Homogeneous tri-additive forms and derivations

Gleason [A.M. Gleason, The definition of a quadratic form, Amer. Math. Monthly 73 (1966) 1049-1066] determined all functionals Q on K-vector spaces satisfying the parallelogram law Q(x+y)+Q(x-y)=2Q(x)+2Q(y) and the homogeneity Q(λx)=λ²Q(x). Associated with Q is a unique symmetric bi-additive form S such that Q(x)=S(x,x) and 4S(x,y)=Q(x+y)-Q(x-y). Homogeneity of Q corresponds to that of S: S(λx,λy)=λ²S(x,y). The associated S is not necessarily bi-linear.Let V be a vector space over a field K, char(K)≠2,3. A tri-additive form T on V is a map of V³ into K that is additive in each of its three variables. T is homogeneous of degree 3 if T(λx,λy,λz)=λ³T(x,y,z) for all .We determine the structure of tri-additive forms that are homogeneous of degree 3. One of the keys to this investigation is to find the general solution of the functional equation

F(t)+t³G(1/t)=0,     

A necessary and sufficient condition for the existence of a spanning tree with specified vertices having large degrees

Let G be a connected simple graph, let X?V (^G) and let f be a mapping from X to the set of integers. When X is an independent set, Frank and Gyárfás, and independently, Kaneko and Yoshimoto gave a necessary and sufficient condition for the existence of spanning tree T in G such that d _T (x) for all x ∈ X, where d _T (x) is the degree of x and T. In this paper, we extend this result to the case where the subgraph induced by X has no induced path of order four, and prove that there exists a spanning tree T in G such that d _T (x) ≥ f(x) for all x ∈ X if and only if for any nonempty subset S ? X, |N _G (S) ? S| ? f(S) + 2|S| ? ω _G (S) ≥, where ω _G (S) is the number of components of the subgraph induced by S.     

A metric characterization of the irrationals via a group operation

Let S be a separable metric space with a compatible metric d that satisfies: For each point x ? S and each nonnegative real number r there exists a unique point y ? S such that d(x,y) = r.In this paper spaces that meet the above criterion are investigated. It is shown that, under the assumption of completeness, this metric property characterizes the space of irrationals.     

A note on Z 3-connected graphs with degree sum condition

A graph G satisfies the Ore-condition if d(x) + d(y) ≥ | V (G) | for any xy ■ E(G). Luo et al. [European J. Combin., 2008] characterized the simple Z3-connected graphs satisfying the Ore-condition. In this paper, we characterize the simple Z3-connected graphs G satisfying d(x) + d(y) ≥ | V (G) |-1 for any xy ■ E(G), which improves the results of Luo et al.     

Score vectors of tournaments

A tournament T on any set X is a dyadic relation such that for any x, y ∈ X (a) (x, x) ? T and (b) if x ≠ y then (x, y) ∈ T iff (y, x) ? T. The score vector of T is the cardinal valued function defined by R(x) = |{y ∈ X : (x, y) ∈ T}|. We present theorems for infinite tournaments analogous to Landau's necessary and sufficient conditions that a vector be the score vector for some finite tournament. Included also is a new proof of Landau's theorem based on a simple application of the “marriage” theorem.     

The asymptotic distribution of lattice points in hyperbolic space

Suppose x and y are two points in the upper half-plane H⁺, and suppose Γ is a discontinuous group of conformal automorphisms of H⁺ having compact fundamental domain S. Denote by N_T(x, y) the number of points of the form γy (γ?Γ) in the closed disc of hyperbolic radius T centered about x, and set $\text{Q}{}_{\mathrm{T}}\text{(x, y) = N}{}_{\mathrm{T}}\text{(x, y) ?}\text{V(T)}\text{A}\text{,}\text{where}\text{V(T)}$ is the hyperbolic area of the disc, and A is the hyperbolic area of S. The asymptotic behavior of the quantity ?_LxL(Q_T(x,y))² is estimated in terms of small eigenvalues of the Laplacian on functions automorphic under Γ.     

