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### Why I still haven’t found a life partner

Yesterday I downloaded an audio book from Audible.com – which is called The Mathematics of Love, written by Hannah Fry who seems to be a pretty charming mathematician, who also from time to time making some YouTube Numberphile videos such as this one.

Never mind… The point is in her book she mentioned a method I used previously on a beer conversation with my lovely colleagues, a method can help you calculate how many potential life partners are out there waiting for you to be find. (I guess every single man or woman with a little of brain function would have probably thought about the same question at least once when he or she was half way intoxicated and there happened to be a young couple across the bar kissing as if they were the main actors of the show that evening…)

Anyway, the method is, actually very simple, just using the number of total potential partners, multiply by a few factors each from a certain perspective can help you estimate how many percentage of them are actually can be your life partner. It said to be the same way scientists try estimate the number of highly evolved civilizations that might exist in our galaxy (Drake equation), mentioned by Hannah in her book when referring a paper called Why I don’t have a girlfriend – by Peter Backus.

$G = N \cdot f_W \cdot f_A \cdot f_S \cdot f_U \cdot f_{B1} \cdot f_{B2} \cdot f_{P1} \cdot f_{P2} \cdot f_f$

I remember I did the calculation, once, for the fun of it, together with my colleagues, and the result was quite shocking. So I am going to do it here again:

Suppose I live in Denmark, and the whole population is around N=6,000,000, and we assume:

• $f_W$ = the fraction of people in DK who are women = 0.5;
• $f_A$ = the fraction of the women in DK who are age-appropriate = 10 years out of 70 years (suppose the age is evenly distributed) = 1/7;
• $f_S$ = the fraction of the women with the right age are single = 0.5;
• $f_U$ = the fraction of the women who have a university education = 0.26 (no offense but I have the same reason with the guy mentioned in the paper Why I don’t have a girlfriend;
• $f_{B1}$ = the fraction of the women with the right age with the right education who I find physically attractive = 2 out of 10 = 0.2;
• $f_{B2}$ = the fraction of the women with the right age with the right education who find I physically attractive = 2 out of 10 = 0.2; (Let’s say I am average looking, if you don’t agree, comments below and I will reduce this value);
• $f_{P1}$ = the fraction of the women with the right age with the right education who we both find attractive and who think my personality is acceptable also enjoy being with me, or you may call it have a word to say between = 0.5;
• $f_{P2}$ = chance that I enjoy being with her, same above = 0.5;
• $f_f$ = sadly because I am a foreigner who don’t speak Danish yet so maybe half of the women in the end decided not to date me = 0.5.

Thus, we can easily calculation G, which is the possible number of ladies in Denmark could become a life partner of mine:

$G = 6e10^6 \cdot 0.5 \cdot {1 \over 7} \cdot 0.5 \cdot 0.26 \cdot 0.2 \cdot 0.2 \cdot 0.5 \cdot 0.5 \cdot 0.5 = 279$

My goodness, only 279 women are out there in DK could be my life partner, based on a positive estimation.

However since I live in a town called Odense, suppose if I don’t go out of the town and just try to meet some local people, with the population of 190,000 in Odense, the number G becomes 8.8. It seems not that bad, but if you look at the numbers again, you will see, statistically, out of every thousand girls at the right age, there’s only a half girl could potentially become my life partner. What a lonely planet ?!

The next question to think about is, how to identify those 279 ladies in Denmark, I mean at least efficiently using my spare time. At the moment, I have no idea 🙂

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# {\int_{{0}}^{{a}}}{3}{x}^{2}\sqrt{{{a}^{2}-{x}^{2}}}{\left.{d}{x}\right.}

$${\int_{{0}}^{{a}}}{3}{x}^{2}\sqrt{{{a}^{2}-{x}^{2}}}{\left.{d}{x}\right.}$$

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Take the integral:
$$\int{3}{x}^{2}\sqrt{{{a}^{2}-{x}^{2}}}{\left.{d}{x}\right.}$$
Factor out constants:
$$\int{3}{x}^{2}\sqrt{{{a}^{2}-{x}^{2}}}{\left.{d}{x}\right.}$$
For the integrand $${x}^{2}\sqrt{{{a}^{2}-{x}^{2}}}$$, (assuming all variables are positive) substitute $${x}={a} \sin{{\left({u}\right)}}{\quad\text{and}\quad}{\left.{d}{x}\right.}={a} \cos{{\left({u}\right)}}{d}{u}$$. Then $$\sqrt{{{a}^{2}-{x}^{2}}}=\sqrt{{{a}^{2}-{a}^{2}{{\sin}^{2}{\left({u}\right)}}}}={a} \cos{{\left({u}\right)}}{\quad\text{and}\quad}{u}={{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}$$:
$$={3}{a}\int{a}^{3}{{\sin}^{2}{\left({u}\right)}}{{\cos}^{2}{\left({u}\right)}}{d}{u}$$
Factor out constants:
$$={3}{a}^{4}\int{{\sin}^{2}{\left({u}\right)}}{{\cos}^{2}{\left({u}\right)}}{d}{u}$$
Write $${{\cos}^{2}{\left({u}\right)}}{a}{s}{1}-{{\sin}^{2}{\left({u}\right)}}$$:
$$={3}{a}^{4}\int{{\sin}^{2}{\left({u}\right)}}{\left({1}-{{\sin}^{2}{\left({u}\right)}}\right)}{d}{u}$$
Expanding the integrand $${{\sin}^{2}{\left({u}\right)}}{\left({1}-{{\sin}^{2}{\left({u}\right)}}\right)}$$ gives $${{\sin}^{2}{\left({u}\right)}}-{{\sin}^{4}{\left({u}\right)}}$$:
$$={3}{a}^{4}\int{\left({{\sin}^{2}{\left({u}\right)}}-{{\sin}^{4}{\left({u}\right)}}\right)}{d}{u}$$
Integrate the sum term by term and factor out constants:
$$=-{3}{a}^{4}\int{{\sin}^{4}{\left({u}\right)}}{d}{u}+{3}{a}^{4}\int{{\sin}^{2}{\left({u}\right)}}{d}{u}$$
Use the reduction formula, $$\int{{\sin}^{m}{\left({u}\right)}}{d}{u}=-\frac{{ \cos{{\left({u}\right)}}{{\sin}^{{{m}-{1}}}{\left({u}\right)}}}}{{m}}+\frac{{{m}-{1}}}{{m}}\int{{\sin}^{{-{2}+{m}}}{\left({u}\right)}}{d}{u}$$, where m = 4:
$$=\frac{3}{{4}}{a}^{4}{{\sin}^{3}{\left({u}\right)}} \cos{{\left({u}\right)}}+\frac{{{3}{a}^{4}}}{{4}}\int{{\sin}^{2}{\left({u}\right)}}{d}{u}$$
Write $${{\sin}^{2}{\left({u}\right)}}{a}{s}\frac{1}{{2}}-\frac{1}{{2}} \cos{{\left({2}{u}\right)}}$$:
$$=\frac{3}{{4}}{a}^{4}{{\sin}^{3}{\left({u}\right)}} \cos{{\left({u}\right)}}+\frac{{{3}{a}^{4}}}{{4}}\int{\left(\frac{1}{{2}}-\frac{1}{{2}} \cos{{\left({2}{u}\right)}}\right)}{d}{u}$$
Integrate the sum term by term and factor out constants:
$$=\frac{3}{{4}}{a}^{4}{{\sin}^{3}{\left({u}\right)}} \cos{{\left({u}\right)}}-\frac{{{3}{a}^{4}}}{{8}}\int \cos{{\left({2}{u}\right)}}{d}{u}+\frac{{{3}{a}^{4}}}{{8}}\int{1}{d}{u}$$
For the integrand cos(2 u), substitute $$s = 2 u$$ and $$ds = 2 du$$:
$$=\frac{3}{{4}}{a}^{4}{{\sin}^{3}{\left({u}\right)}} \cos{{\left({u}\right)}}-\frac{{{3}{a}^{4}}}{{16}}\int \cos{{\left({s}\right)}}{d}{s}+\frac{{{3}{a}^{4}}}{{8}}\int{1}{d}{u}$$
The integral of $$\cos(s)$$ is $$\sin(s)$$:
$$=-\frac{3}{{16}}{a}^{4} \sin{{\left({s}\right)}}+\frac{3}{{4}}{a}^{4}{{\sin}^{3}{\left({u}\right)}} \cos{{\left({u}\right)}}+\frac{{{3}{a}^{4}}}{{8}}\int{1}{d}{u}$$
The integral of 1 is u:
$$=-\frac{3}{{16}}{a}^{4} \sin{{\left({s}\right)}}+\frac{{{3}{a}^{4}{u}}}{{8}}+\frac{3}{{4}}{a}^{4}{{\sin}^{3}{\left({u}\right)}} \cos{{\left({u}\right)}}+\text{constant}$$
Substitute back for $$s = 2 u$$:
$$=\frac{{{3}{a}^{4}{u}}}{{8}}+\frac{3}{{4}}{a}^{4}{{\sin}^{3}{\left({u}\right)}} \cos{{\left({u}\right)}}-\frac{3}{{8}}{a}^{4} \sin{{\left({u}\right)}} \cos{{\left({u}\right)}}+\text{constant}$$
Substitute back for $${u}={{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}$$:
$$=\frac{3}{{8}}{a}^{4}{{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}+\frac{3}{{4}}{a}^{4}{ \sin{{\left({{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}\right)}}^{3} \cos{{\left({{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}\right)}}}-\frac{3}{{8}}{a}^{4} \sin{{\left({{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}\right)}} \cos{{\left({{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}\right)}}+\text{constant}$$
Simplify using $$\cos{{\left({{\sin}^{{-{1}}}{\left({z}\right)}}\right)}}=\sqrt{{{1}-{z}^{2}}}{\quad\text{and}\quad} \sin{{\left({{\sin}^{{-{1}}}{\left({z}\right)}}\right)}}={z}$$:
$$=\frac{3}{{8}}{a}^{4}{{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}-\frac{3}{{8}}{a}^{2}{x}\sqrt{{{a}^{2}-{x}^{2}}}+\frac{3}{{4}}{x}^{3}\sqrt{{{a}^{2}-{x}^{2}}}+\text{constant}$$
Factor the answer a different way:
$$=\frac{1}{{8}}{\left({3}{a}^{4}{{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}+{3}{x}\sqrt{{{a}^{2}-{x}^{2}}}{\left({2}{x}^{2}-{a}^{2}\right)}\right)}+\text{constant}$$
Which is equivalent for restricted x and a values to:
$$=\frac{3}{{8}}\sqrt{{{a}^{2}-{x}^{2}}}{\left(-{a}^{2}{x}+\frac{{{a}^{3}{{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}}}{\sqrt{{{1}-\frac{{x}^{2}}{{a}^{2}}}}}+{2}{x}^{3}\right)}+\text{constant}$$


text:
Remember that Haskell is lazy and therefore pure. This means two primary things:

1. Functions may not have any external effects. For example, a function may not print anything on the screen. Functions may only compute their outputs.

