Article
Mathematics
Wenjie Wang
Summary: In this paper, the geometry of affine Killing and two-Killing vector fields on Riemannian manifolds is investigated. A new characterization of Euclidean space via the affine Killing vector fields is provided. Some conditions for an affine Killing and two-Killing vector field to be a conformal or Killing one are given.
MATHEMATICA SLOVACA
(2022)
Article
Mathematics
Sharief Deshmukh, Olga Belova
Summary: The study focuses on the impact of a unit Killing vector field on the geometry of Riemannian manifolds, revealing the relationship between the vector field and smooth functions, as well as the necessary and sufficient conditions for different dimensions of manifolds under specific constraints. Additionally, it demonstrates the influence of a unit Killing vector field on the structure and properties of manifolds.
Article
Mathematics
Antonio Algaba, Cristobal Garcia, Jaume Gine
Summary: In this study, an algorithm based on normal form theory is developed to determine if a planar vector field is orbitally reversible. Previous works only focused on algorithms for reversibility and conjugate reversibility. The algorithm is particularly useful in solving the center problem, as it can identify nondegenerate and nilpotent centers as well as degenerate centers that are orbitally reversible.
Article
Mathematics, Applied
Shimin Li, Changjian Liu, Jaume Llibre
Summary: This paper extends the Poincare-Bendixson formula to planar piecewise smooth vector fields and provides a method to compute the index of equilibrium points.
JOURNAL OF NONLINEAR SCIENCE
(2023)
Article
Computer Science, Software Engineering
Kacper Pluta, Michal Edelstein, Amir Vaxman, Mirela Ben-Chen
Summary: A new method for computing planar hexagonal meshes is proposed, based on Coordinate Power Fields and an optimization framework, as well as a constraint combination for planar hexagonal meshing addressing challenging meshing problems.
ACM TRANSACTIONS ON GRAPHICS
(2021)
Article
Mathematics, Applied
Nataliya Goncharuk, Yury G. Kudryashov, Nikita Solodovnikov
Summary: This study focuses on global bifurcations in generic three-parameter families of vector fields on S-2. It reveals the structural instability of three-parameter unfoldings of vector fields with the polycycle 'tears of the heart', as well as of those with separatrix graphs 'ears' and 'glasses'. Additionally, the classical bifurcation of a saddle loop is studied in detail and used as a building block in the main example.
Article
Mathematics, Applied
Isaac A. Garcia, Susanna Maza
Summary: This paper analyzes the role of non-autonomous inverse Jacobi multipliers in the problem of nonexistence, existence, localization, and hyperbolic nature of periodic orbits of planar vector fields. It extends and generalizes previous results that focused only on the autonomous or periodic case, providing novel applications of inverse Jacobi multipliers.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Atsushi Katsuda, Takuya Nakamura
Summary: This study proves a rigidity theorem for Killing vector fields on manifolds with almost nonpositive Ricci curvature, which is a generalization of Bochner's classical results.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
J. Davidov, G. Grantcharov, O. Mushkarov
Summary: We investigate the connection between the presence of null conformal Killing vector fields and the presence of compatible complex and para-hypercomplex structures on a pseudo-Riemannian manifold with a metric of signature (2, 2). We first determine the topological types of pseudo-Hermitian surfaces that admit a non-vanishing null vector field. Then, we demonstrate that a pair of orthogonal, pointwise linearly independent, null conformal Killing vector fields defines a para-hyperhermitian structure and utilize this fact to classify smooth compact four-manifolds that possess such a pair of vector fields. We also provide examples of neutral metrics with two orthogonal, pointwise linearly independent, null Killing vector fields on most of these manifolds.
JOURNAL OF GEOMETRIC ANALYSIS
(2023)
Article
Engineering, Geological
Zan Gojcic, Lorenz Schmid, Andreas Wieser
Summary: The novel method accurately estimates 3D displacement vectors from point cloud data, being sensitive to motion and deformations parallel to the underlying surface, and capable of efficiently handling large point cloud datasets.
Article
Mathematics
Amir Babak Aazami, Robert Ream
Summary: This paper presents a complete local classification of Riemannian 3-manifolds (M,g) with a nonvanishing Killing vector field T. The classification is then extended to Lorentzian 3-manifolds with timelike Killing vector fields, which are automatically nonvanishing. The classification is based on the scalar curvature S of g and the function Ric(T,T), with their sum representing the Gaussian curvature. The paper also provides conditions for locally conformally flat g, and discusses the global setting in terms of geodesic completeness and constant positive sectional curvature for compact 3-manifolds M.
MATHEMATISCHE NACHRICHTEN
(2023)
Article
Multidisciplinary Sciences
Maryam Khalid Albuhayr, Ashfaque H. Bokhari, Tahir Hussain
Summary: In this paper, an algebraic approach is used to classify cylindrically symmetric static spacetimes based on their killing vector fields. The approach involves using a maple algorithm to simplify the Killing's equations and integrating them subject to the algorithm's constraints. The results show that this approach yields additional spacetime metrics not obtained by direct integration. The physical implications of these metrics are discussed by applying them to the Einstein equations.
