Q-binomial Research Articles

Manipulation of curved beams using beam-domain optimization.

An efficient scheme for the design of aperture fields (distributed sources) that radiate arbitrary trajectory curved (accelerating) beams, with enhanced controllability of various beam features, is presented. The scheme utilizes a frame-based phase-space representation of aperture fields to overcome the main hurdles in the design for large apertures: First, it uses the a-priory localization of caustic beams to significantly reduce the optimization problem's variable space, to that of few Gaussian window coefficients accurately capturing those beams. Then, the optimization problem is solved in the reduced (local) spectral domain. We adopt a linearization approach that enables the solution by sequential application of conventional convex optimization tools, which are naturally compatible with the proposed phase-space representation. The localized nature of the Gaussian windows' radiation is used also for fast field evaluation at a greatly reduced number of optimization constraint points. The significant enhancement in the controllability over the various beam parameters is demonstrated through a range of examples.

A New q-Hypergeometric Symbolic Calculus in the Spirit of Horn, Borngässer, Debiard and Gaveau

The purpose of this article is to introduce a new complete multiple q-hypergeometric symbolic calculus, which leads to q-Euler integrals and a very similar canonical system of q-difference equations for multiple q-hypergeometric functions. q-analogues of recurrence formulas in Horns paper and Borngässers thesis lead to a more exact way to find these Frobenius solutions. To find the right formulas, the parameters in q-shifted factorials can be changed to negative integers, which give no extra q-factors. In proving these q-formulas, in the limit q→1 we obtain versions of the paper by Debiard and Gaveau for the solution of differential or q-difference equations. The paper is also a correction of some of the statements in the paper by Debiard and Gaveau, e.g., the Euler integrals and other solutions to differential equations for Appell functions, also without references to page numbers in the standard work of Appell and Kampé de Fériet. Sometimes the q-binomial theorem is used to simplify q-integral formulas. By the Horn method, we find another solution to the Appell Φ1 function partial differential equation, which was not mentioned in the thesis by Le Vavasseur 1893.

Function fitting for modeling seasonal normalized difference vegetation index time series and early forecasting of soybean yield

Forecasting crop yields based on remote sensing data is one of the most important tasks in agriculture. Soybean is the main crop in the Russian Far East. It is desirable to forecast soybean yield as early as possible while maintaining high accuracy. This study aimed to investigate seasonal time series of the normalized difference vegetation index (NDVI) to achieve early forecasting of soybean yield. This research used data from the Moderate Resolution Image Spectroradiometer (MODIS), an arable-land mask obtained from the VEGA-Science web service, and soybean yield data for 2008–2017 for the Jewish Autonomous Region (JAR) districts. Four approximating functions were fitted to model the NDVI time series: Gaussian, double logistic (DL), and quadratic and cubic polynomials. In the period from calendar weeks 22–42 (end of May to mid-October), averaged over two districts, the model using the DL function showed the highest accuracy (mean absolute percentage error –4.0%, root mean square error (RMSE) –0.029, P < 0.01). The yield forecast accuracy of prediction in the period of weeks 25–30 in JAR municipalities using the parameters of the Gaussian function was higher (P < 0.05) than that using the other functions. The mean forecast error for the Gaussian function was 14.9% in week 25 (RMSE was 0.21 t ha−1) and 5.1%−12.9% in weeks 26–30 (RMSE varied from 0.06 to 0.15 t ha−1) according to the 2013–2017 data. In weeks 31–32, the error was 5.0%−5.4% (RMSE was 0.07 t ha−1) using the Gaussian parameters and 7.4%−7.7% (RMSE was 0.09–0.11 t ha−1) for the DL function. When the method was applied to municipal districts of other soy-producing regions of the Russian Far East. RMSE was 0.14–0.32 t ha−1 in weeks 25–26 and did not exceed 0.20 t ha−1 in subsequent weeks.

The q-log-concavity and unimodality of q-Kaplansky numbers

q-Kaplansky numbers were considered by Chen and Rota. We find that q-Kaplansky numbers are connected to the symmetric differences of Gaussian polynomials introduced by Reiner and Stanton. Based on the work of Reiner and Stanton, we establish the unimodality of q-Kaplansky numbers. We also show that q-Kaplansky numbers are the generating functions for the inversion number and the major index of two special kinds of (0,1)-sequences. Furthermore, we show that q-Kaplansky numbers are strongly q-log-concave.

