Search results for "Jacobian"
showing 10 items of 60 documents
The Lie algebra of polynomial vector fields and the Jacobian conjecture
1998
The Jacobian conjecture for polynomial maps ϕ:Kn→Kn is shown to be equivalent to a certain Lie algebra theoretic property of the Lie algebra\(\mathbb{D}\) of formal vector fields inn variables. To be precise, let\(\mathbb{D}_0 \) be the unique subalgebra of codimensionn (consisting of the singular vector fields),H a Cartan subalgebra of\(\mathbb{D}_0 \),Hλ the root spaces corresponding to linear forms λ onH and\(A = \oplus _{\lambda \in {\rm H}^ * } H_\lambda \). Then every polynomial map ϕ:Kn→Kn with invertible Jacobian matrix is an automorphism if and only if every automorphism Φ of\(\mathbb{D}\) with Φ(A)\( \subseteq A\) satisfies Φ(A)=A.
Jacobian-free approximate solvers for hyperbolic systems: Application to relativistic magnetohydrodynamics
2017
Abstract We present recent advances in PVM (Polynomial Viscosity Matrix) methods based on internal approximations to the absolute value function, and compare them with Chebyshev-based PVM solvers. These solvers only require a bound on the maximum wave speed, so no spectral decomposition is needed. Another important feature of the proposed methods is that they are suitable to be written in Jacobian-free form, in which only evaluations of the physical flux are used. This is particularly interesting when considering systems for which the Jacobians involve complex expressions, e.g., the relativistic magnetohydrodynamics (RMHD) equations. On the other hand, the proposed Jacobian-free solvers hav…
Jacobian-Free Incomplete Riemann Solvers
2018
The purpose of this work is to present some recent developments about incomplete Riemann solvers for general hyperbolic systems. Polynomial Viscosity Matrix (PVM) methods based on internal approximations to the absolute value function are introduced, and they are compared with Chebyshev-based PVM solvers. These solvers only require a bound on the maximum wave speed, so no spectral decomposition is needed. Moreover, they can be written in Jacobian-free form, in which only evaluations of the physical flux are used. This is particularly interesting when considering systems for which the Jacobians involve complex expressions. Some numerical experiments involving the relativistic magnetohydrodyn…
On GIT quotients of Hilbert and Chow schemes of curves
2011
The aim of this note is to announce some results on the GIT problem for the Hilbert and Chow scheme of curves of degree d and genus g in P^{d-g}, whose full details will appear in a subsequent paper. In particular, we extend the previous results of L. Caporaso up to d>4(2g-2) and we observe that this is sharp. In the range 2(2g-2)<d<7/2(2g-2), we get a complete new description of the GIT quotient. As a corollary, we get a new compactification of the universal Jacobian over the moduli space of pseudo-stable curves.
Jacobian of weak limits of Sobolev homeomorphisms
2016
Abstract Let Ω be a domain in ℝ n {\mathbb{R}^{n}} , where n = 2 , 3 {n=2,3} . Suppose that a sequence of Sobolev homeomorphisms f k : Ω → ℝ n {f_{k}\colon\Omega\to\mathbb{R}^{n}} with positive Jacobian determinants, J ( x , f k ) > 0 {J(x,f_{k})>0} , converges weakly in W 1 , p ( Ω , ℝ n ) {W^{1,p}(\Omega,\mathbb{R}^{n})} , for some p ⩾ 1 {p\geqslant 1} , to a mapping f. We show that J ( x , f ) ⩾ 0 {J(x,f)\geqslant 0} a.e. in Ω. Generalizations to higher dimensions are also given.
On the accurate determination of nonisolated solutions of nonlinear equations
1981
A simple but efficient method to obtain accurate solutions of a system of nonlinear equations with a singular Jacobian at the solution is presented. This is achieved by enlarging the system to a higher dimensional one whose solution in question is isolated. Thus it can be computed e. g. by Newton's method, which is locally at least quadratically convergent and selfcorrecting, so that high accuracy is attainable.
Motives of quadric bundles and relative intermediate jacobians of K3-Fano pairs
2015
This thesis consists of two parts. In the first part we study the Chow motive of a quadric bundle of odd relative dimension over a surface. We show that this motive admits a decomposition which involves the Prym motive of the double covering of the discriminant curve.In the second part, we consider Lagrangian fibrations, obtained as relative intermediate Jacobians of families of Fano threefolds containing a fixed K3 surface, and the existence of a symplectic compactification. In a particular case, we study a partial compactification using calculations with the software system Macaulay2.
Planar Mappings of Finite Distortion
2010
We review recent results on planar mappings of finite distortion. This class of mappings contains all analytic functions and quasiconformal mappings.
The PCHIP subdivision scheme
2016
In this paper we propose and analyze a nonlinear subdivision scheme based on the monotononicity-preserving third order Hermite-type interpolatory technique implemented in the PCHIP package in Matlab. We prove the convergence and the stability of the PCHIP nonlinear subdivision process by employing a novel technique based on the study of the generalized Jacobian of the first difference scheme. MTM2011-22741
Applications to Algebraic Cycles: Nori's Theorem
2017
Deligne cohomology is a tool that makes it possible to unify the study of cycles through an object that classifies extensions of ( p , p )-cycles by points in the p -th intermediate Jacobian (which is the target of the Abel–Jacobi map on cycles of codimension p ). This is treated in Section 10.1 with applications to normal functions. Before giving the proof of Nori's theorem in Section 10.6, we need some results from mixed Hodge theory. These are proven in Section 10.2 where we also state different variants of the theorem. Sections 10.3 and 10.4 treat a localto- global principle and an extension of the method of Jacobian representations of cohomology which are both essential for the proof. …