2. Functions may not depend on external stuff. For example, they may not read from the keyboard, or filesystem, or network. Functions may depend only on their inputs—put another way, functions should give the same output for the same input every time.

But—sometimes we do want to be able to do stuff like this! If the only thing we could do with Haskell is write functions which we can then evaluate at the ghci prompt, it would be theoretically interesting but practically useless.

In fact, it is possible to do these sorts of things with Haskell, but it looks very different than in most other languages.

# The IO type

The solution to the conundrum is a special type called IO. Values of type IO a are descriptions of effectful computations, which, if executed would (possibly) perform some effectful I/O operations and (eventually) produce a value of type a. There is a level of indirection here that’s crucial to understand. A value of type IO a, in and of itself, is just an inert, perfectly safe thing with no effects. It is just a description of an effectful computation. One way to think of it is as a first-class imperative program.

As an illustration, suppose you have

c :: Cake

What do you have? Why, a delicious cake, of course. Plain and simple.

By contrast, suppose you have

r :: Recipe Cake

What do you have? A cake? No, you have some instructions for how to make a cake, just a sheet of paper with some writing on it.

Not only do you not actually have a cake, merely being in possession of the recipe has no effect on anything else whatsoever. Simply holding the recipe in your hand does not cause your oven to get hot or flour to be spilled all over your floor or anything of that sort. To actually produce a cake, the recipe must be followed (causing flour to be spilled, ingredients mixed, the oven to get hot, etc.).

In the same way, a value of type IO a is just a “recipe” for producing a value of type a (and possibly having some effects along the way). Like any other value, it can be passed as an argument, returned as the output of a function, stored in a data structure, or (as we will see shortly) combined with other IO values into more complex recipes.

So, how do values of type IO a actually ever get executed? There is only one way: the Haskell compiler looks for a special value

main :: IO ()

which will actually get handed to the runtime system and executed. That’s it! Think of the Haskell runtime system as a master chef who is the only one allowed to do any cooking.

If you want your recipe to be followed then you had better make it part of the big recipe (main) that gets handed to the master chef. Of course, main can be arbitrarily complicated, and will usually be composed of many smaller IO computations.

So let’s write our first actual, executable Haskell program! We can use the function

putStrLn :: String -> IO ()

which, given a String, returns an IO computation that will (when executed) print out that String on the screen. So we simply put this in a file called Hello.hs:

main = putStrLn "Hello, Haskell!"

Then typing runhaskell Hello.hs at a command-line prompt results in our message getting printed to the screen! We can also use ghc -O Hello.hs to produce an executable version called Hello (or Hello.exe on Windows).

## There is no String “inside” an IO String

Many new Haskell users end up at some point asking a question like “I have an IO String, how do I turn it into a String?”, or, “How do I get the String out of an IO String”? Given the above intuition, it should be clear that these are nonsensical questions: a value of type IO String is a description of some computation, a recipe, for generating a String. There is no String “inside” an IO String, any more than there is a cake “inside” a cake recipe. To produce a String (or a delicious cake) requires actually executing the computation (or recipe). And the only way to do that is to give it (perhaps as part of some larger IO value) to the Haskell runtime system, via main.

## Combining IO

As should be clear by now, we need a way to combine IO computations into larger ones.

### Sequencing

Thinking about our recipe analogy, we could combine two recipes by writing one below the other (and discarding the result of the first one). Does it make sense to perform the steps of a recipe without using the result? Yes, it may: There might be a recipe for “heating up the oven”, and although no food comes out of it, we still want to perform it for the effects it have.

We can do the equivalent thing with IO actions, using the (>>) operator (pronounced “and then”), which has the type

(>>) :: IO a -> IO b -> IO b

This simply creates an IO computation which consists of running the two input computations in sequence. Notice that the result of the first computation is discarded; we only care about it for its effects. For example:

main = putStrLn "Hello" >> putStrLn "world!"

### Tupling

This works fine for code of the form “do this; do this; do this” where the results don’t really matter. However, in general this is insufficient. What if we don’t want to throw away the result from the first computation?

Speaking in terms of recipes, it would be nice to be able to tack two recipes together and get both results, i.e. to create a multi-course menu.

For that, we would need a function of type IO a -> IO b -> IO (a,b), or, more generally, a function

liftM2 :: (a -> b -> r) -> IO a -> IO b -> IO r

where we can give a (pure) function that indicates how to combine the two courses.

### Bind

However, this is still insufficient. The recipe analogy gets a bit stretched here, but imagine you have a recipe, suitable for very late night hunger attacks, of the form “pull some random stuff from the freezer”. That would have type Recipe FrozenStuff. Now how to prepare the frozen stuff depends on what particular thing you pulled out, and a frozen pizza is prepared differently than a frozen stew. So you need to actually look at FrozenStuff to decide what recipe to use to make the stuff yummy. So the second step is actually a function from frozen stuff to a recipe that produces yummy stuff: FrozenStuff -> Recipe YummyStuff, and you will notice that we cannot combine these two recipes using liftM2.

Therefore, there is the operator (>>=) (pronounced “bind”) with the type

(>>=) :: IO a -> (a -> IO b) -> IO b

It takes a computation which will produce a value of type a, and a function which gets to compute a second computation based on this intermediate value of type a. The result of (>>=) is a (description of a) computation which performs the first computation, uses its result to decide what to do next, and then does that.

For example, we can write a program to read a number from the user and print out its successor. Note our use of readLn :: Read a => IO a which is a computation that reads input from the user and converts it into any type which is an instance of Read.

main :: IO ()
main = putStrLn "Please enter a number: " >>
readLn >>= \n ->
let m = n + 1 in
putStrLn (show m)

Of course, this looks kind of ugly, but there are better ways to write it, which we’ll talk about in late today.

Side remark: readLn is not a good citizen, as it throws an exception if the input value is not a number. In real code, better write something like:

import Text.Read
main = putStrLn "Hello World. Please enter a number:" >>
getLine >>= \s ->
case readMaybe s of
Just n ->  let m = n + 1 in
putStrLn (show m)
Nothing -> putStrLn "That’s not a number! Try again"

While introducing the IO combinators, I talked about recipes. So there seems to be some abstract concept here, and because we like abstraction, let us search for more instances of the pattern.

## Maybe a monad?

Consider the following function, which combines two trees if they have the same shape, and returns Nothing if they do not have the same shape:

data Tree a = Node (Tree a) a (Tree a) | Empty deriving (Show)

zipTree :: (a -> b -> c) -> Tree a -> Tree b -> Maybe (Tree c)
zipTree _ (Node _ _ _) Empty = Nothing
zipTree _ Empty (Node _ _ _) = Nothing
zipTree _ Empty Empty        = Just Empty
zipTree f (Node l1 x r1) (Node l2 y r2) =
case zipTree f l1 l2 of
Nothing -> Nothing
Just l  -> case zipTree f r1 r2 of
Nothing -> Nothing
Just r  -> Just $Node l (f x y) r Clearly, there is some repetition in the last case that obscures whats going on. Twice, we look at some value of type Maybe c. If it is Nothing, we return Nothing. If it is Just something, we continue with some code that uses something. But that is precisely the pattern that we saw above when we combined recipes or IO actions. And in fact if we factor out the pattern as a separate function, we find that that function has a type signature quite similar to bind: bindMaybe :: Maybe a -> (a -> Maybe b) -> Maybe b bindMaybe mx f = case mx of Nothing -> Nothing Just x -> f x zipTree :: (a -> b -> c) -> Tree a -> Tree b -> Maybe (Tree c) zipTree _ (Node _ _ _) Empty = Nothing zipTree _ Empty (Node _ _ _) = Nothing zipTree _ Empty Empty = Just Empty zipTree f (Node l1 x r1) (Node l2 y r2) = bindMaybe (zipTree f l1 l2)$ \l ->
bindMaybe (zipTree f r1 r2) \$ \r ->
Just (Node l (f x y) r)

This is already better. But if bindMaybe and the bind of IO have such a similar type signature, and capture such similar concepts, then we would want to use the same name for it.

## Bind is overloaded

Luckily, we can. In fact, we can use (>>=), as it is in fact overloaded using the Monad type class, of which both IO and Maybe are instances:

zipTree :: (a -> b -> c) -> Tree a -> Tree b -> Maybe (Tree c)
zipTree _ (Node _ _ _) Empty = Nothing
zipTree _ Empty (Node _ _ _) = Nothing
zipTree _ Empty Empty        = Just Empty
zipTree f (Node l1 x r1) (Node l2 y r2) =
zipTree f l1 l2 >>= \l ->
zipTree f r1 r2 >>= \r ->
Just (Node l (f x y) r)

Despite their scary reputation, there’s nothing all that frightening about monads. The concept of a monad started its life as an abstract bit of mathematics from the field of Category Theory (Monads are just monoids on the category of endofunctors!). It so happened that functional programmers stumbled upon it as a useful programming construct!

A monad is handy whenever a programmer wants to sequence actions. The details of the monad says exactly how the actions should be sequenced. A monad may also store some information that can be read from and written to while performing actions.

We’ve already learned about the IO monad, which sequences its actions quite naturally, performing them in order, and gives actions access to read and write anything, anywhere. We’ll also see the Maybe and [] (pronounced “list”) monads, which don’t give any access to reading and writing, but do interesting things with sequencing. There are other monads such as the Rand monad, which doesn’t much care about sequencing, but it does allow actions to read from and update a random generator.

One of the beauties of programming with monads is that monads allow programmers to work with mutable state from a pure language. Haskell doesn’t lose its purity when monads come in (although monadic code is often called “impure”). Instead, the degree to which code can be impure is denoted by the choice of monad. For example, the Rand monad means that an action can generate random numbers, but can’t for example, write strings to the user. And the Maybe monad doesn’t give you any extra capabilities at all, but makes writing possibly-erroring computations much easier to write.

