ION-CYCLOTRON ABSORPTION OF FAST WAVES IN A CYLINDRICAL CURRENT-CARRYING PLASMA

Influence of a longitudinal stationary current on the absorption and the radial structure of fast waves in a cylindrical current-carrying plasma is discussed. To evaluate the dispersion equation for fast waves, there was used the dielectric tensor taking into account the radial current structure and geometry of the confining helical magnetic field by the plasma safety factor. It is shown that the damping rate of fast waves in a non-equilibrium current-carrying plasma differ from those for an equilibrium plasma column in a homogeneous magnetic field nearby the cutoffs and resonances due to the rotational transformation (including shear-effects) of the helical magnetic field lines.

Formation Design for Single-Pass GEO InSAR Considering Earth Rotation Based on Coordinate Rotational Transformation

The single-pass geosynchronous synthetic aperture radar interferometry (GEO InSAR) adopts the formation of a slave satellite accompanying the master satellite, which can reduce the temporal decorrelation caused by atmospheric disturbance and observation time gap between repeated tracks. Current formation design methods for spaceborne SAR are based on the Relative Motion Equation (RME) in the Earth-Centered-Inertial (ECI) coordinate system (referred to as ECI-RME). Since the Earth rotation is not taken into account, the methods will lead to a significant error for the baseline calculation while applied to formation design for GEO InSAR. In this paper, a formation design method for single-pass GEO InSAR based on Coordinate Rotational Transformation (CRT) is proposed. Through CRT, the RME in Earth-Centered-Earth-Fixed (ECEF) coordinate system (referred to as ECEF-RME) is derived. The ECEF-RME can be used to describe the accurate baseline of close-flying satellites for different orbital altitudes, but not limited to geosynchronous orbit. Aiming at the problem that ECEF-RME does not have a regular geometry as ECI-RME does, a numerical formation design method based on the minimum baseline error criterion is proposed. Then, an analytical formation design method is proposed for GEO InSAR, based on the Minimum Along-track Baseline Criterion (MABC) subject to a fixed root mean square of the perpendicular baseline. Simulation results verify the validity of the ECEF-RME and the analytical formation design method. The simulation results also show that the proposed method can help alleviate the atmospheric phase impacts and improve the retrieval accuracy of the digital elevation model (DEM) compared with the ECI-RME-based approach.

HybridHAM: A Novel Hybrid Heuristic for Finding Hamiltonian Cycle

Hamiltonian Cycle Problem is one of the most explored combinatorial problems. Being an NP-complete problem, heuristic approaches are found to be more powerful than exponential time exact algorithms. This paper presents an efficient hybrid heuristic that sits in between the complex reliable approaches and simple faster approaches. The proposed algorithm is a combination of greedy, rotational transformation and unreachable vertex heuristics that works in three phases. In the first phase, an initial path is created by using greedy depth first search. This initial path is then extended to a Hamiltonian path in second phase by using rotational transformation and greedy depth first search. Third phase converts the Hamiltonian path into a Hamiltonian cycle by using rotational transformation. The proposed approach could find Hamiltonian cycles from a set of hard graphs collected from the literature, all the Hamiltonian instances (1000 to 5000 vertices) given in TSPLIB, and some instances of FHCP Challenge Set. Moreover, the algorithm has O(n3) worst case time complexity. The performance of the algorithm has been compared with the state-of-the-art algorithms and it was found that HybridHAM outperforms others in terms of running time.

The Unruh effect for eccentric uniformly rotating observers

It is common to use Galilean rotational transformation (GRT) to investigate the Unruh effect for uniformly rotating observers. However, the rotating observer in this subject is an eccentric observer while GRT is only valid for centrally rotating observers. Thus, the reliability of the results of applying GRT to the study of the Unruh effect might be considered as questionable. In this work, the rotational analog of the Unruh effect is investigated by employing two relativistic rotational transformations corresponding to the eccentric rotating observer, and it is shown that in both cases, the detector response function is nonzero. It is also shown that although consecutive Lorentz transformations cannot give a frame within which the canonical construction can be carried out, the expectation value of particle number operator in canonical approach will be zero if we use modified Franklin transformation. These conclusions reinforce the claim that correspondence between vacuum states defined via canonical field theory and a detector is broken for rotating observers. Some previous conclusions are commented on and some controversies are also discussed.

