This work consists of two parts. The first one contains a characterization (localization) of the point spectrum of one sided, infinite and periodic Jacobi matrices with scalar entries. The second one deals with the same questions about one sided, infinite periodic Jacobi matrices with matrix entries. In particular, an example illustrating the difference between the above localization property in scalar and matrix entries cases is given. 相似文献
The slow growing hierarchy is commonly defined as follows: G0(x) = 0, Gx−1(x) := Gx(x) + 1 and Gλ(x) := Gλ[x](x) where λ<0 is a limit and ·[·]:0∩ Lim × ω → 0 is a given assignment of fundamental sequences for the limits below 0. The first obvious question which is encountered when one looks at this definition is: How does this hierarchy depend on the choice of the underlying system of fundamental sequences? Of course, it is well known and easy to prove that for the standard assignment of fundamental sequence the hierarchy (Gx)x<0 is slow growing, i.e. each Gx is majorized by a Kalmar elementary recursive function.
It is shown in this paper that the slow growing hierarchy (Gx)x<0 — when it is defined with respect to the norm-based assignment of fundamental sequences which is defined in the article by Cichon (1992, pp. 173–193) — is actually fast growing, i.e. each PA-provably recursive function is eventually dominated by Gx for some <0. The exact classification of this hierarchy, i.e. the problem whether it is slow or fast growing, has been unsolved since 1992. The somewhat unexpected result of this paper shows that the slow growing hierarchy is extremely sensitive with respect to the choice of the underlying system of fundamental sequences.
The paper is essentially self-contained. Only little knowledge about ordinals less than 0 — like the existence of Cantor normal forms, etc. and the beginnings of subrecursive hierarchy theory as presented, for example, in the 1984 textbook of Rose — is assumed. 相似文献
Chaotic sequences generated by nonlinear difference systems or ‘maps’ where the defining nonlinearities are polynomials, have
been examined from the point of view of the sequential points seeking zeroes of an unknown functionf following the rule of Newton iterations. Following such nonlinear transformation rule, alternative sequences have been constructed
showing monotonie convergence. Evidently, these are maps of the original sequences. For second degree systems, another kind
of possibly less chaotic sequences have been constructed by essentially the same method. Finally, it is shown that the original
chaotic system can be decomposed into a fast monotonically convergent part and a principal oscillatory part showing sharp
oscillations. The methods are exemplified by the well-known logistic map, delayed-logistic map and the Hénon map. 相似文献
This work deals with various finite algorithms that solve two special Structured Inverse Eigenvalue Problems (SIEP). The first
problem we consider is the Jacobi Inverse Eigenvalue Problem (JIEP): given some constraints on two sets of reals, find a Jacobi
matrix J (real, symmetric, tridiagonal, with positive off-diagonal entries) that admits as spectrum and principal subspectrum
the two given sets. Two classes of finite algorithms are considered. The polynomial algorithm which is based on a special
Euclid–Sturm algorithm (Householder's terminology) and has been rediscovered several times. The matrix algorithm which is
a symmetric Lanczos algorithm with a special initial vector. Some characterization of the matrix ensures the equivalence of
the two algorithms in exact arithmetic. The results of the symmetric situation are extended to the nonsymmetric case. This
is the second SIEP to be considered: the Tridiagonal Inverse Eigenvalue Problem (TIEP). Possible breakdowns may occur in the
polynomial algorithm as it may happen with the nonsymmetric Lanczos algorithm. The connection between the two algorithms exhibits
a similarity transformation from the classical Frobenius companion matrix to the tridiagonal matrix. This result is used to
illustrate the fact that, when computing the eigenvalues of a matrix, the nonsymmetric Lanczos algorithm may lead to a slow
convergence, even for a symmetric matrix, since an outer eigenvalue of the tridiagonal matrix of order n − 1 can be arbitrarily
far from the spectrum of the original matrix.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
A probability inequality for Sn and somepth moment (p⩾2) inequalities for |Sn| and max 1⩽k⩽n | Sk| are established. Here Sn is the partial sum of a negatively associated sequence. Based on these inequalities, a weak invariance principle for strictly
stationary negatively associated sequences is proved under some general conditions.
Project supported by the National Natural Science Foundation of China, the Doctoral Program Foundation of the State Education
Commission of China and the High Eductional Natural Science Foundation of Guangdong Province. 相似文献
In this paper we give an explicit construction of the moduli space of the pointed complete Gorenstein curves of arithmetic genus g with a given quasi-symmetric Weierstrass semigroup, that is, a Weierstrass semigroup whose last gap is equal to 2g – 2. We identify such a curve with its image under the canonical embedding in the (g – 1)-dimensional projective space. By normalizing the coefficients of the quadratic relations and by constructing Gröbner bases of the canonical ideal, we obtain the equations of the moduli space in terms of Buchberger's criterion. Moreover, by analyzing syzygies of the canonical ideal we establish criteria that make the computations less expensive. 相似文献
A cyclic code is a cyclic q‐ary code of length n, constant weight w and minimum distance d. Let denote the largest possible size of a cyclic code. The pure and mixed difference method plays an important role in the determination of upper bound on . By analyzing the distribution of odd mixed and pure differences, an improved upper bound on is obtained for . A new construction based on special sequences is provided and the exact value of is almost completely determined for all d and n except when and . Our constructions reveal intimate connections between cyclic constant weight codes and special sequences, particularly Skolem‐type sequences. 相似文献
An error bound for multidimensional quadrature is derived that includes the Koksma-Hlawka inequality as a special case. This error bound takes the form of a product of two terms. One term, which depends only on the integrand, is defined as a generalized variation. The other term, which depends only on the quadrature rule, is defined as a generalized discrepancy. The generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the -star discrepancy and that arises in the study of lattice rules.