scholarly journals Rigidity through a Projective Lens

2021 ◽  
Vol 11 (24) ◽  
pp. 11946
Author(s):  
Anthony Nixon ◽  
Bernd Schulze ◽  
Walter Whiteley
Keyword(s):  
Geometric Reasoning ◽  
Multivariate Splines ◽  
Infinitesimal Rigidity ◽  
Singular Configurations ◽  
Projective Invariance ◽  
Projective Representations ◽  
Discrete Structures ◽  
Static Rigidity

In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar–joint framework in projective d-space and places particular emphasis on the projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body–hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas.

The Stability of Tensegrity Frameworks

1992 ◽  
Vol 7 (2) ◽  
pp. 153-163 ◽  
Author(s):  
Robert Connelly ◽  
Walter Whiteley
Keyword(s):  
Second Order ◽  
Infinitesimal Rigidity ◽  
Mathematical Concepts ◽  
Compression Members ◽  
Static Rigidity ◽  
The Stability ◽  
Tension Members

For a tensegrity framework with bars, cables (unilateral tension members) and struts (unilateral compression members), this paper presents a sequence of four mathematical concepts for detecting its rigidity. In order from the strongest to the weakest, these concepts are called infinitesimal (or static) rigidity, prestress stability, second-order rigidity and rigidity. Emphasis is placed on pre-stress stability, which lies between infinitesimal rigidity and second-order rigidity.

The High Resolution Appearance of Biological Substrates

1969 ◽  
Vol 27 ◽  
pp. 416-417
Author(s):  
Glen B. Haydon
Keyword(s):  
High Resolution ◽  
Phase Image ◽  
Biological Structure ◽  
Electron Microscopic ◽  
Thin Sections ◽  
Resolution Electron ◽  
Discrete Structures ◽  
Large Phase ◽  
Biological Substrates ◽  
High Resolution Electron Micrographs

High resolution electron microscopic study of negatively stained macromolecules and thin sections of tissue embedded in a variety of media are difficult to interpret because of the superimposed phase image granularity. Although all of the information concerning the biological structure of interest may be present in a defocused electron micrograph, the high contrast of large phase image granules produced by the substrate makes it impossible to distinguish the phase ‘points’ from discrete structures of the same dimensions. Theory predicts the findings; however, it does not allow an appreciation of the actual appearance of the image under various conditions. Therefore, though perhaps trivial, training of the cheapest computer produced by mass labor has been undertaken in order to learn to appreciate the factors which affect the appearance of the background in high resolution electron micrographs.

scholarly journals On the N-pion extension of the Lovelace-Shapiro model

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Massimo Bianchi ◽  
Dario Consoli ◽  
Paolo Di Vecchia
Keyword(s):  
Extra Dimensions ◽  
Tree Level ◽  
Negative Norm ◽  
Projective Invariance ◽  
Non Linear ◽  
Point Amplitude

Abstract We reconsider a modification of the N-point amplitude of the Neveu-Schwarz (NS) model in which the tachyon becomes a pion by shifting its mass to zero and keeping the super-projective invariance of the integrand of the amplitude. For the scattering of four particles it reduces to the amplitude written by Lovelace and Shapiro that has Adler zeroes. We confirm that also the N-pion amplitude has Adler zeroes and show that it reduces to that of the non-linear σ-model for α′ → 0 keeping Fπ fixed. The four- and six-point flavour-ordered amplitudes satisfy tree-level unitarity since they can be derived from the correspondent amplitudes of the NS model in ten dimensions by suitably choosing the components of the momenta of the external mesons in the six extra dimensions. Negative norm states (ghosts) are shown to appear instead in higher-point amplitudes. We also discuss several amplitudes involving different external mesons.

scholarly journals Near quantitative synthesis of urea macrocycles enabled by bulky N-substituent

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Yingfeng Yang ◽  
Hanze Ying ◽  
Zhixia Li ◽  
Jiang Wang ◽  
Yingying Chen ◽  
...  
Keyword(s):  
High Efficiency ◽  
Molecular Structures ◽  
High Yield ◽  
Kinetic Control ◽  
High Concentration ◽  
Weak Association ◽  
Quantitative Synthesis ◽  
Thermodynamic Stabilization ◽  
Discrete Structures ◽  
Very High

