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1.
We propose a general study of the convergence of a Hermite subdivision scheme ℋ of degree d>0 in dimension 1. This is done by linking Hermite subdivision schemes and Taylor polynomials and by associating a so-called
Taylor subdivision (vector) scheme
. The main point of investigation is a spectral condition. If the subdivision scheme of the finite differences of
is contractive, then
is C
0 and ℋ is C
d
. We apply this result to two families of Hermite subdivision schemes. The first one is interpolatory; the second one is a
kind of corner cutting. Both of them use the Tchakalov-Obreshkov interpolation polynomial.
相似文献
2.
Parametric factorizations of linear partial operators on the plane are considered for operators of orders two, three and four.
The operators are assumed to have a completely factorable symbol. It is proved that “irreducible” parametric factorizations
may exist only for a few certain types of factorizations. Examples are given of the parametric families for each of the possible
types. For the operators of orders two and three, it is shown that any factorization family is parameterized by a single univariate
function (which can be a constant function).
相似文献
3.
In a recent paper, we investigated factorization properties of Hermite subdivision schemes by means of the so-called Taylor factorization. This decomposition is based on a spectral condition which is satisfied for example by all interpolatory Hermite schemes. Nevertheless, there exist examples of Hermite schemes, especially some based on cardinal splines, which fail the spectral condition. For these schemes (and others) we provide the concept of a generalized Taylor factorization and show how it can be used to obtain convergence criteria for the Hermite scheme by means of factorization and contractivity. 相似文献
4.
In a recent paper, we investigated factorization properties of Hermite subdivision schemes by means of the so-called Taylor factorization. This decomposition is based on a spectral condition which is satisfied for example by all interpolatory Hermite schemes. Nevertheless, there exist examples of Hermite schemes, especially some based on cardinal splines, which fail the spectral condition. For these schemes (and others) we provide the concept of a generalized Taylor factorization and show how it can be used to obtain convergence criteria for the Hermite scheme by means of factorization and contractivity. 相似文献
5.
Motivated by the problem of Hurwitz equivalence of Δ 2 factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, of 1 s
n
factorizations with transposition factors. Looking at the transpositions as the edges in a graph, we show that two factorizations
are Hurwitz equivalent if and only if their graphs have the same weighted connected components. The graph structure allows
us to compute Hurwitz equivalence in the symmetric group. Using this result, one can compute non-Hurwitz equivalence in the
braid group.
This paper is part of the author’s PhD thesis.
This work was partially supported by the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation
of Germany), the Excellency Center “Group Theoretic Methods in the Study of Algebraic Varieties” of the Israel Science Foundation,
and by EAGER (EU network, HPRN-CT-2009-00099). Received December 31, 2001 and in revised form August 6, 2002 相似文献
6.
Motivated by boundary problems for linear differential equations, we define an abstract boundary problem as a pair consisting
of a surjective linear map (“differential operator”) and an orthogonally closed subspace of the dual space (“boundary conditions”).
Defining the composition of boundary problems corresponding to their Green’s operators in reverse order, we characterize and
construct all factorizations of a boundary problem from a given factorization of the defining operator. For the case of ordinary
differential equations, the main results can be made algorithmic. We conclude with a factorization of a boundary problem for
the wave equation.
This work was supported by the Austrian Science Fund (FWF) under the SFB grant F1322. 相似文献
7.
The practical application of graph prime factorization algorithms is limited in practice by unavoidable noise in the data.
A first step towards error-tolerant “approximate” prime factorization, is the development of local approaches that cover the
graph by factorizable patches and then use this information to derive global factors. We present here a local, quasi-linear
algorithm for the prime factorization of “locally unrefined” graphs with respect to the strong product. To this end we introduce
the backbone
\mathbb B ( G)\mathbb{B} (G) for a given graph G and show that the neighborhoods of the backbone vertices provide enough information to determine the global prime factors. 相似文献
8.
For a simplicial complex or more generally Boolean cell complex Δ we study the behavior of the f- and h-vector under barycentric subdivision. We show that if Δ has a non-negative h-vector then the h-polynomial of its barycentric subdivision has only simple and real zeros. As a consequence this implies a strong version
of the Charney–Davis conjecture for spheres that are the subdivision of a Boolean cell complex or the subdivision of the boundary
complex of a simple polytope. For a general ( d − 1)-dimensional simplicial complex Δ the h-polynomial of its n-th iterated subdivision shows convergent behavior. More precisely, we show that among the zeros of this h-polynomial there is one converging to infinity and the other d − 1 converge to a set of d − 1 real numbers which only depends on d.
