Xiaohui Zhou

According to the new wavelet random numbers (WRNs), some applications in numerical computation have been discussed in this paper. First, an approach is introduced to generate WRNs and transform the random numbers into uniform distribution random numbers and exponential random numbers. The randomness of WRN is identified by the white noise Hurst exponent and NIST SP 800-22 tests. Second, Mentor Carlo integration method based on WRNs has been applied to compute different numerical integrations compared to the results by randn, rand and PCG. Four examples are given to illustrate the improvement in computing a numerical integration, including a continuous integrand and a discontinuous oscillating integrand. According to the linear regression test of variance, it is verified that the variance of estimators decreases with the increase of sample size. Third, WRNs can be used to construct a one-time pad (OTP) briefly, which is statistically tested by the frequency (monobit) test and Shannon entropy test. Finally, WRNs can be chosen for bootstrap. An illustrative example indicates that bootstrap based on WRNs can be a case study of improving the accuracy of estimating a confidence interval for skewed exponential distribution, compared to randi.

Based on the Frenet frame, the approximate Local Wavelet Transform (LWT) is investigated on a class of generalized helix curves by their tangent projections at specific points. Initially, parametric equations are formulated for various types of helix curves, including circular helices and generalized helices. A particular helix curve on a unit sphere is selected as a representative instance of a generalized helix, and its tangent projection at a designated point is computed. Subsequently, the dilation and translation operators for a wavelet function are rigorously defined to facilitate the study of the LWT on this curve. A corresponding reconstruction formula is also derived. The Morlet wavelet is projected onto the space curve at the specified point t = 0 by the tangent projection method. To validate the proposed approach, three numerical examples are presented, demonstrating the application of the LWT and its inverse LWT (reconstruction formula) to analyze localized signals. The simulation results are illustrated through graphical representations.

Based on the wavelet image denoising method, a new approach is discussed for generating multi-dimensional normal random numbers in this paper. A thousand groups of multivariate random normal numbers are computed by this wavelet approach. According to Royston’s MVN test (Multivariate Normality test), the distributions of Royston’s test statistic and p-value are shown to illustrate multivariate normality in the boxplot figures. Moreover, based on the Jarque–Bera test, each row and column of the data are shown to illustrate normality in other boxplot figures, respectively. However, the distributions of average correlation and equivalent degrees of freedom are shown to illustrate possible correlation structure in the boxplot figures, compared to randn function and quantum random generator. Finally, the running times of this algorithm are also estimated, compared to the randn function and quantum random generator. The wavelet approach is faster than the randn function and quantum random generator in generating ultra-high-dimensional normal random numbers.

From the perspective of probability theory, this study makes a mathematical model of the proverbs themselves or the background stories reflected by them, carries out experiments and analyses, fully excavating the philosophical, scientific, and cultural connotations contained in the proverbs, and reveals the relationship between “mathematics,” “philosophy,” and “culture.” With the interpretation of probability theory, common sayings show their charm in traditional culture, and the application of probability theory has also been expanded and extended. In this paper, the problems reflected in the proverbs, such as “slanders three to, the mother does not love,” “In repentance, there is salvation., “Three cobblers with their wits combined exceed the mastermind of Zhuge Liang.,“ and “Gain the initiative by preemptive strike, otherwise you will be very passive.” are mathematically and modelically analyzed and tested from the perspective of probability theory and the philosophical culture, and scientific thought behind the proverbs are expounded. “One who sticks to his folly and does nothing to stand by a tree stump waiting for a hare,” “dripping water wears through a stone,” and “iron pestle grinds a needle” are a group of common sayings, which directly reflect the principle of small probability events and its two sides through the story behind them.

Some examples of dialectics philosophy in higher mathematics are illustrated in this paper. Firstly, the principle of interconversion between quality and quantity in dialectics philosophy is quantified by the mathematical definition of the limit theory. Secondly, some natural and social phenomena imply the definition of continuous function in incremental form and it is a new explanation for the Zeno paradox. Finally, the dialectics relationship between the local change and the whole change of some variables is discussed in the differential median theorems.

A multiscale prediction algorithm was presented in this paper based on the monthly data of property insurance income of Zhejiang Province from February 2013 to May 2018, and the correlation of the wavelet coefficients for property insurance income was also discussed at different time scales. By ADF tests, the stability of nonlinear approximation data and detail data were given. Furthermore, the prediction algorithm of nonlinear trend and detail data was studied according to the wavelet decomposition formula and the autoregressive moving average (ARMA) model. Finally, the short-term prediction results from June 2018 to January 2019 were given by the wavelet reconstruction formula. Compared to the prediction results of the traditional ARMA model, the relative error of the prediction results in this paper is smaller and better.