Q-binomial Research Articles

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.

Case Study on the Fitting Method of Typical Objects

This study proposes different fitting methods for different types of targets in the 400–900 nm wavelength range, based on convex optimization algorithms, to achieve the effect of high-precision spectral reconstruction for small space-borne spectrometers. This article first expounds on the mathematical model in the imaging process of the small spectrometer and discretizes it into an AX=B matrix equation. Second, the design basis of the filter transmittance curve is explained. Furthermore, a convex optimization algorithm is used, based on 50 filters, and appropriate constraints are added to solve the target spectrum. First, in terms of spectrum fitting, six different ground object spectra are selected, and Gaussian fitting, polynomial fitting, and Fourier fitting are used to fit the original data and analyze the best fit of each target spectrum. Then the transmittance curve of the filter is equally divided, and the corresponding AX=B discrete equation set is obtained for the specific object target, and a random error of 1% is applied to the equation set to obtain the discrete spectral value. The fitting is performed for each case to determine the best fitting method with errors. Subsequently, the transmittance curve of the filter with the detector characteristics is equally divided, and the corresponding AX=B discrete equation set is obtained for the specific object target. A random error of 1% is applied to the equation set to obtain the error. After the discrete spectral values are obtained, the fitting is performed again, and the best fitting method is determined. In order to evaluate the fitting accuracy of the original spectral data and the reconstruction accuracy of the calculated discrete spectrum, the three evaluation indicators MSE, ARE, and RE are used for evaluation. To measure the stability and accuracy of the spectral reconstruction of the fitting method more accurately, it is necessary to perform 500 cycles of calculations to determine the corresponding MSE value and further analyze the influence of the fitting method on the reconstruction accuracy. The results show that different fitting methods should be adopted for different ground targets under the error conditions. The three indicators, MSE, ARE, and RE, have reached high accuracy and strong stability. The effect of high-precision reconstruction of the target spectrum is achieved. This article provides new ideas for related scholars engaged in hyperspectral reconstruction work and promotes the development of hyperspectral technology.

The minimum modulus of Gaussian trigonometric polynomials

We prove that the minimum of the modulus of a random trigonometric polynomial with Gaussian coefficients, properly normalized, has limiting exponential distribution.

Differential growth and yield responses of local potato cultivars from basic seed tubers

ABSTRACT Potato quality of seed tubers is a determinant factor that allows obtaining high yields, and its use must be a priority for a country or a producing region. Lack of information about how local cultivars perform differentially according to the environment has been identified in order to support potential seeds production programs. The research was carried out in 2018 at the Obonuco Research Center of Corporación Colombiana de Investigación Agropecuaria (AGROSAVIA), with the aim of analyzing the growth and yield of basic tuber seeds of five potato (Solanum tuberosum spp. andigena) cultivars. 1,080 seed tubers from each cultivar (treatments) were planted in a completely randomized block design and three replications. Mean comparison Tukey-Kramer (p≤0.05) test was performed for the yield analysis. From emergency to final harvest, destructive sampling was carried out every fifteen days in order to calculate dry matter and leaf area and establish growth models based on the accumulated degree-days (DD). The yield and proportion of seed tubers per plant were determined according to local resolution for seed certification in Colombia. Growth variables fitted better with the Hoerl, Gaussian, and third-degree polynomial models. The leaf area index of the potato cultivars reached the maximum value at 876 DD, with values ranging from 2.91 to 6.11 DD. The highest yield per plant was obtained by the Ica Única cultivar (2.73 kg plant-1). However, this cultivar showed the lowest percentage of seed tubers, with 50.59% compared to the others, which ranged between 80.82 and 87.40%. Differential potato growth responses through models based on the DD would explain the differences in final yield and seeds tuber production.

Accelerated basis adaptation in homogeneous chaos spaces

Polynomial chaos expansions (PCE) provide an efficient approach to uncertainty quantification (UQ) and have been adapted to diverse applications across the spectrum of science and engineering. For situations involving large stochastic parameterizations, the curse of dimensionality renders PCE-based methods computationally prohibitive. A basis adaptation approach for PCE was proposed by Tipireddy and Ghanem (2014) which transforms the input random variables through an isometry such that sufficient probabilistic characterization of specific quantities of interest (QoI) is concentrated in an algebraic manifold embedded in the linear span of the dominant transformed variables. While quite versatile, that original version of the PCE basis adaptation exhibited slow convergence for a number of problems of practical significance. In the present paper, we propose two novel methods to accelerate the convergence of the original basis adaptation approach, thus expanding its reach while also providing insight into its performance. In the first method, information gained from a pilot PCE representation is used to correct the mean and Gaussian coefficients in the adapted space. By taking advantages of probabilistic information in higher dimensional adaptation gleaned from an initial adaptation, the second method updates the rotation matrix used to identify the dominant transformed variables. In this manner, the new rotation matrix concentrates even more probabilistic information in its first few dimensions. These two method can be combined to achieve even better performance, the combined method is referred to as sequentially optimized adaptation method. The methods are demonstrated on an analytical test function and a model of a space structure with several sub-components and a non-smooth quantity of interest representing the maximum acceleration over time. Both methods achieve accelerated convergence of the basis adaptation approach with negligible additional costs.

