Article
Mathematics, Applied
Hongjia Chen, Kuan Xu
Summary: This study investigates the backward errors of computed eigenpairs using the compact rational Krylov linearization method for solving PEP or REP problems. One-sided factorizations are constructed to relate the eigenpairs of the linearization and those of the original problems, providing upper bounds for the backward error of approximate eigenpairs. Numerical experiments show successful reduction of actual backward errors by scaling, with errors well predicted by the established bounds.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Computer Science, Theory & Methods
Daniele Bartoli, Giuliana Fatabbi, Francesco Ghiandoni
Summary: We investigate APN functions represented as rational functions and provide non-existence results by connecting these functions to algebraic varieties over finite fields. This approach allows for classifying families of functions where previous approaches are not applicable.
DESIGNS CODES AND CRYPTOGRAPHY
(2023)
Article
Mathematics
Zhiguo Ding, Michael E. Zieve
Summary: This article presents a construction of an infinite sequence of exceptional rational functions f(X) in F-q(X), where q is an odd prime power. These functions induce bijections of P-1 (F-qn) for infinitely many n and cannot be decomposed as compositions of lower-degree rational functions in F-q(X). These are the first known examples of wildly ramified indecomposable exceptional rational functions other than linear changes of polynomials.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2023)
Article
Mathematics
Zeyuan Song, Zuoren Sun
Summary: The central problem of this study is to represent any holomorphic and square integrable function on the Kepler manifold in the series form based on Fourier analysis. Three different domains on the Kepler manifold are considered and the weak pre-orthogonal adaptive Fourier decomposition (POAFD) is proposed. The weak maximal selection principle is shown to select the coefficient of the series, and a convergence theorem is proved to demonstrate the accuracy of the method.
Article
Mathematics
Benoit Collins, Tobias Mai, Akihiro Miyagawa, Felix Parraud, Sheng Yin
Summary: This paper investigates the convergence of the spectral distribution of noncommutative rational functions in independent random matrix models using free probability. It answers an open question by Roland Speicher and demonstrates that the approximation of self-adjoint noncommutative rational functions by generic matrices can be upgraded in terms of distribution convergence.
MATHEMATISCHE ANNALEN
(2022)
Article
Mathematics
Daniele Bartoli, Herivelto Borges, Luciane Quoos
Summary: This paper investigates rational functions with relatively small value sets in connection with Galois theory and algebraic curves, and proves under certain circumstances that having a small value set is equivalent to the field extension being Galois.
JOURNAL OF ALGEBRA
(2021)
Article
Mathematics, Applied
Satoshi Tsujimoto, Luc Vinet, Alexei Zhedanov
Summary: The biorthogonal rational functions of F-3(2) type on the uniform grid demonstrate properties similar to classical orthogonal polynomials, with three different operators X, Y, Z describing these properties and generating a quadratic algebra akin to Askey-Wilson type attached to hypergeometric polynomials.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
Gradimir Milovanovic, Abdullah Mir
Summary: Various versions of Bernstein and Turan-type inequalities have been generalized for rational functions in the complex plane, inspired by classical inequalities. The results obtained in this study extend many inequalities for polynomials as special cases.
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
(2021)
Article
Mathematics, Applied
Luc Vinet, Alexei Zhedanov
Summary: An algebra denoted mh with three generators is introduced, which admits embeddings of the Hahn algebra and the rational Hahn algebra. It includes representation bases corresponding to Hahn polynomials, the real version of the deformed Jordan plane, and eigenvalue problems. Overlaps between these bases are shown to be bispectral orthogonal polynomials or biorthogonal rational functions based on mh.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Daniele Bartoli, Marco Timpanella
Summary: This paper investigates the non-existence problem of rational perfect nonlinear functions over a finite field, using deep results about the number of points of algebraic varieties over finite fields.
ANNALI DI MATEMATICA PURA ED APPLICATA
(2023)
Article
Mathematics
Kelly Bickel, James Eldred Pascoe, Alan Sola
Summary: This study examines the boundary behavior of rational inner functions (RIFs) in higher dimensions from both analytic and geometric perspectives. The results highlight the loss of certain favorable behavior observed in the two-variable case.
AMERICAN JOURNAL OF MATHEMATICS
(2022)
Article
Mathematics, Applied
Desong Kong, Shuhuang Xiang, Li Li, Ran Li
Summary: In this paper, two kinds of extended barycentric rational schemes for approximating functions of singularities are proposed. The schemes achieve higher convergence rates as the scaled parameter and the degree of the local approximation polynomial increase. Moreover, an accurate Levin method for dealing with highly oscillatory integrals is derived from the barycentric formula.
