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1. Bivariate Dimension Quasi-polynomials of Difference–Differential Field Extensions with Weighted Basic Operators

   https://doi.org/10.1007/s11786-018-0361-5  

   Levin, Alexander ( June 2019 , Mathematics in Computer Science)
2. 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)

   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 somemore »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.« less
3. Eulerian partial-differential-equation methods for complex-valued eikonals in attenuating media

   https://doi.org/10.1190/GEO2020-0659.1  

   Hu, Jiangtao ; Qian, Jianliang ; Song, Jian ; Ouyang, Min ; Cao, Junxing ; Leung, Shingyu ( July 2021 , GEOPHYSICS)

   Seismic waves in earth media usually undergo attenuation, causing energy losses and phase distortions. In the regime of high-frequency asymptotics, a complex-valued eikonal is an essential ingredient for describing wave propagation in attenuating media, where the real and imaginary parts of the eikonal function capture dispersion effects and amplitude attenuation of seismic waves, respectively. Conventionally, such a complex-valued eikonal is mainly computed either by tracing rays exactly in complex space or by tracing rays approximately in real space so that the resulting eikonal is distributed irregularly in real space. However, seismic data processing methods, such as prestack depth migration andmore »tomography, usually require uniformly distributed complex-valued eikonals. Therefore, we have developed a unified framework to Eulerianize several popular approximate real-space ray-tracing methods for complex-valued eikonals so that the real and imaginary parts of the eikonal function satisfy the classic real-space eikonal equation and a novel real-space advection equation, respectively, and we dub the resulting method the Eulerian partial-differential-equation method. We further develop highly efficient high-order methods to solve these two equations by using the factorization idea and the Lax-Friedrichs weighted essentially nonoscillatory schemes. Numerical examples demonstrate that our method yields highly accurate complex-valued eikonals, analogous to those from ray-tracing methods. Our methods can be useful for migration and tomography in attenuating media.« less
4. CLUE: exact maximal reduction of kinetic models by constrained lumping of differential equations

   https://doi.org/10.1093/bioinformatics/btab010  

   Ovchinnikov, Alexey ; Pérez Verona, Isabel ; Pogudin, Gleb ; Tribastone, Mirco ( February 2021 , Bioinformatics)

   Valencia, Alfonso (Ed.)

   Abstract Motivation Detailed mechanistic models of biological processes can pose significant challenges for analysis and parameter estimations due to the large number of equations used to track the dynamics of all distinct configurations in which each involved biochemical species can be found. Model reduction can help tame such complexity by providing a lower-dimensional model in which each macro-variable can be directly related to the original variables. Results We present CLUE, an algorithm for exact model reduction of systems of polynomial differential equations by constrained linear lumping. It computes the smallest dimensional reduction as a linear mapping of the state spacemore »such that the reduced model preserves the dynamics of user-specified linear combinations of the original variables. Even though CLUE works with non-linear differential equations, it is based on linear algebra tools, which makes it applicable to high-dimensional models. Using case studies from the literature, we show how CLUE can substantially lower model dimensionality and help extract biologically intelligible insights from the reduction. Availabilityand implementation An implementation of the algorithm and relevant resources to replicate the experiments herein reported are freely available for download at https://github.com/pogudingleb/CLUE. Supplementary information Supplementary data are available at Bioinformatics online.« less
5. VC density of definable families over valued fields

   https://doi.org/10.4171/JEMS/1056  

   Basu, Saugata ; Patel, Deepam ( January 2021 , Journal of the European Mathematical Society)