共查询到20条相似文献,搜索用时 31 毫秒
1.
John Lagnese 《Journal of Functional Analysis》1973,13(3):302-316
Let V?, W?, W and X be Hilbert spaces (0 < ? ? 1) with V? ? W? ? W ? X algebraically and topologically, each space being dense in the one that follows it. For each t? [0, T] let a?(t; u, v), b?(t; u, v) and b(t; u, v) be continuous sesqui-linear forms on V?, W? and W, respectively, which satisfy certain ellipticity conditions. Consider the two equations a?(t; u?, v) + b?(t; u?, v) = 〈f?, v〉 (v?V?) and (u′, v)x + b(t; u, v) = 〈f, v〉 (v?W). Estimates are obtained on the rate of convergence of u? to u, assuming a?(t; u, v) → (u, v)x and b?(t; u, v) → b(t; u, v) in an appropriate sense. These results are then applied to singular perturbation of a class of parabolic boundary value problems. 相似文献
2.
P.D. Johnson Jr. 《Discrete Mathematics》2009,309(14):4746-4749
Let χf denote the fractional chromatic number and ρ the Hall ratio, and let the lexicographic product of G and H be denoted GlexH. Main results: (i) ρ(GlexH)≤χf(G)ρ(H); (ii) if ρ(G)=χf(G) then ρ(GlexH)=ρ(G)ρ(H) for all H; (iii) χf−ρ is unbounded. In addition, the question of how big χf/ρ can be is discussed. 相似文献
3.
Oscillation criteria for the class of forced functional differential inequalities x(t){Lnx(t) + f(t, x(t), x[g1(t)],…, x[gm(t)]) ? h(t)} ? 0, for n even, and x(t){Lnx(t) ? f(t, x(t), x[g1(t)],…, x[gm(t)]) ? h(t)} ? 0, for n odd, are established. 相似文献
4.
Changqing Xu Xianli Ma Shouliang Hua 《Journal of Applied Mathematics and Computing》2009,31(1-2):45-50
Let G=(V(G),E(G)) be a simple graph. Given non-negative integers r,s, and t, an [r,s,t]-coloring of G is a mapping c from V(G)∪E(G) to the color set {0,1,…,k?1} such that |c(v i )?c(v j )|≥r for every two adjacent vertices v i ,v j , |c(e i )?c(e j )|≥s for every two adjacent edges e i ,e j , and |c(v i )?c(e j )|≥t for all pairs of incident vertices and edges, respectively. The [r,s,t]-chromatic number χ r,s,t (G) of G is defined to be the minimum k such that G admits an [r,s,t]-coloring. We determine χ r,s,t (K n,n ) in all cases. 相似文献
5.
The Dirichlet problem for the region of the plane inside closed smooth curve C for second-order elliptic equations is considered. It is shown that under certain circumstances the solution u can be written uniquely in the form u(P) = ∝cF(P, Q) g(Q) dsQ, where F(P, Q) is the fundamental solution of the elliptic equation, and g?L2 if the boundary value function f is absolutely continuous with square integrable derivative (f?W); and u(P) = p(F(P, ·)) where p is a unique bounded linear functional on W if f?L2. These representations are valid in the exterior of C also. As special cases with slight modifications, the exterior Dirichlet problems for the Helmholtz and Laplace equations are mentioned.It is shown also that if kernel F(P′, Q), with P′ and Q on C, has a complete set of eigenfunctions {ψk(P′)} then u(P) can be expanded in a series of their extensions {ψk(P)}, where ψk(P) = λk ∝cF(P, Q) ψk(Q) dsQ. 相似文献
6.
R.E White 《Journal of Mathematical Analysis and Applications》1979,68(1):157-170
We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)?1?Lq((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of Lp(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x0)) u(x0), (Fu)(x) = ∫xLf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L2(0, L) function. 相似文献
7.