In the end, the best way to really understand monads is to work with them for a while. After programming using several different monads, you’ll be able to abstract away the essence of what a monad really is.

## Another example: Lists

If lists are an instance of monads, then what does its bind operator do? Its type is

(>>=) :: [a] -> (a -> [b]) -> [b]

From the type signature, we can guess that it replaces every a in the first list with the list produced by the second argument, producing one flat list of bs.

The mental model here is that [a] represents nondeterministic programs that can have multiple possible results.

Here is one example using it, which calculates the cartesian product of the given list and then keeping the ordered pairs:

ordPairs :: Ord t => [a] -> [(a, a)]
ordPairs xs =
xs >>= \x1 ->
xs >>= \x2 ->
if x1 < x2 then [(x1,x2)] else []

(This is the first time you see if…then…else. It does what you expect.)

## The other methods of Monad

Above, we have already seen the >> combinator for IO. Indeed, that operator is also overloaded:

(>>) :: Monad m => m a -> m b -> m b

It is, in fact, just a special case of (>>=):

f >> g == f >>= \_ -> g

What does it do for Maybe? What does it do for the list monad?

The other method that Monad provides is this:

return :: Monad m => a -> m a

The name is a bit misleading, as it is unrelated to return as a control statement in other programming languages. And maybe it should have been called pure instead…

The purpose of this function is to take something you already have, and then create the “computation” that produces this thing. In terms of recipes: Consider a cake with a label attached: “Recipe. Done”.

What is return for Maybe, and what for lists? (For IO the implementation is hidden, so it would not be fair to ask that.)

The point of having this is to write generic monad functions that you can use with IO, or with Maybe or with lists, or with one of the myriads of other types that have a Monad instance. Here is one example:

mapM :: Monad m => (a -> m b) -> [a] -> m [b]
mapM f [] = return []
mapM f (a:as) = f a >>= \b -> mapM f as >>= \bs -> return (b:bs)

## The Monad laws

Two weeks ago, when we introduced type classes, I said that good type classes are those that come with laws, and Monad is one such type class. The following three are called the monad laws:

return a >>= k          = k a
m >>= return            = m
m >>= (\x -> k x >>= h) = (m >>= k) >>= h

These effectively say that return really has no effect (whatever effect is supposed to mean), and that bind is associative, i.e. it does not matter which way these things nest.

## Do notation

Now that we have monads that allow us to write code that do IO (in addition to other fancy stuff, like nondeterministic calculuations), the only thing missing is nice syntax. And we have that. Recall the main function above:

main :: IO ()
main = putStrLn "Please enter a number: " >>
readLn >>= \n ->
let m = n + 1 in
putStrLn (show m)

We can rewrite that using do-notation as follows:

main' :: IO ()
main' = do putStrLn "Please enter a number: "
let m = n + 1
putStrLn (show m)

That looks almost like your good old imperative programming language, doesn’t it? No wonder there are people that say that Haskell is the best imperative programming language…

Keep in mind that this is purely syntactic sugar, and both functions are equivalent. Every line in a do block can be either

• A single expression e, which is then prepended to the rest with e >> …
• A monadic bind p <- e, which is replaced by e >>= (\p -> …). On the left of the arrow you can not only use a single variable, but also a full pattern, for example to take a pair apart.
• A let binding let p = e, which is translated to let e = p in ….

Here, as another example, is the zipTree example, rewritten using do-notation and return:

zipTree :: (a -> b -> c) -> Tree a -> Tree b -> Maybe (Tree c)
zipTree _ (Node _ _ _) Empty = Nothing
zipTree _ Empty (Node _ _ _) = Nothing
zipTree _ Empty Empty        = return Empty
zipTree f (Node l1 x r1) (Node l2 y r2) =
do l <- zipTree f l1 l2
r <- zipTree f r1 r2
return (Node l (f x y) r)


text:
Next: Alternative forms of Lagrange Up: Derivation of Lagrange planetary Previous: Transformation of Lagrange brackets

# Lagrange planetary equations

Now,

 (G.86) (G.87) and (G.88)

with . Hence,

 (G.89) (G.90) (G.91) (G.92) (G.93) and (G.94)

with all other partial derivatives zero. Thus, from Equation (G.85), the only non-zero Lagrange brackets are

 (G.95) (G.96) (G.97) (G.98) (G.99) and (G.100)

Hence, Equations (G.32) yield

 (G.101) (G.102) (G.103) (G.104) (G.105) and (G.106)

Finally, Equations (G.95)-(G.106) can be rearranged to give

 (G.107) (G.108) (G.109) (G.110) (G.111) and (G.112)

Equations (G.107)-(G.112), which specify the time evolution of the osculating orbital elements of our planet under the action of the disturbing function, are known collectively as the Lagrange planetary equations (Brouwer and Clemence 1961).

In fact, the orbital element always appears in the disturbing function in the combination . This combination is known as the mean longitude, and is denote . It follows that

 (G.113) and (G.114)

The integral appearing in the previous equation is problematic. Fortunately, it can easily be eliminated by replacing the variable by . In this case, the Lagrange planetary equations become

 (G.115) (G.116) (G.117) (G.118) (G.119) and (G.120)

where is taken at constant , and at constant (Brouwer and Clemence 1961).

Next: Alternative forms of Lagrange Up: Derivation of Lagrange planetary Previous: Transformation of Lagrange brackets
Richard Fitzpatrick 2016-03-31


text:
# Math Help - show that f has a fixed point

1. ## show that f has a fixed point

If f:[a,b]-->[a,b] is continuous, shwo that f has a fixed point; that is, show that there is some point x in [a,b] with f(x)=x.

I'm really struglling with the continuous functions section with metric space, any help will be appreaciated!

3. ## Re: show that f has a fixed point

If f(a)= a we are done. If not, then, because f(a) must be in [a, b], f(a)> a.
If f(b)= b we are done. If not, then, because f(b) must be in [a, b], f(b)< b

Let $\Phi(x)= f(x)- x$.


text:
Ask question
Question

# Write the equation of the circle described. a. Center at the origin, containing the point (-6, -8) b. Center (7, 5), containing the point (3, -2).

Transformations of functions
ANSWERED
asked 2021-06-26
Write the equation of the circle described.
a. Center at the origin, containing the point (-6, -8)
b. Center (7, 5), containing the point (3, -2).

## Answers (1)

2021-06-27

Use the standard equation of a circle with center (h,k) and radius r:
$$\displaystyle{\left({x}−{h}\right)}^{{2}}+{\left({y}−{k}\right)}^{{2}}={r}^{{2}}$$
a.Using $$(h,k)=(0,0)$$ and $$(x,y)=(−6,−8)$$, solve for $$\displaystyle{r}^{{2}}$$:
$$\displaystyle{\left(−{6}−{0}\right)}^{{2}}+{\left(−{8}−{0}\right)}^{{2}}={r}^{{2}}$$
$$\displaystyle{36}+{64}={r}^{{2}}$$
$$\displaystyle{100}={r}^{{2}}$$
So, the equation of the circle is:
$$\displaystyle{\left({x}−{0}\right)}^{{2}}+{\left({y}−{0}\right)}^{{2}}={100}$$
$$\displaystyle{x}^{{2}}+{y}^{{2}}={100}$$
b.Using $$(h,k)=(7,5)$$ and $$(x,y)=(3,−2)$$, solve for $$\displaystyle{r}^{{2}}$$:
$$\displaystyle{\left({3}−{7}\right)}^{{2}}+{\left(−{2}−{5}\right)}^{{2}}={r}^{{2}}$$
$$\displaystyle{16}+{49}={r}^{{2}}$$
$$\displaystyle{65}={r}^{{2}}$$
So, the equation of the circle is:
$$\displaystyle{\left({x}−{7}\right)}^{{2}}+{\left({y}−{5}\right)}^{{2}}={65}$$

expert advice

...


text:
# How do you evaluate f(x)=7x^3+9x^2+3x at x=10 using direct substitution and synthetic division?

Sep 11, 2016

$7930$

#### Explanation:

We have: $f \left(x\right) = 7 {x}^{3} + 9 {x}^{2} + 3 x$; $x = 10$

Let's substitute $10$ in place of $x$:

$\implies f \left(10\right) = 7 {\left(10\right)}^{3} + 9 {\left(10\right)}^{2} + 3 \left(10\right)$

$\implies f \left(10\right) = 7 \cdot 1000 + 9 \cdot 100 + 30$

$\implies f \left(10\right) = 7000 + 900 + 30$

$\implies f \left(10\right) = 7930$


text:
# zbMATH — the first resource for mathematics

Quotients of Coxeter complexes and $$P$$-partitions. (English) Zbl 0751.06002
Mem. Am. Math. Soc. 460, 134 p. (1992).
One aspect of this study involves finite Coxeter systems $$(W,S)$$, where $$W$$ is a finite group generated by Euclidean reflections acting on a vector space of dimension $$| S|$$, where $$S$$ is the generating set of $$W$$ and the associated Coxeter complex $$\Sigma(W,S)$$ constructed in standard ways, for the study of which there exists a very considerable literature. In particular, if $$G<W$$, then the orbit structure $$\Sigma(W,S)/G$$, depending on the description of $$\Sigma(W,S)$$ can be viewed as a topological or an algebraic object, but in any case one whose combinatorial nature is also an essential aspect to consider. This Memoir is based on the author’s doctoral thesis written under the supervision of R. Stanley and among its contributions it contains some interesting and deft extensions of ideas on $$P$$-partitions and multipartite $$P$$- partitions whose original investigation belongs to Stanley’s own Memoir volume based on his dissertation under G. C. Rota [R. P. Stanley, “Ordered structures and partitions”, Mem. Am. Math. Soc. 119 (1972; Zbl 0246.05007)]. Among the names associated with this entire circle connecting aspects of (combinatorial) topology with corresponding aspects in commutative-algebra-cum-algebraic-geometry and hence to the combinatorics-structure theory of posets, which, in turn, connects directly to order complexes and topology, dealing with shellability, Cohen-Macaulayness in the ordered portion of the theory, Stanley’s contributions touch on all facets in fundamental ways and his point of view is mirrored nicely in this particular work. All the standard actors appear and the circle remains unbroken in this discussion also, which succeeds in extending a considerable variety of important known results to new selections of pairs $$W$$ and $$G$$, e.g., reflection subgroups, in providing alternative and sometimes simpler proofs of results considered classical in this area along with new and interesting conclusions sometimes based on obviously useful notions such as “parsets”, for example. As a consequence, not only is it a useful addition to the literature on this subject, but an interested person can select this little volume as a good starting place whence to trace back to more detailed information in other monographs or to historical precedents on which results obtained here happen to be based even when altered and generalized. There is a sufficient though short bibliography while the introduction provides good orientation both to the material following and to the rationale for pursuing certain particular lines of reasoning.

