Generalized derivations acting on multilinear polynomials in prime rings

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, let F, G and H be three generalized derivations of R, I an ideal of R and f(x₁,..., x _n ) a multilinear polynomial over C which is not central valued on R. If

$$F(f(r))G(f(r)) = H(f(r)^2 )$$

for all r = (r₁,..., r _n ) ∈ I ⁿ , then one of the following conditions holds:

1. (1)
   there exist a ∈ C and b ∈ U such that F(x) = ax, G(x) = xb and H(x) = xab for all x ∈ R
    
2. (2)
   there exist a, b ∈ U such that F(x) = xa, G(x) = bx and H(x) = abx for all x ∈ R, with ab ∈ C
    
3. (3)
   there exist b ∈ C and a ∈ U such that F(x) = ax, G(x) = bx and H(x) = abx for all x ∈ R
    
4. (4)
   f(x₁,..., x _n )² is central valued on R and one of the following conditions holds

   1. (a)
      there exist a, b, p, p’ ∈ U such that F(x) = ax, G(x) = xb and H(x) = px + xp’ for all x ∈ R, with ab = p + p’
       
   2. (b)
      there exist a, b, p, p’ ∈ U such that F(x) = xa, G(x) = bx and H(x) = px + xp’ for all x ∈ R, with p + p’ = ab ∈ C.
       
    

     

Valuation ideals and primary <Emphasis Type="Italic">w</Emphasis>-ideals

Let D be an integral domain, V (D) (resp., t-V (D)) be the set of all valuation (resp., t-valuation) ideals of D, and w-P(D) be the set of primary w-ideals of D. Let D[X] be the polynomial ring over D, c(f) be the ideal of D generated by the coefficients of f ∈ D[X], and N _v = {f ∈ D[X] | c(f) _v = D}. In this paper, we study integral domains D in which w-P(D) ? t-V (D), t-V (D) ? w-P(D), or t-V (D) = w-P(D). We also study the relationship between t-V (D) and \(V\left( {D{{\left[ X \right]}_{{N_v}}}} \right)\), and characterize when t-V (A + XB[X]) ? w-P(A + XB[X]) holds for a proper extension A ? B of integral domains.     

The Upper Connected Vertex Detour Monophonic Number of a Graph

For any vertex x in a connected graph G of order n ≥ 2, a set S _x ? V (G) is an x-detour monophonic set of G if each vertex v ∈ V (G) lies on an x-y detour monophonic path for some element y in S _x . The minimum cardinality of an x-detour monophonic set of G is the x-detour monophonic number of G, denoted by dm _x (G). A connected x-detour monophonic set of G is an x-detour monophonic set S _x such that the subgraph induced by S _x is connected. The minimum cardinality of a connected x-detour monophonic set of G is the connected x-detour monophonic number of G, denoted by cdm _x (G). A connected x-detour monophonic set S _x of G is called a minimal connected x-detour monophonic set if no proper subset of S _x is a connected x-detour monophonic set. The upper connected x-detour monophonic number of G, denoted by cdm⁺ _x (G), is defined to be the maximum cardinality of a minimal connected x-detour monophonic set of G. We determine bounds and exact values of these parameters for some special classes of graphs. We also prove that for positive integers r,d and k with 2 ≤ r ≤ d and k ≥ 2, there exists a connected graph G with monophonic radius r, monophonic diameter d and upper connected x-detour monophonic number k for some vertex x in G. Also, it is shown that for positive integers j,k,l and n with 2 ≤ j ≤ k ≤ l ≤ n - 3, there exists a connected graph G of order n with dm _x (G) = j,dm⁺ _x (G) = k and cdm⁺ _x (G) = l for some vertex x in G.     

