Q-binomial Research Articles

Evaluation of maximum wall shear stress from air impingement to remove food deposits from stainless steel surfaces

The risk of cross-contamination during manufacturing of low water activity foods requires careful attention to cleaning and sanitation. Considering the challenges encountered in such manufacturing, air impingement can provide a unique approach to remove deposits from the food-contacted surface, particularly, the crevices or cracks. The objective of this study was to evaluate the shear stress-induced by air jet impingement for removal of food residues. The wall shear stress distribution on a surface below an air jet was measured using oil-film interferometry. The air impingement was operated at three level of air velocity (Reynolds number = 7756, 13,444, and 18,787) and three different nozzle height (24 (H/D = 6), 32 (H/D = 8), 40 mm (H/D = 10)). The shear stress was found to become higher along with the increase of air velocity and decrease of nozzle height. The observed distribution of wall shear stress on the surface was fitted using the Gaussian model and Polynomial model. The Gaussian model appeared to be preferable according to the statistical analysis, which aligned with previous literature. Based on the Gaussian model, the maximum wall shear stress was estimated. The maximum wall shear stress increased with Reynolds number ( Re ) and the ratio of jet height to jet nozzle diameter. The highest maximum shear stress (10.74 Pa) occurred at a Reynolds number of 18,787 and a height of 24 mm (H/D = 6). This trend was directly correlated with removal efficiency. The greatest removal of the treatment was achieved by 40% at maximum wall shear stress. A linear relationship between percent residue removed and maximum wall shear stress was obtained from the experimental results, and a threshold maximum wall shear stress of 2.73 Pa has been proposed for removal of peanut butter residue from a stainless-steel surface. The current study demonstrates the promising of the air impingement in the cleaning operation for low water activity food. • Wall shear stress increased with an increase in velocity and decrease in height. • Removal of food deposits revealed a similar trend to velocity and height. • Linear relation was found between maximum shear stress and food removal.

Combinatorial solution of the eclectic spin chain

The one-loop dilatation operator in the holomorphic 3-scalar sector of the dynamical fishnet theory is studied. Due to the non-unitary nature of the underlying field theory this operator, dubbed in [1] the eclectic spin chain Hamiltonian, is non-diagonalisable. The corresponding spectrum of Jordan blocks leads to logarithms in the two-point functions, which is characteristic of logarithmic conformal field theories. It was conjectured in [2] that for certain filling conditions and generic couplings the spectrum of the eclectic model is equivalent to the spectrum of a simpler model, the hypereclectic spin chain. We provide further evidence for this conjecture, and introduce a generating function which fully characterises the Jordan block spectrum of the simplified model. This function is found by purely combinatorial means and is simply related to the q-binomial coefficient.

On the divisibility of sums of even powers of q-binomial coefficients

We prove the divisibility conjecture on sums of even powers of q-binomial coefficients, which was recently proposed by Guo, Schlosser and Zudilin. Our proof relies on two q-harmonic series congruences due to Shi and Pan.

The Limit Empirical Spectral Distribution of Gaussian Monic Complex Matrix Polynomials

We define the empirical spectral distribution (ESD) of a random matrix polynomial with invertible leading coefficient, and we study it for complex n times n Gaussian monic matrix polynomials of degree k. We obtain exact formulae for the almost sure limit of the ESD in two distinct scenarios: (1) n rightarrow infty with k constant and (2) k rightarrow infty with n constant. The main tool for our approach is the replacement principle by Tao, Vu and Krishnapur. Along the way, we also develop some auxiliary results of potential independent interest: We slightly extend a result by Bürgisser and Cucker on the tail bound for the norm of the pseudoinverse of a nonzero mean matrix, and we obtain several estimates on the singular values of certain structured random matrices.

Generalized q-Difference Equations for q-Hypergeometric Polynomials with Double q-Binomial Coefficients

In this paper, we apply a general family of basic (or q-) polynomials with double q-binomial coefficients as well as some homogeneous q-operators in order to construct several q-difference equations involving seven variables. We derive the Rogers type and the extended Rogers type formulas as well as the Srivastava-Agarwal-type bilinear generating functions for the general q-polynomials, which generalize the generating functions for the Cigler polynomials. We also derive a class of mixed generating functions by means of the aforementioned q-difference equations. The various results, which we have derived in this paper, are new and sufficiently general in character. Moreover, the generating functions presented here are potentially applicable not only in the study of the general q-polynomials, which they have generated, but indeed also in finding solutions of the associated q-difference equations. Finally, we remark that it will be a rather trivial and inconsequential exercise to produce the so-called (p,q)-variations of the q-results, which we have investigated here, because the additional forced-in parameter p is obviously redundant.

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.