The Jacobian algebras

A new class of algebras (the Jacobian algebras) is introduced and studied in detail. The Jacobian algebras are obtained from the Weyl algebras by inverting (not in the sense of Ore) certain elements. Surprisingly, the Jacobian algebras and the Weyl algebras have little in common. Moreover, they have almost opposite properties.     

Jacobian of meromorphic curves

The contact structure of two meromorphic curves gives a factorization of their jacobian. Abhyankar’s work was partly supported by NSF Grant DMS 91-01424 and NSA grant MDA 904-97-1-0010.     

Jacobian Conditions For p-Bases

We discuss various sufficient conditions, involving derivations and jacobians, for elements of a domain of characteristic p > 0, to form a p-basis of a ring of constants of a derivation. We also prove that in a special case of eigenvectors these conditions are both sufficient and necessary.     

Cubic analogues of the Jacobian theta functions and the Jacobian elliptic functions

In this paper, the Jacobian theta functions, the Jacobian elliptic functions, and their cubic analogues are introduced. Then, the properties satisfied the cubic analogues are discussed.     

Reduced Recurrence Relations for the Chebyshev Method

In this paper, we give sufficient conditions ensuring the convergence of the Chebyshev method in Banach spaces. We use a new system of recurrence relations which simplifies those given by Kantorovich for the Newton method or those given by Candela and Marquina for the Chebyshev and Halley methods.     

Moment Maps and Jacobian Modules

Ginzburg, Guillemin, and Karshon investigated, for compactAbelian groups G, the question: ‘When is anabstract moment map a moment map in the usual pre-symplectic sense?’ Weconsider the analogous question for non-Abelian groups and show that itis closely connected to the problem of determining the representationstructure of certain modules, known as Jacobian modules, which arenaturally attached to a linear action of G.     

Dicritical divisors and Jacobian problem

The dicritical divisor coming out of integrable holonomic systems gets miraculously married to the jacobian problem of algebraic geometry.     

Automorphisms of Jacobian Kummer surfaces

We study automorphisms of a generic Jacobian Kummer surface. First weanalyse the action of classically known automorphisms on the Picard lattice of the surface, then proceed to construct new automorphisms not generated by classical ones. We find 192 such automorphisms, all conjugateby the symmetry group of the (16,6)-configuration.     

Polynomial maps with constant Jacobian

It has been long conjectured that ifn polynomialsf ₁, …,f _n inn variables have a (non-zero) constant Jacobian determinant then every polynomial can be expressed as a polynomial inf ₁, …,f _n. In this paper, various extra assumptions (particularly whenn=2) are shown to imply the conclusion. These conditions are discussed algebraically and geometrically.     

Representations and the Jacobian Conjecture

In this paper we apply the representation theory of the Lie algebra sl₂(C) to the problem of describing Hessian nilpotent polynomials, which are important in the theory of the Jacobian Conjecture. In the two variable case we describe them as the maximal and minimal weight vectors of the irreducible representations of sl₂(C). For the first time this gives a characterization of the Hessian nilpotent polynomials in terms of linear differential operators.     

Optimal Jacobian accumulation is NP-complete

We show that the problem of accumulating Jacobian matrices by using a minimal number of floating-point operations is NP-complete by reduction from Ensemble Computation. The proof makes use of the fact that, deviating from the state-of-the-art assumption, algebraic dependences can exist between the local partial derivatives. It follows immediately that the same problem for directional derivatives, adjoints, and higher derivatives is NP-complete, too.      

Autoduality of the Compactified Jacobian

The following autoduality theorem is proved for an integralprojective curve C in any characteristic. Given an invertiblesheaf L of degree 1, form the corresponding Abel map A_L:C, which maps C into its compactifiedJacobian, and form its pullback map , which carries the connected component of 0 in the Picard schemeback to the Jacobian. If C has, at worst, points of multiplicity2, then is an isomorphism, and forming it commutes with specializing C. Much of the work in the paper is valid, more generally, fora family of curves with, at worst, points of embedding dimension2. In this case, the determinant of cohomology is used to constructa right inverse to . Then a scheme-theoretic version of the theorem of the cube is proved,generalizing Mumford's, and it is used to prove that is independent of the choice of L.Finally, the autoduality theorem is proved. The presentationscheme is used to achieve an induction on the difference betweenthe arithmetic and geometric genera; here, special propertiesof points of multiplicity 2 are used.     

大M法和两阶段法中检验向量间的关系

若大M法中的检验向量为ξ=ζ+Mμ,则μ正是采用两阶段法时同一个基对应的单纯形表中的辅助目标函数g的检验向量,而ζ则是原来目标函数的检验向量.