Article
Mathematics
L. G. S. Duarte, L. A. C. P. da Mota
Summary: The paper introduces an efficient method for computing Darboux polynomials for polynomial vector fields in the plane, specifically for those presenting a Liouvillian first integral or a Liouvillian general solution. The key to this method is to separate the process of solving the algebraic system arising from the condition for the existence of Darboux polynomials into feasible steps. Additionally, a brief performance analysis of the algorithms developed is presented.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Luiz F. S. Gouveia, Joan Torregrosa
Summary: In this work, isolated crossing periodic orbits in planar piecewise polynomial vector fields defined in two zones separated by a straight line are studied, focusing on the number of limit cycles of small amplitude. Lower bounds for local cyclicity are provided for planar piecewise polynomial systems with degrees 2, 3, 4, and 5, using parallelization algorithms for computations.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2021)
Article
Mathematics, Applied
Anna Cima, Armengol Gasull, Victor Manosa
Summary: This paper investigates the probability of occurrence of phase portraits in random planar homogeneous polynomial vector fields, providing complete solutions for n=1,2,3 by determining exact probabilities or estimating them using the Monte Carlo method. The majority of phase portraits are characterized by the index at the origin and the number of invariant straight lines passing through this point, with only two exceptions.
QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
(2021)
Article
Neurosciences
Tanya Glozman, Justin Solomon, Franco Pestilli, Leonidas Guibas
JOURNAL OF ALZHEIMERS DISEASE
(2017)
Article
Mathematics, Applied
Montacer Essid, Justin Solomon
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2018)
Article
Computer Science, Software Engineering
Mikhail Bessmeltsev, Justin Solomon
ACM TRANSACTIONS ON GRAPHICS
(2019)
Article
Computer Science, Software Engineering
Yue Wang, Yongbin Sun, Ziwei Liu, Sanjay E. Sarma, Michael M. Bronstein, Justin M. Solomon
ACM TRANSACTIONS ON GRAPHICS
(2019)
Article
Computer Science, Software Engineering
D. Palmer, O. Stein, J. Solomon
Summary: This paper studies a class of operators that generalize the fourth-order Bilaplacian to support anisotropic behavior, parametrized by a symmetric frame field. The discretization of these operators shows good convergence and allows for fine-grained control of local direction variations.
COMPUTER GRAPHICS FORUM
(2021)
Article
Computer Science, Artificial Intelligence
Oded Stein, Jiajin Li, Justin Solomon
Summary: In this study, we introduce a robust optimization method for eliminating flip-free distortion energies, which can be effectively applied in parametrization, deformation, and volume correspondence. By exploiting the special structure of distortion energies and utilizing the operator splitting technique, we propose a novel alternating direction method of multipliers (ADMM) algorithm that is highly parallelizable. The resulting optimization algorithm exhibits robustness to flipped elements in the data and during the optimization process.
SIAM JOURNAL ON IMAGING SCIENCES
(2022)
Article
Computer Science, Software Engineering
S. Mazdak Abulnaga, Oded Stein, Polina Golland, Justin Solomon
Summary: This paper proposes a method for shape correspondence in volumetric data and selects a preferred energy function that favors isometric correspondences through theoretical discussion. The method is demonstrated to produce boundary-aligned and low-distortion matchings on diverse geometric datasets.
ACM TRANSACTIONS ON GRAPHICS
(2023)
Article
Mathematics, Applied
Justin Solomon, Kristjan Greenewald, Haikady Nagaraja
Summary: We introduce k-variance, a generalization of variance built on random bipartite matchings, which measures the expected cost of matching two sets of k samples from a distribution and captures local information as k increases.
SIAM JOURNAL ON MATHEMATICS OF DATA SCIENCE
(2022)
Article
Physics, Fluids & Plasmas
Elle Najt, Daryl DeFord, Justin Solomon
Summary: This paper explores the connections between redistricting and statistical physics, analyzing key questions using techniques such as self-avoiding walks. It discusses the influences of new factors in redistricting context and assesses the robustness of typical properties of districting plans in relation to score functions and geographic region analysis.
Article
Mathematics, Applied
Tara Abrishami, Nestor Guillen, Parker Rule, Zachary Schutzman, Justin Solomon, Thomas Weighill, Si Wu
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2020)
Article
Mathematics, Applied
Daryl DeFord, Hugo Lavenant, Zachary Schutzman, Justin Solomon
SIAM JOURNAL ON APPLIED ALGEBRA AND GEOMETRY
(2019)
Proceedings Paper
Computer Science, Artificial Intelligence
Matthew Staib, Sebastian Claici, Justin Solomon, Stefanie Jegelka
ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS 30 (NIPS 2017)
(2017)
Article
Computer Science, Software Engineering
S. Claici, M. Bessmeltsev, S. Schaefer, J. Solomon
COMPUTER GRAPHICS FORUM
(2017)
Article
Computer Science, Software Engineering
Sema Berkiten, Maciej Halber, Justin Solomon, Chongyang Ma, Hao Li, Szymon Rusinkiewicz
COMPUTER GRAPHICS FORUM
(2017)
Article
Computer Science, Software Engineering
Etienne Corman, Justin Solomon, Mirela Ben-Chen, Leonidas Guibas, Maks Ovsjanikov
ACM TRANSACTIONS ON GRAPHICS
(2017)