Homework Statement
I have to show that (the question says deduce from the fact that magnetization is monotonically increasing and a concave function for h>0)
\left< \sigma^2_{j} \right> - \left \sigma_j \right>^2 \geq 0
and \left< \left( \sigma^_{j} \right> - \sigma_j \right)^2 \right> \geq 0...
ok thanks for the replies.
I'm having trouble thinking of a suitable function to apply the squeeze theorem with.
If I use \int_{-\infty}^{\infty} \frac{{e}^{ax}}{{e^{x}}} dx = \int_{-\infty}^{\infty} {e}^{x(a-1) dx.
The function works in the \lim_{x\rightarrow\infty} but not as...
Homework Statement
I = \int_{-\infty}^{\infty} \frac{{e}^{ax}}{1+{e}^{x}} dx \; \; 0 < a < 1
a) Show that the improper real integral is absolutely convergent.
b) Integrating around the closed rectangle \boldsymbol{R} with corners -R, R, R+2\pi\iota, -R+2\pi\iota use residue calculus to...
So I have
<[I+T^{*}T]x,x>
and the definition of T^* is <Tx,x>=<x,T^*x>
but I'm not really familiar with <Sx,x>.. this means that S is operating on x, right?
Then another way to say it is
<[I+T^{*}T]x,x>
I'm looking for a way to 'break it up'.. I...
Homework Statement
Let T : V \rightarrow V be a linear operator on a complex inner product space V , and let
S = I + T^{*}T, where I : V \rightarrow V is the identity.
(a) Write <Sx,x> in terms of x and Tx.
(b) Prove that every eigenvalue \lambda of S is real and satisfies \lambda\geq 1.
(c)...
Homework Statement
Let v,w be vectors in a complex inner product space such that ||v|| = 1,
||w|| = 3 and <v,w> = 1 + 2i. Find ||v + iw||.
Homework Equations
The properties of an inner product.
The Attempt at a Solution
I figured
||v+iw||^2 = <v+iw,v+iw>
Then using the...
oops, I see.. but I still get a denominator of
a^{2}+2ab+2*b^{2}
I am really new to this stuff and I'm also having a hard time trying to think of an example which would demonstrate that the field is not algebraically closed. Any hints would be greatly appreciated...
Ok So I have shown that F meets all the field axioms, except I am a little stuck on the multiplicative inverse.
Let x= a+b\omega
So I want some number x^{-1}\times x=1
So I propose that x^{-1} =\frac{1}{a+b \omega}
Let \bar{x} =a - b \omega
I then manipulate it, multiplying by...
Homework Statement
Let \omega = \frac{1}{2} + \frac{\sqrt7}{2}i(a) Verify that \omega^2 = \omega - 2
(b) Prove that F = \{a + b \omega : a, b \in \mathbb{Q} \} is a field, using the usual operations of
addition and multiplication for complex numbers.
(c) Recall that we can think of F as a...
bump: so the centre of the plate moves vertically if one of the springs is different?
and the generalized coordinates are 2 angles (measuring the tilt in each plane) and a displacement coordinate measuring the displacement of the centre of the plane from it's undisturbed position?
how do you...