In nature or societies, the power-law is present ubiquitously, and then it is important to investigate the characteristics of power-laws in the recent era of big data. In this paper we prove the superposition of non-identical stochastic processes with power-laws converges in density to a unique stable distribution. This property can be used to explain the universality of stable laws such that the sums of the logarithmic return of non-identical stock price fluctuations follow stable distributions.

The exponential law has been discovered in various systems around the world. In this study, we introduce two existing and one proposed analytical method for exponential decay time-series predictions. The proposed method is given by a linear regression that is based on rescaling the time axis in terms of exponential decay laws. We confirm that the proposed method has a higher prediction accuracy than existing methods by performance evaluation using random numbers and verification using actual data. The proposed method can be used for analyzing real data modeled with exponential functions, which are ubiquitous in the world.

In recent years, the e-commerce market has grown with the spread of the internet worldwide every year. Accordingly, in service industries, purchasing products with reservations has become common. With the spread of online reservations, the booking curve, which is the concept of the time series in the cumulative number of reservations and has been used for sales optimization in the airline ticket and hotel industries, has been used in various industries. Booking curves in specific industries have been studied, but a universally applicable model across various industries has not been developed. In this study, we show that booking curves can be modeled universally by the exponential decay function, and we also show that the model is valid by using real data from some industries before and after the COVID-19 pandemic, that is, under completely different market conditions. The cross-industry exponential laws of booking curves constitute an important discovery in regard to mathematical laws in the social sciences and can be applied to give leading microeconomic indicators.

We propose a new dynamic pricing algorithm based on the universal exponential law of booking curves in services with reservations. The algorithm includes a parametric learning model which makes it possible to simulate the effect of changes in prices on quantity demanded from historical data continuously for practical use. Furthermore, we show an example, where some real data in a hotel applies for the learning model. Our proposed algorithm with the learning model, which can dynamically update the optimum parameters, is envisaged to be utilized as a practical dynamic pricing strategy.

The exponential law has been discovered in various systems around the world. In this study, we introduce two existing and one proposed analytical method for exponential decay time-series predictions. The proposed method is given by a linear regression that is based on rescaling the time axis in terms of exponential decay laws. We confirm that the proposed method has a higher prediction accuracy than existing methods by performance evaluation using random numbers and verification using actual data. The proposed method can be used for analyzing real data modeled with exponential functions, which are ubiquitous in the world.

The conditional Lyapunov exponent is defined for investigating chaotic synchronization, in particular complete synchronization and generalized synchronization. We find that the conditional Lyapunov exponent is expressed as a formula in terms of ergodic theory. Dealing with this formula, we find what factors characterize the conditional Lyapunov exponent in chaotic systems.

The booking curve time series in perishable asset industries, including hotels, has been studied as a means for daily forecasting in revenue management (RM). Such studies have developed many sophisticated forecasting algorithms for RM practitioners. However, based on the opinion that timing is the key element in pricing, RM professionals have faced challenges in understanding people's booking window shift, which represents macroscopic changes in booking curves due to changing times, e.g., economy and technology. We investigate macroscopic aspects of booking curves with actual sales data across six properties in the hotel and car-rental industries for two years, considering the difference in the economic environment characterized before, middle, and after the COVID-19 epidemic. We explain a new cross-industry and cross-economic-environment universal statistical law: average booking curves draw exponential functions (the ABCDEF law). We provide a basis for the ABCDEF law from three perspectives; data confirmation, modeling in the statistical physics framework, and empirical justification for the causality of the model. The ABCDEF law provides a booking curve with its usefulness besides daily forecasting in RM; it is expected to offer informative statistics about people's booking patterns in the property and to support various industries' RM practitioners in deciding on sales strategies at an appropriate time.

We present a solvable chaotic synchronization model of unidirectionally coupled dynamical systems. We establish a new interpretation of the conditional Lyapunov exponent that characterizes chaotic synchronization completely. Moreover, we newly show how the conditional Lyapunov exponent relates to common noise-induced synchronization phenomena by the new interpretation.