Let R be a noncommutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, and L a noncentral Lie ideal of R. If F and G are generalized derivations of R and k ≥1 a fixed integer such that [F(x), x] k x ? x[G(x), x] k = 0 for any x ∈ L, then one of the following holds:
- either there exists an a ∈ U and an α ∈ C such that F(x) = xa and G(x) = (a + α)x for all x ∈ R
- or R satisfies the standard identity s 4(x 1, …, x 4) and one of the following conclusions occurs
- there exist a, b, c, q ∈ U, such that a ?b + c ?q ∈ C and F(x) = ax + xb, G(x) = cx + xq for all x ∈ R
- there exist a, b, c ∈ U and a derivation d of U such that F(x) = ax+d(x) andG(x) = bx+xc?d(x) for all x ∈ R, with a + b ? c ∈ C.
8.
Daniel A. Klain 《Advances in Mathematics》2010,224(2):601-4601
For n?2 a construction is given for convex bodies K and L in Rn such that the orthogonal projection Lu onto the subspace u⊥ contains a translate of Ku for every direction u, while the volumes of K and L satisfy Vn(K)>Vn(L).A more general construction is then given for n-dimensional convex bodies K and L such that each orthogonal projection Lξ onto a k-dimensional subspace ξ contains a translate of Kξ, while the mth intrinsic volumes of K and L satisfy Vm(K)>Vm(L) for all m>k.For each k=1,…,n, we then define the collection Cn,k to be the closure (under the Hausdorff topology) of all Blaschke combinations of suitably defined cylinder sets (prisms).It is subsequently shown that, if L∈Cn,k, and if the orthogonal projection Lξ contains a translate of Kξ for every k-dimensional subspace ξ of Rn, then Vn(K)?Vn(L).The families Cn,k, called k-cylinder bodies of Rn, form a strictly increasing chain
Cn,1⊂Cn,2⊂?⊂Cn,n−1⊂Cn,n, 相似文献
9.
F.K. Hwang 《Discrete Mathematics》1980,32(2):163-165
Let Tn denote a binary tree with n terminal nodes V={υ1,…,υn} and let li denote the path length from the root to υi. Consider a set of nonnegative numbers W={w1,…,wn} and for a permutation π of {1,…,n} to {1,…,n}, associate the weight wi to the node υπ(i). The cost of Tn is defined as C(Tn∣W)=Minπ∑ni=1wilπ(i).A Huffman tree Hn is a binary tree which minimizes C(Tn∣W) over all possible Tn. In this note, we give an explicit expression for C(Hn∣W) when W assumes the form: wi=k for i=1,…,n?m; wi=x for i=n?m+1,…,n. This simplifies and generalizes earlier results in the literature. 相似文献
10.
Y. Stein 《Israel Journal of Mathematics》1995,89(1-3):301-319
LetP=x n +P n?1(y)x n?1+…+P 0(y),Q=x m +Q m?2(y)x m?2+…+Q 0(y) belong toK[x, y], whereK is a field of characteristic zero. The main result of this paper is the following: Assume thatP x Q y ?P y Q x =1. Then:*
- K[Q m?2(y), …,Q 0(y)]=K[y],
- K[P, Q]=K[x, y] ifQ=x m +Q k (y)x k +Q r (y)x r
11.
R. Balakrishnan 《Discrete Mathematics》2008,308(12):2607-2610
In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), which is called the Mycielskian of G. This paper investigates the vertex-connectivity κ(μ(G)) and edge-connectivity κ′(μ(G)) of μ(G) . We show that κ(μ(G))=min{δ(μ(G)),2κ(G)+1} and κ′(μ(G))=δ(μ(G)). 相似文献
12.