AbstractMacrocycles are unique molecular structures extensively used in the design of catalysts, therapeutics and supramolecular assemblies. Among all reactions reported to date, systems that can produce macrocycles in high yield under high reaction concentrations are rare. Here we report the use of dynamic hindered urea bond (HUB) for the construction of urea macrocycles with very high efficiency. Mixing of equal molar diisocyanate and hindered diamine leads to formation of macrocycles with discrete structures in nearly quantitative yields under high concentration of reactants. The bulky N-tert-butyl plays key roles to facilitate the formation of macrocycles, providing not only the kinetic control due to the formation of the cyclization-promoting cis C = O/tert-butyl conformation, but also possibly the thermodynamic stabilization of macrocycles with weak association interactions. The bulky N-tert-butyl can be readily removed by acid to eliminate the dynamicity of HUB and stabilize the macrocycle structures.

scholarly journals Vibroacoustic Assessment of an Innovative Composite Material for the Roof of a Coupe Car

2021 ◽  
Vol 11 (3) ◽  
pp. 1128
Author(s):  
Nunziante Cascone ◽  
Luca Caivano ◽  
Giuseppe D’Errico ◽  
Roberto Citarella
Keyword(s):  
Composite Material ◽  
Transfer Functions ◽  
Realistic Model ◽  
Point Of View ◽  
Damping Capacity ◽  
Geometry Structure ◽  
Fiberglass Composite ◽  
New Material ◽  
Static Rigidity ◽  
Innovative Material

The objective of this paper is the vibroacoustic evaluation of an innovative material for a sports car roof, aiming at replacing fiberglass composite materials. Such evaluation was carried out using numerical and experimental analysis techniques, with cross-comparison between the corresponding results. The innovative material under analysis is a composite material, with a thermoplastic polypropylene matrix and reinforcement made of cellulose fibers. In order to validate the virtual dynamic modeling of the new material, the inertance on different points of some sheets made of the material under analysis was evaluated by an in-house made experimental activity, performed in the CRF (Fiat Research Center) test room, and cross-compared with corresponding results from a numerical analysis performed with the MSC Nastran software. Then, a realistic model of the car roof of the Alfa Romeo 4C car, made with the new material, was implemented and analyzed from the vibroacoustic point of view. The mere switch to the new material, with no changes in the geometry/structure of the car roof, did not allow preserving the original values of static rigidity, dynamic rigidity, and configuration of modal shapes. For this reason, a geometric/structural optimization of the component was performed. Once the new geometry/structure was defined, a vibroacoustic analysis was carried out, checking for a possible coupling between the fluid cavity modes and the structure car body modes. Finally, the vibroacoustic transfer functions to the driver’s ear node were assessed, considering two different excitation points on the structure. The excellent damping capacity of the proposed material led to an improvement in the vibroacoustic transfer functions and to a reduction in the weight of the pavilion.

scholarly journals Approximation of noisy data using multivariate splines and finite element methods

2021 ◽  
Vol 15 ◽  
pp. 174830262110084
Author(s):  
Bishnu P Lamichhane ◽  
Elizabeth Harris ◽  
Quoc Thong Le Gia
Keyword(s):  
Finite Element ◽  
Differential Operators ◽  
Impulsive Noise ◽  
Noisy Data ◽  
Multivariate Splines ◽  
The Finite Element Method ◽  
Data Smoothing ◽  
Multivariate Spline ◽  
Partial Differential Operators ◽  
Mixture Of Gaussian

We compare a recently proposed multivariate spline based on mixed partial derivatives with two other standard splines for the scattered data smoothing problem. The splines are defined as the minimiser of a penalised least squares functional. The penalties are based on partial differential operators, and are integrated using the finite element method. We compare three methods to two problems: to remove the mixture of Gaussian and impulsive noise from an image, and to recover a continuous function from a set of noisy observations.

Multivariate piecewise polynomials

Acta Numerica ◽  
1993 ◽  
Vol 2 ◽  
pp. 65-109 ◽  
Author(s):  
C. de Boor
Keyword(s):  
Approximation Theory ◽  
Variational Approach ◽  
Polynomial Function ◽  
Piecewise Polynomial ◽  
Multivariate Splines ◽  
Piecewise Polynomial Function ◽  
Multivariate Spline ◽  
Piecewise Polynomials ◽  
The Subject ◽  
Function Of Several Arguments

This article was supposed to be on ‘multivariate splines». An informal survey, taken recently by asking various people in Approximation Theory what they consider to be a ‘multivariate spline’, resulted in the answer that a multivariate spline is a possibly smooth piecewise polynomial function of several arguments. In particular the potentially very useful thin-plate spline was thought to belong more to the subject of radial basis funtions than in the present article. This is all the more surprising to me since I am convinced that the variational approach to splines will play a much greater role in multivariate spline theory than it did or should have in the univariate theory. Still, as there is more than enough material for a survey of multivariate piecewise polynomials, this article is restricted to this topic, as is indicated by the (changed) title.

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