in an earlier work we showed that for ordered fields f not isomorphic to the reals r, there are continuous 1-1 unctions on [0, 1]f which map some interior point to a boundary point of the image (and so are not open). here we show that over closed bounded intervals in the rationals q as well as in all non-archimedean ordered fields of countable cofinality, there are uniformly continuous 1-1 func...

where, in addition to satisfying conditions which insure existence and uniqueness of solutions, g(x) and p(t) satisfy the following conditions: (i) there exist positive constants r and k such that for all t, p(t) = p(t+r) and \p(t)\ k. We show that for r sufficiently small, Equation (1) has a solution x(t)...

In this paper we introduce the cone bounded linear mapping and demonstrate a proof to show that the cone norm is continuous. Among other things, we prove the open mapping theorem and the closed graph theorem in TVS-cone normed spaces. We also show that under some restrictions on the cone, two cone norms are equivalent if and only if the topologies induced by them are the same. In the sequel, we...

A continuous function f(x+ iy) = u(x, y) + iv(x, y) defined in a domain D ⊂ C (Complex plane) is harmonic in D if u and v are real harmonic in D. Clunie and Shiel-Small [3] showed that in a simply connected domain such functions can be written in the form f = h+g, where both h and g are analytic. We call h the analytic part and g, the co-analytic part of f . Let w(z) = g ′(z) h′(z) be the dilat...

ABSTRACT. Let G be a compact connected metric abelian group equipped with its normalised Haar measure. Let Tx = a+A(x) be a continuous surjective affine map of G such that γ ≡ 1 is the only character of G satisfying γ ◦A = γ for some positive integer n. Then if φ is a polynomial with real coefficients mapping the natural numbers to themselves, (pl) ∞ l=1 is the sequence of rational primes and f...

Existing environment mapping techniques include spherical mapping and cube mapping. These techniques have inherent flaws that cause sampling issues and aliasing. Continuous cube mapping is offered as an alternative environment mapping approach that effectively folds the cube onto the sphere, providing a better parameterization of cube mapping. We provide a hardware implementation.

The main result in this paper is the following linearization theorem. For each open set U in a complex Banach space E , there is a complex Banach space G°°(Í7) and a bounded holomorphic mapping gv: U —► G°°(U) with the following universal property: For each complex Banach space F and each bounded holomorphic mapping /: U —> F , there is a unique continuous linear operator T,\ G°°(U) —» F such t...

Let G be a Lie group and § a Banach space. A representation n of G on § is a mapping which assigns to every element x in G a bounded linear operator n(x) on § such that the following two conditions are fulfilled: (1) 7t(xy) = n(x) 7i(y) (x, y e G),n(l) = / and (2) the mapping (x, tp) ->uz(x)y) of G x § into § is continuous. (Here 1 is the unit element of G and I is the unit operator,) In partic...

We study metric spheres ( Z , d ) $(Z, d_{Z} )$ obtained by gluing two hemispheres of S 2 $\mathbb {S}^{2}$ along an orientation-preserving homeomorphism g : 1 → $g \colon \mathbb {S}^{1} \rightarrow {S}^{1}$ where $d_{Z}$ is the canonical distance that locally isometric to off seam. show if quasiconformally equivalent in geometric sense, then $g$ a welding with conformally removable curves. al...