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1.
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
2.
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
3.
If a˜cardinal κ 1, regular in the ground model M, is collapsed in the extension N to a˜cardinal κ 0 and its new cofinality, ρ, is less than κ 0, then, under some additional assumptions, each cardinal λ>κ 1 less than cc( P(κ 1)/[κ 1] <κ1) is collapsed to κ 0 as well. If in addition N= M[ f], where f : ρ→κ 1 is an unbounded mapping, then N is a˜|λ|=κ 0-minimal extension. This and similar results are applied to generalized forcing notions of Bukovský and Namba. 相似文献
4.
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
5.
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
6.
Let X1, X2, … be independent identically distributed random variables. Then, Hsu and Robbins (1947) together with Erdös (1949, 1950) have proved that , if and only if E[X21] < ∞ and E[X1] = 0. We prove that there are absolute constants C1, C2 (0, ∞) such that if X1, X2, … are independent identically distributed mean zero random variables, then c1λ−2 E[X12·1{|X1|λ}]S(λ)C2λ−2 E[X12·1{|X1|λ}] , for every λ > 0. 相似文献
7.
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
8.
Let U be the unitary group over a finite field K where [ K]≠3 and char K≠2. Every transformation π in U with detπ = ±1 is a product of reflections. We determine the length t of every such transformation π i.e. for each π we find reflections σ i in U such that π=σ 1 σ t and so that no factorization of π into fewer than t reflections exists. 相似文献
9.
The following theorem is proved: given square matrices A, D of the same size, D nonnegative, then either the equation Ax + B| x| = b has a unique solution for each B with | B| ≤ D and for each b, or the equation Ax + B0| x| = 0 has a nontrivial solution for some matrix B0 of a very special form, | B0| ≤ D; the two alternatives exclude each other. Some consequences of this result are drawn. In particular, we define a λ to be an absolute eigenvalue of A if | Ax| = λ| x| for some x ≠ 0, and we prove that each square real matrix has an absolute eigenvalue. 相似文献
10.
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
11.
A q × n array with entries from 0, 1,…, q − 1 is said to form a difference matrix if the vector difference (modulo q) of each pair of columns consists of a permutation of [0, 1,… q − 1]; this definition is inverted from the more standard one to be found, e.g., in Colbourn and de Launey (1996). The following idea generalizes this notion: Given an appropriate δ (-[−1, 1] t, a λ q × n array will be said to form a ( t, q, λ, Δ) sign-balanced matrix if for each choice C1, C2,…, Ct of t columns and for each choice = ( 1,…, t) Δ of signs, the linear combination ∑ j=1t jCj contains (mod q) each entry of [0, 1,…, q − 1] exactly λ times. We consider the following extremal problem in this paper: How large does the number k = k( n, t, q, λ, δ) of rows have to be so that for each choice of t columns and for each choice ( 1, …, t) of signs in δ, the linear combination ∑ j=1t jCj contains each entry of [0, 1,…, q t- 1] at least λ times? We use probabilistic methods, in particular the Lovász local lemma and the Stein-Chen method of Poisson approximation to obtain general (logarithmic) upper bounds on the numbers k( n, t, q, λ, δ), and to provide Poisson approximations for the probability distribution of the number W of deficient sets of t columns, given a random array. It is proved, in addition, that arithmetic modulo q yields the smallest array - in a sense to be described. 相似文献
12.
The concept of a ( q, k, λ, t) almost difference family (ADF) has been introduced and studied by C. Ding and J. Yin as a useful generalization of the concept of an almost difference set. In this paper, we consider, more generally, ( q, K, λ, t, Q)-ADFs, where K = { k1, k2, ..., kr} is a set of positive integers and Q = ( q1, q2,..., qr) is a given block-size distribution sequence. A necessary condition for the existence of a ( q, K, λ, t, Q)-ADF is given, and several infinite classes of ( q, K, λ, t, Q)-ADFs are constructed. 相似文献
13.
We present a new data structure for a set of n convex simply-shaped fat objects in the plane, and use it to obtain efficient and rather simple solutions to several problems including (i) vertical ray shooting—preprocess a set of n non-intersecting convex simply-shaped flat objects in 3-space, whose xy-projections are fat, for efficient vertical ray shooting queries, (ii) point enclosure—preprocess a set C of n convex simply-shaped fat objects in the plane, so that the k objects containing a query point p can be reported efficiently, (iii) bounded-size range searching— preprocess a set C of n convex fat polygons, so that the k objects intersecting a “not-too-large” query polygon can be reported efficiently, and (iv) bounded-size segment shooting—preprocess a set C as in (iii), so that the first object (if exists) hit by a “not-too-long” oriented query segment can be found efficiently. For the first three problems we construct data structures of size O(λ s( n)log 3n), where s is the maximum number of intersections between the boundaries of the ( xy-projections) of any pair of objects, and λ s( n) is the maximum length of ( n, s) Davenport-Schinzel sequences. The data structure for the fourth problem is of size O(λ s( n)log 2n). The query time in the first problem is O(log 4n), the query time in the second and third problems is O(log 3n + klog 2n), and the query time in the fourth problem is O(log 3n). We also present a simple algorithm for computing a depth order for a set as in (i), that is based on the solution to the vertical ray shooting problem. (A depth order for , if exists, is a linear order of , such that, if K1, K2 and K1 lies vertically above K2, then K1 precedes K2.) Unlike the algorithm of Agarwal et al. (1995) that might output a false order when a depth order does not exist, the new algorithm is able to determine whether such an order exists, and it is often more efficient in practical situations than the former algorithm. 相似文献
14.
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
15.
A construction is given for a ( p2a( p+1), p2, p2a+1( p+1), p2a+1, p2a( p+1)) ( p a prime) divisible difference set in the group H× Z2pa+1 where H is any abelian group of order p+1. This can be used to generate a symmetric semi-regular divisible design; this is a new set of parameters for λ 1≠0, and those are fairly rare. We also give a construction for a ( pa−1+ pa−2+…+ p+2, pa+2, pa( pa+ pa−1+…+ p+1), pa( pa−1+…+ p+1), pa−1( pa+…+ p2+2)) divisible difference set in the group H× Zp2× Zap. This is another new set of parameters, and it corresponds to a symmetric regular divisible design. For p=2, these parameters have λ 1=λ 2, and this corresponds to the parameters for the ordinary Menon difference sets. 相似文献
16.
We consider a variation of a classical Turán-type extremal problem (F. Chung, R. Graham, Erd
s on Graphs: His Legacy of Unsolved Problems, AK Peters Ltd., Wellesley, 1998, Chapter 3) as follows: Determine the smallest even integer σ( Kr,s, n) such that every n-term graphic sequence π=( d1, d2,…, dn) with term sum σ(π)= d1+ d2++ dnσ( Kr,s, n) is potentially Kr,s-graphic, where Kr,s is a r× s complete bipartite graph, i.e., π has a realization G containing Kr,s as its subgraph. In this paper, we first give sufficient conditions for a graphic sequence being potentially Kr,s-graphic, and then we determine σ( Kr,r, n) for r=3,4. 相似文献
18.
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
19.
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
20.
Let C1,…, Cn and C′ 1,…, C′ n be two collections of equal disks in the plane, with centers c1,…, cn and c′ 1,…, c′ n. According to a well-known conjecture of Klee and Wagon (1991), if | ci − cj| ≥ | c′ i − c′ j| for all i, j, then Area(∩ i Ci) ≤ Area(∩ i C′ i). We prove this statement in the special case when there is a continuous contraction of {c1,…, cn} onto {c′1,…, c′n}. 相似文献
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