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1.
AbstractIn this article, we discuss Jacobi spectral Galerkin and iterated Jacobi spectral Galerkin methods for Volterra-Urysohn integral equations with weakly singular kernels and obtain the convergence results in both the infinity and weighted L2-norm. We show that the order of convergence in iterated Jacobi spectral Galerkin method improves over Jacobi spectral Galerkin method. We obtain the convergence results in two cases when the exact solution is sufficiently smooth and non-smooth. For finding the improved convergence results, we also discuss Jacobi spectral multi-Galerkin and iterated Jacobi spectral multi-Galerkin method and obtain the convergence results in weighted L2-norm. In fact, we prove that the iterated Jacobi spectral multi-Galerkin method improves over iterated Jacobi spectral Galerkin method. We provide numerical results to verify the theoretical results. 相似文献
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Li Shan Liu 《数学学报(英文版)》2001,17(1):103-112
In this paper, we will prove that Ky Fan's Theorem (Math. Z. 112(1969), 234–240) is true for 1-set-contractive maps defined
on a bounded closed convex subset K in a Banach space with intK≠0. This class of 1-set-contractive maps includes condensing maps, nonexpansive maps, semicontractive maps, LANE maps and
others. As applications of our theorems, some fixed point theorems of non-selfmaps are proved under various well-known boundary
conditions. Our results are generalizations and improvements of the recent results obtained by many authors.
Project supported by the National Natural Science Foundation of China and Natural Science Foundation of
Shandong Province of China 相似文献
4.
We study the problem of realization of a given generalized oscillator by a system of N generalized oscillators of a different type. We consider a generalized oscillator related to a fixed system of orthogonal
polynomials that are determined by three-term recurrent relations and the corresponding three-diagonal Jacobi matrix J. The case N =2 was considered in a previous paper. It was shown that in this case the orthogonality measure is symmetric with respect
to rotation at angle π. In this paper, we consider the case N =3. We prove that such a problem has a solution only in two cases. In the first case, the Jacobi matrix related to the given
“composite” generalized oscillator has block-diagonal form and consists of similar 3×3 blocks. In the second (more interesting)
possible case, the Jacobi matrix is not block-diagonal. For this matrix, we construct the corresponding system of orthogonal
polynomials. This system decomposes into three series which are related to Chebyshev polynomials of the second kind. The main
result of the paper is a solution of the moment problem for the corresponding Jacobi matrix. In this case, the constructed
measure is symmetric with respect to rotation at angle 2π/3. Bibliography: 6 titles. 相似文献
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Abedallah Rababah Mohammed Al-Refai 《Numerical Functional Analysis & Optimization》2013,34(5-6):660-673
In this paper, we give a new, simple, and efficient method for evaluating the pth derivative of the Jacobi polynomial of degree n. The Jacobi polynomial is written in terms of the Bernstein basis, and then the pth derivative is obtained. The results are given in terms of both Bernstein basis of degree n ? p and Jacobi basis form of degree n ? p and presented in a matrix form. Numerical examples and comparisons with other well-known methods are presented. 相似文献
7.
Lucyna Szczepanik 《代数通讯》2013,41(9):3359-3367
ABSTRACT In this note it is proved that certain level sets of some real proper polynomial maps are nothing but spheres. As an application of this, we provide new proofs of Theorems 1.1, 1.2 and of the fundamental theorem of algebra. In addition, we show that every strictly convex (concave) polynomial map is proper. The latter implies that every real polynomial map g(x): R n → R n , whose Jacobian matrix is symmetric and has nonzero eigenvalues of the same sign, is a homeomorphism of R n onto R n . 相似文献
8.
Shayne Waldron 《Constructive Approximation》2011,33(3):405-424
Given a suitable weight on ℝ
d
, there exist many (recursive) three-term recurrence relations for the corresponding multivariate orthogonal polynomials.
In principle, these can be obtained by calculating pseudoinverses of a sequence of matrices. Here we give an explicit
recursive three-term recurrence for the multivariate Jacobi polynomials on a simplex. This formula was obtained by seeking the best
possible three-term recurrence. It defines corresponding linear maps, which have the same symmetries as the spaces of Jacobi
polynomials on which they are defined. The key idea behind this formula is that some Jacobi polynomials on a simplex can be
viewed as univariate Jacobi polynomials, and for these the recurrence reduces to the univariate three-term recurrence. 相似文献
9.
We study the facial structures of the cone of all decomposable positive linear maps from the matrix algebra Mm into Mn. Especially, we completely determine the faces of the cone which arise from the dual of positive linear maps.Partially supported by BSRI-MOE and RIM-SNU 相似文献
10.
