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For the precision calculations in perturbative Quantum Chromodynamics (QCD) gigantic expressions (several GB in size) in terms of highly complicated divergent multi-loop Feynman integrals have to be calculated analytically to compact expressions in terms of special functions and constants. In this article we derive new symbolic tools to gain large-scale computer understanding in QCD. Here we exploit the fact that hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package Sigma in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code HypSeries transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code solvePartialLDE is designed. Generalized hypergeometric functions, Appell-, Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton–type functions are considered. We illustrate the algorithms by examples.

Let V be a vector space over a field F and let W be a subspace of V . The semigroup of partial linear transformations on V whose restriction to W belongs to an injective partial linear transformation semigroup I(W) is denoted by PI(W)(V) . In this paper, we describe Green's relations for PI(W)(V) , characterize its regular elements, and give necessary and sufficient conditions for PI(W)(V) to be regular, inverse, or completely regular. We also analyze the ideal structure of PI(W)(V) , identifying its maximal and minimal ideals.

The Kortweg–de Vries equations play an important role to model different physical phenomena in nature. In this research article, we have investigated the analytical solution to system of nonlinear fractional Kortweg–de Vries, partial differential equations. The Caputo operator is used to define fractional derivatives. Some illustrative examples are considered to check the validity and accuracy of the proposed method. The obtained results have shown the best agreement with the exact solution for the problems. The solution graphs are in full support to confirm the authenticity of the present method.

Partial linear models have been widely used as flexible method for modelling linear components in conjunction with non-parametric ones. Despite the presence of the non-parametric part, the linear, parametric part can under certain conditions be estimated with parametric rate. In this paper, we consider a high-dimensional linear part. We show that it can be estimated with oracle rates, using the least absolute shrinkage and selection operator penalty for the linear part and a smoothness penalty for the nonparametric part.

We develop a (global) solvability theory for Euler type linear partial differential equations P(θ)P(\\theta ) on C∞(Ω)C^\\infty (\\Omega ), with Ω\\Omega an open subset of Rd\\mathbb {R}^d, i.e., for a special type of linear partial differential equation with polynomial coefficients. There is a natural closed upper bound CI(P)∞(Ω)C^\\infty _{I(P)}(\\Omega ) for the range of P(θ)P(\\theta ) on C∞(Ω)C^\\infty (\\Omega ). We characterize by P(θ)P(\\theta )-convexity type conditions those Ω\\Omega such that P(θ)P(\\theta ) is surjective on CI(P)∞(Ω)C^\\infty _{I(P)}(\\Omega ). We also clarify when all shifted operators P(c+θ)P(c+\\theta ) are surjective on CI(P(c+ ⋅ ))∞(Ω)C^\\infty _{I(P(c+\\ \\cdot \\ ))}(\\Omega ). We classify in geometric terms those Ω\\Omega with 0∈Ω0\\in \\Omega such that every Euler operator P(θ)P(\\theta ) is surjective on CI(P)∞(Ω)C^\\infty _{I(P)}(\\Omega ). Moreover, we determine the operators P(θ)P(\\theta ) which are surjective onto CI(P)∞(Ω)C^\\infty _{I(P)}(\\Omega ) for every open set Ω⊆Rd\\Omega \\subseteq \\mathbb {R}^d. Under some mild assumptions on Ω\\Omega, we characterize those Euler operators which are invertible on C∞(Ω)C^\\infty (\\Omega ). Under the same assumptions we also calculate the spectrum of P(θ)P(\\theta ) on C∞(Ω)C^\\infty (\\Omega ). The results follow from the solvability theory for Hadamard type operators on the space of smooth functions and from a new general Mellin transform, both developed in this paper.

