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Due to the inherent uncertainty of wind conditions as well as the price unpredictability in the competitive electricity market, wind power producers are exposed to the risk of concurrent fluctuations in both price and volume. Therefore, it is imperative to develop strategies to effectively stabilize their revenues, or cash flows, when trading wind power output in the electricity market. In light of this context, we present a novel endeavor to construct multivariate derivatives for mitigating the risk of fluctuating cash flows that are associated with trading wind power generation in electricity markets. Our approach involves leveraging nonparametric techniques to identify optimal payoff structures or compute the positions of derivatives with fine granularity, utilizing multiple underlying indexes including spot electricity price, area-wide wind power production index, and local wind conditions. These derivatives, referred to as mixed derivatives, offer advantages in terms of hedge effectiveness and contracting efficiency. Notably, we develop a methodology to enhance the hedge effects by modeling multivariate functions of wind speed and wind direction, incorporating periodicity constraints on wind direction via tensor product spline functions. By conducting an empirical analysis using data from Japan, we elucidate the extent to which the hedge effectiveness is improved by constructing mixed derivatives from various perspectives. Furthermore, we compare the hedge performance between high-granular (hourly) and low-granular (daily) formulations, revealing the advantages of utilizing a high-granular hedging approach.

We obtain solution representation formulae for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second-order parabolic PDEs; in this case one can alternatively eliminate the mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant maps. It is shown here that these maps are well defined provided that certain analyticity issues are appropriately addressed.

Using the extended physics-informed neural network with twin subnets to study the coupled mixed derivative nonlinear Schrödinger equation (NLSE), seven types of vector solitons including vector single soliton, vector double solitons, anti-dark vector single soliton, anti-dark vector double solitons, vector rogue wave, vector bright–dark single soliton and vector bright–dark double solitons are predicted. The prediction results from seven types of vector solitons with different angles confirm that the physical neural network can be used to effectively solve the coupled mixed derivative NLSE. The error on the two sides and the falling point of the vector rogue wave solution is significantly greater than the middle, while the error of other six types of vector soliton solution mainly reflects in the peak or valley value of bright or dark soliton, and increases along the transmission distance. In addition, how to improve the prediction of model parameters from two aspects of data set and loss function is also studied. These results have certain reference value for studying the optical soliton transmission process by the machine learning.

In this paper, a coupled mixed derivative nonlinear Schrödinger system, which describes the short pulses in the femtosecond or picosecond regime of a birefringent optical fiber, is investigated. Based on the known N th-order breather solutions, we derive the first-order breathers and investigate their properties, e.g., velocities and peak amplitudes, where N is a positive integer. Then, two kinds of the second-order breathers are presented. We construct the N th-order semirational solutions, with only one spectral parameter involved. Based on the obtained N th-order semirational solutions, we analytically investigate and graphically illustrate the vector degenerate breathers, rogue and breather-rogue waves. We discuss how certain parameters, e.g., the nonlinear coefficients, affect the shapes of the degenerate breathers, rogue and breather-rogue waves.

In recent years, generative adversarial networks(GAN) has achieved great success in generating realistic images. However, the instability of GAN and the lower accuracy of physics-informed neural networks(PINN) in solving highly complex partial differential equations make training models extremely challenging. This paper proposes a novel physics-informed GAN with gradient penalty (PIGAN-GP) and applies it to predict solutions of the 2-coupled mixed derivative nonlinear Schrödinger. The PIGAN-GP integrates PINN as part of the generator in the GAN framework, namely, utilizes PINN to solve the physical equation and generate predictions for the soliton positions and shapes. We predict the positions and shapes of nondegenerate solitons by the real and predicted solutions to demonstrate the high accuracy and stability of this PIGAN-GP network. Additionally, we also discuss the influence of noise levels and different initializations on the model parameter discovery using the PINN.

We consider the problem of nonparametric regression when the covariate is d dimensional, where d ≥ 1. In this paper, we introduce and study two non-parametric least squares estimators (LSEs) in this setting—the entirely monotonic LSE and the constrained Hardy–Krause variation LSE. We show that these two LSEs are natural generalizations of univariate isotonic regression and univariate total variation denoising, respectively, to multiple dimensions. We discuss the characterization and computation of these two LSEs obtained from n data points. We provide a detailed study of their risk properties under the squared error loss and fixed uniform lattice design. We show that the finite sample risk of these LSEs is always bounded from above by n −2/3 modulo logarithmic factors depending on d; thus these nonparametric LSEs avoid the curse of dimensionality to some extent. We also prove nearly matching minimax lower bounds. Further, we illustrate that these LSEs are particularly useful in fitting rectangular piecewise constant functions. Specifically, we show that the risk of the entirely monotonic LSE is almost parametric (at most 1/n up to logarithmic factors) when the true function is well approximable by a rectangular piecewise constant entirely monotone function with not too many constant pieces. A similar result is also shown to hold for the constrained Hardy–Krause variation LSE for a simple subclass of rectangular piecewise constant functions. We believe that the proposed LSEs yield a novel approach to estimating multivariate functions using convex optimization that avoid the curse of dimensionality to some extent.

This study introduces a novel approach for investigating the solvability of boundary value problems for differential equations that incorporate both ordinary and fractional derivatives, specifically within the context of non-autonomous variable order. Unlike traditional methods in the literature, which often rely on generalized intervals and piecewise constant functions, we propose a new fractional operator better suited for this problem. We analyze the existence and uniqueness of solutions, establishing the conditions necessary for these properties to hold using the Krasnoselskii fixed-point theorem and Banach’s contraction principle. Our study also addresses the Ulam–Hyers stability of the proposed problems, examining how variations in boundary conditions influence the solution dynamics. To support our theoretical framework, we provide numerical examples that not only validate our findings but also demonstrate the practical applicability of these mixed derivative equations across various scientific domains. Additionally, concepts such as symmetry may offer further insights into the behavior of solutions. This research contributes to a deeper understanding of complex differential equations and their implications in real-world scenarios.

In this paper, we give known embedding theorems in Sobolev spaces and Sobolev-Morrey spaces with dominant mixed derivatives. And as an application of the embedding theorems we study the problem of existence, uniqueness and smoothness of solutions of p-type equation.

The paper considers spaces of periodic functions of several variables, namely, the Lorentz space Lq,τ(Tm) , the class of functions with bounded mixed fractional derivative Wq,τr¯ , 1

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