1. Use theLorentz transformations to showthatthespacetimeinterval is invariant, s'2 =s2 for relative motion along one direction.
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On his 30th birthday, a twin gets on a spaceship travelling at 4/5c which takes him to a distance star system. When he arrives he immediately turns around and comes home at the same speed. His twin sister stays behind on the Earth. Then first twin arrives home on his 48th birthday (by his own watch).
a) How old is his twin si

My question is in three parts and asks for a relation between two space time intervals, an explanation of length contraction and time dilation, then an application of proper time.
i) Two inertial frames O and O' are in standard configuration. Write down the two equations relating spacetimeintervals Delta x' and Delta t' in

The exercise is as follows:
a) Let E be a linear subspace of R^n. Showthat E is a closed subspace of R^n.
b) Let A be a real n x n matrix. Showthat a linear subspace E of R^n is A-invariant if and only if E is e^{tA} -invariant for all t in R, where e^{tA} is the exponential matrix associated to A.

Can you showthatthe momentum P and the kinetic energy T of a particle of rest mass M can be related by: P^2=2TM+T^2.
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A particle of rest mass M moving with speed 'Beta-not' collides inelastically with (i.e. 'sticks to') a stationary particle of rest mass m. Find the speed of the

Please solve the following numerical analysis problem:
Determine the flow Qt : R^2 into R^2 for the nonlinear system: x' =f(x) with
f(x) = [ -x1 ]
[ x1^2 + 2x2 ]
and showthatthe set S = { x E R^2l x2 = -x^2/4 } is invariant with respect to the flow {Qt}.
Plea

Please see the attached file for the fully formatted problems.
B5. (a) Define a homomorphism between topological spaces X and Y. Define what is meant by a topological invariant.
(b) State what it means for a map f X -?> Y to be open. Showthat a continuous open bijection is a homomorphism.
(c) (i) Recall that Fr E, the fron