Suppose G is a graph and T is a set of non-negative integers that contains 0. A T-coloring of G is an assignment of a non-negative integer f(x) to each vertex x of G such that |f(x)−f(y)|∉T whenever xy∈E(G). The edge span of a T-coloring−f is the maximum value of |f(x) f(y)| over all edges xy, and the T-edge span of a graph G is the minimum value of the edge span of a T-coloring of G. This paper studies the T-edge span of the dth power Cdn of the n-cycle Cn for T={0, 1, 2, …, k−1}. In particular, we find the exact value of the T-edge span of Cnd for n≡0 or (mod d+1), and lower and upper bounds for other cases.
Received: May 13, 1996 Revised: December 8, 1997 相似文献
For x and y vertices of a connected graph G, let TG(x, y) denote the expected time before a random walk starting from x reaches y. We determine, for each n > 0, the n-vertex graph G and vertices x and y for which TG(x, y) is maximized. the extremal graph consists of a clique on ?(2n + 1)/3?) (or ?)(2n ? 2)/3?) vertices, including x, to which a path on the remaining vertices, ending in y, has been attached; the expected time TG(x, y) to reach y from x in this graph is approximately 4n3/27. 相似文献
In this note, we present a simple method for constructing maximal curves defined over by the equation yn=T(x), where T(x) is a q-polynomial over . 相似文献
Let G = (V, E) be an interval graph with n vertices and m edges. A positive integer R(x) is associated with every vertex x ? V{x\in V}. In the conditional covering problem, a vertex x ? V{x \in V} covers a vertex y ? V{y \in V} (x ≠ y) if d(x, y) ≤ R(x) where d(x, y) is the shortest distance between the vertices x and y. The conditional covering problem (CCP) finds a minimum cardinality vertex set C í V{C\subseteq V} so as to cover all the vertices of the graph and every vertex in C is also covered by another vertex of C. This problem is NP-complete for general graphs. In this paper, we propose an efficient algorithm to solve the CCP with nonuniform
coverage radius in O(n2) time, when G is an interval graph containing n vertices. 相似文献
It is known that if a smooth function h in two real variables x and y belongs to the class Σn of all sums of the form Σnk=1ƒk(x) gk(y), then its (n + 1)th order "Wronskian" det[hxiyj]ni,j=0 is identically equal to zero. The present paper deals with the approximation problem h(x, y) Σnk=1ƒk(x) gk(y) with a prescribed n, for general smooth functions h not lying in Σn. Two natural approximation sums T=T(h) Σn, S=S(h) Σn are introduced and the errors |h-T|, |h-S| are estimated by means of the above mentioned Wronskian of the function h. The proofs utilize the technique of ordinary linear differential equations. 相似文献
Let T be a continuous map of the space of complex n×n matrices into itself satisfying T(0)=0 such that the spectrum of T(x)-T(y) is always a subset of the spectrum of x-y. There exists then an invertible n×n matrix u such that either T(a)=uau-1 for all a or T(a)=uatu-1 for all a. We arrive at the same conclusion by supposing that the spectrum of x-y is always a subset of the spectrum of T(x)-Tt(y), without the continuity assumption on T. 相似文献
Let C be a closed, convex subset of a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and let T be an asymptotically nonexpansive mapping from C into itself such that the set F (T) of fixed points of T is nonempty. Let {an} be a sequence of real numbers with 0 £ an £ 10 \leq a_n \leq 1, and let x and x0 be elements of C. In this paper, we study the convergence of the sequence {xn} defined by¶¶xn+1=anx + (1-an) [1/(n+1)] ?j=0nTjxn x_{n+1}=a_n x + (1-a_n) {1\over n+1} \sum\limits_{j=0}^n T^j x_n\quad for n=0,1,2,... . n=0,1,2,\dots \,. 相似文献
Let f ? C(\Bbb Rn,\Bbb Rn) f\in C(\Bbb R^n,\Bbb R^n) be quasimonotone increasing such that Y(f(y)-f(x)) £ -c Y(y-x) (x << y) \Psi (f(y)-f(x)) \!\le -c \Psi (y-x) (x\ll y) for a linear and strictly positive functional Y \Psi and c > 0. We prove that f is a homeomorphism with decreasing and Lipschitz continuous inverse and we prove the global asymptotic stability of the equilibrium solution of x¢=f(x) x'=f(x) . 相似文献
LetT(λ) be a bounded linear operator in a Banach spaceX for eachλ in the scalar fieldS. The characteristic value-vector problemT(λ)x = 0 with a normalization conditionφ x = 1, whereφ ε X*, is formulated as a nonlinear problem inX xS:P(y) ≡ (T(λ)x, φ x - 1) = 0,y= (X, A). Newton's method and the Kantorovič theorem are applied. For this purpose, representations and criteria for existence
ofP′(y)−1 are obtained. The continuous dependence onT of characteristic values and vectors is investigated. A numerical example withT(λ) =A +λB +λ2C is presented.
Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No.: DA-31-124-ARO-D-462. 相似文献
We show that the unsolvability of the Diophantine equation
axn + byn = zn\alpha x^n + \beta y^n = z^n
is equivalent to a good uniform distribution of the set
{ n ?{axn + byn} }\{ \root n \of{\alpha x^n + \beta y^n} \}
. The proof depends on the asymptotic evaluation of the Gauss sum
?x, ye (n ?{axn + byn})\sum_{x, y} e (\root n \of{\alpha x^n + \beta y^n})
. 相似文献
We study the Tikhonov regularization for perturbed inclusions of the form T(x) ' y*{T(x) \ni y^*} where T is a set-valued mapping defined on a Banach space that enjoys metric regularity properties and y* is an element near 0. We investigate the case when T is metrically regular and strongly regular and we show the existence of both a solution x* to the perturbed inclusion and a Tikhonov sequence which converges to x*. Finally, we show that the Tikhonov sequences associated to the perturbed problem inherit the regularity properties of the
inverse of T. 相似文献
A planar mapping was derived from a second order delay differential equation with a piecewise constant argument. Invariant curves for the planar mapping reflects on the dynamics of the differential equation. Results were reported on a planar mapping admitting quadratic invariant curves y=x2+C, except for the case -3/4≥C≤0. This remaining case is now resolved, and we describe the solutions of the functional equation K(x2+C)+k(x)=x by iterations of y. 相似文献
We extend an interesting theorem of Yuan [12] for two quadratic forms to three matrices. Let C1, C2, C3 be three symmetric matrices in ℜn×n, if max{xTC1x,xTC2x,xTC3x}≥0 for all x∈ℜn, it is proved that there exist ti≥0 (i=1,2,3) such that ∑i=13ti=1 and ∑i=13tiCi has at most one negative eigenvalue.
Received February 18, 1997 / Revised version received October 1, 1997? Published online June 11, 1999 相似文献
ABSTRACTLet n≥1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the extended centroid, and L a non-central Lie ideal of R. If F is a generalized skew derivation of R such that (F(x)F(y)?yx)n = 0 for all x,y∈L, then char(R) = 2 and R?M2(C), the ring of 2×2 matrices over C. 相似文献
Let (m, n) ∈ ℕ2, Ω an open bounded domain in ℝm, Y = [0, 1]m; uε in (L2(Ω))n which is two-scale converges to some u in (L2(Ω × Y))n. Let φ: Ω × ℝm × ℝn → ℝ such that: φ(x, ·, ·) is continuous a.e. x ∈ Ω φ(·, y, z) is measurable for all (y, z) in ℝm × ℝn, φ(x, ·, z) is 1-periodic in y, φ(x, y, ·) is convex in z. Assume that there exist a constant C1 > 0 and a function C2 ∈ L2(Ω) such that
Summary Let Abe a semisimple H*-algebra and let T: A→Abe an additive mapping such that T(xn+1)<span lang=EN-US style='font-size:10.0pt;mso-ansi-language:EN-US'>=T(x)xn+xnT(x) holds for all x∈Aand some integer n≥1. In this case Tis a left and a right centralizer. 相似文献
We characterize locally convex topological algebrasA satisfying: a sequence (xn) inA converges to 0 if, and only if, (xn2) converges to 0. We also show that a real Banach algebra such thatxn2+yn2→0 if, and only if,xn → 0 andyn → 0, for every sequences (xn) and (yn) inA, is isomorphic to, whereX is a compact space.
相似文献