Question

Let λ be a positive irrational real number. If n is a positive integer, choose by the Archimedean Property an integer k such that kλ ≤ n < (k + 1)λ. Let φ(n) = n − kλ. Prove that the set of all φ(n), n > 0, is dense in the interval [0, λ]. (Hint: Examine the proof of the density of the rationals in the reals.)

Answer #1

) Let α be a fixed positive real number, α > 0. For a
sequence {xn}, let x1 > √ α, and define x2, x3, x4, · · · by the
following recurrence relation xn+1 = 1 2 xn + α xn (a) Prove that
{xn} decreases monotonically (in other words, xn+1 − xn ≤ 0 for all
n). (b) Prove that {xn} is bounded from below. (Hint: use proof by
induction to show xn > √ α for all...

Discrete Math
6. Prove that for all positive integer n, there exists an even
positive integer k such that
n < k + 3 ≤ n + 2
. (You can use that facts without proof that even plus even is
even or/and even plus odd is odd.)

This problem outlines a proof that the number π is irrational.
Suppose not, Then there are relatively prime positive integers a
and b for which π = a/b. If p is any polynomial let Ip= ∫0a/bp(x)
sinx dx. i. Show that if p is non-negative and not identically 0 on
[0,a/b] then Ip>0; ii. Show that if p and all of its derivatives
are integer-valued at 0 and a/b then Ip is an integer. iii. Let N
be a large...

let's fix a positive integer n. for a nonnegative integer k, let
ak be the number of ways to distribute k
indistinguishable balls into n distinguishable bins so that an even
number of balls are placed in each bin (allowing empty bins). The
generating function for sequence ak is given as 1/F(x).
Find F(x).

Let f(n) be a negligible function and k a positive integer.
Prove the following:
(a) f(√n) is negligible.
(b) f(n/k) is negligible.
(c) f(n^(1/k)) is negligible.

4) Let F be a finite field. Prove that there exists an integer n
≥ 1, such that n.1F = 0F . Show further that the smallest positive
integer with this property is a prime number.

Let n be a positive integer, and let Hn denote the graph whose
vertex set is the set of all n-tuples with coordinates in {0, 1},
such that vertices u and v are adjacent if and only if they differ
in one position. For example, if n = 3, then (0, 0, 1) and (0, 1,
1) are adjacent, but (0, 0, 0) and (0, 1, 1) are not. Answer the
following with brief justification (formal proofs not
necessary):
a....

Let m be a composite positive integer and suppose that m = 4k
+ 3 for some integer k. If m = ab for some integers a and b, then a
= 4l + 3 for some integer l or b = 4l + 3 for some integer l.
1. Write the set up for a proof by contradiction.
2. Write out a careful proof of the assertion by the method of
contradiction.

Let n be a positive integer and p and
r two real numbers in the interval (0,1). Two random
variables X and Y are defined on a the same
sample space. All we know about them is that
X∼Geom(p) and
Y∼Bin(n,r). (In particular, we do not
know whether X and Y are independent.) For each
expectation below, decide whether it can be calculated with this
information, and if it can, give its value (in terms of p,
n, and r)....

For all problems on this page, use the following setup:
Let N be a positive integer random variable with PMF of the
form
pN(n)=1/2⋅n⋅2^(−n),n=1,2,….
Once we see the numerical value of N, we then draw a random
variable K whose (conditional) PMF is uniform on the set
{1,2,…,2n}.
Write down an expression for the joint PMF pN,K(n,k).
For n=1,2,… and k=1,2,…,2n:
pN,K(n,k)=

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