Introduction Of Coordinates Research Articles

A stretchable, mechanically robust polymer exhibiting shape-memory-assisted self-healing and clustering-triggered emission

Self-healing and recyclable polymer materials are being developed through extensive investigations on noncovalent bond interactions. However, they typically exhibit inferior mechanical properties. Therefore, the present study is aimed at synthesizing a polyurethane–urea elastomer with excellent mechanical properties and shape-memory-assisted self-healing behavior. In particular, the introduction of coordination and hydrogen bonds into elastomer leads to the optimal elastomer exhibiting good mechanical properties (strength, 76.37 MPa; elongation at break, 839.10%; toughness, 308.63 MJ m−3) owing to the phased energy dissipation mechanism involving various supramolecular interactions. The elastomer also demonstrates shape-memory properties, whereby the shape recovery force that brings damaged surfaces closer and facilitates self-healing. Surprisingly, all specimens exhibite clustering-triggered emission, with cyan fluorescence is observed under ultraviolet light. The strategy reported herein for developing multifunctional materials with good mechanical properties can be leveraged to yield stimulus-responsive polymers and smart seals.

Ecosystem health assessment in the pearl river estuary of China by considering ecosystem coordination.

Marine ecosystem is a complex nonlinear system. However, ecosystem health assessment conventionally builds on a linear superposition of changes in ecosystem components and probably fails to evaluate nonlinear interactions among various components. To better reflect the intrinsic interactions and their impacts on ecosystem health, an ecosystem coordination index, defined as the matching level of ecosystem structure/services, is proposed and incorporated into the ecosystem health index for a systematic diagnosis in the Pearl River Estuary, China. The analysis results show that the ecosystem health index over the last three decades decreased from 0.91 to 0.50, indicating deteriorating from healthy to unhealthy status. The health index is 3–16% lower than that calculated using the common method without considering ecosystem coordination. Ecosystem health degradation in the Pearl River Estuary manifested as significant decreases in structure/services and somewhat mismatching among them. Overall, the introduction of coordination in ecosystem health assessment could improve the understanding of the mechanism of marine ecosystem change and facilitate effective restoration of ecosystem health.

Some remarks on the paper ‘A unified tensor approach to the analysis of mechanical systems’ by A. A. Fogarasy and M. R. Smith

The tensor calculus is a powerful mathematical tool in the analysis of mechanical problems (or generally, of problems in physics). The aim of Fogarasy and Smith's ‘A unified tensor approach to the analysis of mechanical systems’ is to demonstrate this fact considering a problem of dynamical analysis of mechanisms. Fogarasy and Smith are applying the traditional approach in tensor calculus, which comes originally from the mathematics. This implies the a priori introduction of coordinates, which is usual in the analysis of mechanisms. Later the tensor calculus is used for the discussion of the basic equations and their mathematical handling. In this sense, the results obtained by Fogarasy and Smith are mathematically correct, and the interested reader can now have many fruitful and stimulating ideas.

Peripheral zone of disturbance from a local impulse in a string mesh formed by two systems of strings with different impedances

The work is based on recurrent relations of a new type for squares of normalized orthogonal polynomials. These relations are applied to problems of transverse oscillations of a string mesh. For a homogeneous mesh, the influence function of a local impulse is represented by Kravchuk polynomials whose discrete arguments are two-dimensional numbers of nodes and discrete time that are counted from the point and time of application of the impulse. We apply the Szegö generalization of Laguerre polynomials to find an asymptotic representation of the influence function in the case where strings of two systems have essentially different impedances. Introduction of coordinates that move together with the expanding influence front simplifies the analysis of the displacement diagram for first waves and their dynamics. We derive a linear first order partial differential equation for squares of normalized orthogonal Laguerre polynomials (modified by the authors) that gives a continual description of the process in the whole. Bibliography: 7 titles.

Projective geometry and modern algebra

Historical foreword. Affine geometry: affine planes transformations of the affine plane. Projective planes: completion of the affine plane homogeneous coordinates for the real projective plane. Desargues' theorem and the principle of duality: the axiom P5 of Desargues Moulton's example axioms for projective space principle of duality. A brief introduction to groups: elements of group theory automorphisms of the projective plane of 7 points. Elementary synthetic projective geometry: Fano's axiom P6 harmonic points perpectivities and projectivities. The fundamental theorem for projectivities on a line: the fundamental theorem: axiom P7 geometry of complex numbers Pappu's theorem. A brief introduction to division rings: division rings the quaternions H a noncommutative division ring with characteristics p. Projective planes over division rings: P2(R) the automorphism group of P2 (R) the algebraic meaning of axioms P6 and P7 independence of axioms. Introduction of coordinates in a projective plane: the major and minor Desargues' axioms division ring number lines introducing coordinates in A. Mobius transformations and cross ratio: assessment Mobius transformations of the extended field cross ratio: a projective invariant. Projective collineations: projective collineations elations and homologies the fundamental theorem of projective collineation Ceva's theorem. (Part contents).

Logic Geometry and Algebra in Modeling

An axiomatic foundation for many identification and modeIing methods is presented. Fol1owing Popper's philosophy, we take the experimentally verifiable statements (including the (non-)repeatability) one makes about the system as primordial. The logic of the system is then shown to be representable by a projective geometry. Algebraization follows through the introduction of coordinates. For a class of phenomena, this algebro-geometrical description leads to the usual Hankel base.d system realization procedures, and the geometrical methods of canonical correlations and principal component analysis follow “naturally” in this framework. Hence this foundational or axiomatic approach not only leads to an economy of thought, but also unifies many existing procedures [6, 8].

Geometric algebra for the High School Program

Plane affine geometry is developed from four axioms, up to the construction of a field (in general non-commutative), introduction of coordinates and derivation of equations for lines. The development is intended to be intelligible and rewarding to better high school students.

Compression Wave Surrounding an Expanding Sphere

The method of describing the propagation of finite-amplitude waves based on the introduction of coordinates “accompanying” the wave system is used for the consideration of the radiation of a compression wave by a sphere that is expanding at a constant rate in a perfect medium. To estimate the degree of accuracy of this method, the results are compared with the exact solution of G. I. Taylor. [Proc. Roy. Soc. (London) A186, 273–292 (1946)].