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1.
Thierry Coquand 《Archive for Mathematical Logic》1998,37(3):143-147
Semantical arguments, based on the completeness theorem for first-order logic, give elegant proofs of purely syntactical
results. For instance, for proving a conservativity theorem between two theories, one shows instead that any model of one
theory can be extended to a model of the other theory. This method of proof, because of its use of the completeness theorem,
is a priori not valid constructively. We show here how to give similar arguments, valid constructively, by using Boolean models.
These models are a slight variation of ordinary first-order models, where truth values are now regular ideals of a given Boolean
algebra. Two examples are presented: a simple conservativity result and Herbrand's theorem.
Received December 5, 1995 相似文献
2.
We investigate several versions of a cardinal characteristic defined by Frankiewicz. Vojtáš showed , and Blass showed . We show that all the versions coincide and that is greater than or equal to the splitting number. We prove the consistency of and of .
Received: 2 October 1996 / Revised version: 22 May 1997 相似文献
3.
Colm Art O'Cinneide 《Numerische Mathematik》1996,73(4):507-519
Summary.
Recently the author showed that the Grassmann-Taksar-Heyman
(GTH) algorithm computes the steady-state
distribution of a finite-state
Markov chain with low relative error.
Here it is shown that the
LU decomposition
computed in the course of the
GTH algorithm also has low relative error.
The proof requires a refinement of the methods used in
the earlier paper.
Received September 2, 1994 / Revised version received
July 17, 1995 相似文献
4.
Summary. It is well known that any nonsingular M–matrix admits an LU factorization into M–matrices (with L and U lower and upper triangular respectively) and any singular M–matrix is permutation similar to an M–matrix which admits an
LU factorization into M–matrices. Varga and Cai establish necessary and sufficient conditions for a singular M–matrix (without
permutation) to allow an LU factorization with L nonsingular. We generalize these results in two directions. First, we find necessary and sufficient conditions for the existence
of an LU factorization of a singular M-matrix where L and U are both permitted to be singular. Second, we establish the minimal block structure that a block LU factorization of a singular
M–matrix can have when L and U are M–matrices.
Received November 21, 1994 / Revised version received August 4, 1997 相似文献
5.
Thomas Huckle 《Numerische Mathematik》1998,79(2):213-229
The problem of solving linear equations with a Toeplitz matrix appears in many applications. Often is positive definite but ill-conditioned with many small eigenvalues. In this case fast and superfast algorithms may show
a very poor behavior or even break down. In recent papers the transformation of a Toeplitz matrix into a Cauchy-type matrix
is proposed. The resulting new linear equations can be solved in operations using standard pivoting strategies which leads to very stable fast methods also for ill-conditioned systems. The
basic tool is the formulation of Gaussian elimination for matrices with low displacement rank. In this paper, we will transform
a Hermitian Toeplitz matrix into a Cauchy-type matrix by applying the Fourier transform. We will prove some useful properties of and formulate a symmetric Gaussian elimination algorithm for positive definite . Using the symmetry and persymmetry of we can reduce the total costs of this algorithm compared with unsymmetric Gaussian elimination. For complex Hermitian , the complexity of the new algorithm is then nearly the same as for the Schur algorithm. Furthermore, it is possible to include
some strategies for ill-conditioned positive definite matrices that are well-known in optimization. Numerical examples show
that this new algorithm is fast and reliable.
Received March 24, 1995 / Revised version received December 13, 1995 相似文献
6.
7.
Summary. A new algorithm for triangularizing an Toeplitz matrix is presented. The algorithm is based on the previously developed recursive algorithms that exploit the Toeplitz
structure and compute each row of the triangular factor via updating and downdating steps. A forward error analysis for this
existing recursive algorithm is presented, which allows us to monitor the conditioning of the problem, and use the method
of corrected semi-normal equations to obtain higher accuracy for certain ill-conditioned matrices. Numerical experiments show
that the new algorithm improves the accuracy significantly while the computational complexity stays in .
