/mg/ - Math General

last i checked there was a guy calling everyone redditors and saying that applied math doesnt exist

I have done method four in the course of my own problems to refresh my memory on multiple occasions. so quasi-ascended tier.

Pascal's triangle is no meme. It's useful as fuck, and useful to sketch out on-command without repairing to a calculator.

I'm in second year and I still use Pascale's triangle occasionally for small exponents

Now do Stirling numbers

do lah numbers

>Pascal's triangle instead of just applying the formula
Absolutely pleb tier

the interviews are actually pretty good at being "unpreppable" ie more of an iq test.
but here are some things to think about:

with #2, we are graphing some equation of abs(x) and abs(y), so whatever it is, will be the same in each quadrant.

think in general about abs(x)^n + abs(x)^n = 1.
clearly (0,+1), (0,-1), (+1,0), (-1,0) will always be solutions
for n=2, we have a unit circle
for n=1, we have a "diamond-orientated" square

now think: intuitively as we went from n=2 to n=1, we "squeezed" the corners inwards (keeping the four points fixed). so as n gets smaller again, we do the same.

for larger n, we do the opposite (pull the corners out), till we end up with a unit square (which happens to be Max(abs(x),abs(y))=1).

the other q: consider tan(y)=x (this is just y=tan(x) with the coordinates switched).

tan(y)=abs(x) is the above with the right-hand side of the plane copy and pasted onto the left-hand side (because tan(y)=abs(-x) iff tan(y)=abs(x) ).
tan(y)=x^2 is the above with the graph "smoothed out" because x^2 is the "smooth version" of abs(x).

pic related

but if you want some advice from a 2nd year ox maths student who had shit grades -- FOCUS ON THE MAT. they just use the interview to check you arent somehow retarded.

good luck tho :) what college you thinking of?

>1/x1 + 1/x2 + ... 1/xn = f(m) * (1/(x1 + m*b) + 1/(x2 + m*b) ... 1/(xn + m*b))

btw, are x1 and x2 distinct variables or x^1 and x^2 ?

>tfw probably got an A on my combinatorics exam

Wow, thanks for the input. That's very helpful.

What did you get on the MAT? How do you think your interview went? Did you ask for feedback?

I'm applying to SJC, which college are you at?

I did v well on the day
But I did all the past papers (about a dozen) available and got strictly increasing scores from below 40 to above 80 (i graphed it out somewhere) - so preparation is literally the most important thing desu.

I won't say the college bc there's only like six of us lmao. But I will confess I only so Maths and Philosophy not straight maths.

Distinct variables. I'm thinking about just going into some numerics.

oh and i never asked for feedback but my tutor might still have it on the system so might ask him.

id probs do each past paper twice actually (unless you are getting consistently very very high first time). value per effort the mat is far more important than a levels

Which book is this from?

>I won't say the college bc there's only like six of us lmao
Smart idea, I wouldn't want to expose myself to the normies either.

>I did v well on the day
Which section was hardest for you, if you remember? I've done tests A, B, and 2007 so far, (getting ~49, ~55, 60 on each) and I always get 3 marks max on #4 (the geometry questions). My gap in geometry knowledge is so large that I don't even know where to start in order to improve. Do you have any tips for that, or is it just a matter of needing to git gud?

amazon.com/Quantum-Fields-Strings-Course-Mathematicians/dp/0821820125

Why even use math when you have physical intuitions?

>I'm prepping for the Oxford interview,
Underage B&

Tbf I am a normie. Otherwise I'd've gone Cambridge.
I never had to do geometry lmao but suspect just practice. Look at the tricks and just memorise them. Pray that the plasticity of a young brain will just retain the techniques. Maybe just brush up on basic geometry - but they won't use anything that EXPLICITLY uses more than c2

you are in your senior year of high school when at the age of 18

> using the hypergeometric series instead

>reddit spacing

good to hear

The interviews are this December usually. The earliest birthday (unless they skipped a year somehow) would be September the 1st 2017 for this year's year 13 students.

[math]\mathcal{P}(\mathbb{R})[/math]

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"Lol" as we physicists say.
I just solved this using my physical intuitions.

Right and wrong. Those mathematicians that dislike the supposed "lack of rigor" in physics should also reject statements proven assuming generalized RH/CH.