Deep support vector machine for PolSAR image classification

ABSTRACT The main problem posed by Polarimetric Synthetic Aperture Radar (PolSAR) image classification in remote sensing is the ability to develop classifiers that can substantially discern the different classes inherent in natural and man-made targets. Emphasis has shifted from the use of conventional classifiers to modern non-parametric classifiers such as the Artificial Neural Network (ANN) and Support Vector Machine (SVM), and most recently the hybrid Deep Neural Network (DNN) which is a fusion of Deep Learning (DL) and ANN. This research therefore presents the novel application of Deep Support Vector Machine (DSVM), which is a fusion of DL and SVM to PolSAR image classification. Two PolSAR images of Flevoland region in the Netherlands and Winnipeg in Canada are used as test beds for the experiment. The Lee filter is used to filter the images to suppress the speckle noise in the images. The Pauli decomposition is applied to decompose the images into , , polarimetric channels. Then, the Gray Level Co-occurrence Matrix (GLCM) texture feature for , , are extracted based on correlation, contrast, energy, and homogeneity statistics, using GLCM directions 0°, 45°, 90°, and 135° with an offset distance of 60. To enhance the efficiency of the model 8, 16, 32, 64, 128, and 256 quantization levels are explored. The DSVM classifier is implemented with four kernel functions: Exponential Radial Basis Function (ERBF), Gaussian Radial Basis Function (GRBF), neural, and polynomial. The first set of results is a comparison of the DSVM and SVM. The result of Flevoland image for ERBF, GRBF, neural, and polynomial kernels are 99.17 (73.39), 99.32 (74.62), 98.64 (71.28), and 99.34 (77.21), respectively; for the Winnipeg image for ERBF, GRBF, neural, and polynomial kernels are 98.65 (72.68), 98.67 (73.54), 98.27 (70.15), and 99.46 (75.03), respectively. The second set of results is a comparison of DSVM, SVM, DNN, Gaussian Mixture Model (GMM), K Nearest Neighbour (KNN), and K Means (KM) classifiers; the results for Flevoland image for DSVM, SVM, DNN, GMM, KNN, and KM are 99.12, 74.13, 96.29, 75.06, 75.85, and 21.43, respectively, while the results for Winnipeg image for DSVM, SVM, DNN, GMM, KNN, and KM are 98.76, 72.85, 95.64, 73.20, 73.91, and 25.60, respectively. Since the Kappa coefficient is presumed to be a more accurate measure for accuracy estimation, it is used to evaluate the performances of all the models. The computed Kappa coefficients for of DSVM, SVM, DNN, GMM, KNN, and KM for Flevoland are 92.45, 70.71, 88.76, 68.59, 68.62, and 18.89, respectively; while the computed Kappa coefficients for DSVM, SVM, DNN, GMM, KNN, and KM for Winnepeg are 92.45, 70.71, 88.76, 68.59, 68.62, and 18.89, respectively. Based on the metrics used to evaluate the performances of the experiments; the results show that the DSVM outperformed the other classifiers. The high accuracy obtained with the DSVM shows it is a state-of-the-art algorithm for PolSAR image classification and a significant progress in the latest of DL applications.

Compounds Produced by the Pyrolysis of Powders and Dusts Present in the Alimentary Industry

ABSTRACT Under certain conditions dust explosions occur in the alimentary industry. Following ATEX and other guidelines have not eliminated accidents. Therefore, more knowledge is needed. The current work delivers experimental results describing phase transitions and decomposition of dusts. Dusts from wheat flour, chili powder, corn starch, milk powder, cocoa powder, and by-product of grain are investigated. The temperature of pyrolisation has been identified using TGA to be in the range [250°C, 600°C] in air and [300°C, 450°C] in nitrogen. It was found that the compositions of the pyrolysis gases depend on temperature. Carbon monoxide, carbon dioxide, methane, and hydrogen were the main contributors to the pyrolysis gases. The distributions are described with a polynomial or Gaussian fit. The current paper proposes coefficients for Gaussian polynomials expressing the concentration for the four primary pyrolysis gases.

