共查询到20条相似文献,搜索用时 93 毫秒
1.
R.A. Willoughby 《Linear algebra and its applications》1977,18(1):75-94
One aspect of the inverse M-matrix problem can be posed as follows. Given a positive n × n matrix A=(aij) which has been scaled to have unit diagonal elements and off-diagonal elements which satisfy 0 < y ? aij ? x < 1, what additional element conditions will guarantee that the inverse of A exists and is an M-matrix? That is, if A?1=B=(bij), then bii> 0 and bij ? 0 for i≠j. If n=2 or x=y no further conditions are needed, but if n ? 3 and y < x, then the following is a tight sufficient condition. Define an interpolation parameter s via x2=sy+(1?s)y2; then B is an M-matrix if s?1 ? n?2. Moreover, if all off-diagonal elements of A have the value y except for aij=ajj=x when i=n?1, n and 1 ? j ? n?2, then the condition on both necessary and sufficient for B to be an M-matrix. 相似文献
2.
E. Szemerédi 《Journal of Number Theory》1976,8(3):264-270
Let n be a positive integer and let A = {a1,…, as}, B = {b1,…, bt} be two sets of positive integers such that the product set consists of st distinct numbers. Then, for a certain positive constant c, , establishing a conjecture made by P. Erdös. 相似文献
3.
An axis-parallel b-dimensional box is a Cartesian product R1×R2×?×Rb where each Ri (for 1≤i≤b) is a closed interval of the form [ai,bi] on the real line. The boxicity of any graph G, is the minimum positive integer b such that G can be represented as the intersection graph of axis-parallel b-dimensional boxes. A b-dimensional cube is a Cartesian product R1×R2×?×Rb, where each Ri (for 1≤i≤b) is a closed interval of the form [ai,ai+1] on the real line. When the boxes are restricted to be axis-parallel cubes in b-dimension, the minimum dimension b required to represent the graph is called the cubicity of the graph (denoted by ). In this paper we prove that , where n is the number of vertices in the graph. We also show that this upper bound is tight.Some immediate consequences of the above result are listed below:
- 1.
- Planar graphs have cubicity at most 3⌈log2n⌉.
- 2.
- Outer planar graphs have cubicity at most 2⌈log2n⌉.
- 3.
- Any graph of treewidth tw has cubicity at most (tw+2)⌈log2n⌉. Thus, chordal graphs have cubicity at most (ω+1)⌈log2n⌉ and circular arc graphs have cubicity at most (2ω+1)⌈log2n⌉, where ω is the clique number.
4.
The following result is proved: If A and B are distinct n × n doubly stochastic matrices, then there exists a permutation σ of {1, 2,…, n} such that ∏iaiσ(i) > ∏ibiσ(i). 相似文献
5.
Given a sequence A = (a 1, …, a n ) of real numbers, a block B of A is either a set B = {a i , a i+1, …, a j } where i ≤ j or the empty set. The size b of a block B is the sum of its elements. We show that when each a i ∈ [0, 1] and k is a positive integer, there is a partition of A into k blocks B 1, …, B k with |b i ?b j | ≤ 1 for every i, j. We extend this result in several directions. 相似文献
6.
Singular values, norms, and commutators 总被引:1,自引:0,他引:1
Omar Hirzallah 《Linear algebra and its applications》2010,432(5):1322-1336
Let and Xi, i=1,…,n, be bounded linear operators on a separable Hilbert space such that Xi is compact for i=1,…,n. It is shown that the singular values of are dominated by those of , where ‖·‖ is the usual operator norm. Among other applications of this inequality, we prove that if A and B are self-adjoint operators such that a1?A?a2 and b1?B?b2 for some real numbers and b2, and if X is compact, then the singular values of the generalized commutator AX-XB are dominated by those of max(b2-a1,a2-b1)(X⊕X). This inequality proves a recent conjecture concerning the singular values of commutators. Several inequalities for norms of commutators are also given. 相似文献
7.
A proof is given for the existence and uniqueness of a correspondence between two pairs of sequences {a},{b} and {ω},{μ}, satisfying bi>0 for i=1,…,n?1 and ω1<μ1<?<μn?1<ωn, under which the symmetric Jacobi matrices J(n,a,b) and J(n?1,a,b) have eigenvalues {ω} and {μ} respectively. Here J(m,a,b) is symmetric and tridiagonal with diagonal elements ai (i=1,…,m) and off diagonal elements bi (i=1,…,m?1). A new concise proof is given for the known uniqueness result. The existence result is new. 相似文献
8.
