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1.
The Newton polygon of the implicit equation of a rational plane curve is explicitly determined by the multiplicities of any
of its parametrizations. We give an intersection-theoretical proof of this fact based on a refinement of the Kušnirenko–Bernštein
theorem. We apply this result to the determination of the Newton polygon of a curve parameterized by generic Laurent polynomials
or by generic rational functions, with explicit genericity conditions. We also show that the variety of rational curves with
given Newton polygon is unirational and we compute its dimension. As a consequence, we obtain that any convex lattice polygon with positive area is the Newton polygon of a rational plane curve. 相似文献
2.
We consider rationally parameterized plane curves, where the polynomials in the parameterization have fixed supports and generic
coefficients. We apply sparse (or toric) elimination theory in order to determine the vertex representation of the implicit
equation’s Newton polygon. In particular, we consider mixed subdivisions of the input Newton polygons and regular triangulations
of point sets defined by Cayley’s trick. We consider polynomial and rational parameterizations, where the latter may have
the same or different denominators; the implicit polygon is shown to have, respectively, up to four, five, or six vertices. 相似文献
3.
We give a combinatorial upper bound for the gonality of a curve that is defined by a bivariate Laurent polynomial with given
Newton polygon. We conjecture that this bound is generically attained, and provide proofs in a considerable number of special
cases. One proof technique uses recent work of M. Baker on linear systems on graphs, by means of which we reduce our conjecture
to a purely combinatorial statement. 相似文献
4.
Let f and g be two analytic function germs without common branches.We define the Jacobian quotients of (g, f), which are firstorder invariants of the discriminant curve of (g, f),and we prove that they only depend on the topological type of(g, f). We compute them with the help of the topology of (g,f). If g is a linear form transverse to f, the Jacobian quotientsare exactly the polar quotients of f and we affirm the resultsof D. T. Lê, F. Michel and C. Weber. 相似文献
5.
We consider implicit integration methods for the solution of stiff initial value problems for second-order differential equations
of the special form y' = f(y). In implicit methods, we are faced with the problem of solving systems of implicit relations. This paper focuses on the construction
and analysis of iterative solution methods which are effective in cases where the Jacobian of the right‐hand side of the differential
equation can be split into a sum of matrices with a simple structure. These iterative methods consist of the modified Newton
method and an iterative linear solver to deal with the linear Newton systems. The linear solver is based on the approximate
factorization of the system matrix associated with the linear Newton systems. A number of convergence results are derived
for the linear solver in the case where the Jacobian matrix can be split into commuting matrices. Such problems often arise
in the spatial discretization of time‐dependent partial differential equations. Furthermore, the stability matrix and the
order of accuracy of the integration process are derived in the case of a finite number of iterations.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
6.
We outline a method to compute the type of the induced polarization of an abelian subvariety of a canonically polarized Jacobian of a smooth projective curve. The method works for curves of not too big genus admitting a “large” group of automorphisms. Several examples are given. 相似文献
7.
M. D. Smooke 《Journal of Optimization Theory and Applications》1983,39(4):489-511
The solution of nonlinear, two-point boundary value problems by Newton's method requires the formation and factorization of a Jacobian matrix at every iteration. For problems in which the cost of performing these operations is a significant part of the cost of the total calculation, it is natural to consider using the modified Newton method. In this paper, we derive an error estimate which enables us to determine an upper bound for the size of the sequence of modified Newton iterates, assuming that the Kantorovich hypotheses are satisfied. As a result, we can efficiently determine when to form a new Jacobian and when to continue the modified Newton algorithm. We apply the result to the solution of several nonlinear, two-point boundary value problems occurring in combustion. 相似文献
8.
In this paper, we propose a new distinctive version of a generalized Newton method for solving nonsmooth equations. The iterative formula is not the classic Newton type, but an exponential one. Moreover, it uses matrices from B‐differential instead of generalized Jacobian. We prove local convergence of the method and we present some numerical examples. 相似文献
9.
