Suppose a
c mod n and b
d
mod n.
(a) show that a + b
c + d mod n
(b) show that a * b
c * d mod n.

It is given that
is a right triangle,
degrees and BD is the height to the hypotenuse.
i) Find a similarity to show that
.
ii) Find a similarity to show that
iii) Prove that
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Suppose that
a) show that
is a context free language
b) show that
for every
is also context free
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suppose
prove that 0 is the only eigenvalue of N
(hint: fist show 0 is an eigenvalue of N, and then show if
is any
eigenvalue then =0
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Let a and be be in . Show
the following. If gcd(a,b)=1, then for every n in there
exist x and y in such
that n=ax+by.
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a) Suppose we know that the series
is convergent, where the sequence an is nonzero. Show
that the series
is divergent by applying the appropriate test.
b) Suppose we know that the series
is convergent, where the sequence cn consists of
exclusively positive terms. Show that the series
is convergent by applying the appropriate test.
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With the standard Dirac Hamiltonian plus Coulomb potential
below:
a) Show that
.
b) Show that
, where
.
c) Show that
.
d) Since
all mutually commute, they should have common eigenfunctions, and
thus using (c), find the eigenvalues of K2 and K, in
terms of j.
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Suppose that
is nonempty and bounded above. Then
has a supremum.
Note: Show that there is a least element
such that
is an upper bound for
. if
is not a least upper bound for
, show there is at least
such that
is an upper bound for
. Proceed in this way to find the supremum.
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Suppose
is a sequence and that the numbers
,
,
, ... are limit points. Show that 0 is also a limit point.
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Suppose constitute a random sample drawn from a population N(, ) and constitute a random sample drawn from another population N(, ). The two samples are drawn independently. Derive a generalised likelihood ratio test for testing against where and are positive constants such that > . We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageμ2 We were unable to transcribe this imageWe were unable...
Find the indefinite integrals for the following:
a.)
b.)
c.)
d.)
(the exponent is
)
Please show work, thanks!
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