##### MSC:
 06A11 Algebraic aspects of posets 05A99 Enumerative combinatorics 51F15 Reflection groups, reflection geometries
Full Text:


text:
# PreviousImputer¶

Imputes missing values by using the most recent value.

## Examples¶

>>> from river import preprocessing

>>> imputer = preprocessing.PreviousImputer()

>>> imputer = imputer.learn_one({'x': 1, 'y': 2})
>>> imputer.transform_one({'y': None})
{'y': 2}

>>> imputer.transform_one({'x': None})
{'x': 1}

## Methods¶

learn_one

Update with a set of features x.

A lot of transformers don't actually have to do anything during the learn_one step because they are stateless. For this reason the default behavior of this function is to do nothing. Transformers that however do something during the learn_one can override this method.

Parameters

• x (dict)

Returns

Transformer: self

transform_one

Transform a set of features x.

Parameters

• x (dict)

Returns

dict: The transformed values.


text:
## Tuesday, April 24, 2007

### Edinburgh

Just got back from a workshop on Geometry and Algorithms in Edinburgh. Was there any Fourier analysis? But of course! What would geometry and algorithms be without it?

Unfortunately, though, it's hard to get it into one's talk in 50 minutes, since one has to cover the geometry and algorithms parts. Guy Kindler mentioned some Fourier analysis in connection with UGC-hardness for Max-Cut. Assaf Naor talked a bit about his upcoming magnum opus with Manor Mendel on "Metric Cotype" which will appear in the Annals of Math. That paper is pretty much all Fourier analysis for functions $f : {\mathbb{Z}_m}^n \to \mathbb{R}$, but with $m > 2$. And Robi Krauthgamer came closest to actually doing some Fourier analysis in his slides, while talking about his work on the communication complexity and embeddability of Edit Distance. Unfortunately, his paper (joint with Alexandr Andoni) doesn't seem to be available yet.

I did my best to get there while talking about the parallel repetition/foams stuff I talked about at the ACO Seminar in March, but no dice. I do think the Fourier analysis we used in that result was pretty cute though; basically: (1) Talagrand's Lemma from Homework 5 Problem 2; and (2) the reversed form of the Hypercontractive Inequality, namely:

If $f : \{-1,1\}^n \to \mathbb{R}^{\geq 0}$ and $-\infty < q \leq p \leq 1$, then $\|T_\rho f\|_q \geq \|f\|_p$ so long as $0 \leq \rho \leq \sqrt{(1-p)/(1-q)}$.

You know -- just like the usual inequality, but reversed.

#### 1 comment:

myinvestorsplace said...

Viewing the Fourier transform as a limit process allows us to apply Fourier analysis in these cases and treat these cases as we would more conventional cases.

futures, trading, value, investing, forex, stock


text:
# Revision history [back]

To see it by hand, we can use:

• $(x + 1)^{5/2} = (x + 1) (x + 1)^{3/2} = x (x + 1)^{3/2} + (x + 1)^{3/2}$
• $(x + 1)^{3/2} = (x + 1) (x + 1)^{1/2} = x (x + 1)^{1/2} + (x + 1)^{1/2}$

to express both as linear combinations of (x + 1)^(1/2) and x * (x + 1)^(1/2).

To see it with Sage, we can differentiate both and compare them.

Define the function to integrate:

sage: x = var('x')
sage: f = x*sqrt(x + 1)
sage: f
sqrt(x + 1)*x

Integrate with Sage:

sage: F = integral(f, x)
sage: F
2/5*(x + 1)^(5/2) - 2/3*(x + 1)^(3/2)

The primitive you computed (up to a constant):

sage: G = (2/3)*x*(x + 1)^(3/2) - (4/15)*(x + 1)^(5/2)
sage: G
-4/15*(x + 1)^(5/2) + 2/3*(x + 1)^(3/2)*x

To see that they are both correct:

sage: bool(F == G)
True

Or check they have the same derivative:

sage: g = G.diff(x)
sage: g
sqrt(x + 1)*x

sage: ff = F.diff(x)
sage: ff
(x + 1)^(3/2) - sqrt(x + 1)

sage: bool(g == ff)
True

Or more visually, compare the plots of F and G (or f and ff):

sage: pF = plot(F, (-1, 1.6), color='firebrick')
sage: pG = plot(G, (-1, 1.6), color='steelblue')
sage: p = graphics_array([pF, pG, pF + pG])
sage: p.show(figsize=(7, 3))
Launched png viewer for Graphics Array of size 1 x 3

To see it by hand, we can use:

• $(x + 1)^{5/2} = (x + 1) (x + 1)^{3/2} = x (x + 1)^{3/2} + (x + 1)^{3/2}$
• $(x + 1)^{3/2} = (x + 1) (x + 1)^{1/2} = x (x + 1)^{1/2} + (x + 1)^{1/2}$

to express both as linear combinations of (x + 1)^(1/2) $(x + 1)^{1/2}$, $x (x + 1)^1/2}$ and x * (x + 1)^(1/2).$x^2 (x + 1)^{1/2}$.

To see it with Sage, we can differentiate both and compare them.

Define the function to integrate:

sage: x = var('x')
sage: f = x*sqrt(x + 1)
sage: f
sqrt(x + 1)*x

Integrate Compute a primitive (aka antiderivative) with Sage:

sage: F = integral(f, x)
sage: F
2/5*(x + 1)^(5/2) - 2/3*(x + 1)^(3/2)

The primitive you computed (up to a constant):

sage: G = (2/3)*x*(x + 1)^(3/2) - (4/15)*(x + 1)^(5/2)
sage: G
-4/15*(x + 1)^(5/2) + 2/3*(x + 1)^(3/2)*x

To see Check that they are both correct:agree:

sage: bool(F == G)
True

Or check Check they have the same derivative:

sage: g = G.diff(x)
sage: g
sqrt(x + 1)*x

sage: ff = F.diff(x)
sage: ff
(x + 1)^(3/2) - sqrt(x + 1)

sage: bool(g == ff)
True

Or more visually, compare the plots of F and G (or f and ff):

sage: pF = plot(F, (-1, 1.6), color='firebrick')
sage: pG = plot(G, (-1, 1.6), color='steelblue')
sage: p = graphics_array([pF, pG, pF + pG])
sage: p.show(figsize=(7, 3))
Launched png viewer for Graphics Array of size 1 x 3

To see it by hand, we can use:

• $(x + 1)^{5/2} = (x + 1) (x + 1)^{3/2} = x (x + 1)^{3/2} + (x + 1)^{3/2}$
• $(x + 1)^{3/2} = (x + 1) (x + 1)^{1/2} = x (x + 1)^{1/2} + (x + 1)^{1/2}$

to express both as linear combinations of $(x + 1)^{1/2}$, $x (x + 1)^1/2}$ 1)^{1/2}$and$x^2 (x + 1)^{1/2}\$.

To see it with Sage, we can can

• ask Sage whether the two different expressions in fact agree
• differentiate both and compare them.

them
• plot the functions to visually check whether they agree

Define the function to integrate:

sage: x = var('x')
sage: f = x*sqrt(x + 1)
sage: f
sqrt(x + 1)*x

Compute a primitive (aka antiderivative) with Sage:

sage: F = integral(f, x)
sage: F
2/5*(x + 1)^(5/2) - 2/3*(x + 1)^(3/2)

The primitive you computed (up to a constant):

sage: G = (2/3)*x*(x + 1)^(3/2) - (4/15)*(x + 1)^(5/2)
sage: G
-4/15*(x + 1)^(5/2) + 2/3*(x + 1)^(3/2)*x

Check that they agree:

sage: bool(F == G)
True

Check they have the same derivative:

sage: g = G.diff(x)
sage: g
sqrt(x + 1)*x

sage: ff = F.diff(x)
sage: ff
(x + 1)^(3/2) - sqrt(x + 1)

sage: bool(g == ff)
True

Or more visually, compare the plots of F and G (or f and ff):

sage: pF = plot(F, (-1, 1.6), color='firebrick')
sage: pG = plot(G, (-1, 1.6), color='steelblue')
sage: p = graphics_array([pF, pG, pF + pG])
sage: p.show(figsize=(7, 3))
Launched png viewer for Graphics Array of size 1 x 3


text:
This is not the latest release View the latest release

# National, state and territory population methodology

Reference period
December 2019
Released
18/06/2020

## Explanatory notes

### Introduction

1 This quarterly release contains the most recent estimates of the resident populations (ERP) of Australia and the states and territories based on the results of the 2016 Census of Population and Housing held on 9 August 2016 (with various adjustments described in paragraphs 5 and 6), and the addition of quarterly components of population growth. The ABS has used the 2016 Census to produce final rebased estimates of the resident population (refer to paragraph 6). This release contains the latest available statistics on births, deaths (including infant deaths) and overseas and interstate migration. In addition, the release includes estimates of the resident population by age groups and major population regions. Articles on specific demographic topics will be released on the ABS web site in conjunction with this release.

2 Population estimates commencing from September quarter 1993 include estimates for another category of the state and territory level, known as Other Territories. Other Territories include Jervis Bay Territory (previously included with the Australian Capital Territory), as well as Christmas Island and the Cocos (Keeling) Islands which were previously excluded from population estimates for Australia. From 1 July 2016 Norfolk Island has been included in the Other Territory category following the introduction of the Norfolk Island Legislation Amendment Act 2015. Data for Other and External Territories are detailed separately in table 3.

3 Estimates for Australian External Territories will be updated annually as at 30 June unless a more recent estimate is required for electoral apportionment purposes under the Commonwealth Electoral Act 1918.

4 Australia’s population estimates for the period since 1971 are compiled according to the place of usual residence of the population. An explanation of the place of usual residence conceptual basis for population estimates is given in Information Paper: Population Concepts (cat. no. 3107.0.55.006) and also in Population Estimates: Concepts, Sources and Methods (cat. no. 3228.0.55.001).