Sentiment Analysis of the Indonesian Health Ministry Performance in Covid-19 Crisis using Support Vector Machine (SVM)

Corona Virus Disease or COVID 19 is a new virus disease that originated in 2019 [6], Indonesia has reported first COVID-19 In 2nd March 2020. Various attempts have been made by the government, such as taking strict measures by temporal lockdown or cordoning off the areas that were suspected of having risks of community spread. As a source of information, the internet has changed substantially,. for example, social media. social media is a communication tool that is very popular among internet users today, From social media, users can update status, send messages, even, become a platform for exchanging socio-economic opinions and political views both in their place of residence or their country. This paper deals with the sentiment analysis of Indonesian after the peformance of Indonesian Ministry Of Health. We used the social media platform Twitter for our analysis. Tweets were studied to gauge the opinion of Indonesian towards peformance of Indonesian Ministry Of Health. Tweets were extracted using the two prominent keywords used namely: “terawan ”and “menkes” from June 15th to September 19th 2020. A total of 200 tweets were considered for the analysis. This study has successfully implemented the SVM algorithm for sentiment analysis on tweet data about peformance of Indonesian Ministry Of Health during COVID-19 Crisis. This is shown by the accuracy of using tweet data as much as 200 data, which is 172 data are training data and 28 are testing data. Besides the amount of data that affects accuracy, there are also other factors, namely the use of the kernel and the number of classes used. The results show that the Linear Kernel has the best accuracy, precision and recall rate compared to other kernels, respectively 75% for accuracy, 78.4% for precision and a recall value of 75%. for polynomial kernels, Gaussian and Sigmoid have the same accuracy, precision, and recall rates, namely, respectively. 60.71% for accuracy, 36.86% for precision and 60.71% recall value.

Building fuzzy relationships between compressive strength and 3D microstructural image features for cement hydration using Gaussian mixture model-based polynomial radial basis function neural networks

Generally, chemical compositions and physical properties such as the contents of clinkers, surface area and particle size distribution are used to estimate cement compressive strength (CCS). However, the conventional approaches are limited by the high complexity of physical changes and chemical reactions during cement hydration, which perform poorly facing the complex and uncertain evolution of cement paste. Considering that the cement microstructure can directly reflect the state of cement hydration and the information related to CCS at microscale, microtomography, which can image three-dimensional microstructure of cement paste, offers scientists another way to study CCS. Moreover, it enables us to understand the formation mechanism of the physical properties of cement and helps design high-performance materials. This paper studies the relationship between cement microstructure and compressive strength using fuzzy neural networks. The fuzzy relationships between CCS and microstructural image features are built as the “if-then” format using the polynomial-based radial basis function networks with Gaussian mixture model (GMM-PRBFNNs) from microtomography images. The GMM-PRBFNNs are proposed to improve the performance by using GMM to generate membership functions for constructing the premise of the fuzzy rules. Moreover, four types of polynomials such as constant, linear, quadratic and modified quadratic are considered as the weights between the hidden layer and the output layer for the consequent of fuzzy rules. Experimental results manifest that the built fuzzy relationships not only perform well in approximating CCS but are also easy to comprehend.

Real Lines on Random Cubic Surfaces

We give an explicit formula for the expectation of the number of real lines on a random invariant cubic surface, i.e., a surface Zsubset {mathbb {R}}{mathrm {P}}^3 defined by a random gaussian polynomial whose probability distribution is invariant under the action of the orthogonal group O(4) by change of variables. Such invariant distributions are completely described by one parameter lambda in [0,1] and as a function of this parameter the expected number of real lines equals: Eλ=9(8λ2+(1-λ)2)2λ2+(1-λ)22λ28λ2+(1-λ)2-13+238λ2+(1-λ)220λ2+(1-λ)2.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} E_\\lambda =\\frac{9(8\\lambda ^2+(1-\\lambda )^2)}{2\\lambda ^2+(1-\\lambda )^2}\\left( \\frac{2\\lambda ^2}{8\\lambda ^2+(1-\\lambda )^2}-\\frac{1}{3}+\\frac{2}{3}\\sqrt{\\frac{8\\lambda ^2+(1-\\lambda )^2}{20\\lambda ^2+(1-\\lambda )^2}}\\right) . \\end{aligned}$$\\end{document}This result generalizes previous results by Basu et al. (Math Ann 374(3–4):1773–1810, 2019) for the case of a Kostlan polynomial, which corresponds to lambda =frac{1}{3} and for which E_{frac{1}{3}}=6sqrt{2}-3. Moreover, we show that the expectation of the number of real lines is maximized by random purely harmonic cubic polynomials, which corresponds to the case lambda =1 and for which E_1=24sqrt{frac{2}{5}}-3.

