Approximation of the Jacobian matrix in (<Emphasis Type="Italic">m</Emphasis>, 2)-methods for solving stiff problems

An initial value problem for stiff systems of first-order ordinary differential equations is considered. In the class of (m, k)-methods, two integration algorithms with a variable step size based on second (m = k = 2) and third (k = 2, m = 3) order-accurate schemes are constructed in which both analytical and numerical Jacobian matrices can be frozen. A theorem on the maximum order of accuracy of (m, 2)-methods with a certain approximation of the Jacobian matrix is proved. Numerical results are presented.     

Generalized Jacobians of spectral curves and completely integrable systems

Consider an ordinary differential equation which has a Lax pair representation , where A(x) is a matrix polynomial with a fixed regular leading coefficient and the matrix B(x) depends only on A(x). Such an equation can be considered as a completely integrable complex Hamiltonian system. We show that the generic complex invariant manifold of this Lax pair is an affine part of a non-compact commutative algebraic group – the generalized Jacobian of the spectral curve with its points at “infinity” identified. Moreover, for suitable B(x), the Hamiltonian vector field defined by the Lax pair on the generalized Jacobian is translation-invariant. Received April 29, 1997; in final form September 22, 1997     

Modified Newton’s method for systems of nonlinear equations with singular Jacobian

It is well known that Newton’s method for a nonlinear system has quadratic convergence when the Jacobian is a nonsingular matrix in a neighborhood of the solution. Here we present a modification of this method for nonlinear systems whose Jacobian matrix is singular. We prove, under certain conditions, that this modified Newton’s method has quadratic convergence. Moreover, different numerical tests confirm the theoretical results and allow us to compare this variant with the classical Newton’s method.     

The geometry of the asymptotics of polynomial maps

The aim of this paper is to develop a theory for the asymptotic behavior of polynomials and of polynomial maps overR and overC and to apply it to the Jacobian conjecture. This theory gives a unified frame for some results on polynomial maps that were not related before. A well known theorem of J. Hadamard gives a necessary and sufficient condition on a local diffeomorphismf: R ⁿ →R ⁿ to be a global diffeomorphism. In order to show thatf is a global diffeomorphism it suffices to exclude the existence of asymptotic values forf. The real Jacobian conjecture was shown to be false by S. Pinchuk. Our first application is to understand his construction within the general theory of asymptotic values of polynomial maps and prove that there is no such counterexample for the Jacobian conjecture overC. In a second application we reprove a theorem of Jeffrey Lang which gives an equivalent formulation of the Jacobian conjecture in terms of Newton polygons. This generalizes a result of Abhyankar. A third application is another equivalent formulation of the Jacobian conjecture in terms of finiteness of certain polynomial rings withinC[U, V]. The theory has a geometrical aspect: we define and develop the theory of etale exotic surfaces. The simplest such surface corresponds to Pinchuk's construction in the real case. In fact, we prove one more equivalent formulation of the Jacobian conjecture using etale exotic surfaces. We consider polynomial vector fields on etale exotic surfaces and explore their properties in relation to the Jacobian conjecture. In another application we give the structure of the real variety of the asymptotic values of a polynomial mapf: R ² →R ² .     

Contractivity of θ-method for semi-discrete systems

We consider nonlinear semi-discrete problems that derive by reaction diffusion systems of partial differential equations, when finite difference methods or Faedo Galerkin methods are used for spatial discretization. The aim of this article is to give sufficient conditions for the contractivity of the θ-method, in a norm generated by a positive diagonal matrix G. We show that the numerical contractivity property is obtained if some matrices, constructed by means of the Jacobian matrix of nonlinear term, are M-matrices. © 1996 John Wiley & Sons, Inc.     

An extension of the Kaskosz maximum principle

We strengthen the conventional maximum principle for the optimal control of nonsmooth differential equations with nonsmooth unilateral constraints. This strengthened principle applies, in particular, to any admissible relaxed trajectory whose endpoint lies on the boundary of the attainable set generated by unrelaxed admissible trajectories. In this new principle the generalized Jacobian of the right-hand side can be replaced by the generalized Jacobian of any compatible selectionh(t, x) of the convexified right-hand side that is Lipschitzian inx. This extends a recent result of Barbara Kaskosz that applies to problems without unilateral constraints and with the functionh restricted to a certain form. We also show how our arguments extend to unilateral problems defined by functional-integral equations (and, in particular, delay-differential equations).This work was partially supported by the National Science Foundation under Grant DMS 8619002.     