Maurice J. Dupré James F. Glazebrook Emma Previato 《Complex Analysis and Operator Theory》2013,7(4):739-763
We present an operator-coefficient version of Sato’s infinite-dimensional Grassmann manifold, and τ-function. In this setting the classical Burchnall–Chaundy ring of commuting differential operators can be shown to determine a C*-algebra. For this C*-algebra topological invariants of the spectral ring become readily available, and further, the Brown–Douglas–Fillmore theory of extensions can be applied. We construct KK classes of the spectral curve of the ring and, motivated by the fact that all isospectral Burchnall–Chaundy rings make up the Jacobian of the curve, we compare the (degree-1) K-homology of the curve with that of its Jacobian. We show how the Burchnall–Chaundy C*-algebra extension by the compact operators provides a family of operator τ-functions. 相似文献
10.
Michael Rapoport 《manuscripta mathematica》2000,101(2):153-166
Let G be an unramified reductive group over a local field. We consider the matrix describing the Satake isomorphism in terms of
the natural bases of the source and the target. We prove that all coefficients of this matrix which are not obviously zero
are in fact positive numbers. The result is then applied to an existence problem of F-crystals which is a partial converse to Mazur's theorem relating the Hodge polygon and the Newton polygon.
Received: 29 June 1999 / Revised version: 7 September 1999 相似文献
11.
Consider a family of zero-sum games indexed by a parameter that determines each player’s payoff function and feasible strategies.
Our first main result characterizes continuity assumptions on the payoffs and the constraint correspondence such that the
equilibrium value and strategies depend continuously and upper hemicontinuously (respectively) on the parameter. This characterization
uses two topologies in order to overcome a topological tension that arises when players’ strategy sets are infinite-dimensional.
Our second main result is an application to Bayesian zero-sum games in which each player’s information is viewed as a parameter.
We model each player’s information as a sub-σ-field, so that it determines her feasible strategies: those that are measurable with respect to the player’s information.
We thereby characterize conditions under which the equilibrium value and strategies depend continuously and upper hemicontinuously
(respectively) on each player’s information. 相似文献
12.
Many applications in science and engineering lead to models that require solving large‐scale fixed point problems, or equivalently, systems of nonlinear equations. Several successful techniques for handling such problems are based on quasi‐Newton methods that implicitly update the approximate Jacobian or inverse Jacobian to satisfy a certain secant condition. We present two classes of multisecant methods which allow to take into account a variable number of secant equations at each iteration. The first is the Broyden‐like class, of which Broyden's family is a subclass, and Anderson mixing is a particular member. The second class is that of the nonlinear Eirola–Nevanlinna‐type methods. This work was motivated by a problem in electronic structure calculations, whereby a fixed point iteration, known as the self‐consistent field (SCF) iteration, is accelerated by various strategies termed ‘mixing’. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
13.
Pavlos Tzermias 《Archiv der Mathematik》2010,95(1):19-24
We prove that the torsion part of the Mordell–Weil group of the Jacobian of a Fermat curve over a cyclotomic field is contained
in the kernel of a certain isogeny. This provides a natural analogue of a similar result on Jacobians of Fermat quotient curves. 相似文献
14.
An orientation reversing involution of a topological compact genus surface induces an antiholomorphic involution of the Teichmüller space of genus g Riemann surfaces. Two such involutions and are conjugate in the mapping class group if and only if the corresponding orientation reversing involutions and of are conjugate in the automorphism group of . This is equivalent to saying that the quotient surfaces and are homeomorphic. Hence the Teichmüller space has distinct antiholomorphic involutions, which are also called real structures of ([7]). This result is a simple fact that follows from Royden's theorem ([4]) stating that the the mapping class group is
the full group of holomorphic automorphisms of the Teichmüller space (). Let and be two real structures that are not conjugate in the mapping class group. In this paper we construct a real analytic diffeomorphism
such that
This mapping d is a product of full and half Dehn–twists around certain simple closed curves on the surface . This has applications to the moduli spaces of real algebraic curves. A compact Riemann surface admitting an antiholomorphic involution is a real algebraic curve of the topological type . All fixed–points of the real structure of the Teichmüller space , are real curves of the above topological type and every real curve of that topological type is represented by an element
of the fixed–point set of . The fixed–point set is the Teichmüller space of real algebraic curves of the corresponding topological type. Given two different real structures
and , let d the the real analytic mapping satisfying (1). It follows that d maps onto and is an explicit real analytic diffeomorphism between these Teichmüller spaces.
Received 8 December 1997; accepted 12 August 1998 相似文献
15.
