And That Is The Rest Of The Story

Ask me anything   throw idea's my way!   Hello everyone, my name is Aidan Patterson, and I am the owner of my own thoughts and actions. My hobbies include Mathematics, Karate, Physics and anything music related. Never be afraid to tell me I am wrong, but be prepared to prove it, and may I say; good luck proving it to me!

Proofsareart

I’ve you’ve never been on proofsareart’s blog, you have not lived to see amazing mathematics!

Mathematics presentations which present new material always end with a section called “Future Work” which is exactly what it sounds like. Questions beget questions; this is the wonderful paradox of knowledge, and when you put 100+ hours into a single question, you are uniquely qualified to say what variants, generalizations, subquestions, and related concepts are most interesting.
I haven’t put anywhere near 100 hours into horn-torus pseudoknots, but I’d like to think that I have some intuition for questions that might be tractable, if not interesting.
I have made some effort to trace back the questions which were inspired by others back to their source. However, there have been many more of you who have supported my effort; I have read all of the reblogs and I thank you for your kind words! And of course, a special thanks to bigblueboo, without whose excellent gif, this investigation would never have been imagined :)

Future work:
Determine a maximal linearly independent subfamily F of the HTPs
Determine a maximal orthonormal subfamily G of the HTPs
Characterize span(F) and span(G). 
In the space of integrable / L^p / bounded / continuous / smooth / analytic functions, are they open / closed / dense ?
Search the literature for notions of the complexity of a parametrized curve and evaluate these measures for HTPs and their higher-order derivatives.
More generally, investigate the higher-order derivatives of HTPs
Observe patterns in the equations of motion caused by appending additional circles. (Also, see the “Edit.” here)
Is there a maximally interesting or natural algorithm for choosing the planes of rotation for these circles? 
Following a particular algorithm is there a behavior as the number of circles goes to infinity?
Following a randomized algorithm, is there an expected behavior as the number of circles goes to infinity?
The generalization from circles to ellipses is relatively clear: is there a generalization to other conics that induce similar equations of motion?
Can we gain any information by passing to projective space?
Explain the geometry that happens when “trig functions”  on other closed curves are used instead of sin and cos.
Is there any advantage to making theta and phi use the same curve type?
Is there any advantage to curves with convex interior?
Is there any advantage to closed curves?
Investigate minimal surfaces with HTPs as their boundaries.
Find solutions, numerically, empirically or analytically, to the differential equation || x’ || = a^2 + b^2 || x ||, where x is a function from the reals into some Hilbert/Banach/normed space.

— 2 weeks ago with 17 notes
#mathematics  #maths  #mathema  #physics  #math 
Reblog or like if you have synesthesia!

a-synesthetic-world:

That way, you can meet other synesthetes by looking at the notes :)

Numbers to colors, letters to colors, music to colors, instrument timbers to colors and textures, weekdays and other nonphysical objects including equations to spatial projections and personalities to colors. I’m still not sure how that last one works though :P

(via emsloe)

— 2 weeks ago with 176 notes
#Synesthesia 

I hate when people are like: “You’re gay? So that’s why people like you!”. Like, no, people don’t like me more than you because I’m gay, I have several other personality traits that make me better than you.

— 2 weeks ago with 35 notes
#gay  #anger  #funny  #this happened today  #people  #suck  #dicks 

airagorncharda:

cutebabe:

embrace-your-earth:

send-a-smile:

"The Rape Poem to End All Rape Poems."

One of the best pieces of group spoken word poetry I’ve ever seen. WATCH IT. 

THIS NEEDS TO BE WATCHED CHILLS FIRING THE WHOLE TIME POWER TO THE PEOPLE OF GOOD INTENTION AND LOVE

i started crying

this is what I mean when I say that anger is a powerful tool

(via can-i-leave-this-blank)

— 2 weeks ago with 214137 notes

hyrodium:

The proof of Sum of Square Numbers!

This is so much cooler than the proof for the Pythagorean theorem :)

(via cyclopentadiene)

— 3 weeks ago with 2328 notes
Pi is Irrational

So, a few days ago I found a proof for the irrationality of πand I thought I would share it with you all. The proof was originally done by Adrien-Marie Legendre in the 7th edition of his book elements de geometrie (1808). The version of this proof that I’ll be posting is more modern, and was completed in 1946 by Carl Siegel and published in his book: Transcendental Numbers. In case you don’t want to read it or the math gets too harry to read (I’ve been there with this proof), the proof is one by contradiction, where if we assume that π2 is a rational number and sub it in rational form into an equation with integer coefficients, we would expect an integer value to come out, but instead get  number between 0 and 1 which is obviously not an integer.This proof is pretty long, so here we go!