F. Brenti and V. Welker are partially supported by EU Research Training Network “Algebraic Combinatorics in Europe”, grant
HPRN-CT-2001-00272 and the program on “Algebraic Combinatorics” at the Mittag-Leffler Institut in Spring 2005. 相似文献
9.
We introduce a general definition of refinable Hermite interpolants and investigate their general properties. We also study a notion of symmetry of these refinable interpolants. Results and ideas from the extensive theory of general refinement equations are applied to obtain results on refinable Hermite interpolants. The theory developed here is constructive and yields an easy-to-use construction method for multivariate refinable Hermite interpolants. Using this method, several new refinable Hermite interpolants with respect to different dilation matrices and symmetry groups are constructed and analyzed. Some of the Hermite interpolants constructed here are related to well-known spline interpolation schemes developed in the computer-aided geometric design community (e.g., the Powell-Sabin scheme). We make some of these connections precise. A spline connection allows us to determine critical Hölder regularity in a trivial way (as opposed to the case of general refinable functions, whose critical Hölder regularity exponents are often difficult to compute). While it is often mentioned in published articles that ``refinable functions are important for subdivision surfaces in CAGD applications", it is rather unclear whether an arbitrary refinable function vector can be meaningfully applied to build free-form subdivision surfaces. The bivariate symmetric refinable Hermite interpolants constructed in this article, along with algorithmic developments elsewhere, give an application of vector refinability to subdivision surfaces. We briefly discuss several potential advantages offered by such Hermite subdivision surfaces. 相似文献
10.
In this paper, we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact, real-analytic Riemannian manifold. Motivated by
the “complexifier” approach of T. Thiemann as well as certain formulas of V. Guillemin and M. Stenzel, we obtain the polarization
associated to the adapted complex structure by applying the “imaginary-time geodesic flow” to the vertical polarization. Meanwhile,
at the level of functions, we show that every holomorphic function is obtained from a function that is constant along the
fibers by “composition with the imaginary-time geodesic flow.” We give several equivalent interpretations of this composition,
including a convergent power series in the vector field generating the geodesic flow. 相似文献
11.
We study line configurations in 3-space by means of “line diagrams”, projections into a plane with an indication of over and
under crossing at the vertices. If we orient such a diagram, we can associate a “contracted tensor” T with it in the same spirit as is done in Knot Theory. We give conditions to make T independent of the orientation, and invariant under isotopy. The Yang-Baxter equation is one such condition. Afterwards we
restrict ourselves to Yang-Baxter invariants with a topological state model, and give some new invariants for line isotopy. 相似文献
12.
We introduce new classes of valid inequalities, called wheel inequalities, for the stable set polytope P
G
of a graph G. Each “wheel configuration” gives rise to two such inequalities. The simplest wheel configuration is an “odd” subdivision W of a wheel, and for these we give necessary and sufficient conditions for the wheel inequality to be facet-inducing for P
W
. Generalizations arise by allowing subdivision paths to intersect, and by replacing the “hub” of the wheel by a clique. The
separation problem for these inequalities can be solved in polynomial time.
Research partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
Research partially supported by scholarships from the Ontario Ministry of Colleges and Universities. 相似文献
13.
Bivariate interpolatory Hermite subdivision schemes have recently been applied to build free-form subdivision surfaces. It is well known to geometric modelling practitioners that interpolatory schemes typically lead to ``unfair" surfaces--surfaces with unwanted wiggles or undulations--and noninterpolatory (a.k.a. approximating in the CAGD community) schemes are much preferred in geometric modelling applications. In this article, we introduce, analyze and construct noninterpolatory Hermite subdivision schemes, a class of vector subdivision schemes which can be applied to iteratively refine Hermite data in a not necessarily interpolatory fashion. We also study symmetry properties of such subdivision schemes which are crucial for application in free-form subdivision surfaces. A key step in our mathematical analysis of Hermite type subdivision schemes is that we make use of the strong convergence theory of refinement equations to convert a prescribed geometric condition on the subdivision scheme--namely, the subdivision scheme is of Hermite type--to an algebraic condition on the subdivision mask. The latter algebraic condition can then be used in a computational framework to construct specific schemes. 相似文献
14.
