3-wave interactions and 3-webs is established. All wave systems, related to the inverse scattering tranform method, correspond to the simplest (the so called coordinate) web. Each extra invariant of a 3-wave interaction defines new conservation law of the corresponding wave kinetic equation (which describes turbulence of the waves). The results are applied to the nonlinear theory of decay instability.