Assume

image 

Where A(x) and B(x) are polynomials of degree n, such that for x≈0 the ratio becomes close to exact. 

Now, assume we can also write this equation

image

Since both A(x) and B(x) are of degree n, we can write:

image

image

By the direct multiplication of B(x) with the power series of ex we get

image

Suppose we want R(x) to start with the 2n+1 term, that is let us assume the first 2n+1 coefficients of R(x) vanish. This will allow R(x) to vanish as fast as x2n+1, which is faster than x itself. 

We now have the three equations

image

image

image

Notice that we have 2n+1 equations, but 2n+2 coefficients so despite the restriction of R(x) we can still have nontrivial solutions. Now, we begin our exploration of differential operators. We let

image

image  

Observe that the condition for R(x) to approach zero when x approaches zero is vital in that

image

image

So taking R(x) to zero for R(0) will allow the operators to commute. We now have to find a way to extend the definition for the inverse operator for all n as opposed to the singular definition we have now. Just to get a feel for the inverse operator for other values of n, let’s see how the n=2 case looks. 

 image

This integral has a nice geometric interpenetration. With t on the y-axis and s on the x-axis, the region we are integrating is under the line s=t, meaning we can make the substitution:

image

Which changes the integral to:

image

Which is again one dimensional. You can do the induction proof on your own, but this is already going to be too long, so I’ll just give the generalization for all n. 

image

We can use this to investigate integrals we’ll come across later. Now, we must investigate functions in the form:

image

We can see that taking the first derivative gives

image

Taking the second derivative begins to show a pattern for nth derivatives, as:

image

image

image

image

image

image

Again, I’ll skip the induction and just say that

image

Back to our formula for R(x),

image

We can say this because A(x) was assumed to be degree n, and therefore disappears when differentiated n+1 times. From before, we can get a formula for the term containing B(x) by using the formula we just derived and setting λ=1.

image

Which means we can immediately write 

image

Since R(x) starts with the 2n+1 term, we can say that:

image

Plus all of the other terms of degree n+1 and higher, where ris the first coefficient of R(x). Because B(x) was assumed to be a polynomial of degree n, 

image

Will have to be degree n as well, as one term will contain 1B(x), and the others will all contain some constant times some derivative of B(x) which will be a lower degree polynomial. If we Say that

image

Then we may only keep the leading term of R(x) in our expansion. If we let e^x take on its power expansion again, we see the only term that need be considered again is image.So,

image

We can use a similar method to derive A(x) (multiply through the original expression for R(x) by e^(-x) and then do the work with derivatives we did here) which works out to:

image

Now, recall that 

image

Using our solution for B(x) that we just derived, we can see that

image

Or,

image

From before

image

So, 

image

Letting u=(s/x), du=1/x, we see that

image

If we let t=1-u and dt=-du then,

image

If we change the dummy variable of integration in the line 

image

To t, add it to the previous expression and then divide by 2, we see that:

image

Changing the form a bit, we get

image

We can now sub in a value for x to make further simplifications on this integral, let x=iπ.

image

Some quick simplification yields:

image

Now, we’re not evaluating this. I just want you to note that depending on the parity of n, the expression for image, we can get either a positive or negative number, but not zero. Therefore, image. Now, we have to do a bit of manipulation with the equation that we had for R(x) in the beginning. 

image

Immediately it is clear that

image

and

image

Remember, our original first alternate expression for R(x),

image

Altering it to fit the previous expression gives

image

For t=-s and ds=-dt,

image

again for u=x-t, du=-dt

image

Which makes it clear that 

image

Now we can get two sets of equations for A(x) and B(x)

image

image

Subtracting the second from the first give

image

This can be true for all x if and only if the brackets vanish identically, that is:

image

image

These are equivalent statements, and they can be brought together by again setting image.

image

So,

image 

as 

image

Since A(x) is a polynomial in x, we can expand it from here and see where this fact leaves us. 