Assuming anything that is not proven (except axioms lol) cannot yield a proof. Any mathematician thinking otherwise is an idiot.

in case of integration of functions of multiple real variables, does the riemann way allow the integration of function that cannot be integrated in lebesgue theory

Riemann implies Lebesgue.

and is there a relevant improper riemann integration in multiple variables

Just use the same idea you would use for an inproper integral in only one variable, I guess. I'm not an analyst, sorry.
>does the riemann way allow the integration of function that cannot be integrated in lebesgue theory
I was just pointing out that this is the other way around, all Riemann integrable functions are Lebesgue integrable.

>Riemann
Assuming anything without proof can only be used to prove that it is wrong.

I know this isn't a homework thread or something, but I came across a problem in my text book I can't seem to figure out. I'd appreciate it if someone could help me out (I don't want people to solve it for me, I really do want an explanation that would help me understand):
Let G = (V, E) be a graph with n nodes, such that the degree of each node is at most d. Prove that there exists in G a set S of independent nodes (meaning, a set that contains no adjacent nodes) so that |S| >= n/(d+1).
I tried proving by induction, but it requires me to split the problem into 3 different cases, and I can only prove it for 2 of them. (the 1st case is when you add a node that doesn't connect with any other node in G, the 2nd is when you add a node that connects with other nodes in G but not nodes in S, and the 3rd which I can't prove is when you add a node that connects with other nodes in G including ones in S).
Any help would be appreciated!

What is the current state of anabelian alebraic geometry

/brainlet/ here. A while ago i fantasized about using traveling waves to represent organisms traveling over evolutionary time. more specifically the waves would represent both genetic and epigenitic codes and they would travel over a gradient that represents the physical landscape, this and the waves interactions would effect the wave function. My idea is too use this to simulate ecological succession, population dynamics, disturbance scenarios, and evolution, it would be applicable in agroecology, conservation, and what not.
the problem is i didnt even pass algebra 2 and can pretty much only fantasize about it, assuming i caught up on math or got help, would this be a good start?
faculty.washington.edu/eeholmes/Files/Holmesetal1994.pdf
what are your thoughts?

Why does G have to be commutative?

Let [math]x = ab[/math] and denote [math]\phi_n:G \rightarrow G[/math] the n-th power map, then [math] \phi_n(x) = \phi_n(ab) = (ab)^n[/math] but [math]\phi_n[/math] is an homomorphism so [math]\phi_n(ab) = \phi_n(a)\phi_n(b) = a^n b^n[/math] and [math](ab)^n = a^n b^n ~\forall a,b\in G[/math] iff [math]G[/math] is commutative.

the waves would split with reproduction to show phylogeny, the interactions would be represented as signs from a biosemiotic perspective allowing to show them as networks of interactions.

>show them as networks of interactions.
that effect the dynamics of traveling waves.

My applied physical intuition says it should be commutative.

It's actually pretty good according to my physical intuition.

Prove that abelian groups exist.

Take the integers with addition

Show that the integers exist.

just consider the group S3 and let n = 3

>physical intuition
What's this meme?

Two thoughts, although I can offer no solution:

1. You usually cannot directly induct by taking one node and adding nodes and vertices. That would require proving that it is actually possible to obtain a Graph with n nodes that satisfies the assumptions in such a way, often not feasible.
Instead, you can take an arbitrary Graph satisfying the assumptions with n nodes, then remove any node such that the assumptions still hold and use the induction hypothesis on the resulting n-1-sized Graph.

2. Inducting as per above, if you remove a node [math]v[/math] and [math]S[/math] on the subgraph [math]G'[/math] with [math]n-1[/math] nodes has cardinality [math]|S| > \frac{n-1}{d+1}[/math], there is nothing to show.
If however [math]|S| = \frac{n-1}{d+1}[/math], you will need to show that [math]v[/math] can be added to [math]S[/math].
Clearly though, this is only the case if [math]N(v) \cap S = \varnothing[/math].
Therefore, you would need to show that [math]N(v) \cap S \neq \varnothing \Rightarrow |S| > \frac{n-1}{d+1}[/math].

which godel proof is this?

dunno
what axiom did godel use to prove that there are no proofs without axioms?

It is true that deductive proof requires axioms.

But there are other forms of proof such as induction. These don't give you 100% certainty but in reality nor does deduction.

Peterson has admitted he can't math and he should stick to his knitting.

That is such a retarded statement. Man, I used to respect this guy when he first went viral but then I found out he was a christcuck and it all went to shit. He is clearly an intelligent person but his christianity has corroded his brain to the point that he will make this nonsensical statement just to justify his own christcuckism.