Some matrix functional equations

We investigate the pair of matrix functional equations G(x)F(y) = G(xy) and G(x)G(y) = F(y/x), featuring the two independent scalar variables x and y and the two N×N matrices F(z) andG(z) (with N an arbitrary positive integer and the elements of these two matrices functions of the scalar variable z). We focus on the simplest class of solutions, i.e., on matrices all of whose elements are analytic functions of the independent variable. While in the scalar (N = 1) case this pair of functional equations only possess altogether trivial constant solutions, in the matrix (N > 1) case there are nontrivial solutions. These solutions satisfy the additional pair of functional equations F(x)G(y) = G(y/x) andF(x)F(y) = F(xy), and an endless hierarchy of other functional equations featuring more than two independent variables.     

Ideal Spaces of Measurable Operators Affiliated to A Semifinite Von Neumann Algebra

Suppose that M is a von Neumann algebra of operators on a Hilbert space H and τ is a faithful normal semifinite trace on M. Let E, F and G be ideal spaces on (M, τ). We find when a τ-measurable operator X belongs to E in terms of the idempotent P of M. The sets E+F and E·F are also ideal spaces on (M, τ); moreover, E·F = F·E and (E+F)·G = E·G+F·G. The structure of ideal spaces is modular. We establish some new properties of the L₁(M, τ) space of integrable operators affiliated to the algebra M. The results are new even for the *-algebra M = B(H) of all bounded linear operators on H which is endowed with the canonical trace τ = tr.     

Annihilating and power-commuting generalized skew derivations on lie ideals in prime rings

Let R be a prime ring of characteristic different from 2 and 3, Q_r its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Let α be an automorphism of the ring R. An additive map D: R → R is called an α-derivation (or a skew derivation) on R if D(xy) = D(x)y + α(x)D(y) for all x, y ∈ R. An additive mapping F: R → R is called a generalized α-derivation (or a generalized skew derivation) on R if there exists a skew derivation D on R such that F(xy) = F(x)y + α(x)D(y) for all x, y ∈ R.     

On Generalized Derivations and Centralizers of Operator Algebras with Involution

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H and A(H) ? B(H) be a standard operator algebra which is closed under the adjoint operation. Let F: A(H)→ B(H) be a linear mapping satisfying F(AA*A) = F(A)A*A + Ad(A*)A + AA*d(A) for all A ∈ A(H), where the associated linear mapping d: A(H) → B(H) satisfies the relation d(AA*A) = d(A)A*A + Ad(A*)A + AA*d(A) for all A ∈ A(H). Then F is of the form F(A) = SA ? AT for all A ∈ A(H) and some S, T ∈ B(H), that is, F is a generalized derivation. We also prove some results concerning centralizers on A(H) and semisimple H^*-algebras.     

Koszul property of a class of graded algebras with nonpure resolutions

Given any integers a, b, c, and d with a > 1, c ≥ 0, b ≥ a + c, and d ≥ b + c, the notion of (a, b, c, d)-Koszul algebra is introduced, which is another class of standard graded algebras with “nonpure” resolutions, and includes many Artin-Schelter regular algebras of low global dimension as specific examples. Some basic properties of (a, b, c, d)-Koszul algebras/modules are given, and several criteria for a standard graded algebra to be (a, b, c, d)-Koszul are provided.     

Distance domination of generalized de Bruijn and Kautz digraphs

Let G = (V,A) be a digraph and k ≥ 1 an integer. For u, v ∈ V, we say that the vertex u distance k-dominate v if the distance from u to v at most k. A set D of vertices in G is a distance k-dominating set if each vertex of V D is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γ _k (G), is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs G _B (n, d) and generalized Kautz digraphs G _K (n, d) are good candidates for interconnection networks. Denote Δ _k := (∑ _j=0 ^k d ^j )^?1. F. Tian and J. Xu showed that ?nΔ _k ? γ _k (G _B (n, d)) ≤?n/d ^k? and ?nΔ _k ? ≤ γ _k (G _K (n, d)) ≤ ?n/d ^k ?. In this paper, we prove that every generalized de Bruijn digraph G _B (n, d) has the distance k-domination number ?nΔ _k ? or ?nΔ _k ?+1, and the distance k-domination number of every generalized Kautz digraph G _K (n, d) bounded above by ?n/(d ^k?1+d ^k )?. Additionally, we present various sufficient conditions for γ _k (G _B (n, d)) = ?nΔ _k ? and γ _k (G _K (n, d)) = ?nΔ _k ?.     