Philippe Baptiste 《Discrete Applied Mathematics》2010,158(5):583-587
The coupled tasks problem consists in scheduling n jobs on a single machine. Each job i is made of two operations with processing times ai and bi and a fixed required delay Li between them. Operations cannot overlap in time but operations of different jobs can be interleaved. The objective is to minimize the makespan of the schedule. In this note we show that the problem with identical jobs (∀i,ai=a,bi=b,Li=L) can be solved in O(logn) time when a,b,L are fixed. This problem is motivated by radar scheduling applications where tasks corresponding to transmitting radiowaves and listening to potential echoes are coupled. 相似文献
9.
Let A=(aij) be a real symmetric matrix of order n. We characterize all nonnegative vectors x=(x1,...,xn) and y=(y1,...,yn) such that any real symmetric matrix B=(bij), with bij=aij, i≠jhas its eigenvalues in the union of the intervals [bij?yi, bij+ xi]. Moreover, given such a set of intervals, we derive better bounds for the eigenvalues of B using the 2n quantities {bii?y, bii+xi}, i=1,..., n. 相似文献
10.
For a string A=a1…an, a reversalρ(i,j), 1?i?j?n, transforms the string A into a string A′=a1…ai-1ajaj-1…aiaj+1…an, that is, the reversal ρ(i,j) reverses the order of symbols in the substring ai…aj of A. In the case of signed strings, where each symbol is given a sign + or -, the reversal operation also flips the sign of each symbol in the reversed substring. Given two strings, A and B, signed or unsigned, sorting by reversals (SBR) is the problem of finding the minimum number of reversals that transform the string A into the string B.Traditionally, the problem was studied for permutations, that is, for strings in which every symbol appears exactly once. We consider a generalization of the problem, k-SBR, and allow each symbol to appear at most k times in each string, for some k?1. The main result of the paper is an O(k2)-approximation algorithm running in time O(n). For instances with , this is the best known approximation algorithm for k-SBR and, moreover, it is faster than the previous best approximation algorithm. 相似文献
11.
A matrix A∈Rn×n is called a bisymmetric matrix if its elements ai,j satisfy the properties ai,j=aj,i and ai,j=an-j+1,n-i+1 for 1?i,j?n. This paper considers least squares solutions to the matrix equation AX=B for A under a central principal submatrix constraint and the optimal approximation. A central principal submatrix is a submatrix obtained by deleting the same number of rows and columns in edges of a given matrix. We first discuss the specified structure of bisymmetric matrices and their central principal submatrices. Then we give some necessary and sufficient conditions for the solvability of the least squares problem, and derive the general representation of the solutions. Moreover, we also obtain the expression of the solution to the corresponding optimal approximation problem. 相似文献
12.
Gideon Nettler 《Journal of Number Theory》1981,13(4):456-462
In a previous paper it was proven that given the continued fractions where the a's and b's are positive integers, then A, B, A ± B, and AB are irrational numbers if for all n sufficiently large, and transcendental numbers if for all n sufficiently large. Using a more direct approach it is proven in this paper that A, B, A ± B, and AB are transcendental numbers if an > bn > an?1(n?1)2 for all n sufficiently large. 相似文献
13.
A circular string A = (a1,…,an) is a string that has n equivalent linear representations Ai = ai,…,an,a1,…,ai?1 for i = 1,…,n. The ai's are assumed to be well ordered. We say that Ai < Aj if the word ai…ana1…ai?1 precedes the word aj…ana1…aj?1 in the lexicographic order, Ai ? Aj if either Ai < Aj or Ai = Aj. Ai0 is a minimal representation of A if Ai0 ? Ai for all 1 ≤ i ≤ n. The index i0 is called a minimal starting point (m.s.p.). In this paper we discuss the problem of finding m.s.p. of a given circular string. Our algorithm finds, in fact, all the m.s.p.'s of a given circular string A of length n by using at most comparisons. The number d denotes the difference between two successive m.s.p.'s (see Lemma 1.1) and is equal to n if A has a unique m.s.p. Our algorithm improves the result of 3n comparisons given by K. S. Booth. Only constant auxiliary storage is used. 相似文献
14.
