SDIV = [0..5] × Z× Z× Z× Z (where [m..n] = {m,m+1, . . . , n}) R̂DIV (pc, x, y, r, q) (pc ′, x′, y′, r′, q′) = (pc = 0) ⇒ ((pc′, x′, y′, r′, q′) = (1, x, y, x, q)) ∧ (pc = 1) ⇒ ((pc′, x′, y′, r′, q′) = (2, x, y, r, 0)) ∧ (pc = 2) ⇒ ((pc′, x′, y′, r′, q′) = if y≤r then (3, x, y, r, q) else (5, x, y, r, q)) ∧ (pc = 3) ⇒ ((pc′, x′, y′, r′, q′) = (4, x, y, (r−y), q)) ∧ (pc = 4) ⇒ ((pc′, x′, y′, r′, ...

and Applied Analysis 3 Throughout this paper, all topological spaces are assumed to be Hausdorff. In order to prove our main theorems, we need the following results. Lemma 2.1. Let X and Y be two topological spaces and G : X → 2 a set-valued mapping with nonempty values. Then the following conditions are equivalent: I G has the compactly local intersection property, II for each nonempty compact...

Suppose M is a topological m-manifold, X is a generalized nmanifold satisfying the disjoint disks property (DDP ), m > n ≥ 5, f : M → X is an approximate fibration, with fiber the shape of a closed topological manifold F , and Y is a closed, 1-LCC, codimension three subset of X. We examine conditions under which f is controlled homeomorphic to an approximate fibration g : M → X such that g|g−1(...

Equation 1 appeared incorrectly. It should read as: () () y t y t y t y t = + − − + − − − − − µ φ µ φ µ φ φ µ 1 1 12 12 1 12 13 () + + − + − + α γ ε 1 4 13 T t a MEI t t The equations in the legend of Table 1 were also incorrect. The full model should be: () () () y t y t y t y t = + − − + − − − − − µ φ µ φ µ φ φ µ 1 1 12 12 1 12 13 + + − + − + − + − + − α α α γ γ 1 4 2 13 3 21 13 4 T t T t T t...

This is the simplest form of exponential smoothing and can be used only for data without any systematic trend or seasonal components. Given such a time series, a sensible approach is to take a weighted average of past values. So for a series, y 1 , y 2 ,…, y n , the estimate of the value of y n+1 , given the information available up to time n, is: y w y w y w y or i n i i n n y w y − ∞ = + ∑ = ...

Sufficient conditions are given, under which the equation y = f(t, y, y′, . . . , y(l))g(y(n−1)) has a singular solution y [T, τ) → R, τ < ∞ satisfying lim t→τ− y(t) = ci ∈ R, i = 0, 1, . . . , l and lim t→τ− |y(j)(t)| = ∞ for j = l + 1, . . . , n− 1 where l ∈ {0, 1, . . . , n− 2}. 2000 Mathematics Subject Classification: 34C11.

SDIV = [0..5] × Z× Z× Z× Z (where [m..n] = {m,m+1, . . . , n}) R̂DIV (pc, x, y, r, q) (pc ′, x′, y′, r′, q′) = (pc = 0) ⇒ ((pc′, x′, y′, r′, q′) = (1, x, y, x, q)) ∧ (pc = 1) ⇒ ((pc′, x′, y′, r′, q′) = (2, x, y, r, 0)) ∧ (pc = 2) ⇒ ((pc′, x′, y′, r′, q′) = if y≤r then (3, x, y, r, q) else (5, x, y, r, q)) ∧ (pc = 3) ⇒ ((pc′, x′, y′, r′, q′) = (4, x, y, (r−y), q)) ∧ (pc = 4) ⇒ ((pc′, x′, y′, r′, ...