Formal Calculation uses an auxiliary form to calculate various nested sums and provides results in three forms. In addition to computation, it is also a powerful tool for analysis, allowing one to study various numbers in a unified way. This article contains many results of two types of Stirling numbers, associated Stirling numbers, and Eulerian numbers, making a great generalization of Euler polynomials, Wilson's theorem, and Wolstenholme's theorem, showing that they are just special cases. Formal Calculation provides a novel method for obtaining combinatorial identities and analyzing q-binomial.This article has obtained a large number of results in q-analogues, including inversion formulas for q-binomial coefficients. This article also introduces a theorem on symmetry.

The guided ultrasonic wave (GUW) is extensively employed in non-destructive testing (NDT) for the purpose of detecting defects in aerospace vehicles, oil pipelines and mechanical equipment. The filtration of GUW signals, which often contain substantial environmental noise, is a crucial procedure in signal processing. This paper presents a novel denoising approach that combines Variational Mode Decomposition (VMD) with an enhanced Singular Value Decomposition (SVD). The VMD method is employed to preprocess the initial signal, thereby segregating the signal component from the noise component. Subsequently, the SampEn-SVD (Sample Entropy-SVD) method is utilized to extract the effective component from the VMD-preprocessed noise component. Finally, the VMD-preprocessed signal component and the SampEn-SVD-processed effective component are combined to yield the resultant signal, which effectively filters out the noise. The efficacy of this integrated denoising approach is substantiated through the examination of experimental signals. Furthermore, a comparative analysis is conducted to evaluate the efficacy of this method in relation to other denoising techniques. The results indicate that the SampEn-SVD method yields a superior signal-to-noise ratio (SNR) when processing the GUW signal transmitted through the steel strand. Moreover, the denoising procedure significantly reduces the discretization of characteristic parameters in the signal waveform, thereby addressing the issue of inadequate reproducibility in testing outcomes. Consequently, the denoised signal exhibits high fidelity and demonstrates a strong correlation between its combined eigenvector and strand stress.

Abstract: Plant types and soil bacterial communities had a close relationship, understanding the profound association between them contributes to better learn bacterial ecological function for plant growth. In this study, rhizosphere soil of six different chemotype Cinnamomum camphora trees were collected, including C. bodinieri var. citralifera, [C. camphora (Linn.) Presl], camphora-type, cineole-type, linalool-type and isoborneol-type. Soil properties content and bacterial communities were analyzed. Two chemotype C. camphora, including [C. camphora (Linn.) Presl] and linalool-type, shaped similar bacterial community structure, decreased Firmcutes relative abundance. richness estimators (Chao1 index and Ace index) of [C. camphora (Linn.) Presl] were decreased compared with the others. Furthermore, soil bacterial community structure was also similar among bodinieri var. citralifera, camphora-type, cineole-type and isoborneol-type. Hence, different chemotype C. camphora altered soil nutrient and shaped rhizosphere bacterial communities.

Potato tubers are susceptible to wounding during post-harvest processes, leading to quality decline, perishability and large economic losses. In this study, the potato cultivar, ‘Longshu No.7’, was foliar-sprayed with 3% chitosan (w/v) three times during the pre-harvest period after flowering to evaluate the effect of foliar spraying with chitosan on suberization processing at wounds of harvested potato tubers. Our results demonstrate that foliar-sprayed with chitosan significantly reduced wound-induced fresh weight loss and dry rot disease index by 37.34% and 41.60% on the 28 day after wounding, respectively. Foliar-sprayed with chitosan accelerated the deposition of suberin polyphenolics and lignin at the wound sites of potato tubers with the formation of thicker cell layers. This occurred with increased localized activities of key enzymes in the suberin polyphenolics and lignin pathways, including phenylalanine ammonia lyase, 4-coumaryl-coenzyme A ligase, cinnamyl alcohol dehydrogenase and peroxidase (33.90–64.32%), as well as the contents of cinnamic acid, sinapic acid, flavonoids, lignins and total phenolics (19.70–23.46%) at wound sites of potato tubers on the 7 day after wounding. Our results indicated that foliar application of chitosan accelerated wound-induced suberization of potato tubers and could mitigate post-harvest product damages.