An evolving ensemble model of multi-stream convolutional neural networks for human action recognition in still images

Still image human action recognition (HAR) is a challenging problem owing to limited sources of information and large intra-class and small inter-class variations which requires highly discriminative features. Transfer learning offers the necessary capabilities in producing such features by preserving prior knowledge while learning new representations. However, optimally identifying dynamic numbers of re-trainable layers in the transfer learning process poses a challenge. In this study, we aim to automate the process of optimal configuration identification. Specifically, we propose a novel particle swarm optimisation (PSO) variant, denoted as EnvPSO, for optimal hyper-parameter selection in the transfer learning process with respect to HAR tasks with still images. It incorporates Gaussian fitness surface prediction and exponential search coefficients to overcome stagnation. It optimises the learning rate, batch size, and number of re-trained layers of a pre-trained convolutional neural network (CNN). To overcome bias of single optimised networks, an ensemble model with three optimised CNN streams is introduced. The first and second streams employ raw images and segmentation masks yielded by mask R-CNN as inputs, while the third stream fuses a pair of networks with raw image and saliency maps as inputs, respectively. The final prediction results are obtained by computing the average of class predictions from all three streams. By leveraging differences between learned representations within optimised streams, our ensemble model outperforms counterparts devised by PSO and other state-of-the-art methods for HAR. In addition, evaluated using diverse artificial landscape functions, EnvPSO performs better than other search methods with statistically significant difference in performance.

Variance of real zeros of random orthogonal polynomials for varying and exponential weights

We determine the asymptotics for the variance of the number of zeros of random linear combinations of orthogonal polynomials of degree at most n associated with varying weights e−2nQn, with Gaussian coefficients. We deduce asymptotics of the variance for fixed exponential weights e−2Q. In particular, we show that very generally, the variance is asymptotic to Cn, where the constant C involves a universal constant and an equilibrium density associated with the weight(s).

On GCED matrices over UFDs

An extension of the GCED matrices from the domain of natural integers to the unique factorization domain is given. The structure of these type of matrices defined on both arbitrary sets and GCED-closed sets are presented. Moreover, we present exact expressions for the determinant and the inverse of such matrices. The domains of Gaussian integers and polynomials over finite fields are used to illustrate the work.

Compression Method of NMR Echo Data Obtained From Complex Pore Structure Formation

To reduce the redundancy of nuclear magnetic resonance (NMR) echo data, kernel principal component analysis (KPCA) is used in the compression. The algorithm of data compression using KPCA is introduced in detail, and the specific calculation process is given. Four commonly used kernel functions are selected: Gaussian kernel, linear kernel, polynomial kernel, and exponential kernel. The compression effect of KPCA based on these four kernel functions on the NMR echo data obtained from complex pore structure formation is compared and analyzed. The study found that: Principal component analysis (PCA) is not suitable for the compression of NMR echo data obtained from the formation with complex pore structure. Compared with PCA, KPCA based on all the four kernels has obvious advantages in computational efficiency. The compression capabilities of KPCA based on Gaussian kernel and exponential kernel are all lower than PCA. The compression capability (CA) of KPCA based on linear kernel is equivalent to PCA. KPCA based on polynomial kernel, when the kernel parameter <inline-formula> <tex-math notation="LaTeX">$\sigma \ge 2$ </tex-math></inline-formula>, its CA is higher than PCA. The compression accuracy of KPCA based on Gaussian kernel, linear kernel, and exponential kernel is stable in the formations with complex pore structure. KPCA based on polynomial kernel is sensitive to the kernel parameter, only when the kernel parameter <inline-formula> <tex-math notation="LaTeX">$\sigma =5$ </tex-math></inline-formula>, the optimal compression accuracy can be obtained. At this time, the optimal compression effect is obtained in the complex pore formation.

Optimized Load Frequency Control for Single Area Power System Using Linear Quadratic Gaussian Technique and Coefficient Diagram Method

Optimized Load Frequency Control for Single Area Power System Using Linear Quadratic Gaussian Technique and Coefficient Diagram Method

Some identities of Gaussian binomial coefficients

In this paper, we present some identities of Gaussian binomial coefficients with respect to recursive sequences, Fibonomial coefficients, and complete functions by use of their relationships.