Graph diameter in long‐range percolation

We study the asymptotic growth of the diameter of a graph obtained by adding sparse “long” edges to a square box in ${\mathbb Z}^dWe study the asymptotic growth of the diameter of a graph obtained by adding sparse “long” edges to a square box in ${\mathbb Z}^d$. We focus on the cases when an edge between x and y is added with probability decaying with the Euclidean distance as |x ? y|^?s+o(1) when |x ? y| → ∞. For s ∈ (d, 2d) we show that the graph diameter for the graph reduced to a box of side L scales like (log L)^Δ+o(1) where Δ^?1 := log₂(2d/s). In particular, the diameter grows about as fast as the typical graph distance between two vertices at distance L. We also show that a ball of radius r in the intrinsic metric on the (infinite) graph will roughly coincide with a ball of radius exp{r^1/Δ+o(1)} in the Euclidean metric. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 210‐227, 2011     

Best constants for metric space inversion inequalities

For every metric space (X, d) and origin o ∈ X, we show the inequality I _o (x, y) ≤ 2d _o (x, y), where I _o (x, y) = d(x, y)/d(x, o)d(y, o) is the metric space inversion semimetric, d _o is a metric subordinate to I _o , and x, y ∈ X \ {o} The constant 2 is best possible.     

Engel type identities with derivations for right ideals in prime rings

Let R be a prime ring of char R≠2, d a non-zero derivation of R and ρ a non-zero right ideal of R such that [[d(x),d(y)]_n [y,x]_m] = 0 for all x,y ∈ ρ or [[d(x),d(y)]_n d[y,x]_m] = 0 for all x,y ∈ ρ, n, m ≥ 0 are fixed integers. If [ρ,ρ]ρ ≠ 0, then d(ρ)ρ = 0.     

Generalized weak peripheral multiplicativity in algebras of Lipschitz functions

Let (X, d _X ) and (Y,d _Y ) be pointed compact metric spaces with distinguished base points e _X and e _Y . The Banach algebra of all $\mathbb{K}$ -valued Lipschitz functions on X — where $\mathbb{K}$ is either?or ? — that map the base point e _X to 0 is denoted by Lip₀(X). The peripheral range of a function f ∈ Lip₀(X) is the set Ran_µ(f) = {f(x): |f(x)| = ‖f‖_∞} of range values of maximum modulus. We prove that if T ₁, T ₂: Lip₀(X) → Lip₀(Y) and S ₁, S ₂: Lip₀(X) → Lip₀(X) are surjective mappings such that $Ran_\pi (T_1 (f)T_2 (g)) \cap Ran_\pi (S_1 (f)S_2 (g)) \ne \emptyset $ for all f, g ∈ Lip₀(X), then there are mappings φ₁φ₂: Y → $\mathbb{K}$ with φ₁(y)φ₂(y) = 1 for all y ∈ Y and a base point-preserving Lipschitz homeomorphism ψ: Y → X such that T _j (f)(y) = φ _j (y)S _j (f)(ψ(y)) for all f ∈ Lip₀(X), y ∈ Y, and j = 1, 2. In particular, if S ₁ and S ₂ are identity functions, then T ₁ and T ₂ are weighted composition operators.     

Forward and inverse scattering on manifolds with asymptotically cylindrical ends

We study an inverse problem for a non-compact Riemannian manifold whose ends have the following properties: On each end, the Riemannian metric is assumed to be a short-range perturbation of the metric of the form ²(dy)+h(x,dx), h(x,dx) being the metric of some compact manifold of codimension 1. Moreover one end is exactly cylindrical, i.e. the metric is equal to ²(dy)+h(x,dx). Given two such manifolds having the same scattering matrix on that exactly cylindrical end for all energies, we show that these two manifolds are isometric.     

Common best proximity points: global minimal solutions

Let us suppose that A and B are nonempty subsets of a metric space. Let S:A?B and T:A?B be nonself-mappings. Considering the fact S and T are nonself-mappings, it is feasible that the equations Sx=x and Tx=x have no common solution, designated as a common fixed point of the mappings S and T. Eventually, when the equations have no common solution, one contemplates to figure out an element x that is in close proximity to Sx and Tx in the sense that d(x,Sx) and d(x,Tx) are minimum. In fact, common best proximity point theorems scrutinize the existence of such optimal approximate solutions, known as common best proximity points, to the equations Sx=x and Tx=x in the event that the equations have no common solution. Further, one can perceive that the real-valued functions x?d(x,Sx) and x?d(x,Tx) estimate the magnitude of the error involved for any common approximate solution of the equations Sx=x and Tx=x. In light of the fact that the distance between x and Sx, and the distance between x and Tx are at least the distance between A and B for all x in A, a common best proximity point theorem ascertains global minimum of both functions x?d(x,Sx) and x?d(x,Tx) by limiting a common approximate solution of the equations Sx=x and Tx=x to fulfil the requirement that d(x,Sx)=d(A,B) and d(x,Tx)=d(A,B). This article discusses a common best proximity point theorem for a pair of nonself-mappings, one of which dominates the other proximally, thereby yielding common optimal approximate solutions of some fixed point equations when there is no common solution.     

