Enhanced study of complex systems by unveiling hidden symmetries with Dynamical Visibility

One of the great challenges in complex and chaotic dynamics is to reveal its deterministic structures. These temporal dynamical structures are sometimes a consequence of hidden symmetries. Detecting and understanding them can allow the study of complex systems even without knowing the full underlying mathematical description of the system. Here we introduce a new technique, called Dynamical Visibility, that quantifies temporal correlations of the dynamics based upon some symmetry conditions. This visibility measures the departure of the dynamics from internal symmetries. We apply this technique to well-known chaotic systems, such as the logistic map and the circle map, as well as to experimental data from a diode laser with optical feedback and external modulation. Our results show the robustness of the method in characterizing dynamics and highlighting transitions in the dynamical behavior of a complex system. We envision that this can be a useful tool in experimental data, as it can extract key features of the deterministic laws that govern the dynamics of a system, despite the lack of knowledge of those specific quantitative descriptions

On An Entropy Estimator Based On a Non-parametric Density Estimator For Non-negative Data

In the recent decades, entropy has become more and more essential in statistics and machine learning. It features in many applications involving data transmission, cryptography, signal processing, network theory, bio-informatics, and so on. A large number of estimators for entropy have been proposed in the past ten years. Here we focus on entropy estimation for non-negative random variables. Specifically, the use of entropy estimator based on Poisson-weights density estimator is found to be of interest. We establish some asymptotic properties of the resulting estimators and present a simulation study comparing these with well known estimators in literature

Spiritual factors in Vietnamese consumers' purchasing decisions

Consumer demand is a complex and dynamic phenomenon that exerts a profound influence on economic decision-making, market trends, and overall societal well-being. In this context, comprehending the determinants of consumer demand assumes paramount significance for various stakeholders, including businesses, policymakers, and scholars. Furthermore, the acknowledgment of spiritual factors as influential determinants in shaping the values and behaviors of a population introduces a nuanced dimension, particularly within culturally diverse societies such as Vietnam. This research endeavor seeks to engage with these pivotal dimensions by elucidating the role of spiritual factors in tandem with the determinants of consumer demand, with specific reference to the distinctive cultural milieu of Vietnam. Employing a comprehensive multidimensional methodology, encompassing qualitative interviews, surveys, quantitative data analysis, historical and cultural inquiries, logical reasoning, and empirical case studies, this study strives to unravel the intricate interplay between spirituality and consumer behavior. It aims to explicate the influence of deeply ingrained spiritual beliefs, firmly entrenched within Vietnamese culture, on various spheres of decision-making. These spheres span from the observance of auspicious birth rituals and naming conventions to the orchestration of significant life events and the valuation of high-value assets such as real estate and automobiles. Moreover, this research seeks to illuminate the enduring nexus between spirituality, cultural heritage, and the choices made by consumers. It endeavors to provide insights into the multifaceted dynamics governing decision-making processes within the Vietnamese context. Additionally, this inquiry extends its purview to the exploration of the economic implications and marketing strategies predicated upon spiritual underpinnings, underscoring the substantial role played by these spiritual beliefs in shaping the consumer landscape. Ultimately, this research aspires to furnish a holistic comprehension of how spirituality is interwoven into the fabric of daily existence in Vietnam, significantly molding the values and behaviors of its citizenry

A Numerical Study of Entropy and Residual Entropy Estimators Based on Smooth Density Estimators for Non-negative Random Variables

In this paper, we are interested in the entropy of a non-negative random variable. Since the underlying probability density function is unknown, we propose the use of Poisson smoothed histogram density estimator in order to estimate the entropy. To study the performance of our estimator, we run simulations on a wide range of densities and compare our entropy estimators with the existing estimators that based on different approaches such as spacing estimators. Furthermore, we extend our study to residual entropy estimators which is the entropy of a random variable given that it has been survived up to time t

Some Fluctuation Results Related to Draw-down Times for Spectrally Negative Levy processes And On Estimation of Entropy and residual Entropy for Nonnegative Random Variable

Part I In this thesis, we first introduce and review some fluctuation theory of Levy processes, especially for general spectrally negative Levy processes and for spectrally negative Levy taxed processes. Then we consider a more realistic model by introducing draw-down time, which is the first time a process falls below a predetermined draw-down level which is a function of the running maximum. Particularly, we present the expressions for the classical two-sided exit problems for these processes with draw-down times in terms of scale functions. We also find the expressions for the discounted present values of tax payments with draw-down time in place of ruin time. Finally, we obtain the expression of the occupation times for the general spectrally negative Levy processes to spend in draw-down interval killed on either exiting a fix upper level or a draw-down lower level. Part II Entropy has become more and more essential in statistics and machine learning. A large number of its applications can be found in data transmission, cryptography, signal processing, network theory, bio-informatics, and so on. Therefore, the question of entropy estimation comes naturally. Generally, if we consider the entropy of a random variable knowing that it has survived up to time $t$, then it is defined as the residual entropy. In this thesis we focus on entropy and residual entropy estimation for nonnegative random variable. We first present a quick review on properties of popular existing estimators. Then we propose some candidates for entropy and residual entropy estimator along with simulation study and comparison among estimators

Some Fluctuation Identities of Hyper-Exponential Jump-Diffusion Processes

Meromorphic L´evy processes have attracted the attention of a lot of researchers recently due to its special structure of the Wiener-Hopf factors as rational functions of infinite degree written in terms of poles and roots of the Laplace exponent, all of which are real numbers. With these Wiener-Hopf factors in hand, we can explicitly derive the expression of fluctuation identities that concern the first passage problems for finite and infinite intervals for the meromorphic L´evy process and the resulting process reflected at its infimum. In this thesis, we consider some fluctuation identities of some classes of meromorphic jump-diffusion processes with either the double exponential jumps or more general the hyper-exponential jumps. We study solutions to the one-sided and two-sided exit problems, and potential measure of the process killed on exiting a finite or infinite intervals. Also, we obtain some results to the process reflected at its infimum