Understanding of Eight Grade Students about Transformation Geometry: Perspectives on Students’ Mistakes

People need the idea of transformation geometry in order to understand the nature and environment they live in. The teachers should provide learning environments towards perceptual understanding in symmetry training and development practice skills of the students. In order to make up such a learning environment, teachers should have information about the mathematical structure of the concept of symmetry, the difficulties the students encounter while learning, misconceptions and the causes. Therefore, the challenges and the common mistakes the 8th grade middle school students encounter about the transformation geometry was analysed in the study. The study was conducted using mixed method designs with 125 8th grade students. At the end of the study, it was observed that the students understood that the translation transformation is a movement of replacement, but they had difficulty in the topics such as the direction of the transformation and the position of the shape within the transformation. A misconception was developed for the reflection by confusing the similarity with the congruence property of the shapes. It was detected that the students had difficulty in identifying the equation of the axis of symmetry for the images of the shapes under reflection, confused the rule that the points intersecting with the symmetry of the shape within the reflection should intersect with the image under transformation and they made mistakes since they couldn’t explore the relationship between symmetry axis in regular polygons and sides. They had problems in finding and practicing the angle of rotation about rotational transformation and also. In the study, learning environments were recommended towards overcoming these challenges for teachers and coursebook writers and improving conceptual information and the skill to practice these concepts.

Role of division algebra in seven-dimensional gauge theory

The algebra of octonions 𝕆 forms the largest normed division algebra over the real numbers ℝ, complex numbers ℂ and quaternions ℍ. The usual three-dimensional vector product is given by quaternions, while octonions produce seven-dimensional vector product. Thus, octonionic algebra is closely related to the seven-dimensional algebra, therefore one can extend generalization of rotations in three dimensions to seven dimensions using octonions. An explicit algebraic description of octonions has been given to describe rotational transformation in seven-dimensional space. We have also constructed a gauge theory based on non-associative algebra to discuss Yang–Mills theory and field equation in seven-dimensional space.

A Closed-Form Second-Order Reliability Method Using Noncentral Chi-Squared Distributions

In the second-order reliability method (SORM), the probability of failure is computed for an arbitrary performance function in arbitrarily distributed random variables. This probability is approximated by the probability of failure computed using a general quadratic fit made at the most probable point (MPP). However, an easy-to-use, accurate, and efficient closed-form expression for the probability content of the general quadratic surface in normalized standard variables has not yet been presented. Instead, the most commonly used SORM approaches start with a relatively complicated rotational transformation. Thereafter, the last row and column of the rotationally transformed Hessian are neglected in the computation of the probability. This is equivalent to approximating the probability content of the general quadratic surface by the probability content of a hyperparabola in a rotationally transformed space. The error made by this approximation may introduce unknown inaccuracies. Furthermore, the most commonly used closed-form expressions have one or more of the following drawbacks: They neither do work well for small curvatures at the MPP and/or large number of random variables nor do they work well for negative or strongly uneven curvatures at the MPP. The expressions may even present singularities. The purpose of this work is to present a simple, efficient, and accurate closed-form expression for the probability of failure, which does not neglect any component of the Hessian and does not necessitate the rotational transformation performed in the most common SORM approaches. Furthermore, when applied to industrial examples where quadratic response surfaces of the real performance functions are used, the proposed formulas can be applied directly to compute the probability of failure without locating the MPP, as opposed to the other first-order reliability method (FORM) and the other SORM approaches. The method is based on an asymptotic expansion of the sum of noncentral chi-squared variables taken from the literature. The two most widely used SORM approaches, an empirical SORM formula as well as FORM, are compared to the proposed method with regards to accuracy and computational efficiency. All methods have also been compared when applied to a wide range of hyperparabolic limit-state functions as well as to general quadratic limit-state functions in the rotationally transformed space, in order to quantify the error made by the approximation of the Hessian indicated above. In general, the presented method was the most accurate for almost all studied curvatures and number of random variables.