Finite element approximation of nonlocal parabolic problem

In this article, we present a finite element scheme combined with backward Euler method to solve a nonlocal parabolic problem. An important issue in the numerical solution of nonlocal problems while using Newton's method is related to its structure. In fact differently from the local case where the Jacobian matrix is sparse and banded, in the nonlocal case the Jacobian matrix is dense and computations are much more onerous compared to that for differential equations. In order to avoid this difficulty, we use the technique given by Gudi (SIAM J Numer Anal 50 (2012), 657–668) for elliptic nonlocal problem of Kirchhoff type. We discuss the well‐posedness of the weak formulation at continuous as well as at discrete levels. We also derive a priori error estimates for semidiscrete and fully discrete formulations in L² and H¹ norms. Results based on the usual finite element method are provided to confirm the theoretical estimates. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 786–813, 2017     

A criterion for the existence of a nonlinear mapping whose Jacobian matrix commutes with a matrix ring

Let Q be a ring of constant square matrices of orderm over the field of complex numbers. We consider the problem on the existence of a nonlinear mapping u: C ^m → C ^m , m ≥ 2, whose Jacobian matrix commutes with each matrix of Q. We prove that such a mapping exists if and only if Q possesses an (r, l)-pair.     

Universal algebraic equivalences between tautological cycles on Jacobians of curves

We present a collection of algebraic equivalences between tautological cycles on the Jacobian J of a curve, i.e., cycles in the subring of the Chow ring of J generated by the classes of certain standard subvarieties of J. These equivalences are universal in the sense that they hold for all curves of given genus. We show also that they are compatible with the action of the Fourier transform on tautological cycles and compute this action explicitly. Supported in part by NSF grant DMS-0302215.     

Endomorphism Algebras of Maximal Rigid Objects in Cluster Tubes

Given a maximal rigid object T of the cluster tube, we determine the objects finitely presented by T. We then use the method of Keller and Reiten to show that the endomorphism algebra of T is Gorenstein and of finite representation type, as first shown by Vatne. This algebra turns out to be the Jacobian algebra of a certain quiver with potential, when the characteristic of the base field is not 3. We study how this quiver with potential changes when T is mutated. We also provide a derived equivalence classification for the endomorphism algebras of maximal rigid objects.     

Jacobians among abelian threefolds: A formula of klein and a question of serre

In this paper we give a criterion when an indecomposable principally polarized abelian threefold (A, a) defined over a field k = ℂ is a Jacobian over k. More precisely, (A, a) is a Jacobian over k if and only if the value of a certain geometric Siegel modular form χ₁₈(A, a) is a square over k. This answers a question of J.-P. Serre.     

An application of Newton–Puiseux charts to the Jacobian problem

We study 2-dimensional Jacobian maps using so-called Newton–Puiseux charts. These are multi-valued coordinates near divisors of resolutions of indeterminacies at infinity of the Jacobian map in the source space as well as in the target space. The map expressed in these charts takes a very simple form, which allows us to detect a series of new analytical and topological properties. We prove that the Jacobian Conjecture holds true for maps (f,g) whose topological degree is ≤5, for maps with gcd(degf,degg)≤16 and for maps with. gcd(degf,degg) equal to 2 times a prime.     

On differentiable area-preserving maps of the plane

F: ℝ² → ℝ² is an almost-area-preserving map if: (a) F is a topological embedding, not necessarily surjective; and (b) there exists a constant s > 0 such that for every measurable set B, μ(F(B)) = sμ(B) where μ is the Lebesgue measure. We study when a differentiable map whose Jacobian determinant is nonzero constant to be an almost-area-preserving map. In particular, if for all z, the eigenvalues of the Jacobian matrix DF _z are constant, F is an almost-area-preserving map with convex image.     

