Show that there is no continuous map g:D^n to S^(n-1) so that g restricted to S^(n-1) is homotopic to id(S^(n-1))
Show that there is no continuous map g:D^n to S^(n-1) so that g restricted to S^(n-1)...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
f(x), a continuous probability function, is equal to 1/3 and the function is restricted to 1 ? x ? 4. Describe P(x >3/2).
Let G be a group and let g ∈ G. Show that the map ig : G → G given by ig(a) = gag−1 for all a ∈ G is an automorphism.
3. A sequence is a map a N°R, typically written (an) = (ao, a1, a2, a3, a4,) As an example, the sequence (an) = 1/(n2 +1) begins (1, 1/2, 1/5, 1/10, 1/17,..) Here is a useful fact relating sequences and continuity: A function f(x) is continuous at x c if and only if for every sequence (an) that converges to c, written anc, then f(x,) f(c). Alternatively, if you and f(yn)L" with L' L", then f is not continuous at...
Let G be a finite group of order n. Let φ : G → G be the function given by φ(x) = z'n where rn E N. If gcd(rn, n) = 1, show that φ s an injective map. Let G be a finite group of order n. Let φ : G → G be the function given by φ(x) = z'n where rn E N. If gcd(rn, n) = 1, show that φ s an injective map.
mk-G) ( m+1 ). 2. let 1 k m and let f : Iml+ Ikl be a surjective map. Show that ΣmifO mk-G) ( m+1 ). 2. let 1 k m and let f : Iml+ Ikl be a surjective map. Show that ΣmifO
topology Problem 1. (1) Suppose Ti and Tz are two different topologies on a set X. When is the identity map id X X given by id(r) (2) Show that the subspace topology Ty is the smallest topology on YcX for which the inclusion : Y+X is a continuous map. = ra continuous map from (X, Ti) to (X, T2)?
Suppose that (Wn: n 2 0) is an autoregressive sequence of order 1, so that for n 2 0, where the Z's are i.i.d. and independent of Wo. a) Express Wn as a function of Wo, Z1, , Zn Suppose that |ρ| 〈 1 and var(WD+var(ZI) 〈 OO b) Compute cov(Wm, Wn) for m, n 2 0 c) Prove that there exists a deterministic constant a for which as n -oo and compute a. (Hint: Compute var(Wn)) Suppose, in addition,...
m and let f : Iml-Ik] be a surjective map. Show that Σ,f()s mik-()s (":1). 2. Let 1 k m+1
mk-()s (m2'). m+1 [k] be a surjective map. Show that Σ'ıf(j) 2. Let 1 kS m and let f : [m] mk-()s (m2'). m+1 [k] be a surjective map. Show that Σ'ıf(j) 2. Let 1 kS m and let f : [m]