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1.
We consider the functional equation of the second kind \gf − λ K\gf = f with K a compact self-adjoint linear operator on a Hilbert space: a Fredholm integral equation of the second kind, for example. We establish the simple bound
where λ is any regular value of K; ø is the solution of the equation corresponding to λ; λ 1 is the characteristic value of K smallest in absolute value; and N = 0, 1, 2, .... For \s|λ\s| < \s|λ 1\s|, this is an estimate for the remainder of the partial sums of the Neumann series. 相似文献
2.
Let A be a positive definite, symmetric matrix. We wish to determine the largest eigenvalue, λ 1. We consider the power method, i.e. that of choosing a vector v0 and setting vk = Akv0; then the Rayleigh quotients Rk = ( Avk, vk)/( vk, vk) usually converge to λ 1 as k → ∞ (here ( u, v) denotes their inner product). In this paper we give two methods for determining how close Rk is to λ 1. They are both based on a bound on λ 1 − Rk involving the difference of two consecutive Rayleigh quotients and a quantity ω k. While we do not know how to directly calculate ω k, we can given an algorithm for giving a good upper bound on it, at least with high probability. This leads to an upper bound for λ 1 − Rk which is proportional to (λ 2/λ 1) 2k, which holds with a prescribed probability (the prescribed probability being an arbitrary δ > 0, with the upper bound depending on δ). 相似文献
3.
It was proved by Dow and Simon that there are 2 ω1 (as many as possible) pairwise nonhomeomorphic compact, T2, scattered spaces of height ω 1 and width ω. In this paper, we prove that if is an ordinal withω 1 < ω 2 and θ = κ ξ: ξ < is a sequence of cardinals such that either κ ξ = ω or κ ξ = ω 1 for every ξ < , then there are 2 ω1 pairwise nonhomeomorphic compact, T2, scattered spaces whose cardinal sequence is θ. 相似文献
4.
Let π i : Ei→ M, i=1,2, be oriented, smooth vector bundles of rank k over a closed, oriented n-manifold with zero sections si : M→ Ei. Suppose that U is an open neighborhood of s1( M) in E1 and F : U→ E2 a smooth embedding so that π 2Fs1 : M→ M is homotopic to a diffeomorphism f. We show that if k>[( n+1)/2]+1 then E1 and the induced bundle f*E2 are isomorphic as oriented bundles provided that f have degree +1; the same conclusion holds if f has degree −1 except in the case where k is even and one of the bundles does not have a nowhere-zero cross-section. For n≡0(4) and [( n+1)/2]+1< kn we give examples of nonisomorphic oriented bundles E1 and E2 of rank k over a homotopy n-sphere with total spaces diffeomorphic with orientation preserved, but such that E1 and f*E2 are not isomorphic oriented bundles. We obtain similar results and counterexamples in the more difficult limiting case where k=[( n+1)/2]+1 and M is a homotopy n-sphere. 相似文献
5.
It is shown that there is no satisfactory first-order characterization of those subsets of ω 2 that have closed unbounded subsets in ω 1,ω 2 and GCH preserving outer models. These “anticharacterization” results generalize to subsets of successors of uncountable regular cardinals. Similar results are proved for trees of height and cardinality κ + and for partitions of [κ +] 2, when κ is an infinite cardinal. 相似文献
6.
We study the concept of strong equality of domination parameters. Let P1 and P2 be properties of vertex subsets of a graph, and assume that every subset of V( G) with property P2 also has property P1. Let ψ 1( G) and ψ 2( G), respectively, denote the minimum cardinalities of sets with properties P1 and P2, respectively. Then ψ 1( G)ψ 2( G). If ψ 1( G)=ψ 2( G) and every ψ 1( G)-set is also a ψ 2( G)-set, then we say ψ 1( G) strongly equals ψ 2( G), written ψ 1( G)≡ψ 2( G). We provide a constructive characterization of the trees T such that γ( T)≡ i( T), where γ( T) and i( T) are the domination and independent domination numbers, respectively. A constructive characterization of the trees T for which γ( T)=γ t( T), where γ t( T) denotes the total domination number of T, is also presented. 相似文献
7.
For a connected graph G with n vertices, let {λ 1,λ 2,…,λ r} be the set of distinct positive eigenvalues of the Laplacian matrix of G. The Hoffman number μ( G) of G is defined by μ( G)=λ 1λ 2…λ r/ n. In this paper, we study some properties and applications of the Hoffman number. 相似文献
8.
This paper presents the finding that the invocation of new words in human language samples is governed by a slowly changing Poisson process. The time dependent rate constant for this process has the form λ(t) = λ1(1−λ2t)e-λ2t+λ3(1−λ4t)e-λ4t+λ5 , where . This form implies that there are opening, middle and final phases to the introduction of new words, each distinguished by a dominant rate constant, or equivalently, rate of decay. With the occasional exception of the phase transition from beginning to middle, the rate λ(t) decays monotonically. Thus, λ(t) quantifies how the penchant of humans to introduce new words declines with the progression of their narratives, written or spoken. 相似文献
9.
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
10.
