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1. Hilbert-Type Dimension Polynomials of Intermediate Difference-Differential Field Extensions

   https://doi.org/10.1007/978-3-030-43120-4_7  

   Levin, Alexander ( December 2019 , Lecture notes in computer science)

   null (Ed.)

   Let K be an inversive difference-differential field and L a (not necessarily inversive) finitely generated difference-differential field extension of K. We consider the natural filtration of the extension L/K associated with a finite system \eta of its difference-differential generators and prove that for any intermediate difference-differential field F, the transcendence degrees of the components of the induced filtration of F are expressed by a certain numerical polynomial \chi_{K, F,\eta}(t). This polynomial is closely connected with the dimension Hilbert-type polynomial of a submodule of the module of K\"ahler differentials $\Omega_{L^{\ast}|K} where L^{\ast} is the inversive closure of L. We prove some properties of polynomials \chi_{K, F,\eta}(t) and use them for the study of the Krull-type dimension of the extension L/K. In the last part of the paper, we present a generalization of the obtained results to multidimensional filtrations of L/K associated with partitions of the sets of basic derivations and translations. 

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2. Some Properties and Invariants of Multivariate Difference-Differential Dimension Polynomials.

   Levin, A. ( June 2018 , Proceedings of the 24th Conference on Applications of Computer Algebra - ACA 2018)

   Multivariate dimension polynomials associated with finitely generated differential and difference field extensions arise as natural generalizations of the univariate differential and difference dimension polynomials. It turns out, however, that they carry more information about the corresponding extensions than their univariate counterparts. We extend the known results on multivariate dimension polynomials to the case of difference-differential field extensions with arbitrary partitions of sets of basic operators. We also describe some properties of multivariate dimension polynomials and their invariants. 

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3. Envelopes in multivariate regression models with nonlinearity and heteroscedasticity

   https://doi.org/10.1093/biomet/asaa036  

   Zhang, X ; Lee, C E ; Shao, X ( June 2020 , Biometrika)

   Summary Envelopes have been proposed in recent years as a nascent methodology for sufficient dimension reduction and efficient parameter estimation in multivariate linear models. We extend the classical definition of envelopes in Cook et al. (2010) to incorporate a nonlinear conditional mean function and a heteroscedastic error. Given any two random vectors ${X}\in\mathbb{R}^{p}$ and ${Y}\in\mathbb{R}^{r}$, we propose two new model-free envelopes, called the martingale difference divergence envelope and the central mean envelope, and study their relationships to the standard envelope in the context of response reduction in multivariate linear models. The martingale difference divergence envelope effectively captures the nonlinearity in the conditional mean without imposing any parametric structure or requiring any tuning in estimation. Heteroscedasticity, or nonconstant conditional covariance of ${Y}\mid{X}$, is further detected by the central mean envelope based on a slicing scheme for the data. We reveal the nested structure of different envelopes: (i) the central mean envelope contains the martingale difference divergence envelope, with equality when ${Y}\mid{X}$ has a constant conditional covariance; and (ii) the martingale difference divergence envelope contains the standard envelope, with equality when ${Y}\mid{X}$ has a linear conditional mean. We develop an estimation procedure that first obtains the martingale difference divergence envelope and then estimates the additional envelope components in the central mean envelope. We establish consistency in envelope estimation of the martingale difference divergence envelope and central mean envelope without stringent model assumptions. Simulations and real-data analysis demonstrate the advantages of the martingale difference divergence envelope and the central mean envelope over the standard envelope in dimension reduction. 

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4. Dimension Reduction and Coefficient Estimation in Multivariate Linear Regression

   https://doi.org/10.1111/j.1467-9868.2007.00591.x  

   Yuan, Ming ; Ekici, Ali ; Lu, Zhaosong ; Monteiro, Renato ( May 2007 , Journal of the Royal Statistical Society Series B: Statistical Methodology)

   We introduce a general formulation for dimension reduction and coefficient estimation in the multivariate linear model. We argue that many of the existing methods that are commonly used in practice can be formulated in this framework and have various restrictions. We continue to propose a new method that is more flexible and more generally applicable. The method proposed can be formulated as a novel penalized least squares estimate. The penalty that we employ is the coefficient matrix's Ky Fan norm. Such a penalty encourages the sparsity among singular values and at the same time gives shrinkage coefficient estimates and thus conducts dimension reduction and coefficient estimation simultaneously in the multivariate linear model. We also propose a generalized cross-validation type of criterion for the selection of the tuning parameter in the penalized least squares. Simulations and an application in financial econometrics demonstrate competitive performance of the new method. An extension to the non-parametric factor model is also discussed.

    

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5. Connections between finite difference and finite element approximations

   https://doi.org/https://doi.org/10.1080/00036811.2021.2005785  

   Constantin Bacuta ; Cristina Bacuta ( October 2021 , Applicable analysis)

   Taylor And Francis Online (Ed.)

   We present useful connections between the finite difference and the finite element methods for a model boundary value problem. We start from the observation that, in the finite element context, the interpolant of the solution in one dimension coincides with the finite element approximation of the solution. This result can be viewed as an extension of the Green function formula for the solution at the continuous level. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices and compare the right hand side load vectors for the two methods. Using evaluation of the Green function, a formula for the inverse of the stiffness matrix is extended to the case of non-uniformly distributed mesh points. We provide an error analysis based on the connection between the two methods and estimate the energy norm of the difference of the two solutions. Interesting extensions to the 2D case are provided. 

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