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1.
A real m×n matrix A is said to be semipositive if there is a nonnegative vector λ such that Ax exists and is componentwise positive. A is said to be minimally semipositive if it is semipositive and no proper m×p submatrix of A is semipositive. Minimal semipositivity is characterized in this paper and is related to rectangular monotonicity and weak r-monotonicity. P-matrices and nonnegative matrices will also be considered.  相似文献   

2.
An L(2,1)-coloring of a graph G is a coloring of G's vertices with integers in {0,1,…,k} so that adjacent vertices’ colors differ by at least two and colors of distance-two vertices differ. We refer to an L(2,1)-coloring as a coloring. The span λ(G) of G is the smallest k for which G has a coloring, a span coloring is a coloring whose greatest color is λ(G), and the hole index ρ(G) of G is the minimum number of colors in {0,1,…,λ(G)} not used in a span coloring. We say that G is full-colorable if ρ(G)=0. More generally, a coloring of G is a no-hole coloring if it uses all colors between 0 and its maximum color. Both colorings and no-hole colorings were motivated by channel assignment problems. We define the no-hole span μ(G) of G as ∞ if G has no no-hole coloring; otherwise μ(G) is the minimum k for which G has a no-hole coloring using colors in {0,1,…,k}.

Let n denote the number of vertices of G, and let Δ be the maximum degree of vertices of G. Prior work shows that all non-star trees with Δ3 are full-colorable, all graphs G with n=λ(G)+1 are full-colorable, μ(G)λ(G)+ρ(G) if G is not full-colorable and nλ(G)+2, and G has a no-hole coloring if and only if nλ(G)+1. We prove two extremal results for colorings. First, for every m1 there is a G with ρ(G)=m and μ(G)=λ(G)+m. Second, for every m2 there is a connected G with λ(G)=2m, n=λ(G)+2 and ρ(G)=m.  相似文献   


3.
The construction of most reliable networks is investigated. In particular, the study of restricted edge connectivity shows that general Harary graphs are max λ–min mi for all i=λ, λ+1,…,2λ−3. As a consequence, this implies that for each pair of positive integers n and e, there is a graph of n vertices and e edges that is max λ–min mi for all i=λ,λ+1,…,2λ−3. General Harary graphs that are max λ–min mi for all i=λ,λ+1,…,2λ−2 are also constructed.  相似文献   

4.
Let T be a linear operator on the space of all m×n matrices over any field. we prove that if T maps rank-2 matrices to rank-2 matrices then there exist nonsingular matrices U and V such that either T(X)=UXV for all matrices X, or m=n and T(X)=UXtV for all matrices X where Xt denotes the transpose of X.  相似文献   

5.
We prove the following result. Let F be an infinite field of characteristic other than two. Let k be a positive integer. Let Sn(F) denote the space of all n × n symmetric matrices with entries in F, and let T:Sn(F)→Sn(F) be a linear operator. Suppose that T is rank-k nonincreasing and its image contains a matrix with rank higher than K. Then, there exist λεF and PεFn,n such that T(A)=λPAPt for all AεSn(F). λ can be chosen to be 1 if F is algebraically closed and ±1 if F=R, the real field.  相似文献   

6.
Let A be a matrixp(x) a polynomial. Put B=p(A). It is shown that necessary and sufficient conditions for A to be a polynomial in B are (i) if λ is any eigenvalue of A, and if some elementary divisor of A corresponding to λ is nonlinear, thenp'(λ)≠0;and (ii) if λ,μ are distinct eigenvalues of A, then p(λ)p(μ) are also distinct. Here all computations are over some algebraically closed field.  相似文献   

7.
In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r> 0, b (− ∞, 2]. We establish a full quadrature sum estimate
1 p < ∞, for every polynomial P of degree at most n + rn1/3, where W2 is a Freud weight such as exp(−¦x¦), > 1, λjn are the Christoffel numbers, xjn are the zeros of the orthonormal polynomials for the weight W2, and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree M = m(n) if m(n) = n + ξnn1/3, where ξn → ∞ as n → ∞. Previous estimates could sum only over those xjn with ¦xjn¦ σx1n, some fixed 0 < σ < 1.  相似文献   

8.
The problem of constructing (m, n) cages suggests the following class of problems. For a graph parameter θ, determine the minimum or maximum value of p for which there exists a k-regular graph on p points having a given value of θ. The minimization problem is solved here when θ is the achromatic number, denoted by ψ. This result follows from the following main theorem. Let M(p, k) be the maximum value of ψ(G) over all k-regular graphs G with p points, let {x} be the least integer of size at least x, and let be given by ω(k) = {i(ik+1)+1:1i<∞}. Define the function ƒ(p, k) by . Then for fixed k2 we have M(p, K=ƒ(p, k) if pω(k) and M(p, k)=ƒ(p,k-1 if pε ω(k) for all p sufficiently large with respect to k.  相似文献   

9.
Let A be an mn- by - mn symmetric matrix. Partition A into m2n - by - n blocks and suppose that each of these blocks is also symmetric. Suppose that for every decomposable (rank one) tensor ν ⊗ w, we have (ν ⊗ w)t A(ν otimes; w) ≥ 0. Here, ν is a column m-tuple and w is a column n-tuple. We study the maximum number of negative eigenvalues such a matrix can have, as well as obtaining alternative characterizations of such matrices.  相似文献   

10.
Let F be a field and let A and n × n matrices over F. We study some properties of A' + B' and A'B', when A' and B' run over the sets of the matrices similar to A and B, respectively.  相似文献   

