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Based on the definitions of poor data, an unascertained model and four axioms, condition functions and range etc. were analyzed then induced second‐order condition function, complemental condition function, connection function and the rule set of some signs concludes with the forecast method, which consists of four theorems and ten inferences, in the condition of data number m (m≥2) in Rⁿ.

In this paper, the authors prove that the Douglas space of second kind with a generalised form of special (α, β)-metric F, is conformally invariant.

For, the authors have used the notion of conformal transformation and Douglas space.

The authors found some results to show that the Douglas space of second kind with certain (α, β)-metrics such as Randers metric, first approximate Matsumoto metric along with some special (α, β)-metrics, is invariant under a conformal change.

The authors introduced Douglas space of second kind and established conditions under which it can be transformed to a Douglas space of second kind.

The purpose of this paper is to find out some new rational non-traveling wave solutions and to study localized structures for (2+1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation.

Along with some special transformations, the Lie group method and the rational function method are applied to the (2+1)-dimensional AKNS equation.

Some new non-traveling wave solutions are obtained, including generalized rational solutions with two arbitrary functions of time variable.

As a typical nonlinear evolution equation, some dynamical behaviors are also discussed.

With the help of the Lie group method, special transformations and the rational function method, new non-traveling wave solutions are derived for the AKNS equation by Maple software. These results are much useful for investigating some new localized structures and the interaction of waves in high-dimensional models, and enrich dynamical features of solutions for the higher dimensional systems.

This paper aims to explore the promotion decisions of heterogeneous sellers on a decentralized platform under competitive conditions and analyze how seller behaviors impact platform profit, seller revenue, buyer surplus and social welfare.

This paper considers a Cournot model consisting of a platform charging a commission rate and two sellers with different conversion rates and browsing costs. Promotion efforts by sellers can increase traffic, but they also incur promotion costs for sellers. The sellers decide on promotion effort by weighing these two effects. The authors also explore the equilibrium when the platform charges a fixed usage fee.

The seller’s profit improves as its conversion rate increases and worsens as browsing costs increase. Also, increasing the commission rate charged by the platform makes the seller invest less in promotional efforts. Therefore, the platform must consider this trade-off to determine an optimal rate. The analysis shows that the seller with a high conversion rate and high browsing cost plays a greater role in generating more overall revenue. When the market favors such a seller, the platform tends to charge less in order not to impair its profitability.

This paper incorporates conversion rate, buyer’s browsing cost, unit promotion cost and the fee charged by the platform into the model to study sellers’ promotion decisions on decentralized platforms.

The discrete Fourier transform (dft) of a fractional process is studied. An exact representation of the dft is given in terms of the component data, leading to the frequency domain form of the model for a fractional process. This representation is particularly useful in analyzing the asymptotic behavior of the dft and periodogram in the nonstationary case when the memory parameter d≥12. Various asymptotic approximations are established including some new hypergeometric function representations that are of independent interest. It is shown that smoothed periodogram spectral estimates remain consistent for frequencies away from the origin in the nonstationary case provided the memory parameter d < 1. When d = 1, the spectral estimates are inconsistent and converge weakly to random variates. Applications of the theory to log periodogram regression and local Whittle estimation of the memory parameter are discussed and some modified versions of these procedures are suggested for nonstationary cases.

Due to the cross-network effect, two-sided users communicate with each other, producing a coupling network. To study the spread of platform self-operation in two-sided users' marketing and purchasing tactics, this paper considers the differences in reputation acquired by platform-owned and third-party operating channels.

This study proposes a two-layer network with cross-network links: one layer represents the social network of consumers, while the other layer represents the competitive network of buyers. A closed system of differential equations, based on the binary dynamics of the stochastic network, is developed to study the trend and stability points of the platform self-operation dissemination. Then the overall benefits of platform are analyzed to unify the platform diffusion and pricing strategies.

The degree of difference in social influence and cross-network effects affect diffusion synergistically. Cross-network effects hinder diffusion when there is a significant difference of social influence between consumers and sellers but promote diffusion when there is little difference of social influence between consumers and sellers. Additionally, the network weights and reputation gap exhibit a nonlinear correlation with diffusion. For pricing strategy of the platform, it can achieve maximum profit when the pricing of self-operated goods and third-party-operated goods is equal.

This study considers the complex network architecture created by bilateral markets and the dynamic influence of group interactions on product. Additionally, this study takes reputation into account when considering the price and dissemination tactics of various operating channels, offering guidelines for platforms to control merchants and mediate disputes between various operating channels.