The relationships between piecewise-Koszul algebras and other “Koszul-type” algebras are discussed. The Yoneda-Ext algebra
and the dual algebra of a piecewise-Koszul algebra are studied, and a sufficient condition for the dual algebra A
! to be piecewise-Koszul is given. Finally, by studying the trivial extension algebras of the path algebras of Dynkin quivers
in bipartite orientation, we give explicit constructions for piecewise-Koszul algebras with arbitrary “period” and piecewise-Koszul
algebras with arbitrary “jump-degree”. 相似文献
15.
Hermite subdivision schemes have been studied by Merrien, Dyn, and Levin
and they appear to be very different from subdivision schemes analyzed before since the rules depend on the subdivision level.
As suggested by Dyn and Levin, it is possible to transform the initial scheme into a uniform stationary vector subdivision
scheme which can be handled more easily.With this transformation, the study of convergence of Hermite subdivision schemes
is reduced to that of vector stationary subdivision schemes. We propose a first criterion for C 0-convergence for a large class of vector subdivision schemes. This gives a criterion for C 1-convergence of Hermite subdivision schemes. It can be noticed that these schemes do not have to be interpolatory. We conclude
by investigating spectral properties of Hermite schemes and other necessary/sufficient conditions of convergence. 相似文献
16.
We extend the classical Siegel-Brjuno-Rüssmann linearization theorem to the resonant case by showing that under A. D. Brjuno's
diophantine condition, any resonant local analytic vector field (resp. diffeomorphism) possesses a well-defined correction which (1) depends on the chart but, in any given chart, is unique (2) consists solely of resonant terms and (3) has the property
that, when substracted from the vector field (resp. when factored out of the diffeomorphism), the vector field or diffeomorphism
thus “corrected” becomes analytically linearizable (with a privileged or “canonical” linearizing map). Moreover, in spite
of the small denominators and contrary to a hitherto prevalent opinion, the correction's analyticity can be established by
pure combinatorics, without any analysis.
Received January 7, 1997; in final form April 22, 1997 相似文献
17.
The paper develops a necessary condition for the regularity of a multivariate refinable function in terms of a factorization
property of the associated subdivision mask. The extension to arbitrary isotropic dilation matrices necessitates the introduction
of the concepts of restricted and renormalized convergence of a subdivision scheme as well as the notion of subconvergence, i.e., the convergence of only a subsequence of the iterations of the subdivision scheme. Since, in addition, factorization
methods pass even from scalar to matrix valued refinable functions, those results have to be formulated in terms of matrix
refinable functions or vector subdivision schemes, respectively, in order to be suitable for iterated application. Moreover,
it is shown for a particular case that the the condition is not only a necessary but also a sufficient one.
Dedicated to Charles A. Micchelli, a unique person, friend, mathematician and collaborator, on the occasion of his sixtieth
birthday
Mathematics subject classifications (2000) 65T60, 65D99. 相似文献
18.
In this paper we focus on Hermite subdivision operators that act on vector valued data interpreting their components as function values and associated consecutive derivatives. We are mainly interested in studying the exponential and polynomial preservation capability of such kind of operators, which can be expressed in terms of a generalization of the spectral condition property in the spaces generated by polynomials and exponential functions. The main tool for our investigation are convolution operators that annihilate the aforementioned spaces, which apparently is a general concept in the study of various types of subdivision operators. Based on these annihilators, we characterize the spectral condition in terms of factorization of the subdivision operator. 相似文献
19.
Let X be a germ of holomorphic vector field at the origin of Cn and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system.
The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first
integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets
in a neighborhood of the origin. These are biholomorphic to the intersection of a polydisc with an analytic set of the form
“resonant monomials = constants”. Such a biholomorphism conjugates the restriction of X to one of its invariant varieties
to the restriction of a linear diagonal vector field to a toric variety. Moreover, we show that the set of “frequencies” defining
the invariant sets is of positive measure. 相似文献
20.
An order topology in vector lattices and Boolean algebras is studied under the additional condition of “closure by one step”
that generalizes the well-known “regularity” property of Boolean algebras and K-spaces. It is proved that in a vector lattice
or a Boolean algebra possessing such a property there exists a basis of solid neighborhoods of zero with respect to an order
topology. An example of a Boolean algebra without basis of solid neighborhoods of zero (an algebra of regular open subsets
of the interval ( 0, 1)) is given. Bibliography: 3 titles.
Translated from Problemy Matematicheskogo Analiza, No. 15 1995, pp. 213–220. 相似文献
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