Article
Mathematics, Applied
Joe Kramer-Miller
Summary: This article proves a 'Newton over Hodge' result for finite characters on curves in finite fields. It provides a lower bound on the Newton polygon of the L-function L(rho, S) based on monodromy invariants, which agrees with Deligne's irregular Hodge filtration under certain assumptions.
FORUM OF MATHEMATICS SIGMA
(2022)
Article
Mathematics
Santiago Molina
Summary: In this note, a new construction of cyclotomic p-adic L-functions attached to classical modular cuspidal eigenforms is proposed, covering most known cases and allowing for generalizations to automorphic forms on arbitrary groups. The method also applies to the construction of p-adic L-functions in extremal cases for Hilbert cusp forms over totally real number fields of even degree. The admissibility and interpolation properties of the extremal p-adic L-functions are studied, and their relation to two-variable p-adic L-functions interpolating cyclotomic p-adic L-functions along a Coleman family is discussed.
Article
Mathematics, Applied
Byeongseon Jeong, Scott N. Kersey, Jungho Yoon
Summary: This study introduces a new class of quasi-interpolation schemes for approximating multivariate functions on sparse grids, utilizing shifted kernels from one-dimensional radial basis functions. The schemes achieve high convergence order and reduce data requirements compared to full grid methods. Single-level and multilevel implementations show similar performance with reduced computation time, providing significantly better approximation rates than existing methods.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Chemistry, Physical
Vladimir V. Rybkin
Summary: The research on embedding potential using products of atomic orbital basis functions offers a new approach in the context of density functional embedding theory, allowing for the treatment of pseudopotential and all-electron calculations in a compact matrix form. With cost reduction procedures and population analysis based potential reduction, the method provides a simplified way to handle basis sets and potentials. Implemented for various systems, including proton-transfer reactions and density of states calculations, the method shows potential for large-scale applications to extended systems.
JOURNAL OF CHEMICAL THEORY AND COMPUTATION
(2021)
Article
Mathematics, Applied
Fuat Usta
Summary: This paper introduces a generalization of Bernstein-Chlodowsky type operators depending on function tau through two sequences of functions. The newly defined operators fix a specific test function set and demonstrate their approximation properties and effectiveness in function approximation.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Interdisciplinary Applications
Megha Pandey, Vishal Agrawal, Tanmoy Som
Summary: In this paper, we investigate the dimension preserving approximation of continuous multivariate functions defined on the domain [0,1](q). We establish well-known constrained approximation results in terms of dimension preserving approximants and propose a method using alpha-fractal interpolation functions for constructing multivariate dimension preserving approximants. We also prove the existence of one-sided approximation of multivariate functions using fractal functions and provide an upper bound for the fractal dimension of the graph of the alpha-fractal function. Furthermore, we study the approximation aspects of alpha-fractal functions and prove the existence of Schauder basis consisting of multivariate fractal functions for the space of all real valued continuous functions defined on [0,1](q), as well as the existence of multivariate fractal polynomials for approximation.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2022)
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
Eugene Shargorodsky, Teo Sharia
Summary: This article investigates the relationship between the conditional expectation operator of a random variable and its moments in a probability space, as well as the optimal constant in the bounded compact approximation property of Lp([0,1]) space.
MATHEMATISCHE NACHRICHTEN
(2023)
Article
Mathematics, Applied
Dhiraj Patel, Yuan Xu
Summary: This paper demonstrates that a set of random points uniformly distributed over Sn-1 provides a reliable sampling strategy for a specific class of localized functions in L2(Sn-1) with a high probability of accuracy. The probability bound for the random sampling inequality of the localized function is derived using the matrix Bernstein inequality on independent random matrices.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics, Applied
Gradimir Milovanovic, Abdullah Mir, Abrar Ahmad
Summary: This paper obtains sharp estimates for the maximal modulus of a rational function with prescribed poles, based on a new version of the Schwarz lemma for regular functions suggested by Osserman. The results also lead to several inequalities for polynomials.
Article
Mathematics
Mariusz Mirek, Tomasz Z. Szarek, Blazej Wrobel
Summary: We prove that the discrete spherical maximal functions corresponding to the Euclidean spheres in Z(d) with dyadic radii have bounded norms in l(p)(Z(d)) for all p in [2, infinity], independent of the dimensions d >= 5. The asymptotic formula in the Waring problem for the squares with a dimension-free multiplicative error term plays a crucial role in our argument. By introducing new approximating multipliers, we demonstrate how to absorb exponential growth in norms arising from the sampling principle and obtain dimension-free estimates for the discrete spherical maximal functions.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2023)
Article
Mathematics
Yunyi Fu, Yuanpeng Zhu
Summary: In this paper, four new generalized quasi cubic trigonometric Bernstein basis functions are constructed under the framework of Extended Chebyshev space. Sufficient conditions are given for the two shape functions to guarantee the new construction of Bernstein basis functions. Specific examples of shape functions and related applications are shown, along with the corresponding curves and corner cutting algorithm.
Article
Chemistry, Physical
Saroj Kumar Kushvaha, Sai Manoj N. V. T. Gorantla, Kartik Chandra Mondal
Summary: The structure and properties of donor ligands with bent Si2C units have been revealed through theoretical calculations. The results suggest that the Si2C unit can be stabilized by a pair of donor base ligands, with electrostatic and covalent orbital interactions playing a major role in the interaction energy.
JOURNAL OF PHYSICAL CHEMISTRY A
(2022)
Article
Mathematics, Applied
Karel Segeth
Summary: The article discusses spherical interpolation and approximation for measured data processing on a unit 2D sphere surface in 3D Euclidean space. It presents the advantages of using spherical radial basis function interpolation and a second degree polynomial in Cartesian coordinates. The chosen formulas have wide applications in geosciences and can be useful in interpreting various physical measurements. However, in practical computation with a high number of sampling nodes, ill-conditioning in the system matrix requires more sophisticated methods for solving the system.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Ales Vavpetic, Emil Zagar
Summary: This paper points out that the results on the optimal approximation of symmetric surfaces in [1] are incorrect and proves it by considering the optimal approximation of spherical squares. The paper provides a detailed analysis and a numerical algorithm, offering the best approximant based on the (simplified) radial error, which differs from the result obtained in [1]. The paper also examines the approximation of the sphere using continuous splines of two and six tensor product quadratic Bezier patches and discusses the problem of approximating spherical rectangles, indicating that multiple optimal approximants might exist in some cases.
APPLIED MATHEMATICS AND COMPUTATION
(2023)