A class of strong limit theorems for nonhomogeneous Markov chains indexed by a generalized Bethe tree on a generalized random selection system

We study strong limit theorems for a bivariate function sequence of an nonhomogeneous Markov chain indexed by a generalized Bethe tree on a generalized random selection system by constructing a nonnegative martingale. As corollaries, we generalize results of Yang and Ye and obtain some limit theorems for frequencies of states, ordered couples of states, and the conditional expectation of a bivariate function on Cayley tree.Вивчаються сильнi граничнi теореми для послiдовностi функцiй двох змiнних неоднорiдного марковського ланцюжка, що проiндексований узагальненим деревом Бете на узагальненiй системi випадкового вибору, шляхом побудови невiд’ємного мартингала. Як наслiдок, узагальнено результати Янга та Є i отримано деякi граничнi теореми для частот станiв, упорядкованих пар та умовного сподiвання функцiї двох змiнних на деревi Келi

Equilibrium Implications of Social Utility and Reference Dependence

It is well-known that consumers can deviate from the assumptions of standard economic theory in many ways. My dissertation examines the strategic implications of some behavioral anomalies of consumers in an equilibrium framework. In particular, I focus on how these behavioral anomalies could affect the optimal strategy and profitability of firms. The dissertation is comprised of two essays. In the first essay, I study the implications of consumer utility having a social component for competitive strategy of the firms and propose a model to explain the randomness of fashion hits observed in the fashion market. In the second essay, I investigate how firm pricing decision and profitability are affected by the presence of reference prices and consumer loss aversion. The first essay looks at the fashion industry, where product demand is largely driven by consumers\u27 desire to signal their social status through product usage, and theoretically explains why fashion hits appear to be randomly created. The fashion industry is characterized by its unpredictability and apparent randomness of fashion hits, i.e., no one is able to predict ahead of time which product will become the fashion statement about one\u27s social status. In this essay, I consider fashion as a means consumers use to signal belonging to the high class and propose an analytical model of fashion hits in the presence of competition and consumers able to coordinate on which product to use. I show that consistently with the observed market phenomenon, in equilibrium, consumer coordination involves randomization between products chosen, i.e., in randomness of fashion hits. Analyzing optimal consumer choice, I derive that whenever low-type consumer demand for a product is positive, a price increase results in a higher probability of high-type consumers choosing this product but lower low-type consumer demand. I also show that although high-type consumers may prefer: higher) prices that would lead to complete separation of the high- and the low-type consumers through product use, in equilibrium, firms always price as to attract positive demand from low-type consumers. The equilibrium price and profits turn out to be non-monotone in the low-type consumer valuation of being recognized as belonging to the high class. Equilibrium profits first increase and then decrease in this valuation. The second essay focuses on another consumer deviation from standard economic theory -- consumers evaluate prices relative to a reference point and exhibit loss aversion, i.e., consumer propensity to buy is more negatively affected by prices above the reference point than it is positively affected by prices below the reference point. Consumer loss aversion is a well-established phenomenon in marketing. The objective of this essay is to analytically examine how the competitive strategy and profitability are affected by the presence of consumer loss aversion. I develop a two-period model of competing firms and consumer search behavior affected by the reference price formed from past prices. While assuming that consumer loss aversion increases consumer propensity to search for lower prices, I find that it does not necessarily lead to lower prices and profits in a competitive environment. Specifically, consumer loss aversion could lead to higher prices and profits when consumer valuation is sufficiently high relative to their search costs and the proportion of consumers with positive search costs is in an intermediate range. I also show that even when forward-looking firms fully incorporate the negative effect of current price promotions on future profits, not only they still find promotional pricing optimal, but the optimal magnitude of promotions should not necessarily be reduced