11.
We are concerned with the behavior of the minimum (maximum) eigenvalue λ0(n) (λn(n)) of an (n + 1) × (n + 1) Hermitian Toeplitz matrix Tn(ƒ) where ƒ is an integrable real-valued function. Kac, Murdoch, and Szegö, Widom, Parter, and R. H. Chan obtained that λ0(n) — min ƒ = O(1/n2k) in the case where ƒ C2k, at least locally, and ƒ — inf ƒ has a zero of order 2k. We obtain the same result under the second hypothesis alone. Moreover we develop a new tool in order to estimate the extreme eigenvalues of the mentioned matrices, proving that the rate of convergence of λ0(n) to inf ƒ depends only on the order ρ (not necessarily even or integer or finite) of the zero of ƒ — inf ƒ. With the help of this tool, we derive an absolute lower bound for the minimal eigenvalues of Toeplitz matrices generated by nonnegative L1 functions and also an upper bound for the associated Euclidean condition numbers. Finally, these results are extended to the case of Hermitian block Toeplitz matrices with Toeplitz blocks generated by a bivariate integrable function ƒ.  相似文献   

12.
A pair of m×n matrices (A,B) is called rank-sum-maximal if rank(A+B)=rank(A)+rank(B), and rank-sum-minimal if rank(A+B)=|rank(A)−rank(B)|. We characterize the linear operators that preserve the set of rank-sum-minimal matrix pairs, and the linear operators that preserve the set of rank-sum-maximal matrix pairs over any field with at least min(m,n)+2 elements and of characteristic not 2.  相似文献   

13.
We construct the polynomial pm,n* of degree m which interpolates a given real-valued function f L2[a, b] at pre-assigned n distinct nodes and is the best approximant to f in the L2-sense over all polynomials of degree m with the same interpolatory character. It is shown that the L2-error pm,n*f → 0 as m → ∞ if f C[a, b].  相似文献   

14.
For an m × n matrix A over a field F we consider the following quantities: μ(A), the maximum multiplicity of a field element as a component of a nonzero vector in the range of A, and δ(A), the minimum number of distinct entries in a nonzero vector in the range of A. In terms of ramk(A), we describe the set of possible values of μand δ and discuss the possible relations between them. We also develop a general affine geometric structure in which the sets of values of μ and δ may be characterized linear algebraically.  相似文献   

15.
Let G be a simple graph on n vertices and let L=L(G) be the Laplacian matrix of G corresponding to some ordering of the vertices. It is known that λ≤n for any eigenvalue λ of L. In this note we characterize when n is an eigenvalue of L with multiplicity m.  相似文献   

16.
The thermal equilibrium state of two oppositely charged gases confined to a bounded domain , m = 1,2 or m = 3, is entirely described by the gases' particle densities p, n minimizing the total energy (p, n). it is shown that for given P, N > 0 the energy functional admits a unique minimizer in {(p, n) ε L2(Ω) x L 2(Ω) : p, n ≥ 0, Ωp = P, Ωn = N} and that p, n ε C(Ω) ∩ L(Ω).

The analysis is applied to the hydrodynamic semiconductor device equations. These equations in general possess more than one thermal equilibrium solution, but only the unique solution of the corresponding variational problem minimizes the total energy. It is equivalent to prescribe boundary data for electrostatic potential and particle densities satisfying the usual compatibility relations and to prescribe Ve and P, N for the variational problem.  相似文献   


17.
LetA(x) be a differentiable family of k × k symmetric matrices where x runs through a domain D in RnWe prove that if λ is a continuous function onDsuch that, for every x εD,λ(x) is a characteristic root of A(x) of constant multiplicity m, then λ is a differentiable function and there exists, locally, a differentiable family of ortho-normal bases for the eigenspace. The case n = 1 has been known in the standard treatises on the perturbation theory for linear operators.  相似文献   

18.
For a 1-dependent stationary sequence {Xn} we first show that if u satisfies p1=p1(u)=P(X1>u)0.025 and n>3 is such that 88np131, then
P{max(X1,…,Xn)u}=ν·μn+O{p13(88n(1+124np13)+561)}, n>3,
where
ν=1−p2+2p3−3p4+p12+6p22−6p1p2,μ=(1+p1p2+p3p4+2p12+3p22−5p1p2)−1
with
pk=pk(u)=P{min(X1,…,Xk)>u}, k1
and
|O(x)||x|.
From this result we deduce, for a stationary T-dependent process with a.s. continuous path {Ys}, a similar, in terms of P{max0skTYs<u}, k=1,2 formula for P{max0stYsu}, t>3T and apply this formula to the process Ys=W(s+1)−W(s), s0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities.  相似文献   

19.
Let F be an algebraically closed field. We denote by i(A) the number of invariant polynomials of a square matrix A, which are different from 1. For A,B any n×n matrices over F, we calculate the maximum of i(XAX-1+B), where X runs over the set of all non-singular n×n matrices over F.  相似文献   

20.
We present an extension of the Combination Lemma of Guibas et al. (1983) that expresses the complexity of one or several faces in the overlay of many arrangements (as opposed to just two arrangements in (Guibas et al. 1989)), as a function of the number of arrangements, the number of faces, and the complexities of these faces in the separate arrangements. Several applications of the new Combination Lemma are presented. We first show that the complexity of a single face in an arrangement of k simple polygons with a total of n sides is Θ(n(k)), where (·) is the inverse of Ackermann's function. We also give a new and simpler proof of the bound on the total number of edges of m faces in an arrangement of n Jordan arcs, each pair of which intersect in at most s points, where λs(n) is the maximum length of a Davenport–Schinzel sequence of order s with n symbols. We extend this result, showing that the total number of edges of m faces in a sparse arrangement of n Jordan arcs is , where w is the total complexity of the arrangement. Several other related results are also obtained.  相似文献   

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