Recent content by tylerc1991

Homework Statement Let f : ℝ2 -> ℝ be some function that is defined on a neighborhood of a point c in ℝ2. If D1f (the derivative of f in the direction of e1) exists and is continuous on a neighborhood of c, and D2f exists at c, prove that f is differentiable at c. Homework Equations...

I see. So there is an electron left over in the first reaction. Now if I calculate the mass deficit using the atomic masses I find 2(1.007825 u) - 2.014102 u - 2(0.0005486 u) = 0.0004508 u, which corresponds to an energy release of 0.4199 MeV. Thank you for your help!

I'm not exactly sure what you mean by this. I've calculated the energies of the other reactions and they seem to correspond to the correct values. It is only the first reaction that is getting me confused. That is, I found a mass deficit of 2(0.0005486 u) = 0.0010972 u for the second...

Homework Statement Find the energy released for the reactions in the Proton-Proton chain. Homework Equations Proton-Proton Chain: 1H + 1H -> 2H + e+ + v e+ + e- -> γ + γ 2H + 1H -> 3He + γ 3He + 3He -> 4He + 2 1H The Attempt at a Solution To find the energy released in each...

Homework Statement I have attached the problem below. Homework Equations The Attempt at a Solution I have tried to use the natural epimorphism from G x G x ... x G to (G x G x ... x G)/(K1 x K2 x ... x Kn), but I do not believe that this is an injective function. Then I tried...

If some root A has multiplicity m > 1, then A will be a root of p' with multiplicity m - 1. Also, I understand that between two simple roots is a root of p'. Now it becomes about counting how many roots p' has. We know that p has n roots by the FTA, and by assumption these are all real. Say...

If all the roots are real, then the number of minima and maxima depend on the multiplicity of the roots. After thinking about a couple of examples, I'll try to generalize here. I'm not 100% sure this is right. If all the roots of a polynomial of degree n are real, then there are either n-1...

Homework Statement Let p be a polynomial. Show that the roots of p' are real if the roots of p are real. Homework Equations The Attempt at a Solution So we start with a root of p', call it r. We want to show that r is real. Judging by the condition given, I am assuming that...

Homework Statement Show that if p is an odd prime of the form 4k + 3 and a is a positive integer such that 1 < a < p - 1, then p does not divide a^2 + 1 Homework Equations If a divides b, then there exists an integer c such that ac = b. The Attempt at a Solution We have to do this proof by...

How about this: Since (d(x, y_n)) is convergent, for all \varepsilon > 0, there exists an N > 0 such that d(x, y_n) < \frac{\varepsilon}{2} when n > N. Therefore, when m, n > N, we have d(y_n, y_m) \leq d(y_n, x) + d(x, y_m) < \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon...

I think so. That definition states that the limit of a convergent sequence is contained in A if and only if A is closed. So to show that A is closed, I would start with an arbitrary convergent sequence of points of A, say (p_n) \to p \in \mathbb{R}, where (p_n) \subset A. I then need to show...

Homework Statement Suppose S is a nonempty closed subset of \mathbb{R}^n, and let x \in \mathbb{R}^n be fixed. Show that A = \{d(x, y) : y \in S\} is closed. Homework Equations A set is closed if its complement is open, or if it contains all of its limit points. The Attempt at a Solution I...

Homework Statement This is a problem from K & K, but I changed it very slightly. A light plane weighing 2,500 lb makes an emergency landing on a short runway. With its engine off, it lands on the runway at 120 ft/s. A hook on the plane snags a cable attached to a 250 lb sandbag and drags...

Exactly. Thank you LCKurtz for your patience! :)

But then doesn't this imply \lim_{n \to \infty} x_n^2 = \infty? completing the problem?