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.
Vijay Kumar Bhat 《Czechoslovak Mathematical Journal》2013,63(4):1049-1056
Let R be a ring. We recall that R is called a near pseudo-valuation ring if every minimal prime ideal of R is strongly prime. Let now σ be an automorphism of R and δ a σ-derivation of R. Then R is said to be an almost δ-divided ring if every minimal prime ideal of R is δ-divided. Let R be a Noetherian ring which is also an algebra over ? (? is the field of rational numbers). Let σ be an automorphism of R such that R is a σ(*)-ring and δ a σ-derivation of R such that σ(δ(a)) = δ(σ(a)) for all a ∈ R. Further, if for any strongly prime ideal U of R with σ(U) = U and δ(U) ? δ, U[x; σ, δ] is a strongly prime ideal of R[x; σ, δ], then we prove the following:
- R is a near pseudo valuation ring if and only if the Ore extension R[x; σ, δ] is a near pseudo valuation ring.
- R is an almost δ-divided ring if and only if R[x; σ, δ] is an almost δ-divided ring.
3.
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
4.
Put Zn = {1, 2,…, n} and let π denote an arbitrary permutation of Zn. Problem I. Let π = (π(1), π(2), …, π(n)). π has an up, down, or fixed point at a according as a < π(a), a > π(a), or a = π(a). Let be the number of π ∈ Zn with r ups, s downs, and t fixed points. Problem II. Consider the triple π?1(a), a, π(a). Let R denote an up and F a down of π and let B(n, r, s) denote the number of π ∈ Zn with r occurrences of π?1(a)RaRπ(a) and s occurrences of π?1(a)FaFπ(a). Generating functions are obtained for each enumerant as well as for a refinement of the second. In each case use is made of the cycle structure of permutations. 相似文献
5.
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
6.
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
7.
Xiaofei Song 《Linear algebra and its applications》2008,429(7):1579-1586
Let Mn be the algebra of all n×n complex matrices and Γn the set of all k-potent matrices in Mn. Suppose ?:Mn→Mn is a map satisfying A-λB∈Γn implies ?(A)-λ?(B)∈Γn, where A, B∈Mn, λ∈C. Then either ? is of the form ?(A)=cTAT-1, A∈Mn, or ? is of the form ?(A)=cTAtT-1, A∈Mn, where T∈Mn is an invertible matrix, c∈C satisfies ck=c. 相似文献
8.
For a connected graph G = (V, E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A set S of vertices of G is a monophonic set of G if each vertex v of G lies on an x ? y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is defined as the monophonic number of G, denoted by m(G). A connected monophonic set of G is a monophonic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected monophonic set of G is the connected monophonic number of G and is denoted by m c (G). We determine bounds for it and characterize graphs which realize these bounds. For any two vertices u and v in G, the monophonic distance d m (u, v) from u to v is defined as the length of a longest u ? v monophonic path in G. The monophonic eccentricity e m (v) of a vertex v in G is the maximum monophonic distance from v to a vertex of G. The monophonic radius rad m G of G is the minimum monophonic eccentricity among the vertices of G, while the monophonic diameter diam m G of G is the maximum monophonic eccentricity among the vertices of G. It is shown that for positive integers r, d and n ≥ 5 with r < d, there exists a connected graph G with rad m G = r, diam m G = d and m c (G) = n. Also, if a,b and p are positive integers such that 2 ≤ a < b ≤ p, then there exists a connected graph G of order p, m(G) = a and m c (G) = b. 相似文献
9.
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
10.
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
11.
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
12.
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
13.
Let Mm,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of Mm,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PAtQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k2. Here the weight of a Boolean matrix is the number of its nonzero entries. 相似文献
14.
