1. Let ω be a k-form in Rn , Π 〉 k. If k is odd, show that ωΛω 0 4. L et ω be a k-form and let )...
et ω be a k-form and let be a /-form. Fin
et ω be a k-form and let be a /-form. Fin
proof:
Let be a k-form and η be a 1-form on Rn
Let be a k-form and η be a 1-form on Rn
Let ω be a k-form and let η be a 1-form. Find d(don η-ω^ dn)
(1) Let w1, be a k-form and w2 be an l- form, both defined in an open subset UC R3. Let d : /\k (U)-ל ЛК +1 (U) be the exterior derivative of differential forms. (a) Show that d is a linear transformation of vector spaces. (b) Show that (c) Show that (d) Show that d(w) -d(d(w)) 0 for every k-form w, i.e. the map is the zero map
(1) Let w1, be a k-form and w2 be an l-...
Let L1 = {ω|ω begins with a 1 and ends with a 0}, L2 = {ω|ω has
length at least 3 and its third symbol is a 0}, and L3 = {ω| every
odd position of ω is a 1} where L1, L2, and L3 are all languages
over the alphabet {0, 1}. Draw finite automata (may be NFA) for L1,
L2, and L3 and for each of the following (note: L means complement
of L):
Let L w begins...
1. Let U с Rn be open, f : U-> Rm be a function, a є U and 0 exists. Show that DAwf(a) exists for every 0メλ R, and DAwf(a) Rn such that Duf(a) λDuf(a). 3 marks
1. Let U с Rn be open, f : U-> Rm be a function, a є U and 0 exists. Show that DAwf(a) exists for every 0メλ R, and DAwf(a) Rn such that Duf(a) λDuf(a). 3 marks
4. Let L2(-π, π)) be the Lebesgue space of square integrable functions f: [-π, π] → C with inner-product, (f,g) =| f(t)g(t)dt (a) Show thatkt k e is an orthonormal system 2rZ s an orthonormal system (b) Let M be the linear span of (1, et, e). Find the point in M closest to the function [4 marks] 2π f(t) = t. [6 marks]
4. Let L2(-π, π)) be the Lebesgue space of square integrable functions f: [-π, π] →...
p-1 mod 4, prove that Σ k ( )-0. Let p be an odd prıme. Suppose that p k=1
p-1 mod 4, prove that Σ k ( )-0. Let p be an odd prıme. Suppose that p k=1
Σ.ix Problem 8. I et Xi,Xy ,x, be independert (0, 1) randoua variables. Let rn.- for k 1,2, 3. Are there numbers mi, m2, ms such that Arethmann (三),(三) rrn2 rng holds? If so, calculate these numbers.
Problem 4. Let K be a field and let f ∈ K[x]. Show that if 1+f2 has a factor of odd degree in K[x] then there is an a ∈ K such that a2 = −1.