Joint SPX–VIX Calibration With Gaussian Polynomial Volatility Models: Deep Pricing With Quantization Hints

Joint SPX–VIX Calibration With Gaussian Polynomial Volatility Models: Deep Pricing With Quantization Hints

A multi-peak detection algorithm for FBG based on WPD-HT

An accurate multi-peak detection algorithm based on wavelet packet decomposition (WPD) denoising and Hilbert transform (HT) is proposed. WPD and wavelet threshold method are used to denoise the high-frequency part of the spectrum. HT and parabola fitting are used to divide the peak area adaptively and calculate the central wavelength. Simulation results show that compared with the other four denoising methods, this denoising algorithm can solve the problem of background noise more effectively. Compared with the Gaussian LM (Levenberg-Marquardt) algorithm, Centroid method, and polynomial fitting method, this method has higher precision. Experimental results of real-time temperature detection show that the stability is better than 0.5 pm at 0℃ and the temperature monitoring accuracy is better than 0.352℃ in the range of −20℃∼ 40℃, which is the best among the four algorithms. The computational speed is 81.2% higher than the Gauss LM algorithm. In general, this method can be applied to FBG real-time temperature monitoring and demodulation system under the condition of background noise.

On Solution of First-Order Linear Systems of Partial Differential Equations

Explicit formulas for the first-order partial differential equations system solving were obtained. Solution found for the system with initial conditions. Calculation examples establishing statements truth mentioned. Searching for partial differential equations system solution mathematical expectation became more difficult issue as partial differential equations system with random processes coefficients were covered. Gaussian coefficients and uniformly distributed random process cases examples has been reviewed.

Generation of an Airy-related beam from the sinusoidal dark hollow beam

In this paper, an Airy transformation of sinusoidal dark hollow beam (SDHB) is introduced and used to generate a unique Airy-related beam. The analytical expression for the SDHB passing through an optical Airy transformation system (OATS) is derived based on the Collins diffraction integral formula. Furthermore, dependence of the characteristics of the generated Airy-related beam on several factors, mainly including the control parameter of the OATS and the incident beam’s Gaussian coefficient and order, is investigated. Results show that the intensity distribution, orientation, size, position and number of side lobes of the generated Airy-related beams are modulated and controlled by the self-parameters of the incident SDHB and the parameters of the OATS. The results obtained here will enrich the research on the characteristics of the hollow beam and the generation method of Airy-related beam, and also expand the application of SDHB.

Study of large axial space rebinning algorithm in γ-photon industrial tomography image reconstruction

The accuracy of the existing single slice and Fourier rebinning algorithms depends on the projection angle of the line of response. The increase of such projection angle with the detector size, typical in the large axial space of γ-photon industrial detection, and the loss of some projection data after rebinning, result in the degradation of the image quality. In addition, those algorithms consider the probability of positron annihilation equally distributed along the line of response, which prevents to estimate accurately the positions of the annihilation point, and can originate artifacts and noise in the reconstructed image. In this work, we propose an alternative large axial space rebinning algorithm. In that algorithm, initially the line of response is divided into transverse and axial components. Then, each line of response is uniformly rebinned into all the 2D sinogram data intersecting with it. To improve the accuracy of the estimate of the annihilation point location and suppress the noise effectively, we assign a Gaussian weight coefficient to the projection data, and optimise the rebinning algorithm with it. Finally, we reconstruct the image on the basis of the 2D sinograms with the optimised weights. On the computational side, the algorithm is also accelerated by making use of parallel computing. Both simulation and experimental results show that the proposed method improves the contrast and spatial resolution of 2D reconstructed images. Furthermore, the reconstruction time is not affected by the new method, which is therefore expected to meet the demand of γ-photon industrial inspection imaging.

Factors of certain sums involving central q-binomial coefficients

Recently, Ni and Pan proved a q-congruence on certain sums involving central q-binomial coefficients, which was conjectured by Guo. In this paper, we give a generalization of this q-congruence and confirm another q-congruence, also conjectured by Guo. Our proof uses Ni and Pan’s technique and a simple q-congruence observed by Guo and Schlosser.