Some Existence Theorems on All Fractional (<Emphasis Type="Italic">g</Emphasis>, <Emphasis Type="Italic">f</Emphasis>)-factors with Prescribed Properties

Let G be a graph, and g, f: V (G) → Z⁺ with g(x) ≤ f(x) for each x ∈ V (G). We say that G admits all fractional (g, f)-factors if G contains an fractional r-factor for every r: V (G) → Z⁺ with g(x) ≤ r(x) ≤ f(x) for any x ∈ V (G). Let H be a subgraph of G. We say that G has all fractional (g, f)-factors excluding H if for every r: V (G) → Z⁺ with g(x) ≤ r(x) ≤ f(x) for all x ∈ V (G), G has a fractional r-factor F _h such that E(H) ∩ E(F _h ) = θ, where h: E(G) → [0, 1] is a function. In this paper, we show a characterization for the existence of all fractional (g, f)-factors excluding H and obtain two sufficient conditions for a graph to have all fractional (g, f)-factors excluding H.     

On a subclass of the class of generalized Douglas-Weyl metrics

We study a class of Finsler metrics whose Douglas curvature is constant along any Finslerian geodesics. This class of Finsler metrics is a subclass of the class of generalized Douglas-Weyl metrics and contains the class of Douglas metrics as a special case. We find a condition under which this class of Finsler metrics reduces to the class of Landsberg metrics. Then we show this class of metrics contains the class of R-quadratic metrics.     

Equations in finite fields with restricted solution sets. I (Character sums)

In earlier papers, for “large” (but otherwise unspecified) subsets A, B of Z _p and for h(x) ∈ Z _p [x], Gyarmati studied the solvability of the equations a + b = h(x), resp. ab = h(x) with a ∈ A, b ∈ B, x ∈ Z _p , and for large subsets A, B, C, D of Z _p Sárközy showed the solvability of the equations a + b = cd, resp. ab + 1 = cd with a ∈ A, b ∈ B, c ∈ C, d ∈ D. In this series of papers equations of this type will be studied in finite fields. In particular, in Part I of the series we will prove the necessary character sum estimates of independent interest some of which generalize earlier results.     

<Emphasis Type="Italic">M</Emphasis> <Subscript> <Emphasis Type="Italic">p</Emphasis> </Subscript>-supplemented subgroups of finite groups

A subgroup K of G is M_p-supplemented in G if there exists a subgroup B of G such that G = KB and TB < G for every maximal subgroup T of K with |K: T| = p^α. In this paper we prove the following: Let p be a prime divisor of |G| and let H be ap-nilpotent subgroup having a Sylow p-subgroup of G. Suppose that H has a subgroup D with D_p ≠ 1 and |H: D| = p_α. Then G is p-nilpotent if and only if every subgroup T of H with |T| = |D| is M_p-supplemented in G and N_G(T_p)/C_G(T_p) is a p-group.     

Equicontinuity of maps on a dendrite with finite branch points

Let(T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote byω(x,f) and P(f) the ω-limit set of x under f and the set of periodic points of,respectively. Write Ω(x,f) = {y| there exist a sequence of points x_k E T and a sequence of positive integers n_1 n_2 … such that lim_(k→∞)x_k=x and lim_(k→∞)f~(n_k)(x_k) =y}. In this paper, we show that the following statements are equivalent:(1) f is equicontinuous.(2) ω(x, f) = Ω(x,f) for any x∈T.(3) ∩_(n=1)~∞f~n(T) = P(f),and ω(x,f)is a periodic orbit for every x ∈ T and map h : x→ω(x,f)(x ET)is continuous.(4) Ω(x,f) is a periodic orbit for any x∈T.     