If
are maximal nests on a finite-dimensional Hilbert space H, the dimension of the intersection of the corresponding nest algebras is at least dim H. On the other hand, there are three maximal nests whose nest algebras intersect in the scalar operators. The dimension of the intersection of two nest algebras (corresponding to maximal nests) can be of any integer value from n to n( n+1)/2, where n=dim H. For any two maximal nests
there exists a basis { f1, f2,…, fn} of H and a permutation π such that
and
where Mi= span{ f1, f2,…, fi} and Ni= span{ fπ(1), fπ(2),…, fπ(i)}. The intersection of the corresponding nest algebras has minimum dimension, namely dim H, precisely when π( j)= n− j+1,1 jn. Those algebras which are upper-triangular matrix incidence algebras, relative to some basis, can be characterised as intersections of certain nest algebras. 相似文献
11.
Let H be a graph with κ 1 components and κ 2 blocks, and let G be a minor-minimal 2-connected graph having H as a minor. This paper proves that | E( G)|−| E( H)|(κ 1−1)+β(κ 2−1) for all (,β) such that +β5,2+5β20, and β3. Moreover, if one of the last three inequalities fails, then there are graphs G and H for which the first inequality fails. 相似文献
13.
We define an arithmetic invariant for the congruence subgroups Γ 0( N), denoted by b(Γ 0( N)) and pose the problem of finding good asymptotic upper bonds for b(Γ 0( N)) as N approaches X especially when N is prime or the product of two (not necessarily distinct) primes. 相似文献
14.
We study the problem of designing fault-tolerant routings with small routing tables for a k-connected network of n processors in the surviving route graph model. The surviving route graph R( G,ρ)/ F for a graph G, a routing ρ and a set of faults F is a directed graph consisting of nonfaulty nodes of G with a directed edge from a node x to a node y iff there are no faults on the route from x to y. The diameter of the surviving route graph could be one of the fault-tolerance measures for the graph G and the routing ρ and it is denoted by D( R( G,ρ)/ F). We want to reduce the total number of routes defined in the routing, and the maximum of the number of routes defined for a node (called route degree) as least as possible. In this paper, we show that we can construct a routing λ for every n-node k-connected graph such that n2 k2, in which the route degree is
, the total number of routes is O( k2n) and D( R( G,λ)/ F)3 for any fault set F (| F|< k). In particular, in the case that k=2 we can construct a routing λ′ for every biconnected graph in which the route degree is
, the total number of routes is O( n) and D( R( G,λ′)/{ f})3 for any fault f. We also show that we can construct a routing ρ 1 for every n-node biconnected graph, in which the total number of routes is O( n) and D( R( G,ρ 1)/{ f})2 for any fault f, and a routing ρ 2 (using ρ 1) for every n-node biconnected graph, in which the route degree is
, the total number of routes is
and D( R( G,ρ 2)/{ f})2 for any fault f. 相似文献
15.
The parametric resource allocation problem asks to minimize the sum of separable single-variable convex functions containing a parameter λ, Σ i = 1n (ƒ i( xi + λ gi( xi)), under simple constraints Σ i = 1n xi = M, li≤ xi≤ ui and xi: nonnegative integers for i = 1, 2, …, n, where M is a given positive integer, and li and ui are given lower and upper bounds on xi. This paper presents an efficient algorithm for computing the sequence of all optimal solutions when λ is continuously changed from 0 to ∞. The required time is O( G√ Mlog 2 n + n log n + n log( M/ n)), where G = Σ i = 1n ui − Σ i = 1n li and an evaluation of ƒ i(·) or gi(·) is assumed to be done in constant time. 相似文献
16.
It is known that the equivariant diffeomorphism group DiffG∞( M) 0 of a principal G-manifold M is perfect. If M has at least two orbit types, then it is not true. The purpose of this paper is to determine the first homology group of DiffG∞( M) 0 when M is a G-manifold with codimension one orbit. 相似文献
17.
The following results are proved: (a) In a model obtained by adding 2 Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of □μ and cof([μ]0,)=μ+ hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of
is equivalent to the weak Freese-Nation property of any of
or
for uncountable κ. (d) Modulo the consistency of (ω+1,ω)(1,0), it is consistent with GCH that
does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding ω Cohen reals destroys the weak Freese-Nation property of
. These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke. 相似文献
18.
We present a characterization of those Euclidean distance matrices (EDMs) D which can be expressed as D=λ( E− C) for some nonnegative scalar λ and some correlation matrix C, where E is the matrix of all ones. This shows that the cones where
is the elliptope (set of correlation matrices) and
is the (closed convex) cone of EDMs. The characterization is given using the Gale transform of the points generating D. We also show that given points
, for any scalars λ1,λ2,…,λn such that we have ∑j=1nλjpi−pj2= forall i=1,…,n, for some scalar independent of i. 相似文献
19.
Using the integral average method, we establish some oscillation criteria of Kamenev type and Yan type for the nonlinear system of differential equation where the functions bi( t) ( i = 1, 2) are nonnegative and summable on each finite segment of the interval Z0, ∞), λ i > 0 ( i = 1,2) with λ 1 λ 2 = 1. 相似文献
20.
Let C be a planar region. Choose n points p1,,p nI.I.D. from the uniform distribution over C. Let MCn be the number of these points that are maximal. If C is convex it is known that either E( MCn)=Θ(√ n)> or E( MCn)=O( log n). In this paper we will show that, for general C, there is very little that can be said, a-priori, about E( MCn). More specifically we will show that if g is a member of a large class of functions then there is always a region C such that E( MCn)=Θ( g( n)). This class contains, for example, all monotically increasing functions of the form g( n)= nlnβn, where 0<<1 and β0. This class also contains nondecreasing functions like g( n)= ln*n. The results in this paper remain valid in higher dimensions. 相似文献
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