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The effect of multiplicity correlation between two bins to the dynamical fluctuations is investigated for a second-order phase transition from quark–gluon plasma (QGP) to hadrons, within the Ginzburg–Landau description for the transition. Normalized factorial correlators are used to characterize the dynamical fluctuations. A scaling behavior among the correlators is found, and an approximate universal exponent is obtained with very weak dependence on the details of the phase transition.

For a long time, the mismatch between material flow and capital flow in the supply chain operation management practice is very prominent, which has led to the inefficiency of supply chain operation and hindered the exertion of supply chain’s advantages. In the field of supply chain management theory research, the information completeness in capital market has long been a hypothetical premise, which has led to the separation of corporate financing and operational decision-making research. In addition to production, inventory, procurement, pricing and other strategies in supply chain operations, payment options and credit incentives are also important decisions for both parties, especially for products with long production and sales cycles; different payment methods directly affect corporate capital flows and the enterprise’s long-term development. On the basis of summarizing and analyzing previous works, this paper analyzed the research status and significance of supply chain financing and operational comprehensive decision-making, expounded the development background, current situation, and future challenges of the fractal dimension of fractional Brownian motion; elaborated the principles and methods of the scaling properties of fractional Brownian motion and the phase space reconstruction of time series, established a financial management analysis model based on the fractal dimension of fractional Brownian motion, performed the analysis of the agglomeration degree, time series and multi-fractal characteristics of supply chain financing, explored the coupling relationship between the comprehensive operational decision-making and Brownian motion’s scaling properties. The final empirical analysis showed that when own funds are sufficient, the production should be carried out with the goal of maximizing profits, and full consideration should be given customer channel stickiness, relative costs of offline and online channel products, and product profitability; the proposed analysis model can achieve the optimal order quantity in supply chain, and reach risk and benefit sharing among financial institutions, retailers, and suppliers by setting the financing interest rate, wholesale price, repurchase price and other parameters, thereby improving supply chain performance. This study results of this paper provided a reference for further researches on the application of fractal dimension of the fractional Brownian motion to the supply chain financing and operational comprehensive decision-making.

Financial markets are characterized by complex and often unpredictable dynamics, presenting significant challenges for investors, analysts, and policymakers. In recent years, fractal theory has emerged as a powerful tool for understanding the intricate patterns and behaviors exhibited by financial time series data. This paper provides a comprehensive review of the application of fractal theory in financial time series analysis, examining its theoretical foundations, empirical applications, and practical implications. Through a synthesis of relevant literature, we explore the utility of fractal techniques such as fractal dimension estimation, detrended fluctuation analysis (DFA), and multifractal analysis in quantifying the long-range dependence, self-similarity, and scaling properties of financial time series. Additionally, we discuss the implications of fractal dynamics for risk management, portfolio optimization, and market microstructure analysis, highlighting opportunities for future research and innovation in this evolving field.

We construct the Multifractal Fractional Random Walks with Hurst index 0 < H < 1/2 as the limits in C[0, 1], when l goes to 0, of the processes being the continuous cascade defined by Bacry and Muzy. This work completes the available literature, mainly dedicated to the case 1/2 ≤ H < 1. It is of particular interest for applications in a variety of areas, such as turbulence or finance.