Viscosity approximation methods for nonexpansive mappings and inverse-strongly monotone mappings in Hilbert spaces

Let C be a closed convex subset of a Hilbert space H. Let f is a contraction on C. Let S be a nonexpansive mapping of C into itself and A be an α-inverse-strongly monotone mapping of C into H. Assuming that F(S)∩VI(C,A)≠φ, and x ₀=x∈C, in this paper we introduce the iterative process x _n+1=α _n f(x _n )+β _n x _n +γ _n (μ Sx _n +(1?μ)(P _C (I?λ _n A)y _n )), where y _n =P _C (I?λ _n A)x _n . We prove that {x _n } and {y _n } converge strongly to the same point z∈F(S)∩VI(C,A). As its application, we give a strong convergence theorem for nonexpansive mapping and strictly pseudo-contractive mapping in a Hilbert space.     

The range of a vector measure

Let X be a completely regular Hausdorff space and E be a locally convex Hausdorff space. Then C_b(X) ? E is dense in (C_b(X, E), β₀), (C_b(X), β) ?_?E = (C_b(X) ? E, β) and (C_b(X), β₁) ?_?E = (C_b(X) ? E, β₁). For a separable space E, (C_b(X, E), β₀) is separable if and only if X is separably submetrizable. As a corollary, for a locally compact paracompact space X, if (C_b(X, E), β₀) is separable, then X is metrizable.     

An elementary operator with log-hyponormal, p-hyponormal entries

A Hilbert space operator A∈B(H) is p-hyponormal, A∈(p-H), if |A^∗|^2p?|A|^2p; an invertible operator A∈B(H) is log-hyponormal, A∈(?-H), if log(TT^∗)?log(T^∗T). Let d_AB=δ_AB or ?_AB, where δ_AB∈B(B(H)) is the generalised derivation δ_AB(X)=AX-XB and ?_AB∈B(B(H)) is the elementary operator ?_AB(X)=AXB-X. It is proved that if A,B^∗∈(?-H)∪(p-H), then, for all complex λ, , the ascent of (d_AB-λ)?1, and d_AB satisfies the range-kernel orthogonality inequality ‖X‖?‖X-(d_AB-λ)Y‖ for all X∈(d_AB-λ)^-1(0) and Y∈B(H). Furthermore, isolated points of σ(d_AB) are simple poles of the resolvent of d_AB. A version of the elementary operator E(X)=A₁XA₂-B₁XB₂ and perturbations of d_AB by quasi-nilpotent operators are considered, and Weyl’s theorem is proved for d_AB.     

Parameter Dependence of Stable Manifolds for Nonuniform (μ, ν)-dichotomies

We construct stable invariant manifolds for semiflows generated by the nonlinear impulsive differential equation with parameters x'= A(t)x + f(t, x, λ), t≠τi and x(τ+i) = Bix(τi) + gi(x(τi), λ), i ∈ N in Banach spaces, assuming that the linear impulsive differential equation x'= A(t)x, t≠τi and x(τ+i) = Bix(τi), i ∈ N admits a nonuniform (μ, ν)-dichotomy. It is shown that the stable invariant manifolds are Lipschitz continuous in the parameter λ and the initial values provided that the nonlinear perturbations f, g are sufficiently small Lipschitz perturbations.     

RELATIONS AMONG SOME PARAMETERS OF HYPERGRAPHS

The relations among the dominating number, independence number and covering number of hypergraphs are investigated. Main results are as follows：Dv（H）≤min{α≤（H）, p（H）, p（H）, T（H）}; De（H）≤min{v（H）, T（H）, p（H）}; DT（H） ≤αT（H）; S（H）≤ Dv （H） ＋ α（H）≤n; 2≤ Dv （H） ＋ T（H） ≤n; 2 〈 Dv （H） ＋ v（H）≤n/2 ＋ [n/r]; Dv （H） ＋ p（H） 〈_n;2≤De（H） ＋ Dv（H）≤n/2 ＋ [n/r];α（H） ＋ De（H）≤n;2 ≤ De（H） ＋ v（H）≤2[n/r]; 2 De（H） ＋ p（H）≤n-r ＋ 2.     

