More Like this

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. 

   more » « less
2. Dimension Quasi-polynomials of Inversive Difference Field Extensions with Weighted Translations

   Levin, A. (November 2017, Mathematical Aspects of Computer and Information Sciences)

   We consider Hilbert-type functions associated with finitely generated inversive difference field extensions and systems of algebraic difference equations in the case when the translations are assigned positive integer weights. We prove that such functions are quasi-polynomials that can be represented as alternating sums of Ehrhart quasi-polynomials of rational conic polytopes. In particular, we generalize the author's results on difference dimension polynomials and their invariants to the case of inversive difference fields with weighted basic automorphisms. 

   more » « less
3. Generalized characteristic sets and new multivariate difference dimension polynomials

   https://doi.org/10.1007/s00200-023-00628-0  

   Levin, Alexander (October 2023, Applicable Algebra in Engineering, Communication and Computing)

   We introduce a new type of characteristic sets of difference polynomials using a generalization of the concept of effective order to the case of partial difference polynomials and a partition of the basic set of translations σ. Using properties of these characteristic sets, we prove the existence and outline a method of computation of a multivariate dimension polynomial of a finitely generated difference field extension that describes the transcendence degrees of intermediate fields obtained by adjoining transforms of the generators whose orders with respect to the components of the partition of σ are bounded by two sequences of natural numbers. We show that such dimension polynomials carry essentially more invariants (that is, characteristics of the extension that do not depend on the set of its difference generators) than previously known difference dimension polynomials. In particular, a dimension polynomial of the new type associated with a system of algebraic difference equations gives more information about the system than the classical univariate difference dimension polynomial. 

   more » « less
4. 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. 

   more » « less
5. 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. 

   more » « less