### Method of estimation

5 Estimates of the resident population are based on Census counts by place of usual residence, to which are added the estimated Census net undercount and the number of Australian residents estimated to have been temporarily overseas on Census night. Overseas visitors in Australia on Census night are excluded in this calculation. Post-Census ERP is obtained by adding to the estimated population at the beginning of each period the component of natural increase (on a usual residence basis) and the component of net overseas migration. For the states and territories, estimated interstate movements involving a change of usual residence are also taken into account.

6 After each Census, estimates for the preceding intercensal period are finalised (rebased) by incorporating an additional adjustment (intercensal difference) to ensure that the difference between the ERPs at the two 30 June dates in the respective Census years agrees with the total intercensal change.

### Status of quarterly ERP data

7 The status of quarterly ERP data changes over time from preliminary to revised to final as new component data becomes available. ERP is updated most quarters due to revisions to the component data for earlier quarters, but will only change status to revised once both natural increase and net overseas migration for that quarter have a status of revised. The table below shows the current status of ERP and the components of population change: natural increase, overseas migration and interstate migration.

### Status of quarterly Estimated Resident Population (ERP) data, as at 18 June 2020

Census baseNatural increaseOverseas migrationInterstate migrationERP STATUS
Sep.1991-Jun. 2016Based to 1996, 2001, 2006, 2011 &
2016 Censuses as applicable
FinalFinalFinalFINAL
Sep. 2016-Jun. 20182016 CensusRevised - based on date of occurrenceFinal - based on actual traveller behaviourPreliminary - based on modelled expansion factors from 2016 CensusREVISED
Sep. 2018-Dec. 20182016 CensusPreliminary - based on date of registrationFinal - based on actual traveller behaviourPreliminary - based on modelled expansion factors from 2016 CensusPRELIMINARY - updated due to revised component data
Mar. 2019-Sep. 20192016 CensusPreliminary - based on date of registrationPreliminary - based on modelled traveller behaviourPreliminary - based on modelled expansion factors from 2016 CensusPRELIMINARY - updated due to revised component data
Dec. 20192016 CensusPreliminary - based on date of registrationPreliminary - based on modelled traveller behaviourPreliminary - based on modelled expansion factors from 2016 CensusPRELIMINARY

### Natural increase: births and deaths

8 Natural increase is a major component of ABS quarterly state and territory population estimates and is calculated using the estimated number of births and deaths. The births and deaths data in this release are shown by state and territory of usual residence, using year/quarter of registration for preliminary data and year/quarter of occurrence for both revised and final data. This may affect time series comparisons within relevant tables.

9 The timeliness and accuracy of ABS quarterly population estimates depend in part on the timeliness and accuracy of estimates of births and deaths which are based on registrations. To provide timely estimates, the ABS produces preliminary estimates using births and deaths by quarter of registration as a proxy for quarter of occurrence. For revised estimates, a factor has been applied to the number of occurrences to allow for those occurrences which were yet to be registered at the time of revision. The major difficulty in this area is that while the vast majority of births and deaths are registered promptly, a small proportion of registrations are delayed for months or even years.

10 Preliminary birth and death estimates are subject to fluctuations caused by lags or accumulations in the reporting of births and deaths registrations. Accumulations can result from the eventual processing of lagged registrations in a later quarter. As a result, preliminary quarterly estimates can be an underestimate or an overestimate of the true numbers of births and deaths occurring in a reference period. Lags or accumulations in births and deaths registrations can be caused by:

• late notification of a birth or death event to a state or territory registry;
• delays arising from incomplete information supplied for a registration;
• procedural changes affecting the processing cycles in any of the state and territory registries; or
• resolution of issues that may arise within the ABS or registry processing systems.

11 Birth and death registration data contributing to preliminary estimates which are higher or lower than usual are noted below along with any explanations provided by the relevant state or territory registrars:

• December 2019: Victoria. As a result of joint investigations between the ABS and the Registry additional death registrations from 2017, 2018 and 2019 were identified that had not previously been provided to the ABS. An issue associated with the Registry's previous processing system (replaced in 2019) has resulted in delays to the provision of some death registrations to the ABS. Approximately 570 additional deaths registrations have now been included in the December 2019 ERP, with the deaths being added across the five previous quarters. The remaining additional deaths (approximately 2200) occurred in 2016-17 and 2017-18, and will be included as part of the final revision of intercensal components in June 2023.
• June 2019: Victoria. There are higher than usual numbers of births and deaths registrations as a result of catch up in processing from March 2019, see below.
• March 2019: Victoria. The Victorian Registry of Births, Deaths and Marriages implemented a new registration system in February 2019. This large scale system change caused processing delays which have led to lower than usual numbers of birth and death registrations in the March quarter 2019. The ABS has worked closely with the registry through this system change and anticipate that lagged registrations will be caught up through the remainder of 2019.
• March 2019: Australia. The delay in processing of birth and death registrations in Victoria is the main reason for low national birth and death counts.

### Overseas migration

12 For the purposes of NOM, a person is regarded as a usual resident if they have been (or expect to be) residing in Australia for a period of 12 months or more. These 12 months do not need to be continuous and are measured over a period of 16 months.

13 The ABS introduced the 12/16 month rule for calculating NOM in September quarter 2006. Consequently, this point marks a break in series and NOM estimates from earlier periods are not directly comparable. For further information on the 12/16 month rule see the Technical Note: '12/16 month rule' Methodology for Calculating Net Overseas Migration from September quarter 2006 onwards in Migration, Australia, 2008-09 (cat. no. 3412.0).

14 Preliminary estimates of NOM are required less than six months after the reference quarter for the production of quarterly estimates of the population of Australia and its states and territories. At that time, complete traveller histories for the 16 months following a reference quarter are not available and as a result preliminary estimates are produced.

15 Final NOM estimates are produced once data on overseas movements become available for the 16 months following the reference period. This enables each traveller’s actual duration of stay to be calculated and the 12/16 month rule to be fully applied. Final NOM estimation methods use a traveller's actual duration of stay in or out of Australia to determine their contribution to NOM estimates and consequently their inclusion or exclusion from the ERP.

16 In an initiative to create a more efficient and streamlined process for travellers departing Australia, the requirement for international travellers to complete an outgoing passenger card was removed by the Department of Home Affairs from 1 July 2017. Due to the removal of the card the ABS has reviewed its net overseas migration (NOM) statistics, methodology and processing systems. Historical NOM data from September quarter 2011 onwards has been produced based on the new methods and has been used in the 2011-2016 intercensal period as part of the final rebasing of population estimates. As NOM estimates cannot be finalised until 16 months after the reference period, NOM is subject to revision. Testing has shown that the revision between preliminary and final NOM using the new methodology has improved when compared to the previous method. For further information see the Information Paper: Improvements to estimation of net overseas migration (cat. no. 3412.0.55.004).

17 From the March quarter 2020, the COVID-19 pandemic caused extreme disruption to ordinarily expected travel patterns and volumes to and from Australia. The Australian Government placed restrictions on travel to and from Australia resulting in vast reductions in the volume of overseas movements. In order to mitigate any adverse effects these travel restrictions and resulting reductions may have on the preliminary NOM estimates, the ABS implemented a more frequent revision cycle. From December quarter 2019, all NOM estimates which have not already been made final shall be revised each quarterly processing cycle. These revisions will utilise all available overseas movements data in order to incorporate the most up to date information in the NOM estimates. Prior to this, due to the stability of travel patterns and the accuracy of the preliminary estimates, NOM estimates were ordinarily revised only once. This occurred one year after production of the preliminary estimate when they were made final.

18 Statistics on migration and related data are also published regularly by the Department of Home Affairs http://www.homeaffairs.gov.au.

### Diplomatic personnel

19 Australia's ERP includes all people, regardless of nationality or citizenship, who usually live in Australia, with the exception of foreign diplomatic personnel and their families. Foreign diplomatic personnel and their families were removed from overseas migration estimates from 1 July 2006. The previous methodology for estimating overseas migration was unable to exclude diplomatic personnel and their families.

### Interstate migration

20 Quarterly interstate migration cannot be directly measured and is estimated using administrative data. To do this the ABS uses information on interstate changes of address from Medicare records (produced by the Department of Human Services) and the Department of Defence in the case of the military.

21 The Medicare-based model is calibrated using migration data from the most recent Census (that data is available for), from which updated expansion factors are calculated. Expansion factors account for undercoverage of Medicare data by age and sex. The current model includes the following characteristics:

• Medicare data is lagged by three months (both for calculating expansion factors and for estimating progressive quarters of interstate migration);
• All single year of age data were smoothed to produce expansion factors;
• capping was applied to expansion factors; and
• expansion factors were applied to males aged 17 to 35 years and females aged 17 to 30 years (this differs to the age range used in the 2011-16 method).

22 The Medicare system theoretically covers all Australian citizens and permanent residents, as well as some temporary visa holders. However, some Australian usual residents do not access the Medicare system, such as temporary migrants or those who have access to other health services. One group is the military. Interstate defence force movements not covered by Medicare are estimated and then added to the Medicare-based interstate movement estimates. Quarterly counts of defence force personnel by age, sex and state/territory, supplied by the Department of Defence, form the number of interstate defence force movements, and 70% are assumed to be not covered by the Medicare-based estimates.

23 Preliminary interstate migration estimates for Other Territories are not available. These movements are included in the data for New South Wales (Jervis Bay Territory and Norfolk Island) and Western Australia (Christmas Island and Cocos (Keeling) Islands). Revised interstate migration estimates will become available following the 2021 Census.

24 For further information see Australian Demographic Statistics, December quarter 2017 (cat. no. 3101.0) Technical Note 2: 2016 Census update of the net interstate migration model.

### Confidentiality

25 The Census and Statistics Act, 1905 provides the authority for the ABS to collect statistical information, and requires that statistical output shall not be published or disseminated in a manner that is likely to enable the identification of a particular person or organisation. This requirement means that the ABS must take care and make assurances that any statistical information about individual respondents cannot be derived from published data.

26 Some techniques used to guard against identification or disclosure of confidential information in statistical tables are suppression of sensitive cells, and random adjustments to cells with very small values. To protect confidentiality within this release, some cell values may have been suppressed and are not available for publication (np) but included in totals where applicable. In these cases, data may not sum to totals due to the confidentialisation of individual cells.

### Rounding

27 In this release, population estimates and their components have sometimes been rounded. Where figures have been rounded, discrepancies may occur between sums of component items and totals.

