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ABSTRACT The need to use rotations occurs very often in different domains. We present a basic extensive treatment of rotations in 3D. The results are presented and derived in a coordinate‐free setting, where no frames are required and no components of any matrix are manipulated. We start with the direct problem of establishing the finite rotation formula. Then we consider the composition and the vector decomposition of finite rotations. We conclude the paper by considering the inverse problem namely finding the axis of rotation and the angle of rotation from its effect on vectors.

We proposed vector diffraction equations based on 3D Huygens-Fresnel principle and vector decomposition in Cartesian coordinate. Our expression should more accurate than classical Richards-Wolf diffraction, because the deduction of Richards-Wolf diffraction used a number of approximation, while our expression is based on two fundamental physical principles: vector decomposition and Huygens-Fresnel principle. Our expressions mathematical strictly satisfy conservation of energy, while Richards-Wolf’s method is based on the geometrical law of conservation of energy and RCWA(Rigorous Coupled Wave Analysis) is also an approximation. The simulated results of our vector diffraction equations with different NA (numerical aperture) are almost the same as Richards-Wolf diffraction results. Traditional Richards-Wolf vector diffraction equations, which are based on angular spectrum of plane wave, have no clear geometrical meaning, while our vector diffraction equations here have clear geometrical meaning. Our approach of vector diffraction to scalar diffraction is much simpler than Richards-Wolf’s approach. The generation of x/y/z components and their properties in the image plane can be investigated clearly by vector decomposition and synthesis. Their intensity ratios can be explained using paraxial approximation very well. The integral region of exit pupil could be reduced because of the symmetric property of the light vector decomposition and synthesis, thus our expressions have advantages on numerical simulation. Our analysis also indicate that light must ‘polarize’ in other extra dimensions and photon spin is a ‘polarize/resonate’ state in extra dimensions. Also, there is a potential convenient ability to study vector optical transfer function (OTF) analytically in Cartesian coordinate.

Arbitrarily oriented object detection is one of the most-popular research fields in remote sensing image processing. In this paper, we propose an approach to predict object angles indirectly, thereby avoiding issues related to angular periodicity and boundary discontinuity. Our method involves representing the long edge and angle of an object as a vector, which we then decompose into horizontal and vertical components. By predicting the two components of the vector, we can obtain the angle information of the object indirectly. To facilitate the transformation between angle-based representation and the proposed vector-decomposition-based representation, we introduced two novel techniques: angle-to-vector encode (ATVEncode) and vector-to-angle decode (VTADecode). These techniques not only improve the efficiency of data processing, but also accelerate the training process. Furthermore, we propose an adaptive coarse-to-fine positive–negative-sample-selection (AdaCFPS) method based on the vector-decomposition-based representation of the object. This method utilizes the Kullback–Leibler divergence loss as a matching degree to dynamically select the most-suitable positive samples. Finally, we modified the YOLOX model to transform it into an arbitrarily oriented object detector that aligns with our proposed vector-decomposition-based representation and positive–negative-sample-selection method. We refer to this redesigned model as the vector-decomposition-based object detector (VODet). In our experiments on the HRSC2016, DIOR-R, and DOTA datasets, VODet demonstrated notable advantages, including fewer parameters, faster processing speed, and higher precision. These results highlighted the significant potential of VODet in the context of arbitrarily oriented object detection.

Vector decomposition of P- and S-wave modes from elastic seismic wavefields is a key step in elastic reverse-time migration (ERTM) to effectively improve the multi-wave imaging accuracy. Most previously developed methods based on the apparent velocities or the polarization characteristics of different wave modes are unable to accurately achieve the vector decomposition of P- and S-wave modes. To effectively overcome the shortcomings of conventional methods, we develop a vector decomposition method of P- and S-wave modes using self-attention deep convolutional generative adversarial networks (SADCGANs) to effectively separate the horizontal and vertical components of P- and S-wave modes from elastic seismic wavefields and accurately preserve their amplitude and phase characteristics for isotropic elastic media. For an elastic model, we use many time slices for a given source position to train the neural network, and use other time slices not in this training dataset to test the neural network. Numerical examples of different models demonstrate the effectiveness and feasibility of our developed method and indicate that it provides an effective intelligent data-driven vector decomposition method of P- and S-wave modes.

This paper presents two novel neural network models for radio-frequency (RF) power amplifiers (PAs): vector decomposed time-delay neural network (VDTDNN) model and augmented vector decomposed time-delay neural network (AVDTDNN) model. In contrast to conventional neural network-based models, VDTDNN and AVDTDNN comply with the physical characteristics of RF PAs by employing carefully designed network structures. In particular, the nonlinear operations are conducted only on the magnitude of the input signals, while the phase information is recovered with the linear weighting. Linear terms with shortcut connection, as well as high-order terms, can be used to further boost the modeling performance. The complexity analysis shows that the proposed models have significantly lower complexity than the existing neural network models. A wideband GaN RF PA excited by the 40- and 60-MHz OFDM signals were employed to evaluate the performance. The extensive experimental results reveal that the proposed VDTDNN and AVDTDNN models can achieve better linearization performance with lower computational complexity compared with the existing neural network-based models.

The stock market of any country indicates the health of a country. It also indicates the confidence of the population of the country in its economy. India, being an emerging country, has exhibited a substantial rate of growth over past years. One of the parameters that determine the growth rate and health of the economy of a nation is its stock market. The aim of this research is to find the volatility in the global oil price, its impact on various sectoral indices listed on the Indian Stock Exchange. BSE SENSEX and other sectoral Indices listed on the Bombay Stock Exchange (BSE) such as BSE Realty, BSE Power, BSE Oil and Gas, BSE Metal, BSE Consumer Durables, BSE Capital Goods, BSE BANKEX, BSE AUTO, BSE Utilities, BSE telecom, BSE Information Technology, BSE Industrials, BSE Finance, BSE Fast Moving Consumer Goods, BSE Energy, BSE Basic Materials are considered to determine the impact of fluctuation of oil price in the stocks of various sectors. The index BSE SENSEX has been considered to determine the performance of the Indian Equity market with respect to the oil price fluctuations. Log normal of the daily closing prices of each of the stock indices and the oil prices have been considered to carry out the research. Techniques such as unit root test, Jarque beta test, correlation test, Granger Causality test, Impulse response and Vector decomposition test have been applied on the closing prices of indices between 1st April, 2010 to 31st December 2017 (1st Financial quarter of the year 2010, in India to the 3rd Financial quarter of the year 2017). The tests show that the average stock returns for the sectors Auto, Bankex, Basic Materials, Capital Goods, Energy, Finance, FMCG, IT, Oil and Gas, Sensex, Industrials is positive whereas for others sectors, average return is negative. Although oil is one of the largest consumers of oil in the world, its stock market returns does not depend on the fluctuation in the oil prices.