On an Algebraical Computation of the Tensor and the Curvature for 3-Web

ISRN Geometry ◽

10.5402/2012/804051 ◽

2012 ◽

Vol 2012 ◽

pp. 1-25

Author(s):

Thomas B. Bouetou

Keyword(s):

Lie Groups ◽

Algebraic Method ◽

Algebraic Methods ◽

Web Geometry ◽

Local Loops ◽

The Web

The algebraic methods are used in the web geometry, in particular in the 3-web. Along the line, we suggest a new, alternative algebraic method for computation of the quantities ∇l1ajki, ∇l2ajki, and djklmi by means of the embedding of local loops into Lie groups.

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Olfactory cues from conspecifics inhibit the web-invasion behavior of Portia, web-invading araneophagic jumping spiders (Araneae: Salticidae)

Canadian Journal of Zoology ◽

10.1139/z93-195 ◽

1993 ◽

Vol 71 (7) ◽

pp. 1415-1420 ◽

Cited By ~ 12

Author(s):

Marianne B. Willey ◽

Robert R. Jackson

Keyword(s):

Olfactory Cues ◽

Adult Males ◽

Aggressive Mimicry ◽

Jumping Spiders ◽

Web Geometry ◽

Adult Females ◽

Conspecific Adult ◽

Invasion Behavior ◽

Courtship Displays ◽

The Web

Portia is a genus of web-invading araneophagic spiders that use aggressive mimicry to capture their spider prey. In an experimental study, we demonstrate that adult females of Portia africana, P. fimbriata, P. labiata, and P. schultzi produce olfactory cues that affect the behavior of conspecific adult males, adult females, and juveniles. The olfactory cues of Portia spp. inhibit aggressive mimicry of conspecific spiders that are on a prey spider's web even if the prey spider is visible. This inhibition occurs regardless of the prey spider's web geometry. Prey pursuit by Portia is also inhibited when conspecific females provide olfactory cues in cases where the prey is a spider inhabiting a web. Olfactory cues from adult females elicit courtship displays of conspecific males when males are on the prey spider's web. Portia spp. do not alter their behavior when exposed to olfactory cues of heterospecifics.

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The Betti Numbers of the Simple Lie Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1958-034-2 ◽

1958 ◽

Vol 10 ◽

pp. 349-356 ◽

Cited By ~ 27

Author(s):

A. J. Coleman

Keyword(s):

Lie Groups ◽

Orthogonal Group ◽

Betti Numbers ◽

Algebraic Methods ◽

Classical Groups ◽

Compact Lie Groups ◽

Poincaré Polynomial ◽

Unimodular Group ◽

Simple Lie Groups

The purpose of the present paper1 is to simplify the calculation of the Betti numbers of the simple compact Lie groups. For the unimodular group and the orthogonal group on a space of odd dimension the form of the Poincaré polynomial was correctly guessed by E. Cartan in 1929 (5, p. 183). The proof of his conjecture and its extension to the four classes of classical groups was given by L. Pontrjagin (13) using topological arguments and then by R. Brauer (2) using algebraic methods.

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In situ three-dimensional spider web construction and mechanics

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.2101296118 ◽

2021 ◽

Vol 118 (33) ◽

pp. e2101296118

Author(s):

Isabelle Su ◽

Neosha Narayanan ◽

Marcos A. Logrono ◽

Kai Guo ◽

Ally Bisshop ◽

...

Keyword(s):

Self Assembly ◽

High Performance ◽

Mechanical Performance ◽

Three Dimensional ◽

Hierarchical Structures ◽

Spider Webs ◽

Spider Web ◽

Web Geometry ◽

Under Construction ◽

The Web

Spiders are nature’s engineers that build lightweight and high-performance web architectures often several times their size and with very few supports; however, little is known about web mechanics and geometries throughout construction, especially for three-dimensional (3D) spider webs. In this work, we investigate the structure and mechanics for a Tidarren sisyphoides spider web at varying stages of construction. This is accomplished by imaging, modeling, and simulations throughout the web-building process to capture changes in the natural web geometry and the mechanical properties. We show that the foundation of the web geometry, strength, and functionality is created during the first 2 d of construction, after which the spider reinforces the existing network with limited expansion of the structure within the frame. A better understanding of the biological and mechanical performance of the 3D spider web under construction could inspire sustainable robust and resilient fiber networks, complex materials, structures, scaffolding, and self-assembly strategies for hierarchical structures and inspire additive manufacturing methods such as 3D printing as well as inspire artistic and architectural and engineering applications.

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The Web Geometry Laboratory Project

Lecture Notes in Computer Science - Intelligent Computer Mathematics ◽

10.1007/978-3-642-39320-4_30 ◽

2013 ◽

pp. 364-368 ◽

Cited By ~ 6

Author(s):

Pedro Quaresma ◽

Vanda Santos ◽

Seifeddine Bouallegue

Keyword(s):

Web Geometry ◽

The Web

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Infinitely divisible projective representations of the Lie algebras

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100047186 ◽

1972 ◽

Vol 72 (3) ◽

pp. 357-368 ◽

Cited By ~ 1

Author(s):

D. Mathon

Keyword(s):

Lie Groups ◽

Lie Algebras ◽

Algebraic Method ◽

Group Cohomology ◽

Continuity Condition ◽

Group Representations ◽

Infinitely Divisible ◽

Projective Representations ◽

One To One ◽

Representations Of Lie Groups

Infinitely divisible group representations were first defined by Streater(1) as an important concept closely related to continuous tensor product. Araki(2) analysed the factorizable representations of Lie groups and obtained a generalization of the Levy–Khinchine formula. A similar concept for Lie algebras was defined and studied by Streater in (3). Although the definition is not strictly an infinitesimal analogue of infinitely divisible representations of Lie groups, the results of (3) in the cohomological formulation are very similar to Araki's main theorem. Parthasarathy and Schmidt(4) generalized the concept of infinite divisibifity to the projective representations of locally compact groups and obtained a one-to-one correspondence between infinitely divisible projective representations and 1-co-cycles in the group cohomology with coefficients in a Hubert space. A similar generalization for Lie algebras is studied in the present paper. Infinitely divisible projective representations of Lie algebras are studied by a purely algebraic method, independently of (4) (since not all our projective representations are necessarily integrable). As expected, a one-to-one relation is obtained between the infinitely divisible projective representations and 1-co-cycles in the cohomology on the corresponding enveloping algebra with coefficients in a Hilbert space. The present problem is simpler than the group case since there is no continuity condition on the multiplier in a Lie algebra. A similar algebraic method was used in a discussion of infinitely divisible representations of canonical anticommutation relations (9).

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