Reduced limits for nonlinear equations with measures

We consider equations (E) −Δu+g(u)=μ in smooth bounded domains Ω⊂RN, where g is a continuous nondecreasing function and μ is a finite measure in Ω. Given a bounded sequence of measures (μk), assume that for each k?1 there exists a solution uk of (E) with datum μk and zero boundary data. We show that if uk→u# in L¹(Ω), then u# is a solution of (E) relative to some finite measure μ#. We call μ# the reduced limit of (μk). This reduced limit has the remarkable property that it does not depend on the boundary data, but only on (μk) and on g. For power nonlinearities g(t)=|t|^q−1t, ∀t∈R, we show that if (μk) is nonnegative and bounded in W−2,q(Ω), then μ and μ# are absolutely continuous with respect to each other; we then produce an example where μ#≠μ.     

Ordinal remainders of ψ-spaces

We consider which ordinals, with the order topology, can be Stone-?ech remainders of which spaces of the form ψ(κ,M), where ω?κ is a cardinal number and M⊂ω[κ] is a maximal almost disjoint family of countable subsets of κ (MADF). The cardinality of the continuum, denoted c, and its successor cardinal, c⁺, play important roles. We show that if κ>c⁺, then no ψ(κ,M) has any ordinal as a Stone-?ech remainder. If κ?c then for every ordinal δ<κ⁺ there exists Mδ⊂ω[κ], a MADF, such that βψ(κ,Mδ)?ψ(κ,Mδ) is homeomorphic to δ+1. For κ=c⁺, βψ(κ,Mδ)?ψ(κ,Mδ) is homeomorphic to δ+1 if and only if c⁺?δ<c⁺⋅ω.     

The eccentric connectivity index of nanotubes and nanotori

Let G be a molecular graph. The eccentric connectivity index ξ^c(G) is defined as ξ^c(G)=∑_u∈V(G)deg_G(u)ε_G(u), where deg_G(u) denotes the degree of vertex u and ε_G(u) is the largest distance between u and any other vertex v of G. In this paper exact formulas for the eccentric connectivity index of TUC₄C₈(S) nanotube and TC₄C₈(S) nanotorus are given.     

Selection over classes of ordinals expanded by monadic predicates

A monadic formula ψ(Y) is a selector for a monadic formula φ(Y) in a structure M if ψ defines in M a unique subset P of the domain and this P also satisfies φ in M. If C is a class of structures and φ is a selector for ψ in every M∈C, we say that φ is a selector for φ over C.For a monadic formula φ(X,Y) and ordinals α≤ω₁ and δ<ω^ω, we decide whether there exists a monadic formula ψ(X,Y) such that for every P⊆αof order-type smaller thanδ, ψ(P,Y) selects φ(P,Y) in (α,<). If so, we construct such a ψ.We introduce a criterion for a class C of ordinals to have the property that every monadic formula φ has a selector over it. We deduce the existence of S⊆ω^ω such that in the structure (ω^ω,<,S) every formula has a selector.Given a monadic sentence π and a monadic formula φ(Y), we decide whether φ has a selector over the class of countable ordinals satisfying π, and if so, construct one for it.     

Small cycles and 2-factor passing through any given vertices in graphs

The theory of vertex-disjoint cycles and 2-factor of graphs has important applications in computer science and network communication. For a graph G, let σ ₂(G):=min?{d(u)+d(v)|uv ? E(G),u≠v}. In the paper, the main results of this paper are as follows:

1. Let k≥2 be an integer and G be a graph of order n≥3k, if σ ₂(G)≥n+2k?2, then for any set of k distinct vertices v ₁,…,v _k , G has k vertex-disjoint cycles C ₁,C ₂,…,C _k of length at most four such that v _i ∈V(C _i ) for all 1≤i≤k.
2. Let k≥1 be an integer and G be a graph of order n≥3k, if σ ₂(G)≥n+2k?2, then for any set of k distinct vertices v ₁,…,v _k , G has k vertex-disjoint cycles C ₁,C ₂,…,C _k such that:
   1. v _i ∈V(C _i ) for all 1≤i≤k.
   2. V(C ₁)∪???∪V(C _k )=V(G), and
   3. |C _i |≤4, 1≤i≤k?1.

Moreover, the condition on σ ₂(G)≥n+2k?2 is sharp.     

