A system of simultaneous multi-variable nonlinear equations can be solved by the Newton’s method with local q-quadratic convergence if the Jacobian is analytically available. If this is not the case, then quasi-Newton methods with local q-superlinear convergence give solutions by approximating the Jacobian in some way. Unfortunately, the quasi-Newton condition (secant equation) doesn’t completely specify the Jacobian approximate in multi-dimensional case, so its full-rank update is not possible with classic variants of methods. The suggested new iteration strategy (“T-Secant”) allows full-rank update of the Jacobian approximate in each iteration by determining two independent approximates for the solution. They are used to generate a set of new independent trial approximates, then the Jacobian approximate can fully be updated. It is shown, that the T-Secant approximate is in the vicinity of the classic quasi-Newton approximate, providing that the solution is evenly surrounded by the new trial approximates. The suggested procedure increases the super-linear convergence of the secant method (φS = 1.618...) to super-quadratic (φT = φS + 1 = 2.618...) and the quadratic convergence of the Newton-method (φN = 2) to cubic (φT = φN + 1 = 3) in one dimensional case. The Broyden-type efficiency (mean convergence rate) of the suggested method in multi-dimensional case is an order higher than the efficiency of other classic low-rank update quasi-Newton methods as shown by numerical examples on a Rosenbrock-type test-function with up to 1000 variables. The geometrical representation (hyperbolic approximation) in single variable case helps explaining the basic operations and a vector-space description is also given in multi-variable case.

The level of development of the innovation factor plays a crucial role in supporting the high-quality development of China's economy. To advance the development of regional innovation factor allocation efficiency along the Belt and Road. This article introduces the super-efficiency SBM-DEA-Malmquist model for the static and dynamic analysis of innovation factor allocation efficiency in 17 provinces along the Belt and Road from 2012 to 2021. The Moran index model analyzes the spatial correlation. The results show that: (1) The overall innovation factor allocation efficiency along the Belt and Road is not high, and there are obvious differences among different regions. The eastern region is the highest compared to other regions. (2) According to the efficiency decomposition results, pure technical efficiency is the main reason for the low innovation factor allocation efficiency. (3) Through the Malmquist index and decomposition index, it is found that pure technical efficiency and scale efficiency are key factors in improving technical efficiency. (4) The analysis of spatial correlation reveals a strong spatial agglomeration feature among the provinces along the Belt and Road. The innovation factor allocation efficiency is mainly manifested in the L-L type. Finally, according to the analysis results, suggestions and policies are put forward to improve the innovation factor allocation efficiency in the regions along the Belt and Road.

We will apply Pascal’s hydrostatic pressure law to black holes and also to the Hubble sphere. Haug has recently demonstrated how hydrostatic pressure in water can be utilized to determine the Planck length. One can conceptualize the energy in the Hubble sphere as a type of superfluid, and if this is correct, then Blaise Pascal’s hydrostatic pressure formula may also be applicable to such a superfluid, even within the Hubble sphere. By employing the same method on the Hubble sphere and considering it as a Schwarzschild black hole with the critical Friedmann mass, we can estimate the Planck length based on its hypothetical hydrostatic pressure.

Super-resolution technology is important not only in bio-related fields but also in nanotechnology, particularly in the semiconductor industry, where fine patterning is required and super-resolution is essential. However, observing microstructures beyond the diffraction limit proposed by Abbe and Rayleigh is considered impossible because of diffraction in traditional optical microscopy ob-servation techniques. However, in recent years, it has been possible to observe microstructures beyond the Rayleigh criterion by analysing the phase distribution of light. This study investigated the physical reasons why phase analysis makes this new analysis technique possible using simula-tions. The results confirmed that the phase component of the zero-order diffracted light reflected from the microstructure and able to pass through the lens system contained the phase information related to the shape of the measured object. Analysis of this information demonstrates the possi-bility of realising super-resolution based on speckle interferometry.

As an important support for power system with high penetration of renewable energy, energy storage system (ESS) has changed the traditional model of simultaneous implementation of electricity production and consumption, and its installed capacity under the source-grid-load scenario is rising year by year, but it faces the problems of insufficient utilization and benefits. This study analyzes the functional combination of ESS under the source-grid-load scenarios, and proposes a comprehensive benefit evaluation method of energy storage projects based on fuzzy decision-making trial and evaluation laboratory (DEMATEL) and super efficiency data envelopment analysis (DEA) model, which considers the perspective of economic and social environment. Firstly, the functional requirements of energy storage on the source-grid-load scenarios are explored respectively, and the characteristics of various functions are analyzed to form functional combination schemes. Secondly, index modeling is carried out from three aspects: the whole life cycle cost, functional combination benefit and social environmental benefit, and the comprehensive benefit evaluation index system of energy storage project is proposed, which includes both economic and social environment. Then, the intuitionistic trapezoidal fuzzy number is combined with DEMATEL to form an effective analysis method for the input-output relationship of the indexes, and the comprehensive evaluation is realized based on the super efficiency DEA model. Finally, the effectiveness of the proposed evaluation method and the rationality of the functional combination are verified under source-grid-load scenarios respectively.

Here, using the theory of electrical-quantum modelling of the neutron and proton as a three-phase alternating current electrical generator, we will analyse the proton decay and the inverse neutron decay and determine the following subatomic particles, Dproton, Protoniu, Dneutron and Neutroniumd.