image

image

Where the final term in A(-x) depends on the parity of n. Adding these expressions together as before gives:

image

where the final term is either of degree n or n-1. If we let [m] be the integer part of a number m, and change the variable from x^2 to u, we can see:

image

Now, here comes the end, suppose that π2 is rational, then with x=iπ and u=-π, it follows that

image

Since all of the coefficients a_i were assumed to be integers, then

image

where j is some integer, positive or negative. Since we know that   

image

then,

image

Finally, from before, 

image

Taking limits, we can see that both the coefficient and the integral approach zero as n approaches infinity, so we can always chose sufficient n such that

image

Which of course implies

image

So,

image

This means that there should be an integer between zero and one, which is an obvious contradiction. This is sufficient to show that π2  is irrational, and it follows that π is irrational

image

Have a happy pi month :)

— 1 month ago with 29 notes
#math  #mathematics  #maths  #mathema  #physics  #integrals  #inequalities  #pi  #e  #euler  #irrational  #number theory  #calculus  #operators  #proof 
Transcendental Pi

I’m going to get a proof that pi squared is irrational (and therefore pi is irrational) on here soon, but in the meantime, have a simple proof that pi is transcendental :)

The Lindermann-Weierstrass theorem states that for the equation

image

b will always be transcendental if a is a non-zero algebraic number. 

From this, we make note that

Meaning either i or pi have to be transcendental. Because i satisfies the relation:

image

Pi must be transcendental.

The real gem of this proof is the Lindermann-Weierstrass theorem, which I will look into finding a proof for soon. I’m about halfway through the proof that pi squared is irrational, so stay tuned, because it should be up by Friday :)

— 1 month ago with 31 notes
#math  #mathematics  #maths  #physics  #mathema  #pi  #i  #theorem  #transcendental  #numbers  #number theory 
Quantum states and superposition →

scientiststhesis:

Sit down, kids, you’re about to learn a thing.

(This will have significant overlaps with the Quantum Physics Sequence, albeit with more mathematics and symbols and what I hope will be a different angle. I will also shamelessly steal pictures from there.)

Let’s start…

If you want to do these experiments, the new 3-D glasses they give out at movie theaters have polarized lenses in them. You could just pop them out and start doing science :D

— 1 month ago with 23 notes

bobbyhoying:

giantspacefetus:

My math binders are always red every year I feel like math is just a red subject

Math is a blue subject and I’m prepared to fight you over this

OKAY NO! Math is red. Red. RED. RED!!! Why? 

Capital M’s have a deep red color to them, (The M in Monarch feels the exact same way)

the “ath” part of Math is actually a light blue color (kind of like a summer sky, the a is white, the t is blue, and the h is an indigo), which is where you might be getting the blue from, but with a “M” at the beginning, the word is overpowered by the red hue that changes the a from white to a pink, making the word feel more red with the addition of the h. 

I wonder if the difference here is the part of the word people focus on when they’re reading?

Mathematics on the other hand is a very intricate word. 

the “M” at the beginning is a strong red again, but the e after “ath” is black, which makes the second m seem bright yellow. The “at” is white again, but the “ic” is strongly red, and the s is bright yellow. 

This doesn’t mean that the overall feel of the word is orange though, and I actually feel that the word is distinctly red in the “Math” section, black on the “e” and finishes yellow with “matics”. Shoved together, I get a pinkish feeling, but I can very clearly see yellow and black. 

And yes, my math binders are always red or pink.

On a side note, this is reason I hate the word maths, because in the presence of a yellow s, the red math turns to pink, and makes the word look like an Easter egg, which I don’t feel is very representative of mathematics. 

I love synesthesia  :)

(via i-am-allergic-to-your-stupidness)

— 1 month ago with 108411 notes
#math  #mathematics  #maths  #physics 
Cauchy-Schwarz Inequality, Time, and Space

Say you’re sitting at home, and you thought to yourself, if I threw myself off the 40th floor balcony in my apartment building, what would my average speed be when I hit the ground? You don’t really intend to jump to find out, so you start doing some math. At this point, 9,999 out of 10,000 people would say that since the acceleration of gravity is 9.8 meters per second per second, and assuming each floor was about ten feet long (total of 122 meters), you would probably guess that your average speed would be about 24.5 m/s and leave it at that. BUT you are a mathematician (which is probably why you were contemplating jumping off a building in the first place), and you want to make sure you’re right. What you just calculated was a time average, that is:

But, what if you wanted a different viewpoint? What if you wanted to look at the speed as a function of the distance fallen just to make sure the number you got was the right one? The equation is remarkably similar,

but, gives you strange answers. Since L=122 and y=4.9t^2, we can see that:

That’s obviously quite different from 24.5 m/s. You could argue (probably for hours) about the meaning of this result, but since you’re a mathematician and not a physicist, you’re more interested in whether

regardless of the boundary conditions. You need a way to compare the two, so you think to yourself, what kind of math could I do with two integrated functions? 

