Spline Approximation, Part 2: From Polynomials in the Monomial Basis to B-splines—A Derivation

In a series of three articles, spline approximation is presented from a geodetic point of view. In part 1, an introduction to spline approximation of 2D curves was given and the basic methodology of spline approximation was demonstrated using splines constructed from ordinary polynomials. In this article (part 2), the notion of B-spline is explained by means of the transition from a representation of a polynomial in the monomial basis (ordinary polynomial) to the Lagrangian form, and from it to the Bernstein form, which finally yields the B-spline representation. Moreover, the direct relation between the B-spline parameters and the parameters of a polynomial in the monomial basis is derived. The numerical stability of the spline approximation approaches discussed in part 1 and in this paper, as well as the potential of splines in deformation detection, will be investigated on numerical examples in the forthcoming part 3.

On the conformal symmetry of exceptional scalar theories

The DBI and special galileon theories exhibit a conformal symmetry at unphysical values of the spacetime dimension. We find the Lagrangian form of this symmetry. The special conformal transformations are non-linearly realized on the fields, even though conformal symmetry is unbroken. Commuting the conformal transformations with the extended shift symmetries, we find new symmetries, which when taken together with the conformal and shift symmetries close into a larger algebra. For DBI this larger algebra is the conformal algebra of the higher dimensional bulk in the brane embedding view of DBI. For the special galileon it is a real form of the special linear algebra. We also find the Weyl transformations corresponding to the conformal symmetries, as well as the necessary improvement terms to make the theories Weyl invariant, to second order in the coupling in the DBI case and to lowest order in the coupling in the special galileon case.

Superposition of Motions in The Space of Lagrange Variables

The technique of superposition of motions in the space of Lagrange variables is described, which allows us to obtain the equations of combined motion by replacing the Lagrange variables of superimposed (external) motion with Euler variables of nested (internal) motion. The components of velocity and acceleration in the combined motion obtained as a result of differentiating the equations of motion in time coincide with the results of vector addition of the velocities and accelerations of the particles involved in the superimposed motions at each moment of time. Examples of motion and superposition descriptions for absolutely solid and deformable bodies with equations for the main kinematic characteristics of motion, including for robot manipulators with three independent drives, pressing with torsion, bending with tension, and cross– helical rolling, are given. For example, given the fragment of calculation of forces in the kinematic pairs shown the advantages of the description of motion in Lagrangian form for the dynamic analysis of lever mechanisms, allows to determine the required external exposure when performing the energy conservation law at any time in any part of the mechanism.

On a cotangent form of a nonholonomic lagrangian dynamics

A Lagrangian form of dynamic equations for nonlinear nonholonomic constraints was studied by the first author in a previous paper. The aim of this paper is to put these equations in a cotangent form, according to some regularity conditions. It is particularized as an example to a decelerated motion of a free particle, when some dual simple equations are obtained.

Unified particle method with application to the 2013 Huangping landslides and tsunamis event of China

In this study, a unified particle method is presented to simulate the fluid-solid coupling problem in broad range of scales such as landslide and tsunamis. First, a general overview of the method is addressed, and the governing equations are solved in a Lagrangian form. Second, the method is used to simulate the Scott Russell wave generator experiment containing simple fluid-solid coupling, and the relationship between the simulation results and the experimental results is analyzed to verify the validity of the model. Then, the method is applied to the 2D processes simulation of Huangping landslide generated impulse waves in large scale. The results show that: in the verification test, the wave amplitude error between the simulation and the experimental data is almost 0. In the application case of Huangping landslides and tsunamis, the maximum impulse wave height obtained by the simulation is close to the local observations, which indicates that the method has high accuracy and credibility.

A model for magneto-elastic behaviour

In this paper, a coupled magnetoelastic model for isotropic ferromagnetic materialsis presented. The constitutive equations are written on the basis of the total energy in whichthe right Cauchy-Green strain tensor and the Lagrangian form of the magnetic eld strengthare used as the basic state variables. It is also applied to ferromacnetic electric steel for whichthe material parameters are calibrated.

Models, structure, and generality in Clerk Maxwell’s theory of electromagnetism

This article examines the gradual development of James Clerk Maxwell’s electromagnetic theory, arguing that he aimed at general structures through his models, illustrations, formal analogies, and scientific metaphors. It also considers a few texts in which Maxwell expounds his conception of physical theories and their relation to mathematics. Following a discussion of Maxwell’s extension of an analogy invented by William Thomson in 1842, the article analyzes Maxwell’s geometrical expression of Michael Faraday’s notion of lines of force. It then revisits Maxwell’s honeycomb model that he used to obtain his system of equations and the concomitant unification of electricity, magnetism, and optics. It also explores Maxwell’s view about the Lagrangian form of the fundamental equations of a physical theory. It shows that Maxwell was guided by general structural requirements that were inspired by partial and temporary models; these requirements were systematically detailed in Maxwell’s 1873 Treatise on electricity and magnetism.