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1.
We prove that for λ ≥ 0, p ≥ 3, there exists an open ball B L2(0,1) such that the problem
− (|u′|p−2 u′)′ − λ|u|p−2u = f, in (0,1)
, subject to certain separated boundary conditions on (0,1), has a solution for f B.  相似文献   

2.
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   

3.
Let Q(x) be a nonnegative definite, symmetric matrix such that (Q(x))(1/2) is Lipschitz continuous. Given a real-valued function b(x) and a weak solution u(x) of div(Q▽u) = b, we find sufficient conditions in order that Q(1/2)▽u has some first order smoothness. Specifically, if Ω is a bounded open set in R~n, we study when the components of Q(1/2)▽u belong to the first order Sobolev space W_Q~(1,2)(Ω)defined by Sawyer and Wheeden. Alternately, we study when each of n first order Lipschitz vector field derivatives X_iu has some first order smoothness if u is a weak solution in Ω of ∑_(i=1)~n X′_iX_(iu) + b = 0.We do not assume that {X_i} is a Hormander collection of vector fields in Ω. The results signal ones for more general equations.  相似文献   

4.
An irredundant set of vertices VV in a graph G=(V,E) has the property that for every vertex uV′, N[V′−{u}] is a proper subset of N[V′]. We investigate the parameterized complexity of determining whether a graph has an irredundant set of size k, where k is the parameter. The interest of this problem is that while most “k-element vertex set” problems are NP-complete, several are known to be fixed-parameter tractable, and others are hard for various levels of the parameterized complexity hierarchy. Complexity classification of vertex set problems in this framework has proved to be both more interesting and more difficult. We prove that the k-element irredundant set problem is complete for W[1], and thus has the same parameterized complexity as the problem of determining whether a graph has a k-clique. We also show that the “parametric dual” problem of determining whether a graph has an irredundant set of size nk is fixed-parameter tractable.  相似文献   

5.
The parametric resource allocation problem asks to minimize the sum of separable single-variable convex functions containing a parameter λ, Σi = 1ni(xi + λgi(xi)), under simple constraints Σi = 1n xi = M, lixiui 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(GMlog2 n + n log n + n log(M/n)), where G = Σi = 1n ui − Σi = 1n li and an evaluation of ƒi(·) or gi(·) is assumed to be done in constant time.  相似文献   

6.
Sufficient conditions for the uniqueness of positive solutions of boundary value problems for quasilinear differential equations of the type
(|u′|m−2u′)′ + f(t,u,u′)=0, m 2
are established. These problems arise, for example, in the study of the m-Laplace equation in annular regions.  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.
In this work we present a numerical approach for finding positive solutions of the type −Δu = λf(u) for x  Ω, with Dirichlet boundary condition, where f is a superlinear function of u. We will show in which range of λ, this problem achieves multiple numerical solutions and what is the behavior of the branches of solutions.  相似文献   

9.
Let G be an infinite locally finite connected graph. We study the reconstructibility of G in relation to the structure of its end set . We prove that an infinite locally finite connected graph G is reconstructible if there exists a finite family i)0i (n2) of pairwise finitely separable subsets of such that, for all x,y,x′,yV(G) and every isomorphism f of G−{x,y} onto G−{x′,y′} there is a permutation π of {0,…,n−1} such that for 0i<n. From this theorem we deduce, as particular consequences, that G is reconstructible if it satisfies one of the following properties: (i) G contains no end-respecting subdivision of the dyadic tree and has at least two ends of maximal order; (ii) the set of thick ends or the one of thin ends of G is finite and of cardinality greater than one. We also prove that if almost all vertices of G are cutvertices, then G is reconstructible if it contains a free end or if it has at least a vertex which is not a cutvertex.  相似文献   

10.
The usual construction of (v,q+1,1)−BIBD's from vector spaces over GF(q) is generalized to the class of near vector spaces over GF(q). It is shown that every (v,q+1,1)−BIBD can be constructed from a near vector space over GF(q). Some corollaries are: Given a (v1,q+1,1)−BIBD P1,B1 and a (v2,q+1,1)−BIBD P2,B2, there is a ((q−1)v1v2+v1+v2,q+1,1)−BIBD P3,B3 containing P1,B1 and P2,B2 as disjoint subdesigns. If there is a (v,q+1,1)−BIBD then there is a ((q−1)v+1,q,1)−BIBD. Every finite partial (v,q,1)−BIBD can be embedded in a finite (v′,q+1,1)−BIBD.  相似文献   

11.
We consider a sequence of integer-valued random variables Xn, n 1, representing a special Markov process with transition probability λn, l, satisfying Pn, l = (1 − λn, l) Pn−1, l + λn, l−1 Pn−1, l−1. Whenever the transition probability is given by λn, l = qn + βl + γ and λn, l = 1 − qnl, we can find closed forms for the distribution and the moments of the corresponding random variables, showing that they involve functions such as the q-binomial coefficients and the q-Stirling numbers. In general, it turns out that the q-notation, up to now mainly used in the theory of q-hypergeometrical series, represents a powerful tool to deal with these kinds of problems. In this context we speak therefore about q-distributions. Finally, we present some possible, mainly graph theoretical interpretations of these random variables for special choices of , β and γ.  相似文献   