Chaos analysis of ship rolling motion in stochastic beam seas

In this paper, the chaotic motion of ship roll in stochastic beam seas, which is regarded as a bounded noise is investigated in detail. The stochastic Melnikov approach is applied to the model and the criterion for the chaos in the mean-square sense is derived. The chaotic thresholds of noise parameters obtained by means of the stochastic Melnikov process are verified by the numerical safe basin. Besides of the factors of noise disturbances, the effects of the other parameters in the system on safe basin are also discussed systematically. The varieties of coefficients of the restoring moment can also induce the erosion of safe basin and lead to the occurrence of the chaos. On the other hand, the increase of damping coefficients can enhance the safe domain of ship sailing

Decentralized Douglas-Rachford splitting methods for smooth optimization over compact submanifolds

We study decentralized smooth optimization problems over compact submanifolds. Recasting it as a composite optimization problem, we propose a decentralized Douglas-Rachford splitting algorithm, DDRS. When the proximal operator of the local loss function does not have a closed-form solution, an inexact version of DDRS, iDDRS, is also presented. Both algorithms rely on an ingenious integration of the nonconvex Douglas-Rachford splitting algorithm with gradient tracking and manifold optimization. We show that our DDRS and iDDRS achieve the best-known convergence rate of $\mathcal{O}(1/K)$. The main challenge in the proof is how to handle the nonconvexity of the manifold constraint. To address this issue, we utilize the concept of proximal smoothness for compact submanifolds. This ensures that the projection onto the submanifold exhibits convexity-like properties, which allows us to control the consensus error across agents. Numerical experiments on the principal component analysis are conducted to demonstrate the effectiveness of our decentralized DRS compared with the state-of-the-art ones

Adsorption–desorption behavior of malachite green by potassium permanganate pre-oxidation polyvinyl chloride microplastics

Microplastics (MPs) and the typical hydrophilic organic pollutant Malachite green (MG) are frequently detected in sewage treatment plants. Potassium permanganate (KMnO4) pre-oxidation is an economical and effective technology in wastewater treatment. It is important to study the surface physicochemical characteristics of MPs and understand their fate in wastewater treatment plants after pre-oxidation. In this study, Polyvinyl chloride (PVC) MPs were treated by single and composite KMnO4 pre-oxidation with different pH values. After the pre-oxidation treatment, the appearance of Osingle bondMn spectra and surface nanoparticles indicated the oxides (MnO2) were produced on the MPs surface. Moreover, the adhesion of MnO2 is helpful to improve the hydrophilicity and adsorption capacity of MG. The adsorption capacity of pristine PVC for MG was 2.6 mg/g. But the adsorption capacity increased to 7.0 mg/g for single oxidation and 140.7 mg/g for composite oxidation, respectively. The desorption experiment results indicate the pre-oxidation process could reduce the release efficiency of MG from the PVC MPs due to the better binding of surface MnO2 nanoparticles to MG. However, the total desorption capacity is still high. which illustrates that there is a high potential risk of MG which can transfer from the surface of the PVC MPs to the gastrointestinal fluids.publishedVersio

Desorption of sulfamethoxazole from polyamide 6 microplastics: Environmental factors, simulated gastrointestinal fluids, and desorption mechanisms

Microplastics (MPs) can enrich pollutants after being released into the environment, and the contaminants-loaded MPs are usually ingested by organisms, resulting in a potential dual biotoxic effect. In this paper, the adsorption behavior of Sulfamethoxazole (SMX) on Polyamide 6 (PA6) MPs was systematically investigated and simulated by the kinetic and isotherm models. The effect of environmental conditions (pH, salinity) on the adsorption process was studied, and the desorption behavior of SMX-loaded PA6 MPs was focused on simulating the seawater, ultrapure water, gastric and intestinal fluids. We found that lower pH and solubilization of SMX by gastrointestinal components (bovine serum albumin (BSA), sodium taurocholate (NaT), and pepsin) can reduce the electrostatic interaction between the surface charge of PA6 MPs and SMX. The result will lead to an increase in the desorption capacity of SMX-loaded PA6 MPs in gastrointestinal fluids and therefore will provide a reasonable mechanism for the desorption of SMX-loaded PA6 MPs in the gastrointestinal fluids. This study will provide a theoretical reference for studying the desorption behavior of SMX-loaded PA6 MPs under gastrointestinal conditions.publishedVersio