Some variations on Tverberg’s theorem

Define T(d, r) = (d + 1)(r - 1) + 1. A well known theorem of Tverberg states that if n ≥ T(d, r), then one can partition any set of n points in R^d into r pairwise disjoint subsets whose convex hulls have a common point. The numbers T(d, r) are known as Tverberg numbers. Reay added another parameter k (2 ≤ k ≤ r) and asked: what is the smallest number n, such that every set of n points in R^d admits an r-partition, in such a way that each k of the convex hulls of the r parts meet. Call this number T(d, r, k). Reay conjectured that T(d, r, k) = T(d, r) for all d, r and k. In this paper we prove Reay’s conjecture in the following cases: when k ≥ [d+3/2], and also when d < rk/r-k - 1. The conjecture also holds for the specific values d = 3, r = 4, k = 2 and d = 5, r = 3, k = 2.     

Expanding curves in T1(Hn) under geodesic flow and equidistribution in homogeneous spaces

Let H = SO(n, 1) and A = {a(t): t ∈ R} be a maximal R-split Cartan subgroup of H. Let G be a Lie group containing H and Γ be a lattice of G. Let φ = gΓ ∈ G/Γ be a point of G/Γ such that its H-orbit Hx is dense in G/Γ. Let φ: I = [a, b] → H be an analytic curve. Then φ(I)x gives an analytic curve in G/Γ. In this article, we will prove the following result: if φ(I) satisfies some explicit geometric condition, then a(t)φ(I)x tends to be equidistributed in G/Γ as t → ∞. It answers the first question asked by Shah in [Sha09c] and generalizes the main result of that paper.     

Conditional connectivity of bubble sort graphs

A subset F ? V (G) is called an R ^k -vertex-cut of a graph G if G ? F is disconnected and each vertex of G ? F has at least k neighbors in G ? F. The R ^k -vertex-connectivity of G, denoted by κ ^k (G), is the cardinality of a minimum R ^k -vertex-cut of G. Let B _n be the bubble sort graph of dimension n. It is known that κ _k (B _n ) = 2 ^k (n ? k ? 1) for n ≥ 2k and k = 1, 2. In this paper, we prove it for k = 3 and conjecture that it is true for all k ∈ N. We also prove that the connectivity cannot be more than conjectured.     

p-Groups with few conjugacy classes of normalizers

For a group $G$ , denote by $\omega (G)$ the number of conjugacy classes of normalizers of subgroups of $G$ . Clearly, $\omega (G)=1$ if and only if $G$ is a Dedekind group. Hence if $G$ is a 2-group, then $G$ is nilpotent of class $\le 2$ and if $G$ is a $p$ -group, $p>2$ , then $G$ is abelian. We prove a generalization of this. Let $G$ be a finite $p$ -group with $\omega (G)\le p+1$ . If $p=2$ , then $G$ is of class $\le 3$ ; if $p>2$ , then $G$ is of class $\le 2$ .     

Minimum implicit degree condition restricted to claws for hamiltonian cycles

Let id(v) denote the implicit degree of a vertex v in a graph G. We define G to be implicit 1-heavy (implicit 2-heavy) if at least one (two) of the end vertices of each induced claw has (have) implicit degree at least n/2. In this paper, we prove that: (a) Let G be a 2-connected graph of order n ≥ 3. If G is implicit 2-heavy and |N(u) ∩ N(v)| ≥ 2 for every pair of vertices u and v with d(u, v) = 2 and max{id(u), id(v)} < n/2, then G is hamiltonian. (b) Let G be a 3-connected graph of order n ≥ 3. If G is implicit 1-heavy and |N(u) ∩ N(v)| ≥ 2 for each pair of vertices u and v with d(u, v) = 2 and max{id(u), id(v)} < n/2, then G is hamiltonian.     

Finite groups with <Emphasis Type="Italic">C</Emphasis>-quasinormal subgroups

Consider some finite group G and a finite subgroup H of G. Say that H is c-quasinormal in G if G has a quasinormal subgroup T such that HT = G and T ∩ H is quasinormal in G. Given a noncyclic Sylow subgroup P of G, we fix some subgroup D such that 1 < |D| < | P| and study the structure of G under the assumption that all subgroups H of P of the same order as D, having no supersolvable supplement in G, are c-quasinormal in G.