Motivated by a question of Sárközy, we study the gaps in the product sequence B = A · A = {b 1 < b 2 < …} of all products a i a j with a i , a j ∈ A when A has upper Banach density α > 0. We prove that there are infinitely many gaps b n+1 ? b n ? α ?3 and that for t ≥ 2 there are infinitely many t-gaps b n+t ? b n ? t 2 α ?4. Furthermore, we prove that these estimates are best possible.We also discuss a related question about the cardinality of the quotient set A/A = {a i /a j , a i , a j ∈ A} when A ? {1, …, N} and |A| = αN. 相似文献
15.
16.
Kai He 《Bulletin des Sciences Mathématiques》2004,128(2):91-104
In this paper, we prove that the process of the quadratic variation of local times of smooth semimartingales can be constructed as the quasi sure limit of the form ∑Δn(Ltai+1n−Ltain)2, where Δn=(ain,ai+1n) is a sequence of subdivisions of [a,b], ain=i(b−a)/2n+a, i=0,1,…,2n. 相似文献
17.
Asymptotic behavior of solutions of the functional differential equation x′(t) = Ax(λt) + Bx(t), λ>0
Eng-Bin Lim 《Journal of Mathematical Analysis and Applications》1976,55(3):794-806
The author discusses the asymptotic behavior of the solutions of the functional differential equation x′(t) = Ax(λt) + Bx(t), λ>0 (1) where x(t) is an n-dimensional column vector and A, B are n × n matrices with complex constant entries. He obtains the following results for the case 0 < λ < 1: (i) If B is diagonalizable with eigenvalues bi such that Re bi < 0 for all i, then there is a constant α such that every solution of (1) is O(tα) as t → ∞. (ii) If B is diagonalizable with eigenvalues bi such that 0 < Re b1 ? Re b2 ? ··· ? Re bn and λ times Re bn < Re b1, then every solution of (1) is O(ebnt) as t → ∞. For the case λ>1, he has the following results: (i) If B is diagonalizable with eigenvalues bi such that Re bi>0 for all i, then there is a constant α such that no solution x(t) of (1), except the identically zero solution, is 0(tα) as t → ∞. (ii) If B is diagonalizable with eigenvalues bi such that Re b1 ? Re b2 ? ··· ? Re bn < 0 and λ Re bn < Re b1, then no solution x(t) of (1), except the identically zero solution, is 0(eb1t) as t → ∞. 相似文献
18.
R.C. Thompson 《Linear algebra and its applications》1976,14(2):135-177
Given n-square Hermitian matrices A,B, let Ai,Bi denote the principal (n?1)- square submatrices of A,B, respectively, obtained by deleting row i and column i. Let μ, λ be independent indeterminates. The first main result of this paper is the characterization (for fixed i) of the polynomials representable as det(μAi?λBi) in terms of the polynomial det(μA?λB) and the elementary divisors, minimal indices, and inertial signatures of the pencil μA?λB. This result contains, as a special case, the classical interlacing relationship governing the eigenvalues of a principal sub- matrix of a Hermitian matrix. The second main result is the determination of the number of different values of i to which the characterization just described can be simultaneously applied. 相似文献
19.
Michel Weber 《Indagationes Mathematicae》2007,18(2):281-293
Let ?= {?i,i ≥1} be a sequence of independent Bernoulli random variables (P{?i = 0} = P{?i = 1 } = 1/2) with basic probability space (Ω, A, P). Consider the sequence of partial sums Bn=?1+...+?n, n=1,2..... We obtain an asymptotic estimate for the probability P{P-(Bn) > >} for >≤ne/log log n, c a positive constant. 相似文献
20.
C.E.R. Alves 《Discrete Applied Mathematics》2008,156(7):1025-1035
Given two strings A and B of lengths na and nb, na?nb, respectively, the all-substrings longest common subsequence (ALCS) problem obtains, for every substring B′ of B, the length of the longest string that is a subsequence of both A and B′. The ALCS problem has many applications, such as finding approximate tandem repeats in strings, solving the circular alignment of two strings and finding the alignment of one string with several others that have a common substring. We present an algorithm to prepare the basic data structure for ALCS queries that takes O(nanb) time and O(na+nb) space. After this preparation, it is possible to build a matrix of size that allows any LCS length to be retrieved in constant time. Some trade-offs between the space required and the querying time are discussed. To our knowledge, this is the first algorithm in the literature for the ALCS problem. 相似文献