Hey all! I'm having difficulty with how my professor gets to an answer.
The question I'm facing is this.
The answer I'm given is here.
The k is a term from thermodynamics, but the rest ofd the math I get, to a degree. What I am lost on is how the answer reaches is the second term of Na/N ln(Na/N), because from my answer I get this.
How do I combine the first two ln(_) terms in my answer?
I just went through the limits/series/sequences unit of a pre-calculus high school course, and one of the extra credit questions on a test asked for $\lim_{n\rightarrow\infty}\frac{\sqrt{2\pi x}(\frac{n}{e})^n}{n!}$.
According to Wikipedia and MathWorld, the expression evaluates to 1. I can prove this to myself empirically by looking at the graph of the function, but I can't understand how students with our math background were expected to figure this out.
Were we supposed to simply plug in successively larger values for n, or was there some method within our abilities? Post any possible solutions, and I'll answer with whether or not we had that knowledge to work with, I guess.
EDIT: comments caught me being silly and messing up the expression.
Is there an application of Stirling's Approximation? I can't see that Stirling's Approximation would be useful computationally for positive integer values of the Gamma Function. I can't see why even for large non-integer inputs that Stirling's Approximation would be useful. Despite it's accuracy as n! grows, it's absolute error grows as well (ie (.5% of 1000 > 10% of 100). Therefore, it could only give a ball park value of n!, for which an average of the closest integer values of n! might do equally well.
Is Stirling's Approximation general for complex values of Gamma; and therefore; avoids calculating a potentially nasty integral? What is it's practical use?
I'm trying to verify N!/[2^N * (N/2+l)! * (N/2-l)!] ~ (e^((-l^2)/N))/(pi*N/2)^(1/4) Where N is even, and l = -N/2,..., N/2 It was suggested to me to use Stirling's Approximation which got me somewhere, just not where I need to be... I'd really appreciate some help.
I hope you'll all forgive me for writing about T20s so soon after all the test series we've just had, but I've been working on this for a while and would like to share it.
#Introduction - not all strike rates are equal
Dawid Malan's T20I strike rate of 137.2 is very similar, in fact a slightly higher than Moeen Ali's 135.3, yet anyone who's watched much of them would know they are two very different players. Malan is an anchor, he takes his time getting in before, after 15-20 balls, putting the after burners on and playing the type of innings which has earned him his title of Number One T20I Batsman^TM. On the other hand, Moeen operates either as a hit-from-ball-one top order player with a penchant for smashing spinners, or a lower order finisher. Intuitively we know this from watching them play and more generally we bookmark certain players into certain roles. Kohli, in fact all of the Fab Four are anchors, players like Roy and Russell hit from ball one etc etc.
But are these labels accurate and how can these different playing styles be properly represented given traditional metrics and strike rate don't paint an accurate picture?
Strike Rate Curves
One way I have begun to do this is using the ball-by-ball data from the excellent Cricsheet database to draw what I call strike rate curves. These show the strike rate of a batter but only on a specific ball of an innings. That is to say if a player has faced the 20th ball of their innings (the x axis) 10 times and scored 15 runs off those 10 deliveries, their strike rate (the y axis) for that point is 150. To smooth the line somewhat at any given point the mean SRs of the ball in question and the two balls previous are plotted, for balls one and two of an innings, the raw value and the average of balls one and two are plotted respectively.
Here is an example for Malan and his line follows the path we would expect, taking six or seven balls before he's scoring at a run a ball, then eventually flourishing to the loftier regions of the graph. When compared with Moeen, again there are no surprises. Whereas Malan's line takes its time, slowly creeping higher and higher, Moeen's takes a few balls to get set then rockets up after around 5 balls and stays there, only coming down for a strange dip at around 12 balls (more on that later).
To give another example, here is Jason Roy's curve plotted alongside Virat Koh
... keep reading on reddit β‘Another thread gave me the idea to ask this, because I'm having trouble thinking of any. Basically, I am asking for examples of theorems, proved before 1900, that prove that an object with some properties exists, but the proof (or the literature surrounding it) does not give an explicit method of obtaining that object. This includes both theorems that prove the existence of a single object, and those that prove that "for all ... there exist ..." by showing that an object exist for each possible input.
What do I mean by explicit? It's a hard question, but I will tentatively try to provide a guideline.
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If it's a string, or a combinatorial object, the proof would provide, or imply an efficient algorithm (say, polynomial time in term of the size of the object), to find it. So not something like "brute force search through this".