Consistent and Contrasting Aspects of the Geomagnetic Field Across Epochs With Distinct Reversal Frequencies Revealed by Modeling the Kiaman Superchron

AbstractThis work presents an extensive directional paleomagnetic database of the Kiaman reversed superchron. It is composed of 1,459 paleomagnetic directions from igneous rocks corresponding to 91 data sets (or paleomagnetic poles). An almost constant behavior of more concentrated and circular distributions for latitudes higher than 10° was found, which contrasts strongly with predictions of the representative models for the past few million years. We searched for simplified and spatially covariant Giant Gaussian Process (GGP) models that best explain the directional distribution of the Kiaman database. We used the mean strength based on the mean of virtual dipole moment (VDM) results for the period drawn from the available databases. Among the tested models, the one that best explains the directional paleosecular variation of the Kiaman database is the covariant type. According to this model, the correlations between the Gaussian coefficients are valid for the last 10 Myr and the Kiaman superchron. The resulting GGP models have parameters similar to the 0–10 Ma models, which indicates that the relation between symmetric and antisymmetric families appears unchanged in the geological past. The relative variability of the Kiaman field, as inferred from the ratio from GGP models, is lower than for the past 10 Myr. Thus, as well as the paleointensity, seems to be a proxy that can be used for evaluating the geomagnetic development along the geological time.

Brittle Deformation in the Neoproterozoic Basement of Southeast Brazil: Traces of Intraplate Cenozoic Tectonics

The basement of southeast Brazil is traditionally interpreted as the result of Neoproterozoic and early Paleozoic orogenic cycles. Wide regions of the Atlantic Plateau (southeast Brazil) are characterized by rocks and tectonic structures of Precambrian age. According to the classical literature, these regions have not been affected by tectonics since the Miocene, despite the fact that they rest close to Cenozoic basins, which have suffered recent tectonic deformation. The objective of this research is to evaluate the role of neotectonics in the Atlantic Plateau. This task is accomplished through a multiscalar approach which includes lineament domain analysis from regionally sized digital elevation models and structural geology field surveys. Lineaments are automatically detected and statistically analyzed. Azimuthal analyses of data on faults and fractures by a polynomial Gaussian fit enables the identification of the main structural trends. Fault-slip direct inversion by means of the original Monte Carlo approach allows one to compute the multiple paleostresses that produced the measured fault population. The results show the presence of a principal ENE–WSW lineament domain, related to an old shear zone possibly reactivated since the Miocene. One of the paleostresses computed from fault-slip inversion is in agreement with the neotectonic stress-field proposed by other authors.

General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications

This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of well-known proofs of the q-Kummer summation theorem with a key role of the q-binomial theorem. In addition to the q-binomial theorem, the first proof makes use of Thomae’s q-integral representation and the second proof needs Heine’s transformation. Whereas the third proof utilizes only the q-binomial theorem. Subsequently, the applications of these summation formulas in obtaining the general contiguous extensions of the second and the third q-Kummer summation theorems are also presented. Furthermore, the investigated results are specialized to give many of the known as well as presumably new q-summation theorems, which are contiguous to the three q-Kummer summation theorems. This work is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including Number Theory, Theory of Partitions and Combinatorial Analysis as well as in the study of Combinatorial Generating Functions. Just as it is known in the theory of the Gauss, Kummer (or confluent), Clausen and the generalized hypergeometric functions, the parameters in the corresponding basic or quantum (or q-) hypergeometric functions are symmetric in the sense that they remain invariant when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. A case has therefore been made for the symmetry possessed not only by hypergeometric functions and basic or quantum (or q-) hypergeometric functions, which are studied in this paper, but also by the symmetric quantum calculus itself.

Modelling the effect of Covid-19 mortality on the economy of Nigeria

ObjectivesThis paper is aimed at modelling the effect of COVID-19 mortality per population (CMP), a proxy for COVID-19 on the Gross Domestics Product (GDP) per capita per COVID-19 cases (RGDPC), a proxy for the economic wellbeing of a nation. MethodsNine models divided into three groups (Gaussian polynomial, other non-linear, and Gamma generalized polynomial models) were fitted for RGDPC data on CMP, collected from 1st June to 31st December 2020. ResultsThe result showed that the gamma cubic model was selected as the best model out of the 9 competing models to predict the economic wellbeing of Nigeria. Predictions were made for the whole day in the year 2021. ConclusionIt is therefore concluded that there is a non-linear relationship between COVID-19 mortality and the economic wellbeing of Nigerians. Thus, COVID-19 mortality has an adverse effect on the wellbeing of Nigerians. The economic wellbeing of Nigerians can be improved if COVID-19 mortality is stopped.