Maps preserving peripheral spectrum of Jordan semi-triple products of operators

Let A and B be (not necessarily unital or closed) standard operator algebras on complex Banach spaces X and Y, respectively. For a bounded linear operator A on X, the peripheral spectrum σ_π(A) of A is the set σ_π(A)={z∈σ(A):|z|=max_ω∈σ(A)|ω|}, where σ(A) denotes the spectrum of A. Assume that Φ:A→B is a map the range of which contains all operators of rank at most two. It is shown that the map Φ satisfies the condition that σ_π(BAB)=σ_π(Φ(B)Φ(A)Φ(B)) for all A,B∈A if and only if there exists a scalar λ∈C with λ³=1 and either there exists an invertible operator T∈B(X,Y) such that Φ(A)=λTAT^-1 for every A∈A; or there exists an invertible operator T∈B(X^∗,Y) such that Φ(A)=λTA^∗T^-1 for every A∈A. If X=H and Y=K are complex Hilbert spaces, the maps preserving the peripheral spectrum of the Jordan skew semi-triple product BA^∗B are also characterized. Such maps are of the form A?UAU^∗ or A?UA^tU^∗, where U∈B(H,K) is a unitary operator, A^t denotes the transpose of A in an arbitrary but fixed orthonormal basis of H.     

On the maximum number of permutations with given maximal or minimal distance

Let us denote by R(k, ? λ)[R(k, ? λ)] the maximal number $\text{M}$ such that there exist $\text{M}$ different permutations of the set {1,…, k} such that any two of them have at least λ (at most λ, respectively) common positions. We prove the inequalities R(k, ? λ) ? kR(k ? 1, ? λ ? 1), R(k, ? λ) ? R(k, ? λ ? 1) ? k!, R(k, ? λ) ? kR(k ? 1, ? λ ? 1). We show: R(k, ? k ? 2) = 2, R(k, ? 1) = (k ? 1)!, R(p^m, ? 2) = (p^m ? 2)!, R(p^m + 1, ? 3) = (p^m ? 2)!, $\text{R(k, ? k ? 3) =}\text{k!}\text{2}$, R(k, ? 0) = k, R(p^m, ? 1) = p^m(p^m ? 1), R(p^m + 1, ? 2) = (p^m + 1)p^m(p^m ? 1). The exact value of R(k, ? λ) is determined whenever k ? k₀(k ? λ); we conjecture that R(k, ? λ) = (k ? λ)! for k ? k₀(λ). Bounds for the general case are given and are used to determine that the minimum of |R(k, ? λ) ? R(k, ? λ)| is attained for $\text{λ = (}\text{k}\text{2}\text{) + O(}\text{k}\text{log}\text{k}\text{)}$.     

The cardinal characteristic for relative γ-sets

For X a separable metric space define p(X) to be the smallest cardinality of a subset Z of X which is not a relative γ-set in X, i.e., there exists an ω-cover of X with no γ-subcover of Z. We give a characterization of p(ω²) and p(ωω) in terms of definable free filters on ω which is related to the pseudo-intersection number p. We show that for every uncountable standard analytic space X that either p(X)=p(ω²) or p(X)=p(ωω). We show that the following statements are each relatively consistent with ZFC: (a) p=p(ωω)<p(ω²) and (b) p<p(ωω)=p(ω²)     

Inclusion relations involving k-numerical ranges

Let W_k(A) denote the k-numerical range of an n × n matrix A. It is known that W_i(A) ? W_j(A) for 1 ? j? i? n. It this paper we derive more general inclusion relations of the form Σⁿ_iλ_iW_i(A) ? Σⁿ_iμ_iW_i(A), where λ_i, μ_i are real coefficients.     

Partitions of natural numbers with the same representation functions

For a set A of nonnegative integers the representation functions R₂(A,n), R₃(A,n) are defined as the number of solutions of the equation n=a+a^′,a,a^′∈A with a<a^′, a?a^′, respectively. Let D(0)=0 and let D(a) denote the number of ones in the binary representation of a. Let A₀ be the set of all nonnegative integers a with even D(a) and A₁ be the set of all nonnegative integers a with odd D(a). In this paper we show that (a) if R₂(A,n)=R₂(N?A,n) for all n?2N−1, then R₂(A,n)=R₂(N?A,n)?1 for all n?12N²−10N−2 except for A=A₀ or A=A₁; (b) if R₃(A,n)=R₃(N?A,n) for all n?2N−1, then R₃(A,n)=R₃(N?A,n)?1 for all n?12N²+2N. Several problems are posed in this paper.     