C. Balbuena 《Discrete Mathematics》2008,308(16):3526-3536
For a connected graph G, the rth extraconnectivity κr(G) is defined as the minimum cardinality of a cutset X such that all remaining components after the deletion of the vertices of X have at least r+1 vertices. The standard connectivity and superconnectivity correspond to κ0(G) and κ1(G), respectively. The minimum r-tree degree of G, denoted by ξr(G), is the minimum cardinality of N(T) taken over all trees T⊆G of order |V(T)|=r+1, N(T) being the set of vertices not in T that are neighbors of some vertex of T. When r=1, any such considered tree is just an edge of G. Then, ξ1(G) is equal to the so-called minimum edge-degree of G, defined as ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}, where d(u) stands for the degree of vertex u. A graph G is said to be optimally r-extraconnected, for short κr-optimal, if κr(G)?ξr(G). In this paper, we present some sufficient conditions that guarantee κr(G)?ξr(G) for r?2. These results improve some previous related ones, and can be seen as a complement of some others which were obtained by the authors for r=1. 相似文献
15.
We prove that the Nielsen fixed point number N(φ) of an n-valued map φ:X?X of a compact connected triangulated orientable q-manifold without boundary is equal to the Nielsen coincidence number of the projections of the graph of φ, a subset of X×X, to the two factors. For certain q×q integer matrices A, there exist “linear” n-valued maps Φn,A,σ:Tq?Tq of q-tori that generalize the single-valued maps fA:Tq→Tq induced by the linear transformations TA:Rq→Rq defined by TA(v)=Av. By calculating the Nielsen coincidence number of the projections of its graph, we calculate N(Φn,A,σ) for a large class of linear n-valued maps. 相似文献
16.
Given a graph G, a proper labelingf of G is a one-to-one function from V(G) onto {1,2,…,|V(G)|}. For a proper labeling f of G, the profile widthwf(v) of a vertex v is the minimum value of f(v)−f(x), where x belongs to the closed neighborhood of v. The profile of a proper labelingfofG, denoted by Pf(G), is the sum of all the wf(v), where v∈V(G). The profile ofG is the minimum value of Pf(G), where f runs over all proper labeling of G. In this paper, we show that if the vertices of a graph G can be ordered to satisfy a special neighborhood property, then so can the graph G×Qn. This can be used to determine the profile of Qn and Km×Qn. 相似文献
17.
A k-containerC(u,v) of G between u and v is a set of k internally disjoint paths between u and v. A k-container C(u,v) of G is a k*-container if it contains all vertices of G. A graph G is k*-connected if there exists a k*-container between any two distinct vertices. The spanning connectivity of G, κ*(G), is defined to be the largest integer k such that G is w*-connected for all 1?w?k if G is a 1*-connected graph. In this paper, we prove that κ*(G)?2δ(G)-n(G)+2 if (n(G)/2)+1?δ(G)?n(G)-2. Furthermore, we prove that κ*(G-T)?2δ(G)-n(G)+2-|T| if T is a vertex subset with |T|?2δ(G)-n(G)-1. 相似文献
18.
《Finite Fields and Their Applications》2001,7(1):92-109
Let q=pu>1 be a power of a prime p, and let kq be an overfield of GF(q). Let m>0 be an integer, let J* be a subset of {1,…,m}, and let E*m, q(Y)=Yqm+∑j∈J*XjYqm−j where the Xj are indeterminates. Let J3 be the set of all m−ν where ν is either 0 or a divisor of m different from m. Let s(T)=∑0≤i≤nsiTi be an irreducible polynomial of degree n>0 in T with coefficients si in GF(q). Let E*[s]m, q(Y) be the generalized sth iterate of E*m, q(Y); i.e., E*[s]m, q(Y)=∑0≤i≤nsiE*[i]m, q(Y), where E*[i]m, q(Y), is the ordinary ith iterate. We prove that if J3⊂J*, m is square-free, and GCD(m,n)=1=GCD(mnu,2p), then Gal(E*[s]m, q,kq({Xj:j∈j*})=GL(m, qn). The proof is based on CT (=the Classification Theorem of Finite Simple Groups) in its incarnation as CPT (=the Classification of Projectively Transitive Permutation Groups, i.e., subgroups of GL acting transitively on nonzero vectors). 相似文献
19.
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
20.
《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