Alan Dow 《Topology and its Applications》2010,157(8):1379-1857
We consider generalizations of a well-known class of spaces, called by S. Mrówka, N∪R, where R is an infinite maximal almost disjoint family (MADF) of countable subsets of the natural numbers N. We denote these generalizations by ψ=ψ(κ,R) for κ?ω. Mrówka proved the interesting theorem that there exists an R such that |βψ(ω,R)?ψ(ω,R)|=1. In other words there is a unique free z-ultrafilter p0 on the space ψ. We extend this result of Mrówka to uncountable cardinals. We show that for κ?c, Mrówka's MADF R can be used to produce a MADF M⊂ω[κ] such that |βψ(κ,M)?ψ(κ,M)|=1. For κ>c, and every M⊂ω[κ], it is always the case that |βψ(κ,M)?ψ(κ,M)|≠1, yet there exists a special free z-ultrafilter p on ψ(κ,M) retaining some of the properties of p0. In particular both p and p0 have a clopen local base in βψ (although βψ(κ,M) need not be zero-dimensional). A result for κ>c, that does not apply to p0, is that for certain κ>c, p is a P-point in βψ. 相似文献
13.
W.Y. Chan 《Journal of Computational and Applied Mathematics》2011,235(13):3831-3840
For the problem given by uτ=(ξrumuξ)ξ/ξr+f(u) for 0<ξ<a, 0<τ<Λ≤∞, u(ξ,0)=u0(ξ) for 0≤ξ≤a, and u(0,τ)=0=u(a,τ) for 0<τ<Λ, where a and m are positive constants, r is a constant less than 1, f(u) is a positive function such that limu→c−f(u)=∞ for some positive constant c, and u0(ξ) is a given function satisfying u0(0)=0=u0(a), this paper studies quenching of the solution u. 相似文献
14.
Let G be a graph. If u,v∈V(G), a u-vshortest path of G is a path linking u and v with minimum number of edges. The closed interval I[u,v] consists of all vertices lying in some u-v shortest path of G. For S⊆V(G), the set I[S] is the union of all sets I[u,v] for u,v∈S. We say that S is a convex set if I[S]=S. The convex hull of S, denoted Ih[S], is the smallest convex set containing S. A set S is a hull set of G if Ih[S]=V(G). The cardinality of a minimum hull set of G is the hull number of G, denoted by hn(G). In this work we prove that deciding whether hn(G)≤k is NP-complete.We also present polynomial-time algorithms for computing hn(G) when G is a unit interval graph, a cograph or a split graph. 相似文献
15.
Hui-Hsiung Kuo 《Journal of Functional Analysis》1976,21(1):63-75
Some parallel results of Gross' paper (Potential theory on Hilbert space, J. Functional Analysis1 (1967), 123–181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator is defined by f(x) = ?lim∈←0{E[f((τ∈ξ))] ? f(x)}/E[τ∈ξ, where τx? is the first exit time of U(t) starting at x from the ball of radius ? with center x. It is shown that f(x) = ?trace D2f(x)+〈Df(x),x〉 for a large class of functions f. Let rt(x, dy) be the transition probabilities of U(t). The λ-potential Gλf, λ > 0, and normalized potential Rf of f are defined by Gλf(X) = ∫0∞e?λtrtf(x) dt and Rf(x) = ∫0∞ [rtf(x) ? rtf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D2Gλf(x) ? 〈DGλf(x), x〉 = ?f(x) + λGλf(x) and trace D2Rf(x) ? 〈DRf(x), x〉 = ?f(x) + ∫Bf(y)p1(dy), where p1 is the Wiener measure in B with parameter 1. Some approximation theorems are also proved. 相似文献
16.
《Discrete Mathematics》2002,231(1-3):319-324
A graph G is called n-factor-critical if the removal of every set of n vertices results in a~graph with a~1-factor. We prove the following theorem: Let G be a~graph and let x be a~locally n-connected vertex. Let {u,v} be a~pair of vertices in V(G)−{x} such that uv∉E(G), x∈NG(u)∩NG(v), and NG(x)⊂NG(u)∪NG(v)∪{u,v}. Then G is n-factor-critical if and only if G+uv is n-factor-critical. 相似文献
17.
