Article
Astronomy & Astrophysics
Dimitrios Katsinis
Summary: We demonstrate that the integrability of the SO(N)/SO(N - 1) principal chiral model (PCM) originates from the Pohlmeyer reduction of the O(N) nonlinear sigma model (NLSM). Specifically, we show that the Lax pair of the PCM can be related to the zero curvature condition through redefinitions and identification of parameters, which arises from the flatness of the enhanced space used in the Pohlmeyer reduction. This identification provides a solution for the auxiliary system corresponding to any NLSM/PCM solution.
Article
Mathematics, Applied
S. Barbieri, L. Niederman
Summary: By utilizing the structure of complex algebraic curves and compactness arguments, this paper provides a self-contained proof that holomorphic algebraic functions satisfy a uniform Bernstein-Remez inequality. This work extends and generalizes previous research, highlighting a key part that has been overlooked in earlier proofs.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2023)
Article
Computer Science, Software Engineering
Wenbing Shao, Falai Chen, Xuefeng Liu
Summary: In this paper, a robust and efficient algorithm is proposed for calculating the intersection points of two planar algebraic curves with guaranteed tolerance. The method combines fundamental techniques from the fields of CAGD, solution verification for nonlinear equations, and symbolic computation. The algorithm utilizes the subdivision method to quickly exclude regions without intersection points, and then employs Krawczyk's method and Sturm's theorem to determine the sharp and guaranteed bounds for intersection points. Examples and comparisons with other methods demonstrate the robustness and efficiency of the proposed algorithm, which can also be easily extended to compute intersection points of parametric curves.
COMPUTER-AIDED DESIGN
(2022)
Article
Mathematics, Applied
Ryo Ohashi
Summary: This paper studies Ciani curves in characteristic p >= 3 and their properties. By determining whether a Ciani curve is superspecial, it can be determined whether it is a maximal or minimal curve over Fp2.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Computer Science, Theory & Methods
Lin Sok
Summary: The author has constructed families of MDS Euclidean self-dual codes from genus zero algebraic geometry (AG) codes and explored more families of optimal Euclidean self-dual codes from AG codes. New families of MDS Euclidean self-dual codes of odd characteristic and almost MDS Euclidean self-dual codes are explicitly constructed from genus zero and genus one curves, respectively. Additional families of Euclidean self-dual codes are constructed from algebraic curves of higher genus.
DESIGNS CODES AND CRYPTOGRAPHY
(2021)
Article
Mathematics, Applied
Piotr Krason
Summary: This paper investigates a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. It discusses sufficient conditions, counterexamples in some cases, examples of curves and their Jacobians, and proves the dynamical version of the local to global principle for etale K-theory of a curve. Furthermore, it shows that all results remain valid for Quillen K-theory of X if certain conjectures hold true.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2021)
Article
Mathematics, Applied
Max H. H. Weinreich
Summary: This paper studies the properties of the pentagram map over an arbitrary algebraically closed field of characteristic not equal to 2 and proves that the pentagram map on twisted polygons is a discrete integrable system. In the course of the proof, the moduli space of twisted n-gons is constructed, formulas for the pentagram map are derived, and the Lax representation is calculated using characteristic-independent methods.
ERGODIC THEORY AND DYNAMICAL SYSTEMS
(2023)
Article
Mathematics, Applied
Chiara Romanengo, Bianca Falcidieno, Silvia Biasotti
Summary: This paper introduces a method that uses the Hough transform to recognize curves and find the curve that best fits given points. The method is applicable to digital models of 3D objects and combines the recognition of curves expressed in both implicit and parametric form.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Materials Science, Multidisciplinary
P. Sriluckshmy, Max Nusspickel, Edoardo Fertitta, George H. Booth
Summary: Quantum embedding approaches involve optimizing a local fragment of a strongly correlated system with the wider environment. The 'energy-weighted' density matrix embedding theory (EwDMET) allows for true quantum fluctuations over this boundary to be self-consistently optimized within a fully static framework. The improved method demonstrated in this work provides a numerically efficient, systematic convergence to the zero-temperature dynamical mean-field theory limit.
Article
Mathematics, Applied
Daniele Bartoli, Massimo Giulietti, Gaia Peraro, Giovanni Zini
Summary: This paper deals with monomial type GAPN functions, which are a generalization of APN functions in finite fields of odd characteristic p introduced in 2017. By connecting the GAPN property of a monomial function to the existence of suitable rational points on an algebraic curve, necessary conditions for a monomial function to be GAPN are provided.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Mathematics
Jaume Gine, Jaume Llibre
Summary: This study focuses on invariant algebraic curves of generalized Lienard polynomial differential systems. We correct and investigate the degrees of the polynomials f and g in the system.
Article
Physics, Fluids & Plasmas
Yunfeng Jiang, Rui Wen, Yang Zhang
Summary: We develop a systematic approach to compute physical observables of integrable spin chains with finite length. Our method is based on the Bethe ansatz solution of the integrable spin chain and computational algebraic geometry. The final results are analytic and no longer depend on Bethe roots. The computation is purely algebraic and does not rely on further assumptions or numerics. This method can be applied to compute a broad family of physical quantities in integrable quantum spin chains. We demonstrate the power of the method by computing two important quantities in quench dynamics-the diagonal entropy and the Loschmidt echo-and obtain analytic results.
Article
Mathematics, Interdisciplinary Applications
Jin Kai, Cheng Jinsan
Summary: The paper presents an algorithm for computing the topology of an algebraic space curve, which is a modified version of a previous algorithm. The authors also analyze the bit complexity of the algorithm, which has the best bounds among existing work. Additionally, the paper includes content from conference papers CASC 2014 and SNC 2014.
JOURNAL OF SYSTEMS SCIENCE & COMPLEXITY
(2021)
Article
Multidisciplinary Sciences
Susmit Bagchi
Summary: In this paper, we comprehensively revisit the Bezout theorem from a topological perspective and explore the role of topology in algebraic curve intersections and complex root translations.
Article
Mathematics, Applied
Bertrand Toen, Gabriele Vezzosi
Summary: This paper is the first in a series of papers about foliations in derived geometry. The authors introduce derived foliations on arbitrary derived stacks and focus on quasi-smooth and rigid derived foliations on smooth complex algebraic varieties. They discuss their formal and analytic versions and prove several results, including the formal integrability of quasismooth rigid derived foliations and the local integrability of their analytifications. The authors also introduce the derived category of perfect crystals on a quasi-smooth rigid derived foliation and establish a Riemann-Hilbert correspondence for them.
SELECTA MATHEMATICA-NEW SERIES
(2023)