Representation of the Clarke Generalized Jacobian via the Quasidifferential

Two differences of convex compact sets in ^{m× n} are proposed. In the light of these differences, representations of the Clarke generalized Jacobian and the B-differential via the quasidifferential are developed for a certain class of functions. These representations can be used to calculate the Clarke generalized Jacobian and the B-differential via the quasidifferential.     

Difference of two sets and estimation of clarke generalized Jacobian via quasidifferential

The notion of difference for two convex compact sets inR ⁿ , proposed by Rubinovet al, is generalized toR ^m×n . A formula of the difference for the two sets, which are convex hulls of a finite number of points, is developed. In the light of this difference, the relation between Clarke generalized Jacobian and quasidifferential, in the sense of Demyanov and Rubinov, for a nonsnooth function, is established. Based on the relation, the method of estimating Clarke generalized Jacobian via quasidifferential for a certain class of functions, is presented.     

${\cal H}^1$-estimates of Jacobians by subdeterminants

Let be a mapping in the Sobolev space . We assume that the cofactors of the differential matrix Df(x) belong to . Then, among other things, we prove that the Jacobian determinant detDf lies in the Hardy space . Received: 20 November 2000 / Revised version: 17 December 2001 / Published online: 5 September 2002 Iwaniec was supported by NSF grant DMS-0070807. This research was done while Onninen was visiting Mathematics Department at Syracuse University. He wishes to thank SU for the support and hospitality.     

Periodic solutions preserved by nonstandard finite-difference schemes for the Lotka–Volterra system: a different approach

The Lotka–Volterra predator–prey system x′ = x ? xy, y′ = ? y+xy is a good differential equation system for testing numerical methods. This model gives rise to mutually periodic solutions surrounding the positive fixed point (1,1), provided the initial conditions are positive. Standard finite-difference methods produce solutions that spiral into or out of the positive fixed point. Previously, the author [Roeger, J. Diff. Equ. Appl. 12(9) (2006), pp. 937–948], generalized three different classes of nonstandard finite-difference methods that when applied to the predator–prey system produced periodic solutions. These methods preserve weighted area; they are symplectic with respect to a noncanonical structure and have the property that the computed points do not spiral. In this paper, we use a different approach. We apply the Jacobian matrix procedure to find a fourth class of nonstandard finite-difference methods. The Jacobian matrix method gives more general nonstandard methods that also produce periodic solutions for the predator–prey model. These methods also preserve the positivity property of the solutions.     

Optimal accumulation of Jacobian matrices by elimination methods on the dual computational graph

The accumulation of the Jacobian matrix F of a vector function can be regarded as a transformation of its linearized computational graph into a subgraph of the directed complete bipartite graph K_n,m. This transformation can be performed by applying different elimination techniques that may lead to varying costs for computing F. This paper introduces face elimination as the basic technique for accumulating Jacobian matrices by using a minimal number of arithmetic operations. Its superiority over both edge and vertex elimination methods is shown. The intention is to establish the conceptual basis for the ongoing development of algorithms for optimizing the computation of Jacobian matrices.     

Exponential sums and singular hypersurfaces

We give upper bounds for the absolute value of exponential sums in several variables attached to certain polynomials with coefficients in a finite field. This bounds are given in terms of invariants of the singularities of the projective hypersurface defined by its highest degree form. For exponential sums attached to the reduction modulo a power of a large prime of a polynomial f with integer coefficients and veryfying a certain condition on the singularities of its highest degree form, we give a bound in terms of the dimension of the Jacobian quotient . Received: 3 November 1997     

A characterization of graphs which can be approximated in area by smooth graphs

For vector valued maps, convergence in W ^1,1 and of all minors of the Jacobian matrix in L ¹ is equivalent to convergence weakly in the sense of currents and in area for graphs. We show that maps defined on domains of dimension n≥ 3 can be approximated strongly in this sense by smooth maps if and only if the same property holds for the restriction to a.e. 2-dimensional plane intersecting the domain. Received April 29, 1999 / final version received July 21, 2000?Published online September 25, 2000