The tail is cut for Ramsey numbers of cubes

The Ramsey number R(G) of a graph G is the least integer p such that for all bicolorings of the edges of the complete graph K_p, one of the monochromatic subgraphs contains a copy of G. We show that for any positive constant c and bipartite graph G=(U,V;E) of order n where the maximum degree of vertices in U is at most , . Moreover, we show that the Ramsey number of the cube Q_n of dimension n satisfies . In both cases, the small terms are removed from the powers in the upper bounds of a earlier result of the author.     

Enumerative properties and cube polynomials of Tribonacci cubes

Tribonacci cubes ${\Gamma }_n^{\left(3\right)}$are induced subgraphs of $Q_n$, obtained by removing all the vertices that contain more than two consecutive 1’s. In the present work, we give some enumerative properties related to ${\Gamma }_n^{\left(3\right)}$. We show that the number of vertices of weight $w$ in ${\Gamma }_n^{\left(3\right)}$is ${\sum }_{j=0}^{n-w+1}\left(\genfrac{}{}{0ex}{}{n-w+1}{j}\right)\left(\genfrac{}{}{0ex}{}{j}{w-j}\right)$ and express the number of edges of these graphs in terms of convolved Tribonacci numbers. We investigate the cube polynomials of Tribonacci cubes and determine the corresponding generating function. Finally, we give a formula for the number of induced $k$-cubes in ${\Gamma }_n^{\left(3\right)}$.     

On Limit Sets and Discreteness Criteria for Nonelementary Subgroups of M（R^n）

In this paper, we will study the nonelementary groups of MSbius transformations in R^n and some properties are obtained. Also in this paper we will prove several theorems about discreteness criteria and group convergence of nonelementary groups of M（R^n）.     

Isometric composition operators on the analytic Besov spaces

We investigate the isometric composition operators on the analytic Besov spaces. For 1<p<2$1<p<2$ we show that an isometric composition operator is induced only by a rotation of the disk. For p>2$p>2$, we extend previous work on the subject. Finally, we analyze this same problem for the Besov spaces with an equivalent norm.     

Coarse version of the Banach-Stone theorem

We show that if there exists a Lipschitz homeomorphism T between the nets in the Banach spaces C(X) and C(Y) of continuous real valued functions on compact spaces X and Y, then the spaces X and Y are homeomorphic provided . By l(T) and l(T−1) we denote the Lipschitz constants of the maps T and T−1. This improves the classical result of Jarosz and the recent result of Dutrieux and Kalton where the constant obtained is . We also estimate the distance of the map T from the isometry of the spaces C(X) and C(Y).     

The Urysohn space embeds in Banach spaces in just one way

In a Master's thesis in 1985 and a subsequent paper published in 1992, the author discovered that the universal separable metric space (up to isometry) discovered by Urysohn in 1925 has a uniquely determined linear closure (up to linear isometry) when isometrically embedded in a Banach space so as to include the zero of the Banach space. The proof of this result is given in this note and the current status of some related questions is discussed.     

单位球上加权Begrman空间和Hardy空间上的保范复合算子

作者证明了复合算子C_φ是单位球上加权Begrman空间上的保范算子当且仅当φ是旋转变换.对单位球上的Hardy空间上的保范复合算子仅得到部分结果.     

Small into isomorphisms on uniformly smooth spaces

Let X be a uniformly smooth infinite dimensional Banach space, and (Ω,Σ,μ) be a σ-finite measure space. Suppose that T:X→L∞(Ω,Σ,μ) satisfies

(1−ε)‖x‖?‖Tx‖?‖x‖,∀x∈X,     

Hadamard流形上等距子群的离散准则

本文对Hadamard流形上的等距群Isom(X)进行了研究,得到了几个关于离 散准则和代数收敛性的定理.     

Algebraic reflexivity of the isometry group of some spaces of Lipschitz functions

We show that the isometry groups of Lip(X,d) and lip(X,dα) with α∈(0,1), for a compact metric space (X,d), are algebraically reflexive. We also prove that the sets of isometric reflections and generalized bi-circular projections on such spaces are algebraically reflexive. In order to achieve this, we characterize generalized bi-circular projections on these spaces.     

Diameter preserving linear maps and isometries, II

We study linear bijections of simplex spacesA(S) which preserve the diameter of the range, that is, the seminorm ϱ(f)=sup{|f(x)−f(y)|:x,yεS}.     

Isometries and Toeplitz operators of Bergman space of bounded symmetric domains

In this paper we completely characterize the isometries of Bergman space (0?p<∞, p≠2) of bounded symmetric domains. We also prove that a pair of Toeplitz operators Tf and Tg on (0<p<∞, p≠2) is isometric equivalence if and only if there is a τ∈Aut(Ω), such that g=f○τ, where Aut(Ω) is the automorphism group of Ω.     

Isometry Groups of Hyperbolic Three-Manifolds which are Cyclic Branched Coverings

Let M be a hyperbolic three-manifold which is an n-fold cyclic branched covering of a hyperbolic link L in the three-sphere, or more precisely, of the hyperbolic three-orbifold whose underlying topological space is the three-sphere and whose singular set, of branching index n, the link L. We say that M has no hidden symmetries (with respect to the given branched covering) if the isometry group of M is the lift of (a subgroup of) the isometry group of the hyperbolic orbifold (which is isomorphic to the symmetry group of the link L). It follows from Thurston's hyperbolic surgery theorem that M has no hidden symmetries if n is sufficiently large. Our main result is an explicit numerical version of this fact: we give a constant, in terms of the volume of the complement of L, such that M has no hidden symmetries for all n larger than this constant; we show by examples that a universal constant working for all hyperbolic knots or links does not exist. We give also some results on the possible orders and the structure of the isometry group of M. Finally, we construct sets of four different -hyperbolic knots which have the same two-fold branched covering (a hyperbolic three-manifold); it is an interesting question for how many different -hyperbolic knots (or links) this may happen (in the case of hyperbolic knots, for arbitrarily many).     

Minimal solutions of the isometry equation

For a finite vector space $W$ over ${\mathbb{F}}_q$, there are described all the pairs of multisets $\{V_1,\dots ,V_{q+1}\}$ and $\{U_1,\dots ,U_{q+1}\}$ of subspaces in $W$ such that for all $w\in W$ the equality $\left|\{i\mid w\in V_i\}\right|=\left|\{i\mid w\in U_i\}\right|$ holds.     

On the categorical behaviour of V-groups

We consider compatible group structures on a V-category, where V is a quantale, and we study the topological and algebraic properties of such groups. Examples of such structures are preordered groups, metric and ultrametric groups, probabilistic (ultra)metric groups. In particular, we show that, when V is a frame, symmetric V-groups satisfy very strong categorical-algebraic properties, typical of the category of groups. In particular, symmetric V-groups form a protomodular category.     

Problems related to a de Bruijn-Erdös theorem

De Bruijn and Erdös proved that every noncollinear set of n points in the plane determines at least n distinct lines. We suggest a possible generalization of this theorem in the framework of metric spaces and provide partial results on related extremal combinatorial problems.