The use of organic materials has been increasing due to improving soil fertility by affecting phosphorus (P) adsorption and desorption behavior of soils. However, previous studies ignored the influences of increased P concentrations in equilibrium solutions caused by P released from applied organic materials in bath P adsorption experiments. To eliminate these influences, a modified Langmuir equation was applied to describe P adsorption properties in dairy manure composts and biosolids-incubated soils. The adsorption and desorption trends shifted around the initial P concentration of 70 mg/L. The fitted parameter of correlation coefficients showed that the modified Langmuir model fitted well for the adsorption data covering only the lower initial P concentrations (0–70 mg P/L) and covering the whole P concentrations applied in the present study (0–400 mg/L). For the fitted results covering the whole P concentrations, adding organic materials generally decreased both the Γmax (maximum adsorption capacity for P) and the MABC (maximum adsorption buffering capacity), except for the 20-g/kg biosolids incubation, indicating that the number of adsorption sites for P decreased as the organic materials applied, and the binding energy for the P adsorption to the soil also decreased to some extent with some fluctuation as reflected by K values which represent adsorption strength. Higher DOC content, lower molecular weight, and higher humic acid might result in the higher Γmax of 20-g/kg biosolids incubated soil. The linear equations are described well for P desorption. The constant a value of slope in a linear equation reflects the P desorption capacity of soils increased due to the application of organic materials, especially in high P concentrations with a value of 0.45 for the control soil increased to 1.02 for 10-g/kg biosolids treated soil. Thus, it would be important to control the P application amounts when the application of P fertilizers to the soils with organic materials applied, as the effects of organic materials on P adsorption and desorption characteristics were more efficient at high P concentrations.

Cryptocurrency is one of the earliest and the most successful applications of blockchain, and it utilizes the distributed ledger, which is a commonly used technique in blockchain, to make a decentralized transaction within the blockchain of a cryptocurrency. However, how to make a decentralized transaction of cryptocurrencies between parties on different blockchains, that is, the cross‐chain exchange, is not well‐studied. In this paper, we develop a new method to make cross‐chain exchanges based on the classical atomic swap. We first study the optionality embedded into the atomic swap and propose to add a premium into the atomic swap, and then design a new procedure with the premium to guarantee the fairness of the cross‐chain exchange. We also provide an algorithm based on the least‐squares Monte Carlo method to estimate the premium and analyze the convergence of the algorithm. Moreover, we study the cross‐chain exchange with margin trading. We propose an adapted exchange procedure to make a fair cross‐chain exchange and an algorithm to estimate the fair premium under the margin trading. Numerical experiments are provided to show the effectiveness of the algorithms.

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This paper establishes the existence of a solution to the optimization problem. Supposing that risk assets pay continuous dividend regarded as the function of time. It is established that the behaviour model of the stock pricing process is jump-diffusion driven by a count process. We give a characterization of the optimal portfolio by means of the value function and the equivalent martingale measure defined by the utility function. The unique equivalent martingale measure, the unique optimal consumption and portfolio pair and the corresponding wealth process are deduced. We provide a simple characterization of an equilibrium market and discuss existence and uniqueness of equilibrium in the economy.

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