Let (X, ρ), (Y, σ) be metric spaces and f : X → Y an injective mapping. We put ‖f‖Lip = sup{σ(f(x), f(y))/ρ(x, y); x, y ∈ X, x 6= y}, and dist(f) = ‖f‖Lip .‖f ‖Lip (the distortion of the mapping f). Some Ramsey-type questions for mappings of finite metric spaces with bounded distortion are studied; e.g., the following theorem is proved: Let X be a finite metric space, and let ε > 0, K be given ...

The reaction of [PPhJ, NOS,] with CuCl and KSCN (1 :3:3) in acetone produces the yell-ow crystals of [PPh,], [WOS,(CuNCS),]. The crystals are triclinic, space group P1(2), z=2, a=12.4823(7), b=12.9224(7), c=18.6395(10)?, ?=83.907(5), ?= 73.152(4), Y= 65.194(4)'. The crystal structure was determined by single crystal Xray diffraction methods (Mo-K ?) and refined by least-squares calculations ...

We apply Lusztig’s theory of character sheaves to the problem of calculating the spherical functions of G(Fq2 )/G(Fq), where G is a connected reductive algebraic group. We obtain the solution for generic spherical functions for any G, and for all spherical functions when G = GLn. The proof includes a result about convolution of character sheaves and its interaction with the associated two-sided...

A topological space X is said to be completely regular if whenever F is a nonempty closed set and x ∈ X \F , there is a continuous function f : X → [0, 1] such that f(x) = 0 and f(F ) = {1}. A completely regular space need not be Hausdorff. For example, ifX is any set with more than one point, then the trivial topology, in which the only closed sets are ∅ and X, is vacuously completely regular,...

in this paper, we establish and prove the existence of best proximity points for multivalued cyclic $f$- contraction mappings in complete metric spaces. our results improve and extend various results in literature.

By a space we shall mean a measurable space, i.e. an abstract set together with a <r-ring of subsets, called measurable sets, whose union is the whole space. The structure of a space will be the <r-ring of its measurable subsets. A measurable transformation from one space to another is a mapping such that the inverse image of every measurable set is measurable. Let X and F be spaces, F a set of...

and Applied Analysis 3 P:4 p x y ≤ p x p y , for all x, y ∈ X triangle inequality , P:5 if λn is a sequence of scalars, with λn → λ n → ∞ , and xn is a sequence of vectors with p xn −x → 0 n → ∞ , then p λnxn −λx → 0 n → ∞ continuity of multiplication by scalars . A complete linear metric space is said to be a Fréchet space. A Fréchet sequence space X is said to be an FK space, if its metric is...

A number of years ago several investigators f published independently this theorem : Any screw motion—the most general mechanical motion of space —can be decomposed into the product of two semi-rotations. By a semi-rotation is meant that transformation which consists of rotating space through 180° about a fixed axis I. For generalizing however it is more convenient to look at this transformatio...

In this paper, we investigate a new type of random $F$-contraction and obtain a common random fixed point theorem for a pair of self stochastic mappings in a separable Banach space. The existence of a unique solution for nonlinear fractional random differential equation is proved under suitable conditions.

Introduction: About one-third of children are involved in bullying in primary school. Parental mood as a family background plays an important role in bullying. This study aimed to compare the effectiveness of the parent training programs of CALM (coaching approach behavior and leading by modeling) and SPACE (supportive parenting for anxious childhood emotions) on the rumination and anxiety of m...