### Acknowledgment

28 ABS statistics draw extensively on information provided freely by individuals, businesses, governments and other organisations. Their continued cooperation is very much appreciated: without it, the wide range of statistics published in the ABS would not be available. Information received by the ABS is treated in strict confidence as required by the Census and Statistics Act 1905.

### Use of population estimates in reporting

29 The ABS provides regular explanatory information to support users in understanding both population trends and methodological changes, but does not comment on the specific use (or otherwise) of official population estimates by other organisations or individuals.

30 Population estimates are used extensively within the Australian community, including in a range of agreements. Although the ABS acknowledges that the official population estimates and changes in these estimates are specifically referenced in various agreements, it neither endorses nor disapproves of the terms of the agreements and decisions made by parties in relation to those agreements. In addition, the ABS does not provide a position on disputes arising from the interpretation of terms of an agreement that reference official population estimates.

31 As well as the statistics included in this and related products, the ABS may have other relevant data available on request. Inquiries should be made to the National Information and Referral Service on 1300 135 070.

## Glossary

### Show all

#### 12/16 month rule

Under a '12/16 month rule', incoming overseas travellers (who are not currently counted in the population) must be resident in Australia for a total period of 12 months or more, during the 16 month follow-up period to then be added to the estimated resident population. Similarly, those travellers departing Australia (who are currently counted in the population) must be absent from Australia for a total of 12 months or more during the 16 month follow-up period to then be subtracted from the estimated resident population.

The 12/16 month rule does not have to be continuous and takes account of those persons who may have left Australia briefly and returned, while still being resident for 12 months out of 16. Similarly, it takes account of Australians who live most of the time overseas but periodically return to Australia for short periods.

#### Average annual rate of growth

The average annual growth rate, r, is calculated as a percentage using the formula:

$$\large{r=\left[(\frac{P_n}{P_o})^{\frac{1}{n}}-1\right] \times 100}$$

where $$\text{P}_0$$ is the population at the start of the period, $$\text{P}_n$$ is the population at the end of the period and n is the length of the period between $$\text{P}_0$$ and $$\text{P}_n$$ in years.

#### Baby boomer

Refers to people born Post-World War II between the years 1946 and 1964.

#### Birth

The delivery of a child, irrespective of the duration of pregnancy, who, after being born, breathes or shows any other evidence of life such as heartbeat.

#### Census

The complete enumeration of a population at a point in time with respect to well-defined characteristics (e.g. Persons, Manufacturing, etc.). When the word is capitalised, "Census" refers to the national Census of Population and Housing.

#### Death

Death is the permanent disappearance of all evidence of life after birth has taken place. The definition excludes deaths prior to live birth.

For the purposes of the Deaths and Causes of Death collections compiled by the ABS, a death refers to any death which occurs in, or en route to Australia and is registered with a state or territory Registry of Births, Deaths and Marriages.

#### Estimated resident population (ERP)

The official measure of the population of Australia is based on the concept of usual residence. It refers to all people, regardless of nationality, citizenship or legal status, who usually live in Australia, with the exception of foreign diplomatic personnel and their families. It includes usual residents who are overseas for less than 12 months over a 16-month period. It excludes overseas visitors who are in Australia for less than 12 months over a 16-month period.

Estimates of the Australian resident population are generated on a quarterly basis by adding natural increase (the excess of births over deaths) and net overseas migration (NOM) occurring during the period to the population at the beginning of each period. This is known as the cohort component method, and can be represented by the following equation:

$$\text{P}_{t+1} = \text{P}_t + \text{B} - \text{D} + \text{NOM}$$ where:

$$\text{P}_{t}$$ = the estimated resident population at time point $$t$$

$$\text{P}_{t+1}$$ = the estimated resident population at time point $$t+1$$

$$\text{B}$$ = the number of births occurring between $$t$$ and $$t+1$$

$$\text{D}$$ = the number of deaths occurring between $$t$$ and $$t+1$$

$$\text{NOM}$$ = net overseas migration occurring between $$t$$ and $$t+1$$.

For state and territory population estimates, an additional term is added to the equation representing net interstate migration (NIM) occurring between $$t$$ and $$t+1$$, represented by the following equation:

$$\text{P}_{t+1} = \text{P}_t + \text{B} - \text{D} + \text{NOM} +\text{NIM}$$.

#### Greater Capital City Statistical Area (GCCSA)

Represent the socioeconomic area of each of the eight state and territory capital cities. These boundaries are built from aggregations of whole Statistical Areas Level 4. GCCSA boundaries represent a broad socioeconomic definition of each capital city, they contain not only the urban area of the capital city, but also surrounding and non-urban areas where much of the population has strong links to the capital city, through for example, commuting to work. For further information see Australian Statistical Geography Standard (ASGS): Volume 1 - Main Structure and Greater Statistical Areas, July 2016 (cat. no. 1270.0.55.001).

#### Infant death

An infant death is the death of a live-born child who dies before reaching their first birthday.

#### Infant mortality rate (IMR)

The number of deaths of children under one year of age in a financial year per 1,000 live births in the same financial year.

#### Intercensal difference

Intercensal difference is the difference between two estimates at 30 June of a Census year population: the first based on the latest Census, and the second arrived at by updating the 30 June estimate of the previous Census year with intercensal components of population change. It is caused by differences in the start and/or finish population estimates and/or in estimates of births, deaths or migration in the intervening period which cannot be attributed to a particular source. For further information see Population Estimates: Concepts, Sources and Methods (cat. no. 3228.0.55.001).

#### Median age

For any distribution, the median value is that which divides the relevant population into two equal parts, half falling below the value, and half exceeding it. Thus, the median age is the age at which half the population is older and half is younger.

Prior to September quarter 2006, the ABS applied a number of adjustments to overseas arrivals and departures data in order to produce estimates of net overseas migration (NOM). These mainly comprised adjustments designed to reflect differences between stated travel intentions and actual travel behaviour. Until recently, adjustments used by ABS to produce NOM estimates were collectively referred to as 'category jumping adjustments'. They are now referred to more simply as 'migration adjustments'.

#### Natural increase

The number of births minus the number of deaths.

#### Interstate migration

The movement of people over a state or territory boundary for the purpose of changing their place of usual residence. Net interstate migration is the number of arrivals minus the number of departures.

#### Net overseas migration (NOM)

Net overseas migration is the net gain or loss of population through immigration to Australia and emigration from Australia. Under the current method for estimating final net overseas migration this term is based on a traveller's actual duration of stay or absence using the '12/16 month rule'. Preliminary NOM estimates are modelled on patterns of traveller behaviours observed in final NOM estimates for the same period one year earlier. NOM is:

• based on an international traveller's duration of stay being in or out of Australia for 12 months or more over a 16-month period;
• the difference between:

• the number of incoming international travellers who stay in Australia for 12 months or more over a 16-month period, who are not currently counted within the population, and are then added to the population (Overseas migrant arrivals); and
• the number of outgoing international travellers (Australian residents and long-term visitors to Australia) who leave Australia for 12 months or more over a 16-month period, who are currently counted within the population, and are then subtracted from the population (Overseas migrant departures).

#### Overseas migrant arrivals

Overseas migrant arrivals (also referred to as NOM arrivals) are all overseas arrivals that contribute to net overseas migration (NOM). It is the number of incoming international travellers who stay in Australia for 12 months or more over a 16 month period, who are not currently counted within the population, and are then added to the population.

Under the current method for estimating final net overseas migration this term is based on a traveller's actual duration of stay using the '12/16 month rule'.

#### Overseas migrant departures

Overseas migrant departures (also referred to as NOM departures) are all overseas departures that contribute to net overseas migration (NOM). It is the number of outgoing international travellers (Australian residents and long term visitors to Australia) who leave Australia for 12 months or more over a 16 month period, who are currently counted within the population, and are then subtracted from the population.

Under the current method for estimating final net overseas migration this term is based on a traveller's actual duration of absence using the '12/16 month rule'.

#### Net undercount

The difference between the actual Census count (including imputations) and an estimate of the number of people who should have been counted in the Census. This estimate is based on the Post Enumeration Survey (PES) conducted after each Census. For a category of person (e.g. based on age, sex and state of usual residence), net undercount is the result of Census undercount, overcount, differences in classification between the PES and Census and imputation error.

#### Population growth

For Australia, population growth is the sum of natural increase and net overseas migration. For states and territories, population growth also includes net interstate migration. After the Census, intercensal population growth also includes an allowance for intercensal difference.

#### Population growth rate

Population change over a period as a proportion (percentage) of the population at the beginning of the period.

#### Post Enumeration Survey (PES)

The Census Post Enumeration Survey (PES) is a household survey conducted following the Census. The PES allows the ABS to estimate the number of people missed in the Census and the number counted more than once or in error. Historically more people are missed than are counted more than once in Australia, leading to a net undercount. Results from the PES contribute to a more accurate calculation of the estimated resident population (ERP) for Australia and the states and territories, which is then backdated to 30 June of the Census year.

#### Rebasing of population estimates

After each Census, the ABS uses Census counts by place of usual residence which are adjusted for undercount to construct a new base population figure for 30 June of the Census year. Because this new population estimate uses the Census as its main data source, it is said to be 'based' on that Census and is referred to as a population base.

Rebasing refers to the process by which the ABS uses this new base to update all previously published population estimates from the previous census to the most recent census (the intercensal period). For further information on rebasing to the 2016 Census see Australian Demographic Statistics, December quarter 2017 (cat. no. 3101.0) Feature Article: Final Rebasing of Australia's Population Estimates using the 2016 Census.

#### Recasting of population estimates

The recasting of ERP was a one-off process undertaken during the course of rebasing to the 2011 Census. The decision to recast historical ERP data from September 1991 to June 2006 was in response to the unusually high preliminary intercensal difference, resulting from a change in the methodology used to estimate undercount in the 2011 Census. For further information see Australian Demographic Statistics, December quarter 2012 (cat. no. 3101.0) Feature Article: Recasting 20 Years of ERP.

#### Sex ratio

The sex ratio relates to the number of males per 100 females. The sex ratio is defined for the total population, at birth, at death and among age groups by appropriately selecting the numerator and the denominator of the ratio.

#### Significant Urban Area (SUA)

Aggregations of whole Statistical Areas Level 2 to define and contain major urban and near-urban concentrations of over 10,000 people. They include the urban population, any immediately associated populations, and may incorporate together one or more closely associated Urban Centre and Localities and the areas between. They are designed to incorporate any likely growth over the next 20 years. SUAs do not cover the whole of Australia, and may cross State boundaries. For further information see Australian Statistical Geography Standard (ASGS): Volume 4 - Significant Urban Areas, Urban Centres and Localities, Section of State, July 2016 (cat. no. 1270.0.55.004).

