Quaternion methods and models of regular celestial mechanics and astrodynamics

This paper is a review, which focuses on our work, while including an analysis of many works of other researchers in the field of quaternionic regularization. The regular quaternion models of celestial mechanics and astrodynamics in the Kustaanheimo-Stiefel (KS) variables and Euler (Rodrigues-Hamilton) parameters are analyzed. These models are derived by the quaternion methods of mechanics and are based on the differential equations of the perturbed spatial two-body problem and the perturbed spatial central motion of a point particle. This paper also covers some applications of these models. Stiefel and Scheifele are known to have doubted that quaternions and quaternion matrices can be used efficiently to regularize the equations of celestial mechanics. However, the author of this paper and other researchers refuted this point of view and showed that the quaternion approach actually leads to efficient solutions for regularizing the equations of celestial mechanics and astrodynamics.This paper presents convenient geometric and kinematic interpretations of the KS transformation and the KS bilinear relation proposed by the present author. More general (compared with the KS equations) quaternion regular equations of the perturbed spatial two-body problem in the KS variables are presented. These equations are derived with the assumption that the KS bilinear relation was not satisfied. The main stages of the quaternion theory of regularizing the vector differential equation of the perturbed central motion of a point particle are presented, together with regular equations in the KS variables and Euler parameters, derived by the aforementioned theory. We also present the derivation of regular quaternion equations of the perturbed spatial two-body problem in the Levi-Civita variables and the Euler parameters, developed by the ideal rectangular Hansen coordinates and the orientation quaternion of the ideal coordinate frame.This paper also gives new results using quaternionic methods in the perturbed spatial restricted three-body problem.

SU(N) q-Toda equations from mass deformed ABJM theory

It is known that the partition functions of the U(N)k × U(N + M)−k ABJM theory satisfy a set of bilinear relations, which, written in the grand partition function, was recently found to be the q-Painlevé III3 equation. In this paper we have suggested that a similar bilinear relation holds for the ABJM theory with $$ \mathcal{N} $$ N = 6 preserving mass deformation for an arbitrary complex value of mass parameter, to which we have provided several non-trivial checks by using the exact values of the partition function for various N, k, M and the mass parameter. For particular choices of the mass parameters labeled by integers ν, a as m1 = m2 = −πi(ν − 2a)/ν, the bilinear relation corresponds to the q-deformation of the affine SU(ν) Toda equation in τ-form.

Statistical Assessment of Simplified CPTU-Based Hydraulic Conductivity Curves

Classical theoretical approaches to evaluate CPTU-based hydraulic conductivity gave the expression of a bilinear relation defined by K D - B q Q t , for undrained and drained soils. In fact, a dividing line may be selected with a certain degree of subjectivity, and several undrained/drained points occur in opposite sites, which may make us doubt the necessity of two intersecting lines to distinguish the undrained and drained soils. And a uniform and relatively simple curve including an arc, parabola, or ellipse correlation may be more suitable and accessible from a practical point of view. A database in the Yangtze Delta region has been collated to assess the three curves compared with the bilinear line. With the graphical and statistical analyses, the results indicate that the arc, parabola, and ellipse simplified curves could give better performance than the bilinear line. Through the statistical analyses and the number of variables, the overall best curve was the ellipse with only two variables, which is expected to improve the application scope and simplicity of hydraulic conductivity.

Scaling relations for globular cluster systems in early-type galaxies

ABSTRACT The formation and growth of globular cluster systems (GCSs) is closely related to the evolutionary processes experienced by their host galaxies. In particular, their radial distributions scale with several properties of the galaxies and their haloes. We performed a photometric study, by means of HST/ACS archival data of several intermediate luminosity galaxies located in low-density environments. It was supplemented with available photometric data of GCSs from the Virgo and Fornax clusters, resulting in a sample of almost 30 GCSs for which we fitted their radial profiles. The resulting overall properties agree with those from previous studies, as we found that the effective radius, extension, and concentration of the GCS radial profiles correlate with the stellar mass, effective radius, and number of globular clusters, presenting in some cases a bilinear relation. The extension also correlates with the central velocity dispersion for central galaxies, but not for satellites. From a statistical comparison with numerical simulations we obtained good agreement between the effective radius and extension of the GCS scale with the effective and virial radius of the haloes, respectively. Finally, we analysed these results in the literature context.

Parafermion algebra with bilinear relation

In this paper, we find the parafermion algebra of order [Formula: see text] expressed in terms of bilinear relation (commutator form). We find the correct Hamiltonian of parafermion of order [Formula: see text] by using the parasupersymmetry. Finally, we discuss the comparison of specific heats for several parafermions of order [Formula: see text].