Olivier Bernardi 《Journal of Combinatorial Theory, Series B》2011,101(5):315-377
We address the enumeration of properly q-colored planar maps, or more precisely, the enumeration of rooted planar maps M weighted by their chromatic polynomial χM(q) and counted by the number of vertices and faces. We prove that the associated generating function is algebraic when q≠0,4 is of the form 2+2cos(jπ/m), for integers j and m. This includes the two integer values q=2 and q=3. We extend this to planar maps weighted by their Potts polynomial PM(q,ν), which counts all q-colorings (proper or not) by the number of monochromatic edges. We then prove similar results for planar triangulations, thus generalizing some results of Tutte which dealt with their proper q-colorings. In statistical physics terms, the problem we study consists in solving the Potts model on random planar lattices. From a technical viewpoint, this means solving non-linear equations with two “catalytic” variables. To our knowledge, this is the first time such equations are being solved since Tutte?s remarkable solution of properly q-colored triangulations. 相似文献
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A planar map is a 2-cell embedding of a connected planar graph, loops and parallel edges allowed, on the sphere. A plane map is a planar map with a distinguished outside (“infinite”) face. An unrooted map is an equivalence class of maps under orientation-preserving homeomorphism, and a rooted map is a map with a distinguished oriented edge. Previously we obtained formulae for the number of unrooted planar n-edge maps of various classes, including all maps, non-separable maps, eulerian maps and loopless maps. In this article, using the same technique we obtain closed formulae for counting unrooted plane maps of all these classes and their duals. The corresponding formulae for rooted maps are known to be all sum-free; the formulae that we obtain for unrooted maps contain only a sum over the divisors of n. We count also unrooted two-vertex plane maps. 相似文献
12.
A full coloring of a planar map is a face coloring such that all the faces at ech vertex are colored differently. In this paper the planar 4-regular maps which have a full 4-coloring are characterized. This leads to a characterization of the planar maps (not necessarily 4-valent) which have a coupled 4-coloring. 相似文献
13.
Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrixrnew rows and columns, so that the original Jacobi matrix is shifted downward. Thernew rows and columns contain 2rnew parameters and the newly obtained orthogonal polynomials thus correspond to an upward extension of the Jacobi matrix. We give an explicit expression of the new orthogonal polynomials in terms of the original orthogonal polynomials, their associated polynomials, and the 2rnew parameters, and we give a fourth order differential equation for these new polynomials when the original orthogonal polynomials are classical. Furthermore we show how the 1?orthogonalizing measure for these new orthogonal polynomials can be obtained and work out the details for a one-parameter family of Jacobi polynomials for which the associated polynomials are again Jacobi polynomials. 相似文献
14.
Maria Isabel Bueno Cachadina Alfredo Deaño Edward Tavernetti 《Numerical Algorithms》2010,54(1):101-139
A monic Jacobi matrix is a tridiagonal matrix which contains the parameters of the three-term recurrence relation satisfied
by the sequence of monic polynomials orthogonal with respect to a measure. The basic Geronimus transformation with shift α transforms the monic Jacobi matrix associated with a measure dμ into the monic Jacobi matrix associated with dμ/(x − α) + Cδ(x − α), for some constant C. In this paper we examine the algorithms available to compute this transformation and we propose a more accurate algorithm,
estimate its forward errors, and prove that it is forward stable. In particular, we show that for C = 0 the problem is very ill-conditioned, and we present a new algorithm that uses extended precision. 相似文献
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Harmonic maps between two Riemannian manifolds M and N are often constructed as energy minimizing maps. This construction is extended for the Dirichlet problem to the case where the Riemannian energy functional on M is replaced by a more general Dirichlet form. We obtain weakly harmonic maps and prove that these maps send the diffusion to N-valued martingales. The basic tools are the reflected Dirichlet space and the stochastic calculus for Dirichlet processes. 相似文献
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By applying the theory of quasiconformal maps in measure metric spaces that was introduced by Heinonen-Koskela, we characterize
bi-Lipschitz maps by modulus inequalities of rings and maximal, minimal derivatives in Q-regular Loewner spaces. Meanwhile the sufficient and necessary conditions for quasiconformal maps to become bi-Lipschitz
maps are also obtained. These results generalize Rohde’s theorem in ℝ
n
and improve Balogh’s corresponding results in Carnot groups.
This research is supported by China NSF (Grant No. 10271077) 相似文献
18.
XuAnZHAO 《数学学报(英文版)》2004,20(6):1131-1134
In this paper the classification of maps from a simply connected space X to a flag manifold G/T is studied. As an application, the structure of the homotopy set for self-maps of flag manifolds is determined. 相似文献
19.
Mirta Castro Smirnova 《Journal of Approximation Theory》2000,104(2):285
We consider complex Jacobi matrices G which can be decomposed in the form G=J+C, where J is a real Jacobi matrix and C is a complex Jacobi matrix whose entries are uniformly bounded. We prove that the determinacy of the operator defined by G is equivalent to that of J. From this we deduce that the determinacy of G is equivalent to the coincidence between the domains of definition of the operators G and its adjoint G*. 相似文献
20.
Pralay Chatterjee 《Advances in Mathematics》2011,226(6):4639
In this paper we study the surjectivity of the power maps g?gn for real points of algebraic groups defined over reals. The results are also applied to study the exponentiality of such groups. 相似文献