Dictators on blocks: Generalizations of social choice impossibility theorems

This paper extends impossibility theorems of Arrow and others to cases in which social comparisons between alternatives in a set X of social alternatives may be made only for each pair {x, y} in a certain subset of distinct pairs taken from X. With E the set of pairs within which social comparisons may be made, G = (X, E) is an undirected graph without loops. The social comparison between x and y for {x, y} ∈ E is to be based on the preferences of individuals in a finite society S. Each individual is presumed to prefer x to y or prefer y to x (not both) for every {x, y} ∈ E and may hold any preference relation that does not cycle in G. A profile is an assignment of one such relation to each individual in S. The paper examines binary social comparison procedures which map each profile into a social preference relation over the pairs in E, subject to x socially preferred to y whenever {x, y} ∈ E and everyone in S prefers x to y. Individual i ∈ S is a dictator [weak dictator] on {x, y} ∈ E iff x is socially preferred to y [x ranks as high as y socially] whenever i prefers x to y, and similarly with x and y interchanged, regardless of the preferences of the other individuals in S. An individual is a dictator [weak dictator] on a subgraph of G iff he is a dictator [weak dictator] on every edge in the subgraph. Under each of three ordering conditions on social preferences, there is a dictator or weak dictator on every block of G which has three or more points, different blocks can have different dictators, and bridges in E need not have dictators.     

Convexity in finite metric spaces

A subsetS of a metric space (X,d) is calledd-convex if for any pair of pointsx,y S each pointz X withd(x,z) +d(z,y) =d(x,y) belongs toS. We give some results and open questions concerning isometric and convexity-preserving embeddings of finite metric spaces into standard spaces and the number ofd-convex sets of a finite metric space.     

Lipschitz homeomorphisms of the Hilbert cube

We study the question whether the Hilbert cube Q is Lipschitz homogeneous. The answer depends on the metric of Q. For example, setting d(x,y)=sup_j|x_j-y_j|/j we obtain a Lipschitz homogeneous metric, but if the last j is replaced by j!, the answer is negative.     

Nonlinear parabolic equations with p-growth and unbounded data

The purpose of this paper is to prove the existence of a solution for a nonlinear parabolic equation in the form u_t - div(a(t, x, u, Du)) = H(t, x, u, Du) - div(g(t, x)) in Q_T =]0,T[×Ω, Ω ⊂ R^N, with an initial condition u(0) = u₀, where u₀ is not bounded, |H(t,x, u, ξ)⩽ β|ξ|^p + f(t,x) + βe^λ₁^|u|f, |g|^p/(p-1) ∈ L^r(Q_T) for some r = r{N) ⩾ 1, and - div(a(t,x,u, Du)) is the usual Leray-Lions operator.     

Circuits including a given set of vertices

Let G = (V, E) be a digraph of order n, satisfying Woodall's condition ? x, y ∈ V, if (x, y) ? E, then d⁺(x) + d^?(y) ≥ n. Let S be a subset of V of cardinality s. Then there exists a circuit including S and of length at most Min(n, 2s). In the case of oriented graphs we obtain the same result under the weaker condition d⁺(x) + d^?(y) ≥ n – 2 (which implies hamiltonism).     

Graphs with small boundary

For a pair of vertices x and y in a graph G, we denote by d_G(x,y) the distance between x and y in G. We call x a boundary vertex of y if x and y belong to the same component and d_G(y,v)?d_G(y,x) for each neighbor v of x in G. A boundary vertex of some vertex is simply called a boundary vertex, and the set of boundary vertices in G is called the boundary of G, and is denoted by B(G).In this paper, we investigate graphs with a small boundary. Since a pair of farthest vertices are boundary vertices, |B(G)|?2 for every connected graph G of order at least two. We characterize the graphs with boundary of order at most three. We cannot give a characterization of graphs with exactly four boundary vertices, but we prove that such graphs have minimum degree at most six. Finally, we give an upper bound to the minimum degree of a connected graph G in terms of |B(G)|.     

Bounding the diameter of a distance-transitive graph

A graph Γ is distance-transitive if for all vertices u, v, x, y such that d(u, v) = d(x, y) there is an automorphism h of Γ such that uh = x, vh = y. We show how to find a bound for the diameter of a bipartite distance-transitive graph given a bound for the order |G_α| of the stabilizer of a vertex.