Modeling Tangential Contact of Rough Surfaces With Elastic- and Plastic-Deformed Asperities

A new tangential contact model between a rough surface and a smooth rigid flat is proposed in this paper. The model considers the contribution of both elastically deformed asperities and plastically deformed asperities to the total tangential load of rough surface. The method combining the Mindlin partial slip solution with the Hertz solution is used to model the contact formulation of elastically deformed asperities, and for the plastically deformed asperities, the solution combining the fully plastic theory of normal contact with the bilinear relation between the tangential load and deformation developed by Fujimoto is implemented. The total tangential contact load is obtained by Greenwood and Williamson statistical analysis procedure. The proposed model is first compared to the model considering only elastically deformed asperities, and the effect of mean separation and plasticity index on the relationship between the tangential load and deformation is also investigated. It is shown that the present model can be used to describe the stick–slip behavior of the rough surface, and it is a more realistic-based model for the tangential rough contact. A comparison with published experimental results is also made. The proposed model agrees very well with the experimental results when the normal load is small, and shows an error when the normal load is large.

BAXTER EQUATIONS AND DEFORMATION OF ABELIAN DIFFERENTIALS

In this paper the proofs are given of important properties of deformed Abelian differentials introduced earlier in connection with quantum integrable systems. The starting point of the construction is the Baxter equation. In particular, we prove the Riemann bilinear relation. The duality plays an important role in our consideration. The classical limit is considered in details.

Influence of Car Tonnage and Wheel Adhesion on Rail and Wheel Wear: A Laboratory Study

This study presents the results and analysis of a laboratory investigation, of rail and wheel wear under clean and dry tangent track conditions, utilizing the IIT-GMEMD quarter scale simulation facility. Important factors influencing rail degradation are discussed followed by five different load/lubrication classifications of contacts. Influence of two important parameters, viz. wheel load (N) and adhesion coefficient of the tractive wheels (μ), on rail and wheel wear has been studied under conditions of Hertzian simulation. Seven separate experiments were conducted to measure wear of rail and nontractive freight car wheels. These were followed by six additional wear tests, simulating a typical U.S. locomotive, to investigate the effect of adhesion coefficients. The wear rates for tonnages* exceeding 65–70 t car increase at a much higher rate than those for tonnages below 65 t. Nonlinear relationship showing wear rate proportional to N5.4 and a bilinear relation have been developed. Considerations of contact plasticity show that the stress corresponding to 68-t freight load is a threshold stress which when exceeded leads to continual plasticity of new rails thus preventing shakedown. The influence of adhesion coefficient is also quite nonlinear, the wear rates being much higher for μ > 0.3. Photomicrographs of the surfaces of the wheel and rail at the end of the tests showed mild wear for μ ≤ 0.25 and severe wear for μ ≥ 0.35 indicating a transition of wear mechanism from mild to severe slightly above μ = 0.25. Wear rate is found to be approximately proportional to the square of the adhesion coefficient. A bilinear relation of wear rate versus μ, which is more accurate, is also given. It was observed that the effect of adhesion is more severe than the effect of tonnage alone. However, the tonnage effect is of serious consequence regarding plastic shakedown of the rails. A formulation of wear rate as a combined function of tonnage and adhesion coefficient is given. The urgent need for a solution of this problem is pointed out.

ESR Studies on 205Tl Hyperfine Couplings in the Radical Cations of Tl(III)Porphyrins

The isotropic 205Tl hyperfine couplings obtained from the ESR spectra of the radical cations of Tl (III) meso tetraphenyl porphyrin (TPP), octaethyl porphyrin (OEP) and octaethyl chlorin (OEC) are reported. The radical cations were generated by electrooxidation in dichloromethane as solvent. A Karplus-Fraenkel type bilinear relation is used to interpret the 205Tl couplings, taking into account the sigma-pi spin polarization of Tl -N bonds by the spin density at the nitrogen atoms in the ligand as well as the direct pi interaction of the orbitals of Tl with the pi orbitals of the ligand. It is shown that for the cation radicals of Tl porphyrins, both these mechanisms contribute to the Tl couplings whereas for the cation radicals of Co- and Zn-porphyrins the sigma-pi polarization alone is sufficient to account for the metal hyperfine couplings. It is suggested that Tl-hyperfine couplings can be used to estimate the nitrogen spin densities of porphyrin radical systems when the nitrogen splittings are not resolved in the ESR spectra.