10. Prove that a connected surface M is orientable iff there exists a nonvanishing differential 2-form on M


10. Prove that a connected surface M is orientable iff there exists a nonvanishing differential 2-form on M 10. Prove t...
Prove that there exists infinitely many numbers of the form an = n(n+1)/2 , for some positive integer n, such that every pair an, am (for n != m) are relatively prime. [Hint: Assume there exists a finite sequence an1 < an2 < an3 < . . . < anm, where nj are increasing positive integers. Show that using those numbers we can construct a new number that fulfills the requirements.]
Let G be a connected graph with m 2 vertices of odd degree. Prove that once is m/2.
Let G be a connected graph with m 2 vertices of odd degree. Prove that once is m/2.
Consider a linear system of the form d/dt x(t) = Ax. Next suppose there exists a positive definite solution to the matrix inequality: A^TP + PA < 0 where M<0 means M is negative definite. Then prove that all eigenvalues of A have negative real part.
Exercise 3. [10 pts Let n 2 1 be an integer. Prove that there exists an integer k 2 1 and a sequence of positive integers al , a2, . . . , ak such that ai+1 2 + ai for all i-1, 2, . . . , k-1 and The numbers Fo 0, F1 1, F2 1, F3 2 etc. are the Fibonacci numbers
A piece of pure zinc with a surface area of 10 cm2 is connected
to a piece of platinum in 0.1 M HCl. The metals are connected
through an ammeter that reads 5 mA. The zinc was corroding at a
rate of 0.1 mA/cm2 before it was connected to the platinum. Based
on the given information, can we accurately determine the corrosion
rate of the zinc (in mm/year)? If yes, calculate the accurate
corrosion rate; if no, give your best...
4.0 m #2-Prove that the net flux through the closed surface is zero. Hint: Calculate the flux through each surface and add them 30°
6. (10 points) Prove that the language L = {< M 1 , M 2 >: M, , M 2 are T M s and L(M-) = L(M 2)) is undecidable.
6. (10 points) Prove that the language L = {: M, , M 2 are T M s and L(M-) = L(M 2)) is undecidable.
prove the following
1)
2)
3)
4)A parallelogram is a square iff it's diagonals are
perpendicular and congruent.
5) the median of a trapeziod is parallel to each
base
3.7) Corollary (Parallel CT). Let l, and l be coplanar lines and I a transversal. a. (Property C) 4 | l, if and only if a pair of interior angles on the same side of t are supplementary b. (Property T) Ift 1 l and 41 || 12, then t 1...
A differential nitrogen pressure exists across a 2-mm-thick steel furnace wall. After some time, steady-state diffusion of the nitrogen is established across the wall. Given that the nitrogen concentration on the high-pressure surface of the wall is 5 kg/m3kg/m3 and on the low-pressure surface is 0.3 kg/m3 , calculate the flow of nitrogen through the wall (in kg/m2.h) if the diffusion coefficient for nitrogen in this steel is 1.0 ×10−10m2/s at the furnace operating temperature. Express your answer to three...
2. (Connected sums) Recall that the connected sum M #M2 of two (path connected) manifolds M and M2 is obtained from the disjoint union of Mi and M2 by removing the interior of a closed n-ball Bi fron Mi (i = 1,2) and gluing together the two boundary (n 1)-spheres by a homeomorphism π1(M,,p) *n(My, P2), Prove for appropriate base point p provided n 2 3. M, #My, PE M, that π1(M, #My, p)
2. (Connected sums) Recall that the...