Proof. The intersection P of all subfields of F is a field by Exercise 1.4. Consider the ring homomorphism φ : Z → F given by φ(n) = n · 1. Since any subfield contains 1 and is closed under addition, imφ is contained in P . If Char F = p 6= 0 then imφ is isomorphic to Z/pZ = Fp. Since this is a field, we have P = imφ ∼= Fp. If Char F = 0 then φ is injective. Define φ̂ : Q → F by φ̂(m/n) = φ(m)/φ(...

Let F be a closed essential surface in a hyperbolic 3-manifold M with a toroidal cusp N . The depth of F in N is the maximal distance from points of F in N to the boundary of N . It will be shown that if F is an essential pleated surface which is not coannular to the boundary torus of N then the depth of F in N is bounded above by a constant depending only on the genus of F . The result is used...

We analyze quasi-stationary distributions {μ}ε>0 of a family of Markov chains {X}ε>0 that are random perturbations of a bounded, continuous map F :M →M , where M is a closed subset of Rk . Consistent with many models in biology, these Markov chains have a closed absorbing set M0 ⊂ M such that F(M0)=M0 and F(M \M0)=M \M0. Under some large deviations assumptions on the random perturbations, we sh...

the notion of a bead metric space is defined as a nice generalization of the uniformly convex normed space such as $cat(0)$ space, where the curvature is bounded from above by zero. in fact, the bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces and normed bead spaces are identical with uniformly convex spaces. in this paper, w...

the main purpose of this paper is to introduce and study new classes of soft closed sets like soft regular generalized b-closed sets in soft topological spaces (brie y soft rgb-closed set) moreover, soft rg-closed, soft gpr-closed, soft gb-closed, soft gsp-closed, soft g-closed, soft gb-closed, and soft sgb-closed sets in soft topological spaces are introduced in this paper and we investigat...

For a function f:[0,1]→R, we consider the set E(f) of points at which f cuts real axis. Given f:[0,1]→R and Cantor D⊂[0,1] with {0,1}⊂D, obtain conditions equivalent to conjunction f∈C[0,1] (or f∈C∞[0,1]) D⊂E(f). This generalizes some ideas Zabeti. We observe that, if is continuous, then closed nowhere dense subset f−1[{0}]. Additionally, Intf−1[{0}]=0̸, each x∈{0,1}∩E(f) an accumulation point E...

For several graph theoretic parameters such as vertex cover and dominating set, it is known that if their values are bounded by k then the treewidth of the graph is bounded by some function of k. This fact is used as the main tool for the design of several fixed-parameter algorithms on minor-closed graph classes such as planar graphs, single-crossing-minor-free graphs, and graphs of bounded gen...

Nielsen fixed point theory (see [1, 4]) deals with the estimation of the number of fixed points of maps in the homotopy class of any given map f : X → X . The Nielsen number N( f ) provides a lower bound. A classical result in Nielsen fixed point theory is: any map f : X → X is homotopic to a map with exactly N( f ) fixed points if the compact polyhedron X has no local cut point and is not a 2-...

In this paper we describe all gradings by a finite abelian group G on the following Lie algebras over an algebraically closed field F of characteristic p = 2: sln(F ) (n not divisible by p), son(F ) (n ≥ 5, n = 8) and spn(F ) (n ≥ 6, n even).

Given two open unit balls B1 and B2 in complex Banach spaces, we consider a holomorphic mapping f : B1 → B2 such that f(0) = 0 and f ′(0) is an isometry. Under some additional hypotheses on the Banach spaces involved, we prove that f(B1) is a complex closed analytic submanifold of B2.