#### Standardised death rate (SDR)

Standardised death rates enable the comparison of death rates between populations with different age structures by relating them to a standard population. The current standard population is all persons in the Australian population at 30 June 2001 (19,413,240), as published prior to recasting the ERP series. SDRs are expressed per 1,000 or 100,000 persons. There are two methods of calculating SDRs:

• The direct method - this is used when the populations under study are large and the age-specific death rates are reliable. It is the overall death rate that would have prevailed in the standard population if it had experienced at each age the death rates of the population under study; and
• The indirect method - this is used when the populations under study are small and the age-specific death rates are unreliable or not known. It is an adjustment to the crude death rate of the standard population to account for the variation between the actual number of deaths in the population under study and the number of deaths which would have occurred if the population under study had experienced the age-specific death rates of the standard population.

#### State or territory of usual residence

State or territory of usual residence refers to the state or territory of usual residence of:

• the population (estimated resident population);
• the mother (birth collection); and
• the deceased (death collection).

In the case of overseas movements, state or territory of usual residence refers to the state or territory regarded by the traveller as the one in which he/she lives or has lived. State or territory of intended residence is derived from the intended address given by overseas arrivals, and by Australian residents returning after a journey abroad. Particularly in the case of the former, this is not necessarily the state or territory in which the traveller will eventually establish a permanent residence.

#### Total fertility rate (TFR)

The sum of age-specific fertility rates (live births at each age of mother per female population of that age) divided by 1,000. It represents the number of children a female would bear during her lifetime if she experienced current age-specific fertility rates at each age of her reproductive life (ages 15 - 49).

#### Usual residence

Usual residence within Australia refers to that address at which the person has lived or intends to live for a total of six months or more in a given reference year.

## Quality declaration - summary

### Institutional environment

Estimated Resident Population (ERP) uses data sourced from a variety of institutional environments. Much of the data is administrative by-product data collected by other organisations for purposes other than estimating the population. Births and deaths statistics are extracted from registers administered by the various State and Territory Registries of Births, Deaths and Marriages. Medicare Australia client address data and Defence data is used to estimate interstate migration. Information provided by the Department of Home Affairs from their Travel and Immigration Processing System is used to calculate overseas migration.

ABS Census of Population and Housing and Post Enumeration Survey (PES) data are used to determine a base population from which ERP is calculated and to finalise all components of population change. For information on the institutional environment of the Australian Bureau of Statistics, please see ABS Institutional Environment.

### Relevance

Estimates of the resident population (ERP) is the official measure of the population of states and territories of Australia according to a usual residence population concept. ERP is used for a range of key decisions such as resource and funding distribution and apportioning seats in the House of Representatives to each state and territory. ERP for the states and territories of Australia are published by sex and age groups, and estimates and projections of the Aboriginal and Torres Strait Islander population are also available.

### Timeliness

Preliminary ERP data, disaggregated by sex and single year of age, is compiled and published quarterly and is generally made available six months after the end of each reference quarter.

Revised estimates are released once more accurate births, deaths and net overseas migration data becomes available. In the case of births and deaths, the revised data is compiled on a date of occurrence basis annually in the March reference period, released in September for the previous financial year.

In the case of overseas migration, final data is based on actual traveller behaviour. Final estimates for overseas migration are released quarterly and made available 18 months after the end of the reference period.

Final ERP estimates are made available every 5 years after a census when revisions are made to the previous intercensal period. ERP data is not changed once it has been finalised. Releasing preliminary, revised and final ERP involves a balance between timeliness and accuracy.

### Accuracy

All ERP data sources are subject to non-sampling error. Non-sampling error can arise from inaccuracies in collecting, recording and processing the data. In the case of Census and PES data every effort is made to minimise reporting error by the careful design of questionnaires, intensive training and supervision of interviewers, and efficient data processing procedures. The ABS does not have control over any non-sampling error associated with births, deaths and migration data (see institutional environment).

Another dimension of non-sampling error in ERP is the fact that the measures of components of population growth become more accurate as more time elapses after the reference period. As discussed under Timeliness, the tradeoff between timeliness and accuracy means that a user can access more accurate data by using the revised or final ERP data. While the vast majority of births and deaths are registered promptly, a small proportion of registrations are delayed for months or even years. As a result, preliminary quarterly estimates can be an underestimate of the true number of births and deaths occurring in a reference period. Revised figures for a reference period incorporate births and deaths registrations that were received after the preliminary data collection phase as well as the estimated number of registrations that have still not been received for that reference period. For more information see the Demography Working Paper 1998/2 - Quarterly birth and death estimates (cat. no. 3114.0) and Population Estimates: Concepts, Sources and Methods (cat. no. 3228.0.55.001).

After each Census the ABS uses the Census population count to update the original series of published quarterly population estimates since the previous Census. For example, 2016 Census results were used to update quarterly population estimates between the 2011 and 2016 Census. The PES is conducted soon after the Census to estimate the number of Australians not included in the Census. Adding this net undercount of people back into the population is a crucial step in arriving at the most accurate ERP possible. For more information on rebasing see the feature article Rebasing of Australia's Population Estimates using the 2016 Census in the December quarter 2016 issue of Australian Demographic Statistics (cat. no. 3101.0). At the time of rebasing to the 2011 Census, estimates for the period September 1991 to June 2006 were revised in response to a methodological improvement in the estimation of Census undercount, for more information see Feature Article: Recasting 20 Years of ERP in Australian Demographic Statistics, December Quarter 2012 (cat. no. 3101.0).

### Coherence

ERP was introduced in 1981 and backdated to 1971 as Australia's official measure of population based on place of usual residence. ERP is derived from usual residence census counts, to which is added the estimated net census undercount and Australian residents temporarily overseas at the time of the census (overseas visitors in Australia are excluded from this calculation). Before the introduction of ERP, the Australian population was based on unadjusted census counts on actual location basis. It is important to note this break in time series when comparing historical population estimates.

An improved definition for calculating overseas migration was applied from September quarter 2006 onwards. The key change is the introduction of a '12/16 month rule' for measuring a person's residency in Australia replacing the '12/12 month rule'. This change results in a break in time series and therefore it is not advised that overseas migration data calculated using the new method is compared to data previous to this. For further information see Information Paper: Improving Net Overseas Migration Estimation (cat. no. 3412.0.55.001).

The births and deaths data in this publication are not coherent with the data found in ABS births and deaths publications. This is because the revision cycle necessary to produce ERP results in a mix of preliminary births and deaths data, based on date of registration, and revised data which is a modelled estimate of births and deaths by date of occurrence. By contrast, the main tables of data in the births and deaths publications are based wholly on registration in the reference year, with some tables and analysis based wholly on date of occurrence data.

### Interpretability

ERP is generally easy to interpret as the official measure of Australia's population (by state and territory) on a place of usual residence basis. However, there are still some common misconceptions. For example, a population estimate uses the term 'estimate' in a different sense than is commonly used. Generally the word estimate is used to describe a guess, or approximation. Demographers mean that they apply the demographic balancing equation by adding births, subtracting deaths and adding the net of overseas and interstate migration to a base population. Each of the components of ERP is subject to error, but ERP itself is not in any way a guess. It is what the population would be if the components are measured well.

Population estimation is also very different to sample survey-based estimation. This is because population estimation is largely based on a full enumeration of components. In the case of the population base, only the PES used sampled data to adjust for census net undercount. In the case of the components of population growth used to carry population estimates forward, Australia has a theoretically complete measure of each component.

Another example of a common misconception relates to the fact that the population projections presented in this publication are not predictions or forecasts. They are an assessment of what would happen to Australia's population if the assumed levels of components of population change - births, deaths and migration - were to hold into the future.

### Accessibility

ERP data is available in a variety of formats on the ABS website under the 3101.0 product family. The formats available free on the web are:

• The main features which has the key figures commentary,
• Time series spreadsheets on population change, components of change and interstate arrivals and departures,


text:
Revision history [back]

Is there a way to have Context Sensitive Morphisms?

I be been using WordMorphism and I was wondering if there is a way to have context sensitive substitutions?

An example would be bc -> ab so a is replaced iff bac appears.

WordMorphism(“ab->b”)

Is there a way to have Context Sensitive Morphisms?

I be been using WordMorphism and I was wondering if there is a way to have context sensitive substitutions?

An example would be bc b”a”c -> ab so a is replaced iff bac appears.

WordMorphism(“ab->b”)

Is there a way to have Context Sensitive Morphisms?

I be been using WordMorphism and I was wondering if there is a way to have context sensitive substitutions?

An example would be b”a”c -> ab so a is replaced iff bac appears.

WordMorphism(“ab->b”)
WordMorphism(“bac->ab”)


text:
# How do you differentiate f(x)=ln(cos(e^(x) ))  using the chain rule?

Mar 17, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \tan {e}^{x} \cdot {e}^{x}$

#### Explanation:

So, we got three functions here:

$\ln \left(\cos \left({e}^{x}\right)\right)$

$\cos \left({e}^{x}\right)$

and

${e}^{x}$

Let $y = \ln \left(\cos \left({e}^{x}\right)\right)$

differentiating w.r.t. $x$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{d}{\mathrm{dx}} \ln \left(\cos \left({e}^{x}\right)\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{\cos} \left({e}^{x}\right) \cdot \frac{d}{\mathrm{dx}} \cos \left({e}^{x}\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{1}{\cos} \left({e}^{x}\right) \cdot \sin \left({e}^{x}\right) \cdot \frac{d}{\mathrm{dx}} {e}^{x}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{1}{\cos} \left({e}^{x}\right) \cdot \sin \left({e}^{x}\right) \cdot {e}^{x}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \sin \frac{{e}^{x}}{\cos} \left({e}^{x}\right) \cdot {e}^{x}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \tan {e}^{x} \cdot {e}^{x}$

This will be the differentiated function.


text:
# Distributive property of Multiplication over Subtraction

## Formula

$a \times (b-c)$ $\,=\,$ $a \times b -a \times c$

An arithmetic property that distributes the multiplication across the subtraction is called the distributive property of multiplication over subtraction.

### Introduction

$a$, $b$ and $c$ are three literals and denote three terms in algebraic form.

The product of the term $a$ and the difference of the terms $b$ and $c$ can be written in the following mathematical form.

