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such that the equality 
holds for all pairs of real numbers 
.
, whether or not the polynomial 
irreducible over 
?
from a distribution with probability density function (PDF) 
for 
, where the parameter 
has a prior distribution with PDF 
. Consider the following two approaches to Bayesian updating:
be the complete data vector. Denote the posterior PDF as 
, where 
, obtained by applying Bayes' rule to the full dataset at once.
. For each , let 
be the current prior and update it using observation 
to obtain the new posterior:
from part (a) and 
from part (b) the same? Provide a proof or a counterexample.
be a non constant polynomial with real coefficients. Prove that the system of simultaneous equations —
has finitely many solutions 
.
Why we can't write 
as 
if 
is even.
a positive integer . suppose that there exist two automorphisms 
of 
such that 
. is even.
,we get 
. Now just with a simple example we can see that 
goes to infinity. Take 
to get smaller than 
,but sufficiently large sum which can be higher 
.
,we can use the AM-GM theorem, to have ![$3 \geq ab+bc+ca \geq 3\sqrt[3]{a^2b^2c^2}$](http://latex.artofproblemsolving.com/9/0/b/90bb6793e43c09da96db20110614929a5d33cc6f.png)
.So, 
.With the example below,we can get that limit of goes to zero in positive numbers:
and 
is an obvious solution. Now, let 
.
or 
.
.
,
such that
Find the range of values that can be taken by :
are positive reals.
and 
Prove that
and ![$ab+bc+ca \geq 3\sqrt[3]{a^2b^2c^2}$](http://latex.artofproblemsolving.com/f/1/1/f115c481a345f3ef53d5e39d4cca4248e32aab63.png)
finishes.