Semi-invariants of binary forms and Sylvester’s theorem

We obtain a combinatorial formula related to the shear transformation for semi-invariants of binary forms, which implies the classical characterization of semi-invariants in terms of a differential operator. Then, we present a combinatorial proof of an identity of Hilbert, which leads to a relation of Cayley on semi-invariants. This identity plays a crucial role in the original proof of Sylvester’s theorem on semi-invariants in connection with the Gaussian coefficients. Moreover, we show that the additivity lemma of Pak and Panova which yields the strict unimodality of the Gaussian coefficients for \(n,k \ge 8\) can be deduced from the ring property of semi-invariants.

Recent acceleration of the Earth rotation in the summer of 2020: possible causes and effects

The extreme acceleration of the Earth rotation observed in the summer of 2020 is considered. It is concluded that this phenomenon is a consequence of two factors: the longterm acceleration of the Earth rotation, which has been observed since the 1970s, and the extremely strong meteorological excitation of the LOD, which took place in the summer of 2020. The coincidence of the anomaly of the AAM and the geomagnetic Dst index, as well as the correlation between the LOD on the one hand and the solar wind speed and the Gaussian coefficients of the expansion of the Earth’s magnetic field, on the other, are noted. The problem of negative leap second is considered. Preliminary estimates have been made of introduction of a negative leap second, if the current trends in the behavior of UT1-UTC continue. The conclusion is made about the low probability of such an event.

Hovering Helicopter Rotors Modeling Using the Actuator Line Method

An implementation of the actuator line method (ALM) is applied to a hovering helicopter rotor. This method, which is widely used for wind turbine simulations, replaces the rotor blades by momentum source terms in the unsteady Reynolds-averaged Navier–Stokes equations. The removal of the blade mesh significantly reduces the computational mesh size, thus lowering the computational cost. The ALM is presented along with some improvements, notably the choice and treatment of the projection kernel. A parameter sweep is performed showcasing the importance of proper selection of the Gaussian smearing coefficient for accurate rotor performance predictions with a value of scaled around a quarter chord in size. With this value, a new set of simulations on a refined mesh is performed and analyzed covering global rotor performance coefficients, sectional blade loading, tip vortex characteristics in terms of positions, circulation, and core radius. The ALM is benchmarked against an equivalent blade resolved case on the well-known S-76 rotor. Results confirm the appropriateness of the ALM model for a hovering rotor for main flow features and performance metrics, although there was a small loss of accuracy on the tip blade loading in the presence of a blade–vortex interaction. Finally, computational performances indicate an elapsed-time speed-up between 3 and 4× in addition to a greater parallel efficiency in favor of the ALM.

Multi-peak detection algorithm based on wavelength feature recognition in FBG sensor networks

We propose an improved Gaussian curve fitting method based on the Hilbert transformation (HTG) to tackle the ineffectiveness of the traditional peak-seeking algorithm in detecting the multi-peak Fiber Bragg grating (FBG) reflection spectra. A five-point sliding filter is used to process the FBG reflection spectral signal, de-noise and smooth the noise, and select the optimal threshold point by the Hilbert transformation (HT). The sub-spectra of the multiple FBG reflection spectral signals were derived and the initial positioning of the spectral peaks were achieved. The Levenberg–Marquardt (LM) algorithm is used to extract the Bragg wavelength from the segmented sub-spectral signals as well as optimize the Gaussian curve fitting coefficients. The HTG-LM algorithm is then proposed, and is optimized and utilized to achieve precise positioning of the spectral peaks. The theoretical analysis and experimental results showed that the proposed HTG-LM algorithm could dynamically detect the multiple reflection spectra of the FBG sensing system with good stability, and at the same time, reduce the amount of peak-seeking data, which is highly beneficial to improve the signal demodulation rate. The peak detection accuracy of the proposed algorithm is better than 1 pm and the precision is better than 4 × 10 − 7 pm, which indicates that this HTG-LM algorithm provides an accurate demodulation algorithm for the FBG sensor networks. As a result, it is a promising multi-peak detection algorithm proposed by this paper to be applied to the FBG sensing systems.