Received April 30, 1995 / Revised version received February 12, 1996 相似文献
8.
Summary. Let be a complex polynomial of
degree with and Cauchy radius 1 about the
origin. We discuss the order of magnitude
of the minimal number such that
Previous estimates of are improved to . Some other related properties of these polynomials
are also exhibited.
Received March 3, 1993 相似文献
9.
Summary.
In this paper, we introduce the notion of hybrid procedures for solving a
system of linear equations. A hybrid procedure consists in a combination of
two arbitrary approximate solutions with coefficients summing
up to one. Thus
the combination only depends on one parameter whose value is chosen
in order to minimize the Euclidean norm of the residual vector obtained by
the hybrid procedure.
Properties of such procedures are studied in detail. The two approximate
solutions which are combined in a hybrid procedure are usually obtained
by two iterative methods. Several strategies for combining these two
methods together or with the previous iterate of the hybrid procedure itself
are discussed and their properties are analyzed.
Numerical experiments illustrate the various procedures.
Received October 21, 1992/Revised version
received May 28, 1993 相似文献
10.
Summary. In this paper, we are concerned with a matrix equation
where A is an real matrix and x and b are n-vectors. Assume that an approximate solution is given together with an approximate LU decomposition. We will present fast algorithms for proving nonsingularity of A and for calculating rigorous error bounds for . The emphasis is on rigour of the bounds. The purpose of this paper is to propose different algorithms, the fastest with
flops computational cost for the verification step, the same as for the LU decomposition. The presented algorithms exclusively use library routines for LU decomposition and for all other matrix and vector operations.
Received June 16, 1999 / Revised version received January 25, 2001 / Published online June 20, 2001 相似文献
11.
A. Melman 《Numerische Mathematik》1995,69(4):483-493
Summary.
A method is proposed for the solution of a secular equation, arising
in modified symmetric eigenvalue problems and in several other
areas.
This equation has singularities which make the application of standard root-finding
methods difficult.
In order to solve the equation, a class of transformations of variables is
considered, which transform the equation into one for which Newton's method
converges from any point in a certain given interval. In addition, the form of the
transformed equation suggests a convergence accelerating modification of
Newton's method. The same ideas are applied to the secant
method and numerical results are presented.
Received July 1, 1994 相似文献
12.
J.M. Peña 《Numerische Mathematik》1998,81(2):293-304
Pivoting strategies for Gaussian elimination leading to upper triangular matrices which are diagonally dominant by rows are
studied. Forward error analysis of triangular systems whose coefficient matrices are diagonally dominant by rows is performed.
We also obtain small bounds of the backward errors for the pivoting strategies mentioned above. Our examples of matrices include
H-matrices and some generalizations of diagonally dominant matrices, and scaled partial pivoting for the 1-norm is an example
of these pivoting strategies. In the case of an
M-matrix, a pivoting strategy of computational complexity is proposed, which satisfies all the results of the paper.
Received June 6, 1997 / Revised version received October 27, 1997 相似文献
13.
Summary. By providing a matrix version of Koenig's theorem we reduce the problem of evaluating the coefficients of a monic factor
r(z) of degree h of a power series f(z) to that of approximating the first h entries in the first column of the inverse of an Toeplitz matrix in block Hessenberg form for sufficiently large values of n. This matrix is reduced to a band matrix if f(z) is a polynomial. We prove that the factorization problem can be also reduced to solving a matrix equation for an matrix X, where is a matrix power series whose coefficients are Toeplitz matrices. The function is reduced to a matrix polynomial of degree 2 if f(z) is a polynomial of degreeN and . These reductions allow us to devise a suitable algorithm, based on cyclic reduction and on the concept of displacement rank,
for generating a sequence of vectors that quadratically converges to the vector having as components the coefficients of the factor r(z). In the case of a polynomial f(z) of degree N, the cost of computing the entries of given is arithmetic operations, where is the cost of solving an Toeplitz-like system. In the case of analytic functions the cost depends on the numerical degree of the power series involved
in the computation. From the numerical experiments performed with several test polynomials and power series, the algorithm
has shown good numerical properties and promises to be a good candidate for implementing polynomial root-finders based on
recursive splitting strategies. Applications to solving spectral factorization problems and Markov chains are also shown.
