问题驱动下的微分与全微分教学探索

微分概念是高等数学中的核心概念,通过问题教学法和几何图形演示,将一元函数的微分教学和二元函数的全微分教学统一起来,使学生更深刻理解微分这一概念的核心本质,并认识到从一元函数到多元函数许多结论要发生变化,量变引发质变的变化过程.     

微分多项式系统的近微分特征列集

本文对微分多项式系统的近微分特征列集与微分特征列集之间的一些关系进行了研究,给出了在某些条件下近微分特征列集是微分特征列集的结论,从而对微分多项式系统特征列集理论（吴方法）进行了改进,并且建立的算法较大地提高了计算微分特征列集的效率.     

Differential invariants

This paper summarizes recent results on the number and characterization of differential invariants of transformation groups. Generalizations of theorems due to Ovsiannikov and to M. Green are presented, as well as a new approach to finding bounds on the number of independent differential invariants.Supported in part by NSF Grants DMS 91-16672 and DMS 92-04192.     

Differential eigenforms

The aim of this paper is to show how differential characters of Abelian varieties (in the sense of [A. Buium, Differential characters of Abelian varieties over p-adic fields, Invent. Math. 122 (1995) 309-340]) can be used to construct differential modular forms of weight 0 and order 2 (in the sense of [A. Buium, Differential modular forms, Crelle J. 520 (2000) 95-167]) which are eigenvectors of Hecke operators. These differential modular forms will have “essentially the same” eigenvalues as certain classical complex eigenforms of weight 2 (and order 0).     

Differential equations

Book Reviews

Differential equationsC. M. Dafermos, G. Ladas, and G. Papanicolaou (editors): Lecture Notes in Pure and Applied Mathematics 118, Marcel Dekker, New York, Basel, 1989, xiv + 787 pp.     

Algorithms for Differential Invariants of Symmetry Groups of Differential Equations

We develop new computational algorithms, based on the method of equivariant moving frames, for classifying the differential invariants of Lie symmetry pseudo-groups of differential equations and analyzing the structure of the induced differential invariant algebra. The Korteweg-deVries (KdV) and Kadomtsev-Petviashvili (KP) equations serve to illustrate examples. In particular, we deduce the first complete classification of the differential invariants and their syzygies of the KP symmetry pseudo-group.     

Introducing Differential Calculus

In this paper we discuss three ways of introducing calculus all based on concepts which students would either already know or which can be introduced without much difficulty at this stage of their mathematical training. The aim of this paper is to persuade teachers of mathematics that topics in ‘higher mathematics’, specifically calculus, can and should be presented to students in terms of concepts with which they are familiar.     

Differential torsion theory

Differential torsion theories are introduced and it is shown that for a hereditary torsion theory τ every derivation on an R-module M has a unique extension to its module of quotients if and only if τ is a differential torsion theory. Dually, we show that when τ is cohereditary, every derivation on M can be lifed uniquely to its module of coquotients.     

Differential gradient methods

A class of recently developed differential descent methods for function minimization is presented and discussed, and a number of algorithms are derived which minimize a quadratic function in a finite number of steps and rapidly minimize general functions. The main characteristics of our algorithms are that a more general curvilinear search path is used instead of a ray and that the eigensystem of the Hessian matrix is associated with the function minimization problem. The curvilinear search paths are obtained by solving certain initial-value systems of differential equations, which also suggest the development of modifications of known numerical integration techniques for use in function minimization. Results obtained on testing the algorithms on a number of test functions are also given and possible areas for future research indicated.     

Nichtlineares invariantes Differential

Zusammenfassung Das invariante Differential eines Vektors ist selbst ein Vektor, der vom Orte xⁱ, von dxⁱ, und von selbst abh?ngig ist. Es wird nun infolge dieser Definition in einigen F?llen die m?gliche Form des invarianten Differentials bestimmt. Enrico Bompiani zu seinem wissenschaftlichen Jubil?um.     

Feedback Differential Invariants

The problem of feedback equivalence for control systems is considered. An algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found.     

Differential commutator identities

I.N. Herstein proved that if R is a prime ring satisfying a differential identity , with d a nonzero derivation of R, then R embeds isomorphically in M₂(F) for F a field. We consider a natural generalization of this result for the class of polynomials E_n(X)=[E_n-1(x₁,…,x_n-1),x_n]. Using matrix computations, we prove that if R satisfies a differential identity , or with some restrictions, then R must embed in M₂(F), but that differential identities using [[E_n,E_m],E_s] with m,n,s>1 need not force R to embed in M₂(F). These results hold if the expressions are identities for a noncommutative Lie ideal of R, rather than for R itself.     

Differential variational inequalities

This paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems, and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensional variational inequalities. Borrowing results from differential inclusions (DIs) with upper semicontinuous, closed and convex valued multifunctions, we establish the convergence of such a procedure for solving initial-value DVIs. We also present a class of DVIs for which the theory of DIs is not directly applicable, and yet similar convergence can be established. Finally, we extend the method to a boundary-value DVI and provide conditions for the convergence of the method. The results in this paper pertain exclusively to systems with “index” not exceeding two and which have absolutely continuous solutions. The work of J.-S. Pang is supported by the National Science Foundation under grants CCR-0098013 CCR-0353074, and DMS-0508986, by a Focused Research Group Grant DMS-0139715 to the Johns Hopkins University and DMS-0353016 to Rensselaer Polytechnic Institute, and by the Office of Naval Research under grant N00014-02-1-0286. The work of D. E. Stewart is supported by the National Science Foundation under a Focused Research Group grant DMS-0138708.     

Mesonic Differential Forms

When M is a differentiable manifold, the exterior differential k -forms on M are the alternate k -linear forms on the tangent bundle T(M) . The mesonic differential k -forms are the k -linear forms on T(M) that are alternate with respect to the variables of odd rank, and also alternate with respect to the variables of even rank. After a reminder about meson algebras, and after the presentation of elementary properties of mesonic forms, this article introduces the mesonic differentiation of mesonic forms, which can be partially compared to the exterior differentiation of exterior forms. Some applications to riemannian manifolds and flat manifolds follow.     

Abstract Differential Null-Equations

Let A be a closed linear operator in a Banach space, and let n 1 be an integer. If the resolvent I – A)^-1 is an entire function of of order < 1/n or of order 1/n and minimal type, then the equation d ⁿ u(t)/dt ⁿ =Au(t) has only the trivial solution u(t)0. An example for partial differential equations is given. Generalizations are indicated.     

Monogenic Differential Operators

In this paper we consider operators acting on Clifford algebra valued polynomials and, in particular, differential operators with polynomial coefficients. The decomposition of polynomials into homogeneous pieces leads to the classical homogeneous decomposition of operators and the further decomposition of homogeneous polynomials into monogenic polynomials leads to the concept of monogenic operator. Monogenic operators are characterized in terms of commutation relations and the monogenic decomposition of differential operators is studied in detail.