12.
In this paper, we consider the second-order nonlinear differential equation
[a(t)|y′(t)|σ−1y′(t)|′+q(t)f(y(t))=r(t)
where σ > 0 is a constant, a C(R, (0, ∞)), q C(R, R), f C(R, R), xf(x) > 0, f′(x) ≥ 0 for x ≠ 0. Some new sufficient conditions for the oscillation of all solutions of (*) are obtained. Several examples which dwell upon the importance of our results are also included.  相似文献   

13.
In this paper we characterize those linear mappings from a second symmetric product space to another which preserve decomposable elements of the form λuu where u is a vector and λ is a scalar. This leads to the corresponding result concerning linear mappings from one vector space of symmetric matrices to another which preserve rank less than or equal to one. We also discuss some consequences of this characterization theorem.  相似文献   

14.
A dominating set for a graph G = (V, E) is a subset of vertices VV such that for all v ε VV′ there exists some u ε V′ for which {v, u} ε E. The domination number of G is the size of its smallest dominating set(s). For a given graph G with minimum size dominating set D, let m1 (G, D) denote the number of edges that have neither endpoint in D, and let m2 (G, D) denote the number of edges that have at least one endpoint in D. We characterize the possible values that the pair (m1 (G, D), m2 (G, D)) can attain for connected graphs having a given domination number.  相似文献   

15.
We show that there are nonisomorphic ordered sets P and Q that have the same maximal and minimal decks and a rank k such that there is a map B from the elements of rank k in P to the elements of rank k in Q such that P{x} is isomorphic to Q{B(x)} for all x of rank k in P. The examples are preceded by a criterion as to when two nonisomorphic ordered sets will have a rank k and a map B as above.  相似文献   

16.
The second-order differential equation σ(x)y″ + τ(x)y′ + λy = 0 is usually called equation of hypergeometric type, provided that σ, τ are polynomials of degree not higher than two and one, respectively, and λ is a constant. Their solutions are commonly known as hypergeometric-type functions (HTFs). In this work, a study of the spectrum of zeros of those HTFs for which , v , and σ, τ are independent of ν, is done within the so-called semiclassical (or WKB) approximation. Specifically, the semiclassical or WKB density of zeros of the HTFs is obtained analytically in a closed way in terms of the coefficients of the differential equation that they satisfy. Applications to the Gaussian and confluent hypergeometric functions as well as to Hermite functions are shown.  相似文献   

17.
In this paper we consider continuity properties of a stochastic heat equation of the form ∂u(t,x)/∂t = ∂2u(t,x)/∂x2 + f(u(t,x))Wx,t. We prove that the solutions of this equation depend continuously on the function f and give some new estimates for this connection.  相似文献   

18.
Given a graph G and a positive integer d, an L(d,1)-labeling of G is a function f that assigns to each vertex of G a non-negative integer such that if two vertices u and v are adjacent, then |f(u)−f(v)|d; if u and v are not adjacent but there is a two-edge path between them, then |f(u)−f(v)|1. The L(d,1)-number of G, λd(G), is defined as the minimum m such that there is an L(d,1)-labeling f of G with f(V){0,1,2,…,m}. Motivated by the channel assignment problem introduced by Hale (Proc. IEEE 68 (1980) 1497–1514), the L(2,1)-labeling and the L(1,1)-labeling (as d=2 and 1, respectively) have been studied extensively in the past decade. This article extends the study to all positive integers d. We prove that λd(G2+(d−1)Δ for any graph G with maximum degree Δ. Different lower and upper bounds of λd(G) for some families of graphs including trees and chordal graphs are presented. In particular, we show that the lower and the upper bounds for trees are both attainable, and the upper bound for chordal graphs can be improved for several subclasses of chordal graphs.  相似文献   

19.
Let[2+k2n(x1,x3)]u(x1,x2,x3)=−δ(x1,y1δ(x2,y2)δ(x3,y3) in R3+. Assume that u(x1,x2,x3=0,y1,y2=0,y3=0,k) is measured at the plane P {x:x3=0} for all positions of the source on the line y = (y1,y2 = 0,y3 = 0), -∞ < y1 < ∞, and receiver on the plane(x1,x2,x3 − <x1,x2 < ∞, and for low-frequencies 0 < k <k0, k0 > 0 is an arbitrary small wave number. Assume thatn(x1,x3) is an arbitrary bounded piecewise-continuous function. The basic result is: the above low-frequency surface data determinen(x1,x3)uniquely.  相似文献   

20.
Let X be a Banach space, S(X) - x ε X : #x02016; = 1 be the unit sphere of X.The parameter, modulus of W*-convexity, W*(ε) = inf <(xy)/2, fx> : x, y S(X), xy ≥ ε, fx Δx , where 0 ≤ ε ≤ 2 and Δx S(X*) be the set of norm 1 supporting functionals of S(X) at x, is investigated_ The relationship among uniform nonsquareness, uniform normal structure and the parameter W*(ε) are studied, and a known result is improved. The main result is that for a Banach space X, if there is ε, where 0 < ε < 1/2, such that W*(1 + ε) > ε/2 where W*(1 + ε) = lim→ε W* (1 + ), then X has normal structure.  相似文献   

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