Received September 9, 1998 / Revised version received November 14, 1999 / Published online February 5, 2001 相似文献
14.
15.
Numerical solution of constant coefficient linear delay differential equations as abstract Cauchy problems 总被引:2,自引:0,他引:2
Summary. In this paper we present an approach for the numerical solution of delay differential equations
where , and , different from the classical step-by-step method. We restate (1) as an abstract Cauchy problem and then we discretize it
in a system of ordinary differential equations. The scheme of discretization is proved to be convergent. Moreover the asymptotic
stability is investigated for two significant classes of asymptotically stable problems (1).
Received May 4, 1998 / Revised version received January 25, 1999 / Published online November 17, 1999 相似文献
16.
The axiom of choice is equivalent to the shrinking principle: every indexed cover of a set has a refinement with the same
index set which is a partition. A simple and direct proof of this equivalence is given within an elementary fragment of constructive
Zermelo–Fraenkel set theory. Variants of the shrinking principle are discussed, and it is related to a similar but different
principle formulated by Vaught. 相似文献
17.
Henri Lombardi 《Mathematische Zeitschrift》2002,242(1):23-46
Résumé. Nous démontrons un Nullstellensatz qui établit une équivalence entre l'existence d'une identité algébrique d'un certain type,
d'une part, et l'impossibilité de trouver une suite croissante de variétés irréductibles répondant à certaines contraintes
d'autre part. De ce point de vue le Nullstellensatz usuel correspond au cas des variétés réduites à un point. Nous établissons
aussi un Nullstellensatz formel du même type, en relation avec les suites croissantes d'idéaux premiers. Un cas particulier
important est donné par la notion de suite pseudo régulière, plus générale que la notion de suite régulière. Nous obtenons
de cette manière une nouvelle caractérisation de la dimension de Krull d'un anneau : un anneau a une dimension de Krull si et seulement si il existe une suite pseudo régulière de longueur dans l'anneau. Dans les cas où ces résultats peuvent avoir une signification constructive précise, nos preuves y aboutissent
constructivement. Nous pensons avoir donné ainsi quelques éléments en vue d'une interprétation constructive de la théorie
de la dimension de Krull des anneaux commutatifs. Notre méthode utilise la notion de structure algébrique dynamique introduite
dans des articles précédents.
Received: 4 October 1999; in final form: 11 October 2000 / Published online: 25 June 2001 相似文献
18.
Kyriakos Keremedis 《Archive for Mathematical Logic》1999,38(3):153-162
We find a characterization of the covering number , of the real line in terms of trees. We also show that the cofinality of is greater than or equal to for every where ( is the additivity number of the ideal of all Lebesgue measure zero sets) is the least cardinal number k for which the statement: fails.
Received: 19 October 1994 / Revised version: 12 December 1996 相似文献
19.
20.
Osamu Hiwatashi Masaru Nagisa Hiroaki Yoshida 《Probability Theory and Related Fields》1999,113(1):115-133
In usual probability theory, various characterizations of the Gaussian law have been obtained. For instance, independence
of the sample mean and the sample variance of independently identically distributed random variables characterizes the Gaussian
law and the property of remaining independent under rotations characterizes the Gaussian random variables. In this paper,
we consider the free analogue of such a kind of characterizations replacing independence by freeness. We show that freeness
of the certain pair of the linear form and the quadratic form in freely identically distributed noncommutative random variables,
which covers the case for the sample mean and the sample variance, characterizes the semicircle law. Moreover we give the
alternative proof for Nica's result that the property of remaining free under rotations characterizes a semicircular system.
Our proof is more direct and straightforward one.
Received: 12 February 1997 / Revised version: 16 June 1998 相似文献