Comparison some of kernel functions with support vector machines classifier for thalassemia dataset

<span id="docs-internal-guid-9a30056f-7fff-8ff1-59e1-69f89f4280bd"><span>In the medical field, accurate classification of medical data is really important because of its impact on disease detection and patient’s treatment. Technology, machine learning, is needed to help medical staff to improve accuracy to classify disease. This research discussed some kernel functions, such as gaussian radial basis function (RBF) kernel, Polynomial kernel, and linear kernel with support vector machine (SVM) to classify thalassemia data. Thalassemia is a genetic blood disorder which is also one of the major public health problems. In this paper, there is an explanation about thalassemia, SVM, and some of the kernel functions that serve as a comprehensive source for the next research about this topic. Furthermore, there is a comparison result from three kernel functions to find out which one has the best performance. The result is gaussian RBF kernel with SVM is the best method with an average of accuracy 99,63%. </span></span>

High-order Combined Multi-step Scheme for Solving Forward Backward Stochastic Differential Equations

In this work, in order to obtain higher-order schemes for solving forward backward stochastic differential equations, we propose a new multi-step scheme by adopting the high-order multi-step method in Zhao et al. (SIAM J. Sci. Comput., 36(4): A1731-A1751, 2014) with the combination technique. Two reference ordinary differential equations containing the conditional expectations and their derivatives are derived from the backward component. These derivatives are approximated by using the finite difference methods with multi-step combinations. The resulting scheme is a semi-discretization in the temporal direction involving conditional expectations, which are solved by using the Gaussian quadrature rules and polynomial interpolations on the spatial grids. Our new proposed multi-step scheme allows for higher convergence rate up to ninth order, and are more efficient. Finally, we provide a numerical illustration of the convergence of the proposed method.

Semi-invariants of binary forms pertaining to a unimodality theorem of Reiner and Stanton

The symmetric difference of the [Formula: see text]-binomial coefficients [Formula: see text] was introduced by Reiner and Stanton. They proved that [Formula: see text] is symmetric and unimodal for [Formula: see text] and [Formula: see text] even by using the representation theory for Lie algebras. Based on Sylvester’s proof of the unimodality of the Gaussian coefficients, as conjectured by Cayley, we find an interpretation of the unimodality of [Formula: see text] in terms of semi-invariants. In the spirit of the strict unimodality of the Gaussian coefficients due to Pak and Panova, we prove the strict unimodality of the symmetric difference [Formula: see text], except for the two terms at both ends, where [Formula: see text], [Formula: see text] and at least one of [Formula: see text] and [Formula: see text] is even.

A refinement of the binomial distribution using the quantum binomial theorem

q-analogs of special functions, including hypergeometric functions, play a central role in mathematics and have numerous applications in physics. In the theory of probability, q-analogs of various probability distributions have been introduced over the years, including the binomial distribution. Here, I propose a new refinement of the binomial distribution by way of the quantum binomial theorem (also known as the noncommutative q-binomial theorem), where the q is a formal variable in which information related to the sequence of successes and failures in the underlying binomial experiment is encoded in its exponent.

Zeros of derivatives of Gaussian random polynomials on plane domains

In this paper, we study the expected density of zeros of derivatives of Gaussian random polynomials on plane domains of C. The Gaussian random polynomials are of the form RN=∑n=0Nanpn where an are i.i.d. complex Gaussian random variables and {pn} are orthonormal polynomials on the boundary of a bounded simply connected analytic plane domain relative to a weight. The main result is that the expected density of zeros of any k-th derivatives of RN tends to the equilibrium measure of the domain.

Flat-top supercontinuum generation via Gaussian pulse shaping.

We present the flat-top supercontinuum source with high repetition rate over a broad bandwidth. The flatness and high repetition rate are achieved by iterative optical line-by-line spectrum shaping on electro-optic optical frequency combs. By applying Gaussian apodized pulse train to a highly nonlinear medium with optimized Gaussian coefficient and nonlinear polarization rotation techniques, we implemented here a flat-top supercontinuum with a 47.7 nm bandwidth at 3 dB and 30 GHz repetition rate. The generation of high repetition rate supercontinuum sources with smooth and coherent spectrum is the critical challenging task for many applications such as optical communications and the optical arbitrary waveform generation. This work leads us to new possibilities for generating hundreds or thousands of flattened coherent optical carriers with a simple configuration.