Well, if we start with the realization that:

If f(t) and g(t) are real valued functions, and U, and, L are real numbers (possibly infinite) and, lambda is real (for now). 

Expanding, we can see that:

If we label each separate definite integral c,b,a respectively, we get:

This has a simple geometric interpretation, the function of lambda can not cross the lambda-axis, the most it can do is touch it, as we are dealing with the area under functions. This must mean that lambda is complex for h(lambda)=0, or has a double zero at that point.

So, we get the inequality:

This finally gives the inequality

So how are we going to use this to see if your velocity measured with time is less than or equal to your velocity measured using space?

Well, we can let g(t)=v(t)/T, and let f(t)=1. The Cauchy-Schwarz Inequality says that:

and

Taking advantage of the notation, 

We’ve now effectively changed the variable to which we are integrating, so we have to change the limits on the integral, namely T=L and t=y

So, 

We’re really close now! Since 

 

We can change the left side of the inequality to be

Which gives us

and finally,

So,

Strange isn’t it?

I’ve deliberately left a bunch of questions unanswered, in the hopes that you’ll look into them. For one, what is the physical interpretation of the velocity in space that we derived in the first section? Also, would drag affect the answer in any way?

In any case, I hope this at least made you think. Pretty cool right? :)

— 1 month ago with 74 notes
#mathematics  #maths  #math  #mathema  #physics  #integrals  #complex  #numbers  #velocity  #time  #space  #relativity  #calculus  #analysis  #dynamics  #proof 
nelolin asked: Hi there, I searched 'gamma function' on tumblr and found your post about fractional derivatives. I am glad I found your blog. :). On another note, do you use a Latex extension for your posts? Or MathJax?


Answer:

Hey! sorry about talking forever to respond, I have a bunch of messages and tumblr does a terrible job of telling me when I have one -.- I use LaTeX for my posts, but more specifically, I use this website: http://www.codecogs.com/latex/eqneditor.php and then just copy the images that it spits out :)  

— 1 month ago with 6 notes

mindfuckmath:

Brussel  Sprouts - Numberphile

Some brussel sprouts are good for you, others will teach you about topology and the Euler characteristic.

— 1 month ago with 32 notes

the-mathematical-poet:

Solution by Radicals:

Solutions to the general equations

ax+b=0

ax^2+bx+c=0

ax^3+bx^2+cx+d=0

ax^4+bx^3+cx^2+dx+e=0

can be expressed through radicals.

There does not exist a general solution by radicals for equations of degree 5 or more - a fact that can be proved using Galois Theory and the fact that Sn is not solvable for n greater than or equal to 5.

I was reading about this again yesterday! I think I’ll put the proof up on here.

— 1 month ago with 476 notes
thatvegancosplayer:

huffsomepluff:

fandomsandfeminism:

huffsomepluff:

Honey, I don’t think you gotta worry about that anytime soon.
#anti feminism#A bit optimistic are we?#vomit warning#i apologize to all of my followers’ eyes

A few things. 
1) Your TAGS are DISGUSTING. You are a misogynistic, fat-shaming, rape apologist. The idea that a woman, regardless of how she looks, would be OPTIMISTIC to be raped is just stomach churning.
2) The image you have posted is CLEARLY PHOTOSHOPPED. Here is this woman’s original picture with her ACTUAL message:

Thanks for proving her point. 
You gross ass sexist crumblefuck. 

I didn’t even make the picture, I fucking found it on the internet. Thanks for not being able to take a fucking joke.