$a \times (b-c)$

The product of them can be calculated by distributing the multiplication over the subtraction.

$\implies$ $a \times (b-c)$ $\,=\,$ $a \times b -a \times c$

### Proof

Learn how to prove the distributive property of multiplication across subtraction in algebraic form by geometrical method.

#### Verification

$3$, $4$ and $7$ are three numbers. Find the product of number $2$ and subtraction of $4$ from $7$.

$3 \times (7-4)$

Now, calculate the value of the arithmetic expression.

$\implies$ $3 \times (7-4)$ $\,=\,$ $3 \times 3$

$\implies$ $3 \times (7-4) \,=\, 9$

Now, find the difference of the products of $3$ and $4$, and $3$ and $7$.

$3 \times 7 -3 \times 4$ $\,=\,$ $21-12$

$\implies$ $3 \times 7 -3 \times 4$ $\,=\,$ $9$

Now, compare the results of both the arithmetic expressions. Numerically, they are equal.

$\,\,\, \therefore \,\,\,\,\,\,$ $3 \times (7-4)$ $\,=\,$ $3 \times 7 -3 \times 4$ $\,=\,$ $9$

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text:
# TeXnicCenter - hotkeys for user-defined command or AutoHotkey without overwriting marked content?

defined commands which I want to use in fast way. I tried to make a short hotkey, not the long default one for inserting own text blocks, but I did not find an option for that. Then I thought I would use AutoHotkey, but normally in TeXnicCenter when you select some text and use a command like \textit{}the text wont be overwritten. If I use Autohotkey for my customized command, I select my text and it will be overwritten.

Is there a solution I did not see?

-

Well, it is possible at least for the F-keys. The following script copies the selected text, inserts \command{}, moves the cursor left and inserts the text again.

F4::
Send ^c
Send \command{{}{}}{LEFT}
Send ^v
Return

-
why only F-keys? –  user1234567890 Feb 17 at 12:08
Well, F-keys do not mess with the text you entered, for combinations like Alt+<Key> it should probably work as well. But as soon as any keys are printed out, you do not have the original chars to copy to the clipboard. –  Uwe Ziegenhagen Feb 17 at 13:21
Just to go sure: This is an Autohotkey script? Please add a short info. –  Speravir Feb 17 at 18:55
Yes, this is Autohotkey. –  Uwe Ziegenhagen Feb 17 at 19:05


text:
# 7.3 Polar coordinates  (Page 6/16)

 Page 6 / 16

Determine the symmetry of the graph determined by the equation $r=2\phantom{\rule{0.2em}{0ex}}\text{cos}\left(3\theta \right)$ and create a graph.

Symmetric with respect to the polar axis.

## Key concepts

• The polar coordinate system provides an alternative way to locate points in the plane.
• Convert points between rectangular and polar coordinates using the formulas
$x=r\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta \phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=r\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta$

and
$r=\sqrt{{x}^{2}+{y}^{2}}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}\theta =\frac{y}{x}.$
• To sketch a polar curve from a given polar function, make a table of values and take advantage of periodic properties.
• Use the conversion formulas to convert equations between rectangular and polar coordinates.
• Identify symmetry in polar curves, which can occur through the pole, the horizontal axis, or the vertical axis.

In the following exercises, plot the point whose polar coordinates are given by first constructing the angle $\theta$ and then marking off the distance r along the ray.

$\left(3,\frac{\pi }{6}\right)$

$\left(-2,\frac{5\pi }{3}\right)$

$\left(0,\frac{7\pi }{6}\right)$

$\left(-4,\frac{3\pi }{4}\right)$

$\left(1,\frac{\pi }{4}\right)$

$\left(2,\frac{5\pi }{6}\right)$

$\left(1,\frac{\pi }{2}\right)$

For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point.

Coordinates of point A .

Coordinates of point B .

$B\begin{array}{cc}\left(3,\frac{\text{−}\pi }{3}\right)\hfill & B\left(-3,\frac{2\pi }{3}\right)\hfill \end{array}$

Coordinates of point C .

Coordinates of point D .

$D\left(5,\frac{7\pi }{6}\right)D\left(-5,\frac{\pi }{6}\right)$

For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in $\left(0,2\pi \right].$ Round to three decimal places.

$\left(2,\phantom{\rule{0.2em}{0ex}}2\right)$

$\left(3,-4\right)$ (3, −4)

$\begin{array}{cc}\left(5,-0.927\right)\hfill & \left(-5,-0.927+\pi \right)\hfill \end{array}$

$\left(8,\phantom{\rule{0.2em}{0ex}}15\right)$

$\left(-6,\phantom{\rule{0.2em}{0ex}}8\right)$

$\left(10,-0.927\right)\left(-10,-0.927+\pi \right)$

$\left(4,\phantom{\rule{0.2em}{0ex}}3\right)$

$\left(3,\text{−}\sqrt{3}\right)$

$\left(2\sqrt{3},-0.524\right)\left(-2\sqrt{3},-0.524+\pi \right)$

For the following exercises, find rectangular coordinates for the given point in polar coordinates.

$\left(2,\frac{5\pi }{4}\right)$

$\left(-2,\frac{\pi }{6}\right)$

$\left(\begin{array}{cc}\text{−}\sqrt{3},\hfill & -1\hfill \end{array}\right)$

$\left(5,\frac{\pi }{3}\right)$

$\left(1,\frac{7\pi }{6}\right)$

$\left(\begin{array}{cc}-\frac{\sqrt{3}}{2},\hfill & \frac{-1}{2}\hfill \end{array}\right)$

$\left(-3,\frac{3\pi }{4}\right)$

$\left(0,\frac{\pi }{2}\right)$

$\left(\begin{array}{cc}0,\hfill & 0\hfill \end{array}\right)$

$\left(-4.5,6.5\right)$

For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the $x$ -axis, the $y$ -axis, or the origin.

$r=3\phantom{\rule{0.2em}{0ex}}\text{sin}\left(2\theta \right)$

Symmetry with respect to the x -axis, y -axis, and origin.

${r}^{2}=9\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta$

$r=\text{cos}\left(\frac{\theta }{5}\right)$

Symmetric with respect to x -axis only.

$r=2\phantom{\rule{0.2em}{0ex}}\text{sec}\phantom{\rule{0.2em}{0ex}}\theta$

$r=1+\text{cos}\phantom{\rule{0.2em}{0ex}}\theta$

Symmetry with respect to x -axis only.

For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.

$r=3$

$\theta =\frac{\pi }{4}$

Line $y=x$

$r=\text{sec}\phantom{\rule{0.2em}{0ex}}\theta$

$r=\text{csc}\phantom{\rule{0.2em}{0ex}}\theta$

$y=1$

For the following exercises, convert the rectangular equation to polar form and sketch its graph.

${x}^{2}+{y}^{2}=16$

${x}^{2}-{y}^{2}=16$

Hyperbola; polar form ${r}^{2}\text{cos}\left(2\theta \right)=16$ or ${r}^{2}=16\phantom{\rule{0.2em}{0ex}}\text{sec}\phantom{\rule{0.2em}{0ex}}\theta .$

$x=8$

For the following exercises, convert the rectangular equation to polar form and sketch its graph.

$3x-y=2$

$r=\frac{2}{3\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta -\text{sin}\phantom{\rule{0.2em}{0ex}}\theta }$

${y}^{2}=4x$

For the following exercises, convert the polar equation to rectangular form and sketch its graph.

$r=4\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta$

${x}^{2}+{y}^{2}=4y$

$r=6\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta$

$r=\theta$

$x\phantom{\rule{0.2em}{0ex}}\text{tan}\sqrt{{x}^{2}+{y}^{2}}=y$

$r=\text{cot}\phantom{\rule{0.2em}{0ex}}\theta \phantom{\rule{0.2em}{0ex}}\text{csc}\phantom{\rule{0.2em}{0ex}}\theta$

For the following exercises, sketch a graph of the polar equation and identify any symmetry.

$r=1+\text{sin}\phantom{\rule{0.2em}{0ex}}\theta$

y -axis symmetry

$r=3-2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta$

$r=2-2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta$

y -axis symmetry

$r=5-4\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta$

$r=3\phantom{\rule{0.2em}{0ex}}\text{cos}\left(2\theta \right)$

x - and y -axis symmetry and symmetry about the pole

$r=3\phantom{\rule{0.2em}{0ex}}\text{sin}\left(2\theta \right)$

$r=2\phantom{\rule{0.2em}{0ex}}\text{cos}\left(3\theta \right)$

x -axis symmetry

$r=3\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\frac{\theta }{2}\right)$

${r}^{2}=4\phantom{\rule{0.2em}{0ex}}\text{cos}\left(2\theta \right)$

x - and y -axis symmetry and symmetry about the pole

${r}^{2}=4\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta$

$r=2\theta$

no symmetry

[T] The graph of $r=2\phantom{\rule{0.2em}{0ex}}\text{cos}\left(2\theta \right)\text{sec}\left(\theta \right).$ is called a strophoid. Use a graphing utility to sketch the graph, and, from the graph, determine the asymptote.

[T] Use a graphing utility and sketch the graph of $r=\frac{6}{2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta -3\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta }.$

a line

[T] Use a graphing utility to graph $r=\frac{1}{1-\text{cos}\phantom{\rule{0.2em}{0ex}}\theta }.$

[T] Use technology to graph $r={e}^{\text{sin}\left(\theta \right)}-2\phantom{\rule{0.2em}{0ex}}\text{cos}\left(4\theta \right).$

[T] Use technology to plot $r=\text{sin}\left(\frac{3\theta }{7}\right)$ (use the interval $0\le \theta \le 14\pi \right).$

Without using technology, sketch the polar curve $\theta =\frac{2\pi }{3}.$

[T] Use a graphing utility to plot $r=\theta \phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta$ for $\text{−}\pi \le \theta \le \pi .$

[T] Use technology to plot $r={e}^{-0.1\theta }$ for $-10\le \theta \le 10.$

[T] There is a curve known as the “ Black Hole .” Use technology to plot $r={e}^{-0.01\theta }$ for $-100\le \theta \le 100.$

[T] Use the results of the preceding two problems to explore the graphs of $r={e}^{-0.001\theta }$ and $r={e}^{-0.0001\theta }$ for $|\theta |>100.$

Answers vary. One possibility is the spiral lines become closer together and the total number of spirals increases.

what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
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Berger describes sociologists as concerned with
Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
Abdul