Each master iteration of a simplified Newton algorithm for solving a system of equations starts by computing the Jacobian
matrix and then uses this matrix in the computation ofp Newton steps: the first of these steps is exact, and the other are called “simplified”.
In this paper we apply this approach to a large step path following algorithm for monotone linear complementarity problems.
The resulting method generates sequences of objective values (duality gaps) that converge to zero with Q-orderp+1 in the number of master iterations, and with a complexity of
iterations.
Corresponding author. Research done while visiting the Delft Technical University, and supported in part by CAPES — Brazil. 相似文献
16.
PAVLOS TZERMIAS 《Compositio Mathematica》1997,106(1):1-9
We study the torsion in the Mordell-Weil group of the Jacobian of the Fermat curve of exponent p over the cyclotomic field obtained by adjoining a primitive p-th root of 1 to Q. We show that for all (except possibly one) proper subfields of this cyclotomic field, the torsion parts of the corresponding Mordell-Weil groups are elementary abelian p-groups. 相似文献
17.
A new class of boundary value problems for parabolic operators is introduced which is based on the Newton polygon method.
We show unique solvability and a priori estimates in corresponding L
2-Sobolev spaces. As an application, we discuss some linearized free boundary problems arising in crystallization theory which
do not satisfy the classical parabolicity condition. It is shown that these belong to the new class of parabolic boundary
value problems, and two-sided estimates for their solutions are obtained.
The second author was supported by Russian Foundation of Basic Research, grant 06–01–00096. 相似文献
18.
We develop general approximate Newton methods for solving Lipschitz continuous equations by replacing the iteration matrix with a consistently approximated Jacobian, thereby reducing the computation in the generalized Newton method. Locally superlinear convergence results are presented under moderate assumptions. To construct a consistently approximated Jacobian, we introduce two main methods: the classic difference approximation method and the -generalized Jacobian method. The former can be applied to problems with specific structures, while the latter is expected to work well for general problems. Numerical tests show that the two methods are efficient. Finally, a norm-reducing technique for the global convergence of the generalized Newton method is briefly discussed. 相似文献
19.
We present two algorithms to compute m-fold hypergeometric solutions of linear recurrence equations for the classical shift case and for the q-case, respectively. The first is an m-fold generalization and q-generalization of the algorithm by van Hoeij (Appl Algebra Eng Commun Comput 17:83–115, 2005; J. Pure Appl Algebra 139:109–131, 1998) for recurrence equations. The second is a combination of an improved version of the algorithms by Petkovšek (Discrete Math
180:3–22, 1998; J Symb Comput 14(2–3):243–264, 1992) for recurrence and q-recurrence equations and the m-fold algorithm from Petkovšek and Salvy (ISSAC 1993 Proceedings, pp 27–33, 1993) for recurrence equations. We will refer to the classical algorithms as van Hoeij or Petkovšek respectively. To formulate our ideas, we first need to introduce an adapted version of an m-fold Newton polygon and its characteristic polynomials for the classical case and q-case, and to prove the important properties in this case. Using the data from the Newton polygon, we are able to present
efficient m-fold versions of the van Hoeij and Petkovšek algorithms for the classical shift case and for the q-case, respectively. Furthermore, we show how one can use the Newton polygon and our characteristic polynomials to conclude
for which
m ? \mathbbN{m\in \mathbb{N}} there might be an m-fold hypergeometric solution at all. Again by using the information obtained from the Newton polygon, the presentation of
the q-Petkovšek algorithm can be simplified and streamlined. Finally, we give timings for the ‘classical’ q-Petkovšek, our q-van Hoeij and our modified q-Petkovšek algorithm on some classes of problems and we present a Maple implementation of the m-fold algorithms for the q-case. 相似文献
20.
AbstractWe study the value semiring Γ, equipped with the tropical operations, associated to an algebroid curve. As a set, Γ determines and is determined by the well-known value semigroup S and we prove that Γ is always finitely generated in contrast to S. In particular, for a plane curve, we present a straightforward way to obtain Γ in terms of the semiring (or the semigroup) of each branch of the curve and the mutual intersection multiplicity of its branches. In the analytic case, this allows us to relate the results of Zariski and Waldi that characterize the topological type of the curve. 相似文献