Oh wow here we fucking go again, but let’s go in categories this time to make this easy for you.
Rape
1 in 6 American women are raped.
9 out of every 10 rapes are against women. 
1 in 33 American men are raped.
44% of victims in the US are minors. (Under 18.)
80% of US victims are under age 30.
Women of color are FAR more likely to be raped than white women. This page I linked has tons more stats.
Only 46% of all rapes are reported to the police. (Fact 23). 
Out of every 100 rapes, only 3 rapists will see jail time. 40 of those get reported, 10 lead to arrest, 4 lead to felony convictions.
Two-thirds of rapes are done by someone the victim knows.
Joking about rape DIRECTLY SUPPORTS rape culture and rapists and rape apologists.
.6% (yes, point six) of rapes are falsified. So when someone tells you or someone else that they have been raped, there is a 99.3% chance that they are telling the truth.
When women go to jail for beating or killing their spouse, it’s usually because they were abusive and/or rapists.
RAPE CAN HAPPEN TO ANYONE YOU FUCKNUT. IT’S ABOUT POWER, NOT “OH DAMN I CAN’T GET SOME SO I’LL JUST FUCK SOMEONE.”
There is so much more than can be said but fuck you, go look that shit up yourself.
Fat Shaming
Fat shaming is damaging, bullying and wrong period. It is fucked up to make fun of someone for any reason. The only reason why someone should be spoken to in a negative fashion is if they have actually done something wrong and harmful. (Like you did!)
The BMI scale is a crock of shit. It doesn’t actually determine what you as an individual need to be functioning properly. It goes by a shitty, rigid “weight by height” requirement. It doesn’t take into account personal family genetics, for instance. 
Some people are just fat. That’s it. Genetics, disorders, pregnancy, diseases, illness and body composition just fucking happen to some people. Body shaming does not make fat go away.
White culture leads to rigid beauty standards: thin, white, able-bodied and minded, symmetrical face with fat in all the “right places”. Only about 10% of women actually fit this “standard” and are ultimately starving for it.
Fat people suffer from not only peer abuse and media ridicule, but eating disorders and other heinous things to try and match their thinner counterparts.
Fat people are systematically oppressed. IE they don’t get jobs like their thin counterparts based on people thinking they are “lazy” and other things.
A case in Australia about how a couple could not adopt a child because the woman was too fat. (Also, England.)
None of this shit actually has to do with these people as individuals, their skill sets, etc. This is discrimination against them as being fat only, regardless if they can “help it” or not. (IE genetics versus simply lifestyle.)
So fuck you.

THANKS FOR THE SOURCES! I AM USING THEM IN AN ENGLISH ESSAY! 

thatvegancosplayer:

huffsomepluff:

fandomsandfeminism:

huffsomepluff:

Honey, I don’t think you gotta worry about that anytime soon.

A few things. 

1) Your TAGS are DISGUSTING. You are a misogynistic, fat-shaming, rape apologist. The idea that a woman, regardless of how she looks, would be OPTIMISTIC to be raped is just stomach churning.

2) The image you have posted is CLEARLY PHOTOSHOPPED. Here is this woman’s original picture with her ACTUAL message:

image

Thanks for proving her point. 

You gross ass sexist crumblefuck. 

I didn’t even make the picture, I fucking found it on the internet. Thanks for not being able to take a fucking joke.

Oh wow here we fucking go again, but let’s go in categories this time to make this easy for you.

Rape

There is so much more than can be said but fuck you, go look that shit up yourself.

Fat Shaming

  • Fat shaming is damaging, bullying and wrong period. It is fucked up to make fun of someone for any reason. The only reason why someone should be spoken to in a negative fashion is if they have actually done something wrong and harmful. (Like you did!)
  • The BMI scale is a crock of shit. It doesn’t actually determine what you as an individual need to be functioning properly. It goes by a shitty, rigid “weight by height” requirement. It doesn’t take into account personal family genetics, for instance. 
  • Some people are just fat. That’s it. Genetics, disorders, pregnancy, diseases, illness and body composition just fucking happen to some people. Body shaming does not make fat go away.
  • White culture leads to rigid beauty standards: thin, white, able-bodied and minded, symmetrical face with fat in all the “right places”. Only about 10% of women actually fit this “standard” and are ultimately starving for it.
  • Fat people suffer from not only peer abuse and media ridicule, but eating disorders and other heinous things to try and match their thinner counterparts.
  • Fat people are systematically oppressed. IE they don’t get jobs like their thin counterparts based on people thinking they are “lazy” and other things.
  • A case in Australia about how a couple could not adopt a child because the woman was too fat. (Also, England.)
  • None of this shit actually has to do with these people as individuals, their skill sets, etc. This is discrimination against them as being fat only, regardless if they can “help it” or not. (IE genetics versus simply lifestyle.)

So fuck you.

THANKS FOR THE SOURCES! I AM USING THEM IN AN ENGLISH ESSAY! 

(via kougannohime-chan)

— 1 month ago with 3282 notes
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