We study the motion of a test particle in a conservative force-field. Our aim is to find three-dimensional potentials with symmetrical properties, i.e., V(x,y,z)=P(x,y)+Q(z), or, V(x,y,z)=P(x2+y2)+Q(z) and V(x,y,z)=P(x,y)Q(z), where P and Q are arbitrary C2-functions, which are characterized as semi-separable and they produce a pre-assigned two-parametric family of orbits f(x,y,z) = c1, g(x,y,z) = c2 (c1, c2 = const) in 3D space. There exist two linear PDEs which are the basic equations of the Inverse Problem of Newtonian Dynamics and are satisfied by these potentials. Pertinent examples are presented for all the cases. Two-dimensional potentials are also included into our study. Families of straight lines is a special category of curves in 3D space and are examined separately.

We consider a flexible semiparametric quantile regression model for analyzing high dimensional heterogeneous data. This model has several appealing features: (1) By considering different conditional quantiles, we may obtain a more complete picture of the conditional distribution of a response variable given high dimensional covariates. (2) The sparsity level is allowed to be different at different quantile levels. (3) The partially linear additive structure accommodates nonlinearity and circumvents the curse of dimensionality. (4) It is naturally robust to heavy-tailed distributions. In this paper, we approximate the nonlinear components using B-spline basis functions. We first study estimation under this model when the nonzero components are known in advance and the number of covariates in the linear part diverges. We then investigate a nonconvex penalized estimator for simultaneous variable selection and estimation. We derive its oracle property for a general class of nonconvex penalty functions in the presence of ultra-high dimensional covariates under relaxed conditions. To tackle the challenges of nonsmooth loss function, nonconvex penalty function and the presence of nonlinear components, we combine a recently developed convex-differencing method with modern empirical process techniques. Monte Carlo simulations and an application to a microarray study demonstrate the effectiveness of the proposed method. We also discuss how the method for a single quantile of interest can be extended to simultaneous variable selection and estimation at multiple quantiles.

In order to promote the green transformation of agricultural development, we used a partial linear function coefficient panel model to measure the impact of environmental regulations in 30 provinces and cities in China on agricultural green technology innovation and agricultural green total factor productivity. The advantage of this model is that it can take into account the heterogeneity of regional economic development levels, that is, by introducing variables that are functions of regional economic development levels as coefficients of environmental regulation. The research results show that: when the level of regional economic development is low, environmental regulation has a limited impact on agricultural green technology innovation and agricultural green total factor productivity, but as the level of regional economic development gradually increases, environmental regulation has a more significant impact on the two. And environmental regulation has a greater impact on agricultural green total factor productivity than on agricultural green technology innovation. Based on the research results, policy recommendations are suggested.

This paper introduces two new estimators based on the philosophy of unbiased ridge regression estimation, where the parameters are part of a partial linear model suffering from multicollinearity. These proposed estimators are called the Difference-Based Unbiased Ridge Estimator$$ {\\hat{\\beta}}_{DB- URR} $$and the Difference-Based Modified Unbiased Ridge Estimator$$ {\\hat{\\beta}}_{DB- MUR} $$for the regression parameters β . The Mean Squared Error Matrix (MSEM) criterion is employed to compare the proposed estimators against the Difference-Based Ordinary Least Squares estimator$$ {\\hat{\\beta}}_{DB- OLS} $$and the Difference-Based Ordinary Ridge Estimator$$ {\\hat{\\beta}}_{DB- ORR} $$ . Finally, the performance of the new estimators is evaluated through a comprehensive simulation study and a numerical example.

A partial linear model with instrumental variables was developed for longitudinal data. In the partially linear model, the explanatory variable is an endogenous variable, which is correlated with the error term. The endogenous variable was expressed by an instrumental variable and an error item. The endogenous variable was estimated by the instrumental variable through the least square method. B-spline regression combined with QR decomposition was used to approximate the nonparametric function. For the estimation of parametric, the Quadratic inference function and Secant method were applied. Under some conditions, the estimator was consistent and asymptotic normality. Some simulation was conducted to prove the finite sample behavior of the estimator.