In this paper we introduce a generalized vector-valued paranormed sequence space Np(Ek,Δm,f,s) using modulus function f, where p=(pk) is a bounded sequence of positive real numbers such that infkpk>0,(Ek,qk) is a sequence of seminormed spaces with Ek+1⊆Ek for each k ∈ N and s?0. We have also studied sequence space Np(Ek,Δm,fr,s), where fr=f°f°f°,…,f (r-times composition of f with itself) and r∈N={1,2,3,…}. Results regarding completeness, K-space, normality, inclusion relations etc. are derived. Further, a study of multiplier of the set Np(Ek,f,s) is also made by choosing (Ek,‖·‖k) as sequence of normed algebras. 相似文献
18.
Chi-Kwong Li 《Journal of Mathematical Analysis and Applications》2008,348(2):843-855
For a positive integer k, the rank-k numerical range Λk(A) of an operator A acting on a Hilbert space H of dimension at least k is the set of scalars λ such that PAP=λP for some rank k orthogonal projection P. In this paper, a close connection between low rank perturbation of an operator A and Λk(A) is established. In particular, for 1?r<k it is shown that Λk(A)⊆Λk−r(A+F) for any operator F with rank(F)?r. In quantum computing, this result implies that a quantum channel with a k-dimensional error correcting code under a perturbation of rank at most r will still have a (k−r)-dimensional error correcting code. Moreover, it is shown that if A is normal or if the dimension of A is finite, then Λk(A) can be obtained as the intersection of Λk−r(A+F) for a collection of rank r operators F. Examples are given to show that the result fails if A is a general operator. The closure and the interior of the convex set Λk(A) are completely determined. Analogous results are obtained for Λ∞(A) defined as the set of scalars λ such that PAP=λP for an infinite rank orthogonal projection P. It is shown that Λ∞(A) is the intersection of all Λk(A) for k=1,2,…. If A−μI is not compact for all μ∈C, then the closure and the interior of Λ∞(A) coincide with those of the essential numerical range of A. The situation for the special case when A−μI is compact for some μ∈C is also studied. 相似文献
19.
Fan [G. Fan, Distribution of cycle lengths in graphs, J. Combin. Theory Ser. B 84 (2002) 187-202] proved that if G is a graph with minimum degree δ(G)≥3k for any positive integer k, then G contains k+1 cycles C0,C1,…,Ck such that k+1<|E(C0)|<|E(C1)|<?<|E(Ck)|, |E(Ci)−E(Ci−1)|=2, 1≤i≤k−1, and 1≤|E(Ck)|−|E(Ck−1)|≤2, and furthermore, if δ(G)≥3k+1, then |E(Ck)|−|E(Ck−1)|=2. In this paper, we generalize Fan’s result, and show that if we let G be a graph with minimum degree δ(G)≥3, for any positive integer k (if k≥2, then δ(G)≥4), if dG(u)+dG(v)≥6k−1 for every pair of adjacent vertices u,v∈V(G), then G contains k+1 cycles C0,C1,…,Ck such that k+1<|E(C0)|<|E(C1)|<?<|E(Ck)|, |E(Ci)−E(Ci−1)|=2, 1≤i≤k−1, and 1≤|E(Ck)|−|E(Ck−1)|≤2, and furthermore, if dG(u)+dG(v)≥6k+1, then |E(Ck)|−|E(Ck−1)|=2. 相似文献
20.
Let D be a directed graph; the (l,ω)-Independence Number of graph D, denoted by αl,ω(D), is an important performance parameter for interconnection networks. De Bruijn networks and Kautz networks, denoted by B(d,n) and K(d,n) respectively, are versatile and efficient topological structures of interconnection networks. For l=1,2,…,n, this paper shows that αl,d−1(B(d,n))=dn,αl,d−1(K(d,n))=αl,d(K(d,n))=dn+dn−1 if d≥3 and n≤d−2. In particular, the paper shows the exact value of the Independence Number for B(d,1) and B(d,2) for any d. For the generalized situation, the paper obtains a lower bound αl,d−1(B(d,n))≥d2 if n≥3 and d≥5. 相似文献