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1.
Let :=. The following are known: two -sets of power are isomorphic. Let >0. Two ordered divisible Abelian groups that are -sets of power are isomorphic, two real closed fields that are -sets of power are isomorphic. The following is shown: (1) there exist 2 nonisomorphic ordered Abelian groups (respectively ordered fields) that are -sets of power ; (2) there exist 2 nonisomorphic ordered divisible Abelian groups (respectively real closed fields) of power all having the same order type; (3) there exist 2 nonisomorphic ordered divisible Abelian groups (respectively real closed fields) that are -sets having the same order type.  相似文献   

2.
Summary In this paper we perform a round-off error analysis of descent methods for solving a liner systemAx=b, whereA is supposed to be symmetric and positive definite. This leads to a general result on the attainable accuracy of the computed sequence {x i } when the method is performed in floating point arithmetic. The general theory is applied to the Gauss-Southwell method and the gradient method. Both methods appear to be well-behaved which means that these methods compute an approximationx i to the exact solutionA –1 b which is the exact solution of a slightly perturbed linear system, i.e. (A+A)x i =b, A of order A, where is the relative machine precision and · denotes the spectral norm.  相似文献   

3.
Summary In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) t v =v +f(x,v )+(x,v ) . Here is a strongly-elliptic second-order operator with constant coefficients, h:=DH xx-h, and the space variablex takes values on the unit circleS 1. The functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0<mM wherem andM are some finite positive constants. The perturbationW is a Brownian sheet. It is well-known that under some simple assumptions, the solutionv 2 is aC k (S 1)-valued Markov process for each 0<1/2, whereC (S 1) is the Banach space of real-valued continuous functions onS 1 which are Hölder-continuous of exponent . We prove, under some further natural assumptions onf and which imply that the zero element ofC (S 1) is a globally exponentially stable critical point of the unperturbed equation t 0 = 0 +f(x,0), that has a unique stationary distributionv K, on (C (S 1), (C K (S 1))) when the perturbation parameter is small enough. Some further calculations show that as tends to zero,v K, tends tov K,0, the point mass centered on the zero element ofC (S 1). The main goal of this paper is to show that in factv K, is governed by a large deviations principle (LDP). Our starting point in establishing the LDP forv K, is the LDP for the process , which has been shown in an earlier paper. Our methods of deriving the LDP forv K, based on the LDP for are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state spaceC (S 1) is inherently infinite-dimensional.This work was performed while the author was with the Department of Mathematics, University of Maryland, College Park, MD 20742, USA  相似文献   

4.
We give a proof ofTheorem 1. Let be the smallest cardinal such that the free subset property Fr (, 1)holds. Assume is singular. Then there is an inner model with 1 measurable cardinals.  相似文献   

5.
We establish a 2-categorical duality involving the 2-category A of all -accessible categories with wide pullbacks, also known as locally -polypresentable categories, and of functors preserving -filtered colimits and wide pullbacks. Commutation of wide pullbacks with so-called quasi-coproducts in Set is the basic ingredient to this duality, which leads to a full characterization of categories of type Wdpb Filt (A, Set)=A The first author acknowledges financial assistance from a special research grant of the Faculty of Arts at York University. The second author is partially supported by an NSERC operating grant.Presented at the European Colloquium of Category Theory, Tours, France, 25–31 July 1994.  相似文献   

6.
If P is a positive operator on a Hilbert space H whose range is dense, then a theorem of Foias, Ong, and Rosenthal says that: [(P)]–1T[(P)]<-12 max {T, P–1TP} for any bounded operator T on H, where is a continuous, concave, nonnegative, nondecreasing function on [0, P]. This inequality is extended to the class of normal operators with dense range to obtain the inequality [(N)]–1T[(N)]<-12c2 max {tT, N–1TN} where is a complex valued function in a class of functions called vase-like, and c is a constant which is associated with by the definition of vase-like. As a corollary, it is shown that the reflexive lattice of operator ranges generated by the range NH of a normal operator N consists of the ranges of all operators of the form (N), where is vase-like. Similar results are obtained for scalar-type spectral operators on a Hilbert space.This author gratefully acknowledges the support of Central Michigan University in the form of a Research Professorship.  相似文献   

7.
A topological space X whose topology is the order topology of some linear ordering on X, is called an interval space. A space in which every closed subspace is homeomorphic to a clopen subspace, is called a CO space. We regard linear orderings as topological spaces, by equipping them with their order topology. If L and K are linear orderings, then L *, L+K, L·K denote respectively the reverse orderings of L, the ordered sum of L and K and the lexicographic order on L×K (so ·2=+ and 2·=). Ordinals are considered as linear orderings, and cardinals are initial ordinals. For cardinals , 0, let L(, )= + 1 + * . Main theorem. Let X be a compact interval space. Then X is a CO space if and only if X is homeomorphic to a space of the form + 1 + i L( i , i ), where is any ordinal, n, for every ii, i are regular cardinals and i i, and if n>0, then max({ i: i}) · . This first part is devoted to show the following result. Theorem: If X is a compact interval CO space, then X is a scattered space (that means that every subspace of X has an isolated point).Supported by the Université Claude-Bernard (Lyon-1), the Ben Gurion University of the Negev, and the C.N.R.S.: UPR 9016Supported by the City of Lyon  相似文献   

8.
LetA be anM-matrix in standard lower block triangular form, with diagonal blocksA ii irreducible. LetS be the set of indices such that the diagonal blockA is singular. We define the singular graph ofA to be the setS with partial order defined by > if there exists a chain of non-zero blocksA i, Aij, , Al.Let 1 be the set of maximal elements ofS, and define thep-th level p ,p = 2, 3, , inductively as the set of maximal elements ofS \( 1 p-1). Denote by p the number of elements in p . The Weyr characteristic (associated with 0) ofA is defined to be (A) = ( 1, 2,, h ), where 1 + + p = dim KerA p ,p = 1, 2, , and h > 0, h+1 = 0.Using a special type of basis, called anS-basis, for the generalized eigenspaceE(A) of 0 ofA, we associate a matrixD withA. We show that(A) = ( 1, , h) if and only if certain submatricesD p,p+1 ,p = 1, , h – 1, ofD have full column rank. This condition is also necessary and sufficient forE(A) to have a basis consisting of non-negative vectors, which is a Jordan basis for –A. We also consider a given finite partially ordered setS, and we find a necessary and sufficient condition that allM-matricesA with singular graphS have(A) = ( 1, , h). This condition is satisfied ifS is a rooted forest.The work of the second-named author was partly supported by the National Science Foundation, under grant MPS-08618 A02.  相似文献   

9.
We discuss the solution of Hermitian positive definite systemsAx=b by the preconditioned conjugate gradient method with a preconditionerM. In general, the smaller the condition number(M –1/2 AM –1/2 ) is, the faster the convergence rate will be. For a given unitary matrixQ, letM Q = {Q* N Q | n is ann-by-n complex diagonal matrix} andM Q + ={Q* n Q | n is ann-by-n positive definite diagonal matrix}. The preconditionerM b that minimizes(M –1/2 AM –1/2 ) overM Q + is called the best conditioned preconditioner for the matrixA overM Q + . We prove that ifQAQ* has Young's Property A, thenM b is nothing new but the minimizer of MA F overM Q . Here · F denotes the Frobenius norm. Some applications are also given here.  相似文献   

10.
1. If is a weakly compact cardinal then ( +)( ). 2. If is measurable andU a normal ultrafilter then ( +)(U ).

Diese Arbeit ist ein Teil der Dissertation des Autors. Teilweise gefördert von der DFG.  相似文献   

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