For
$\left|r\right|<1$ , these sums converge to the geometric series
$\sum _{k=0}^{\infty}{r}^{k}=\frac{1}{1-r}$ Differentiation yields the following two useful series:
Exponential series .
${e}^{x}=\sum _{k=0}^{\infty}\frac{{x}^{k}}{k!}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{e}^{-x}=\sum _{k=0}^{\infty}{(-1)}^{k}\frac{{x}^{k}}{k!}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{any}\phantom{\rule{0.277778em}{0ex}}x$ Simple algebraic manipulation yields the following equalities usefulfor the Poisson distribution:
Sums of powers of integers$\sum _{i=1}^{n}i=\frac{n(n+1)}{2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\sum _{i=1}^{n}{i}^{2}=\frac{n(n+1)(2n+1)}{6}$
Some useful integrals
The gamma function$\Gamma \left(r\right)={\int}_{0}^{\infty}{t}^{r-1}{e}^{-t}\phantom{\rule{0.166667em}{0ex}}dt\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}r>0$ Integration by parts shows
$\Gamma \left(r\right)=(r-1)\Gamma (r-1)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}r>1$ By induction
$\Gamma \left(r\right)=(r-1)(r-2)\cdots (r-k)\Gamma (r-k)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}r>k$ For a positive integer
$n,\phantom{\rule{0.277778em}{0ex}}\Gamma \left(n\right)=(n-1)!\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{with}\phantom{\rule{0.277778em}{0ex}}\Gamma \left(1\right)=0!=1$
By a change of variable in the gamma integral, we obtain
For nonnegative integers
$m,n\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\int}_{0}^{1}{u}^{m}{(1-u)}^{n}\phantom{\rule{0.166667em}{0ex}}du=\frac{m!n!}{(m+n+1)!}$
Some basic counting problems
We consider three basic counting problems, which are used repeatedly as components of
more complex problems. The first two,
arrangements and
occupancy are
equivalent. The third is a basic
matching problem.
Arrangements of
r objects selected from among
n distinguishable
objects.
The order is significant.
The order is irrelevant.
For each of these, we consider two additional alternative conditions.
No element may be selected more than once.
Repitition is allowed.
Occupancy of
n distinct cells by
r objects. These objects are
Distinguishable.
Indistinguishable.
The occupancy may be
Exclusive.
Nonexclusive (i.e., more than one object per cell)
The results in the four cases may be summarized as follows:
We reduce the problem to determining
$m(n,0)$ , as follows:
Select
k places for matches in
$C(n,k)$ ways.
Order the
$n-k$ remaining elements so that no matches in the other
$n-k$ places.
$$M(n,k)=C(n,k)M(n-k,0)$$
Some algebraic trickery shows that
$M(n,0)$ is the integer nearest
$n!/e$ . These are
easily calculated by the MATLAB command
M = round(gamma(n+1)/exp(1)) For example
>>M = round(gamma([3:10]+1)/exp(1));>>disp([3:6;M(1:4);7:10;M(5:8)]')3 2 7 1854
4 9 8 148335 44 9 133496
6 265 10 1334961
Extended binomial coefficients and the binomial series
The ordinary
binomial coefficient is
$C(n,k)=\frac{n!}{k!(n-k)!}$ for integers
$n>0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0\le k\le n$ For any real
x , any integer
k , we extend the definition by
For
$x=n$ , a positive integer, the series becomes a polynomial of degree
n .
Cauchy's equation
Let
f be a real-valued function defined on
$(0,\infty )$ , such that
$f(t+u)=f\left(t\right)+f\left(u\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}t,\phantom{\rule{0.277778em}{0ex}}u>0$ , and
There is an open interval
I on which
f is bounded
above (or is bounded below).
Then
$f\left(t\right)=f\left(1\right)t\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}t>0$
Let
f be a real-valued function defined on
$(0,\infty )$ such that
$f(t+u)=f\left(t\right)f\left(u\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}t,\phantom{\rule{0.277778em}{0ex}}u>0$ , and
There is an interval on which
f is bounded above.
Then, either
$f\left(t\right)=0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}t>0$ , or there is a constant
a such that
$f\left(t\right)={e}^{at}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{0.277778em}{0ex}}t>0$
[For a proof, see Billingsley,
Probability and Measure , second edition, appendix A20]
Countable and uncountable sets
A set (or class) is
countable iff either it is finite or its members can be
put into a one-to-one correspondence with the natural numbers.
Examples
The set of odd integers is countable.
The finite set
$\{n:1\le n\le 1000\}$ is countable.
The set of all rational numbers is countable. (This is established by
an argument known as diagonalization).
The set of pairs of elements from two countable sets is countable.
The union of a countable class of countable sets is countable.
A set is
uncountable iff it is neither finite nor can be put into a one-to-one
correspondence with the natural numbers.
Examples
The class of positive real numbers is uncountable. A well known operation shows that
the assumption of countability leads to a contradiction.
The set of real numbers in any finite interval is uncountable, since these can be
put into a one-to-one correspondence of the class of all positive reals.
Questions & Answers
anyone know any internet site where one can find nanotechnology papers?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?