Comparing the adsorption of methyl orange and malachite green on similar yet distinct polyamide microplastics: Uncovering hydrogen bond interactions

Microplastics (MPs) and dye pollutants are widespread in aquatic environments. Here, the adsorption characteristics of anionic dye methyl orange (MO) and cationic dye malachite green (MG) on polyamide 6 (PA6) and polyamide 66 (PA66) MPs were investigated, including kinetics, isotherm equilibrium and thermodynamics. The co-adsorption of MO and MG under different pH was also evaluated. The results reveal that the adsorption process of MO and MG is suitably expounded by a pseudo-second-order kinetic model. The process can be characterized by two stages: internal diffusion and external diffusion. The isothermal adsorption equilibrium of MO and MG can be effectively described using the Langmuir model, signifying monolayer adsorption. Furthermore, the thermodynamic results indicated that the adsorption was spontaneous with exothermic and endothermic properties, respectively. The results of binary systems reveal that MO dominates the adsorption at low pH (2–5), while MG dominates at high pH (8–10). Strong competitive adsorption was observed between MO and MG in neutral conditions (pH 6–8). The desorption experiments confirm that PA6 and PA66 could serve as potential carriers of MO and MG. The interaction between dyes and polyamide MPs is primarily mediated through hydrogen bonds and electrostatic attraction. The results reveal that PA6 formed more hydrogen bonds with the dyes, resulting in higher adsorption capacity than that of PA66. This difference can be attributed to the disparities in the synthesis process and polymerization method. Our study uncovered the adsorption mechanism of dye pollutants on PA6 and PA66, and provided a more comprehensive theoretical basis for the risk assessment concerning different types of polyamide MPs in aquatic environments.publishedVersio

Competition adsorption of malachite green and rhodamine B on polyethylene and polyvinyl chloride microplastics in aqueous environment

Microplastics (MPs) will cause compound pollution by combining with organic pollutants in the aqueous environment. It is important for environmental protection to study the adsorption mechanism of different MPs for pollutants. In this study, the adsorption behaviors of malachite green (MG) and rhodamine B (RhB) on polyethylene (PE) and polyvinyl chloride (PVC) were studied in single systems and binary systems, separately. The results show that in single system, the adsorptions of between MPs for pollutants (MG and RhB) are more consistent with the pseudo-second-order kinetics and Freundlich isotherm model, the adsorption capacity of both MPs for MG is greater than that of RhB. The adsorption capacities of MG and RhB were 7.68 mg/g and 2.83 mg/g for PVC, 4.52 mg/g and 1.27 mg/g for PE. In the binary system, there exist competitive adsorption between MG and RhB on MPs. And the adsorption capacities of PVC for the two dyes are stronger than those of PE. This is attributed to the strong halogen-hydrogen bond between the two dyes and PVC, and the larger specific surface area of PVC. This study revealed the interaction and competitive adsorption mechanism between binary dyes and MPs, which is of great significance for understanding the interactions between dyes and MPs in the multi-component systems.publishedVersio

Toward the Tradeoffs between Privacy, Fairness and Utility in Federated Learning

Federated Learning (FL) is a novel privacy-protection distributed machine learning paradigm that guarantees user privacy and prevents the risk of data leakage due to the advantage of the client's local training. Researchers have struggled to design fair FL systems that ensure fairness of results. However, the interplay between fairness and privacy has been less studied. Increasing the fairness of FL systems can have an impact on user privacy, while an increase in user privacy can affect fairness. In this work, on the client side, we use fairness metrics, such as Demographic Parity (DemP), Equalized Odds (EOs), and Disparate Impact (DI), to construct the local fair model. To protect the privacy of the client model, we propose a privacy-protection fairness FL method. The results show that the accuracy of the fair model with privacy increases because privacy breaks the constraints of the fairness metrics. In our experiments, we conclude the relationship between privacy, fairness and utility, and there is a tradeoff between these.Comment: 17 pages, 3 figures, conferenc