Conservation Laws and Associated Noether Type Vector Fields via Partial Lagrangians and Noether’s Theorem for the Liang Equation

We show how one can construct conservation laws of the Liang equation which is not variational but may be regarded as Euler-Lagrange in part. This first requires the determination of the Noether-type symmetries associated with the partial Lagrangian. The final construction of the conservation laws resort to a formula equivalent to Noether’s theorem. A variety of subclasses are given and, for each, a large number of conserved flows are found—the method is usable for any general choice of the variable speed of sound.     

Noether versus Killing symmetry of conformally flat Friedmann metric

In a recent study Noether symmetries of some static spacetime metrics in comparison with Killing vectors of corresponding spacetimes were studied. It was shown that Noether symmetries provide additional conservation laws that are not given by Killing vectors. In an attempt to understand how Noether symmetries compare with conformal Killing vectors, we find the Noether symmetries of the flat Friedmann cosmological model. We show that the conformally transformed flat Friedman model admits additional conservation laws not given by the Killing or conformal Killing vectors. Inter alia, these additional conserved quantities provide a mechanism to twice reduce the geodesic equations via the associated Noether symmetries.     

Experimental verification of response of embedded optical fiber Bragg grating sensors in non-homogeneous strain fields

One of the advantages of optical fiber sensors is their ease of embedment within a structure for non-destructive strain monitoring. In particular, Bragg grating sensors are written directly into an optical fiber hence remaining unobtrusive. In addition, several gratings can be written in series along a single fiber, permitting sensing at discrete points throughout the strain field. However, in regions of strong strain gradients, measuring the strain at discrete points may not be sufficient. One solution is to write a Bragg grating longer than the strain region of interest and use the change in its spectral response to determine the applied strain field as a function of position along the fiber. This paper presents an experimental verification of the response of an embedded optical fiber Bragg grating (OFBG) to applied non-homogeneous strain fields. Optical fiber Bragg grating sensors were embedded in four epoxy specimens of different forms so as to apply known strain functions along the gauge length when the specimen is under uniaxial tension. The complete spectral response of the Bragg gratings was then measured as a function of increasing load. The results are compared with analytical calculations, based on the piecewise-uniform period assumption for chirped gratings. Finally, the use of these spectra is discussed as possible basis functions for the resolution of an arbitrary applied strain distribution.     

Noether Symmetries Versus Killing Vectors and Isometries of Spacetimes

Symmetries of spacetime manifolds which are given by Killing vectors are compared with the symmetries of the Lagrangians of the respective spacetimes. We find the point generators of the one parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian (Noether symmetries). In the examples considered, it is shown that the Noether symmetries obtained by considering the Larangians provide additional symmetries which are not provided by the Killing vectors. It is conjectured that these symmetries would always provide a larger Lie algebra of which the KV symmetres will form a subalgebra. PACS: 04.25.-g, 02.20.Sv, 11.30.-j     

An analysis of the fourth‐order PDEs – ‘patterns’ and ‘di‐block polymers’

We analyze some fourth‐order partial differential equations that model the ‘propagation of hexagonal patterns’ and the ‘microphase separation of di‐block copolymers’. The underlying invariance properties and conservation laws of the models and related partial differential equations are studied. Copyright © 2017 John Wiley & Sons, Ltd.     

Consistent Burgers equation expansion method and its applications to high- dimensional Burgers-type equations

In this paper, a novel method, named the consistent Burgers equation expansion (CBEE) method, is proposed to solve nonlinear evolution equations (NLEEs) by the celebrated Burgers equation. NLEEs are said to be CBEE solvable if they are satisfied by the CBEE method. In order to verify the effectiveness of the CBEE method, we take (2+1)-dimensional Burgers equation as an example. From the (1+1)-dimensional Burgers equation, many new explicit solutions of the (2+1)-dimensional Burgers equation are derived. The obtained results illustrate that this method can be effectively extended to other NLEEs.     

A note on rearrangement of Fourier series

We give an example of a permutation of integers which preserves all convergent Fourier series and makes some divergent Fourier series converge.     

Calculated pre-exponential factors and energetics for adatom hopping on terraces and steps of Cu(1 0 0) and Cu(1 1 0)

We have calculated the vibrational dynamics and thermodynamics for Cu adatom hopping on terraces and near step edges on Cu(1 0 0) and Cu(1 1 0), using the embedded atom method for the interatomic potential. The local vibrational densities of states were calculated using real space Green’s function formalism and the thermodynamical functions were evaluated in the harmonic approximation. The calculated diffusion energy barriers for six specific local environments on Cu(1 0 0) agree well with experimental and previous theoretical results. Contribution of vibrational entropy to the change in the free energy of the system as the adatom moves from the equilibrium configuration (hollow site) to the saddle point, is found to be as much as 55 meV (144 meV) at 300 K (600 K). The prefactors for all 13 cases are found to be of the order of 10⁻³ cm²/s, almost independent of temperature, and the respective activation energy barriers.     

Short review of precipitation systems of colour sols determined by tyndallometer

Summary The short review of an attempt to measure colour sols by tyndallometer is given for some precipitation systems. The data are presented as precipitation diagrams.

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Zusammenfassung Es werden Versuche beschrieben, die Bildung gefärbter Niederschläge tyndallometrisch zu untersuchen und in Präzipitationsdiagrammen darzustellen.

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With 6 figures