Article
Mathematics, Applied
Ismael Bussiere, Julien Gaboriaud, Luc Vinet, Alexei Zhedanov
Summary: In this paper, a family of rational functions biorthogonal to q-hypergeometric distribution is introduced, showing that a triplet of q-difference operators X, Y, Z plays a role analogous to bispectral operators of orthogonal polynomials. The recurrence relation and difference equation are presented as generalized eigenvalue problems involving the three operators, and the algebra generated by X, Y, Z is similar to Askey-Wilson type algebras in the case of orthogonal polynomials. The actions of these operators in three bases and connections with Wilson's (10)phi(9) biorthogonal rational functions are discussed.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics
A. Prymak
Summary: The paper computes the behavior of the Christoffel function in an arbitrary planar convex domain up to a constant factor, using comparison with other simple reference domains. By constructing appropriate ellipse and parallelogram containing the domain, lower and upper bounds are obtained respectively. As an application, a new proof is presented that every planar convex domain possesses optimal polynomial meshes.
JOURNAL OF APPROXIMATION THEORY
(2021)
Article
Computer Science, Software Engineering
Takafumi Saito, Norimasa Yoshida
Summary: In this paper, we have derived functions of the lowest possible degree to evaluate curvature monotonicity for any 2D and 3D rational Bezier curves. We have proven that the degree of the function is at most 8n - 12 for planar rational Bezier curves of degree n, and at most 11n - 18 for space rational Bezier curves of degree n. These functions, derived in the Bernstein basis, allow for efficient checking of curvature monotonicity using subdivision or Bezier clipping. As an application, we have presented real-time visualization of the region of a particular control point that guarantees monotonic variation of curvature over the entire segment of the rational Bezier curve, enabling users to identify where to move the control point to ensure monotonic changes in curvature.
COMPUTERS & GRAPHICS-UK
(2023)
Article
Mathematics, Applied
Junfeng Cao, Ke Chen, Huan Han
Summary: This paper proposes a two-stage image segmentation model based on structure tensor and fractional-order regularization. In the first stage, fractional-order regularization is used to approximate the Hausdorff measure of the MS model. The solution is found using the ADI scheme. In the second stage, thresholding is used for target segmentation. The proposed model demonstrates superior performance compared to state-of-the-art methods.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Dylan J. Oliver, Ian W. Turner, Elliot J. Carr
Summary: This paper discusses a projection-based framework for numerical computation of advection-diffusion-reaction (ADR) equations in heterogeneous media with multiple layers or complex geometric structures. By obtaining approximate solutions on a coarse grid and reconstructing solutions on a fine grid, the computational cost is significantly reduced while accurately approximating complex solutions.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Nathan V. Roberts, Sean T. Miller, Stephen D. Bond, Eric C. Cyr
Summary: In this study, the time-marching discontinuous Petrov-Galerkin (DPG) method is applied to the Vlasov equation for the first time, using backward Euler for a Vlasov-Poisson discretization. Adaptive mesh refinement is demonstrated on two problems: the two-stream instability problem and a cold diode problem.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Yizhi Sun, Zhilin Sun
Summary: This work investigates the convexity of a specific class of positive definite probability measures and demonstrates the preservation of convexity under multiplication and intertwining product. The study reveals that any integrable function on an interval with a polynomial expansion of fast absolute convergence can be decomposed into a pair of positive convex interval probabilities, simplifying the study of interval distributions and discontinuous probabilistic Galerkin schemes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Bhagwan Singh, Komal Jangid, Santwana Mukhopadhyay
Summary: This paper examines the prediction of bending characteristics of nanoscale materials using the Moore-Gibson-Thompson thermoelasticity theory in conjunction with the nonlocal strain gradient theory. The study finds that the stiffness of the materials can be affected by nonlocal and length-scale parameters, and the aspect ratios of the beam structure play a significant role in bending simulations.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Guoliang Wang, Bo Zheng, Yueqiang Shang
Summary: This paper presents and analyzes a parallel finite element post-processing algorithm for the simulation of Stokes equations with a nonlinear damping term, which integrates the algorithmic advantages of the two-level approach, the partition of unity method, and the post-processing technique. The algorithm generates a global continuous approximate solution using the partition of unity method and improves the smoothness of the solution by adding an extra coarse grid correction step. It has good parallel performance and is validated through theoretical error estimates and numerical test examples.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Hao Xu, Zeng-Qi Wang
Summary: Fluid flow control problems are crucial in industrial applications, and solving the optimal control of Navier-Stokes equations is challenging. By using Oseen's approximation and matrix splitting preconditioners, we can efficiently solve the linear systems and improve convergence.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Zhengya Yang, Xuejuan Chen, Yanping Chen, Jing Wang
Summary: This paper focuses on the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for spatial discretization and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. As a result, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Several numerical examples are presented to demonstrate the effectiveness and feasibility of the proposed numerical scheme.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)