The forcing hull and forcing geodetic numbers of graphs

For every pair of vertices u,v in a graph, a u-v geodesic is a shortest path from u to v. For a graph G, let I_G[u,v] denote the set of all vertices lying on a u-v geodesic. Let S⊆V(G) and I_G[S] denote the union of all I_G[u,v] for all u,v∈S. A subset S⊆V(G) is a convex set of G if I_G[S]=S. A convex hull [S]_G of S is a minimum convex set containing S. A subset S of V(G) is a hull set of G if [S]_G=V(G). The hull number h(G) of a graph G is the minimum cardinality of a hull set in G. A subset S of V(G) is a geodetic set if I_G[S]=V(G). The geodetic number g(G) of a graph G is the minimum cardinality of a geodetic set in G. A subset F⊆V(G) is called a forcing hull (or geodetic) subset of G if there exists a unique minimum hull (or geodetic) set containing F. The cardinality of a minimum forcing hull subset in G is called the forcing hull number f_h(G) of G and the cardinality of a minimum forcing geodetic subset in G is called the forcing geodetic number f_g(G) of G. In the paper, we construct some 2-connected graph G with (f_h(G),f_g(G))=(0,0),(1,0), or (0,1), and prove that, for any nonnegative integers a, b, and c with a+b≥2, there exists a 2-connected graph G with (f_h(G),f_g(G),h(G),g(G))=(a,b,a+b+c,a+2b+c) or (a,2a+b,a+b+c,2a+2b+c). These results confirm a conjecture of Chartrand and Zhang proposed in [G. Chartrand, P. Zhang, The forcing hull number of a graph, J. Combin. Math. Combin. Comput. 36 (2001) 81-94].     

Derivations and right ideals of algebras

Let R be a K-algebra acting densely on V_D, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let ρ be a nonzero right ideal of R and let f(X₁,…,X_t) be a nonzero polynomial over K with constant term 0 such that μR≠0 for some coefficient μ of f(X₁,…,X_t). Suppose that d:R→R is a nonzero derivation. It is proved that if rankd(f(x₁,…,x_t))?m for all x₁,…,x_t∈ρ and for some positive integer m, then either ρ is generated by an idempotent of finite rank or d=ad(b) for some b∈End(V_D) of finite rank. In addition, if f(X₁,…,X_t) is multilinear, then b can be chosen such that rank(b)?2(6t+13)m+2.     

A generalized weight for linear codes and a Witt-MacWilliams theorem

Let F be a finite field, H a subgroup of $\text{F}{}^1$ of index ν, and α₁,…, α_ν coset representatives. For each n-tuple u = (u₁,…, u_n) ?Fⁿ define W_H(u) = (w₁(u),…, w_ν(u)), where w_m(u) = #{u_i: u_i?α_mH}. An H-monomial map on Fⁿ is an automorphism of Fⁿ whose matrix with respect to the co-ordinate basis is of the form P · D, where P is a permutation matrix and D is a diagonal matrix with non-zero entries from H. Suppose C is an (n, k) code over F (that is, a k-dimensional subspace of Fⁿ) and that ?: C → Fⁿ is an injective homomorphism which preserves W_H in the sense that W_H(?(u)) = W_H(u) for all u ?C. We prove that ? may be extended to an H-monomial map on Fⁿ. This generalization of a theorem of MacWilliams on the (Hamming) equivalence of codes may be considered an analogue of the Witt theorem of metric vector space theory.     

A method of finding automorphism groups of endomorphism monoids of relational systems

For any set X and any relation ρ on X, let T(X,ρ) be the semigroup of all maps a:X→X that preserve ρ. Let S(X) be the symmetric group on X. If ρ is reflexive, the group of automorphisms of T(X,ρ) is isomorphic to N_S(X)(T(X,ρ)), the normalizer of T(X,ρ) in S(X), that is, the group of permutations on X that preserve T(X,ρ) under conjugation. The elements of N_S(X)(T(X,ρ)) have been described for the class of so-called dense relations ρ. The paper is dedicated to applications of this result.