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If it's a natural number or rational number, it should be described using a formula with standard arithmetic and combinatorial operations, or part of a recurrence sequence where the recurrence relation has such formula. Combinatorial operations are those of the form "count all objects up to this upper bound such that they satisfies this property" where the property can be easily checked by an efficient algorithm (say, polynomial time in term of the size of the object).
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If it's a real number, there is at least a recurrence sequence where the recurrence relation is provided by an explicit formula using standard functions, or a sequence with an explicit formula. The formula should be sufficiently easy to compute, and the sequence converges fast enough, that time taken to compute decimal digits is at most polynomial.
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If it's an algebraic object, or a geometric object, then there is a standard representation of them using real numbers or natural numbers or strings or combinatorial objects, and we reduce to the previous cases.
Of course, the above guideline isn't perfect, and I can already think of problems. For example, the prime counting function seems explicit enough, but we didn't have an efficient method to check for prime. Any geometry that happened before Descartes probably didn't come with explicit representation in real numbers. But I feel like that guideline should be narrow things enough down to a handful of examples.
What are some examples of non-explicit proof of existence before the 20th century?
After wrecking my brain a bit and search around, I can only think of a few.
A convincing example: Minkowski's theore
... keep reading on reddit β‘Hi, this is my second post.
n! can be written as.
https://preview.redd.it/pnmiw5crha881.png?width=400&format=png&auto=webp&s=544a1be909dd5dd6d655eb387f9ca3d38154f816
I really like this one because it uses the factorial number is the formula for the factorial number, now i'll apply the formula to 5.
https://preview.redd.it/q8ex9g7xja881.png?width=580&format=png&auto=webp&s=4117f8b0998a7cb6692337054e1eede23104ea5c
I'll write all the steps:
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5! = 5 β’ (5-1)!
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5! = 5 β’Β (4)! , 4! = 4 β’ (4-1) --> 4! = 4 β’ (3)!
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5! = 5 β’ 4 β’ (3)! , 3! = 3 β’Β (3-1)! --> 3! = 3 β’ (2)!
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5! = 5 β’ 4 β’ 3 β’ (2)! , 2! = 2 β’ (2-1)! --> 2! = 2 β’ (2-1)!
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5! = 5 β’ 4 β’ 3 β’ 2 β’ (1)! , 1! = 1 β’ (1-1)! --> 1! = 1 β’ (0)!
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5! = 5 β’ 4 β’ 3 β’ 2 β’ 1
Now this is the simplest way of writing the factorial number but you can understand the concept a n! is the number multiple by all the numbers before it.
You might be wondering, what is the n! of a negative number?
Answer : it does not exist.
However -(n)! does exist because -(5)! is -(5 β’ 4 β’ 3 β’ 2 β’ 1) and that would be -(120) and since the negative symbol is outside the parenthesis the negative symbol is not gonna affect the operation.
Another really interesting equation is the Stirling approximation
this is the Stirling approximation
Which, if we try with 5 gives us
https://preview.redd.it/2jkc5jt6pa881.png?width=738&format=png&auto=webp&s=4796f561188a5ab1cd02e52f6c37c45ac2f89c80
118.019 which is very close with an approximation error of about 2.
Post.Scriptum : there are many other ways to write the factorial umber but for this post I decided to only use one
Post.Post.Scriptum : the Stirling approximation has many different variation but for this post I decided to only use one
Thank for reading.
I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.
Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.
Do your worst!
I'm surprised it hasn't decade.
For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.
I said "hey look, an escaPEA"
No one near me but it didn't half make me laugh for a good hour or so!
Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies π
It really does, I swear!
Theyβre on standbi
Pilot on me!!
Nothing, he was gladiator.
Because she wanted to see the task manager.
Dad jokes are supposed to be jokes you can tell a kid and they will understand it and find it funny.
This sub is mostly just NSFW puns now.
If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.
Edit* I'm not replying any longer and turning off notifications but to all those that say "no one cares", there sure are a lot of you arguing about it. Maybe I'm wrong but you people don't need to be rude about it. If you really don't care, don't comment.
When I got home, they were still there.
What did 0 say to 8 ?
" Nice Belt "
So What did 3 say to 8 ?
" Hey, you two stop making out "
I won't be doing that today!
[Removed]
Where ever you left it π€·ββοΈπ€
This morning, my 4 year old daughter.
Daughter: I'm hungry
Me: nerves building, smile widening
Me: Hi hungry, I'm dad.
She had no idea what was going on but I finally did it.
Thank you all for listening.
You take away their little brooms
There hasn't been a post all year!