It is true that proof itself without axioms is impossible because to have a proof then you must have a statement that you want to prove, and if you have a statement then you have an underlying theory of which it is a statement of, and that underlying theory is a set of axioms. But to then conclude from this that god is a prerequisite for all proof is retarded because he is basically saying that axioms come from god which is retarded because historically we have been fiddling about with axiomatic systems, even to this day. So unless he is saying that mathematicians are gods then he is retarded.

Books to study set theory and logic?

Anything from en.wikipedia.org/wiki/Saharon_Shelah

i am the OP of this thread asking for help and links i am curious where i can go from here. books on working problems like these? i guess the next step is the quartic.

>ax^3 + bx^2 + cx + d = 0
>t^3 + pt + q = 0
>(u+v)^3 + p(u+v) + q = 0
>(u+v)^3 = -q ; w^3 + wbar^3 = -q
>uv = -p/3 ;; 3wwbar = -p

depress the cubic into an associated quadratic via substitutions of the "complete the cube" flavor.

repeat, condense z^6 into (z^3)^2 and solve with substitutions of the "complete the square" flavor (quadratic equation).

but how do we get the values that satisfy these conditions?

im curious: when do they teach solving cubics in this manner?

i took calc over two years ago and i stumbled upon this problem while brushing up on my algebra a few days ago and have been working it since i encountered it.

Fun question

(may require some knowledge of elementary number theory [en.wikipedia.org/wiki/Pell's_equation], but there may be a solution that works around that)

this can be 1

>this can be 1
There's no 1st Avenue.

Suppose an odd number n is defined by:
n = 2x + 1
Does the following prove that if n is odd, then n^2 is odd?
n^2 = (2x+1)^2
= 4x^2 + 2x + 2x + 1
= 4x^2 + 4x + 1
= 2(2x^2 + 2x) + 1
which falls into the definition of an odd number, 2x + 1.

I'm working in a discrete math book, trying to practise proofing, and in this example problem the book concluded that a direct proof wasn't really a good option, even though the proof i wrote up there seemed easy enough, and now I'm self-doubting myself. Is there anything wrong with the reasoning?

Any odd number times any other odd number gives an odd number, that's trivial

If you are looking to prove this statement, then is the right way to go.

Prove using modular arithmetic (mod 2) for a 1 line proof.

Analytic propositions don't require axioms...

ya sure and

>imaginary numbers arent real
>0 is infinity
>there is no 1st avenue

>But there are other forms of proof such as induction.
The principle of induction itself can't be proved without a much stronger principle.

I'm trying to prove that given an absolute value [math] |\cdot | [/math] (non-negative, multiplicative, satisfies triangle inequality) on a field [math] F [/math] that [math] |n|\leq 1 [/math] for all [math] n\in \mathbb{Z} [/math] implies [math]|\cdot | [/math] is non-archimedian (i.e. [math]|x+y|\leq max(|x|,|y|) [/math] for all [math] x,y\in F[/math]).

So far the best I can do is:

Assume [math] |n|\leq 1 [/math] for all [math] n\in \mathbb{Z} [/math]:
[eqn]
|x+y|^n =|(x+y)^n|\\
= |\sum_{i=0}^n \binom{n}{i} x^iy^{n-i}|\\
\leq \sum_{i=0}^n |\binom{n}{i} x^iy^{n-i}| \\
= \sum_{i=0}^n |\binom{n}{i}| |x|^i|y|^{n-i} \\
\leq \sum_{i=0}^n |\binom{n}{i}| max(|x|^n,|y|^n) \\
\leq \sum_{i=0}^n max(|x|^n,|y|^n) \\
= (n+1) max(|x|^n,|y|^n) \\
= (n+1) max(|x|,|y|)^n \\
[/eqn]

where the first inequality uses the triangle inequality, the second inequality is straightforward, and the third inequality uses the given assumption. How do I do better than the (n+1) factor?

An alternative direction (probably more of a wrong direction since it doesn't even use the hypothesis) is:

[eqn]
|x+y|^n \leq (|x|+|y|)^n\\
= \sum_{i=0}^n \binom{n}{i} |x|^i|y|^{n-i}
\\ \leq \sum_{i=0}^n \binom{n}{i} max(|x|,|y|)^n\\
=2^nmax(|x|,|y|)^n \\
[/eqn]

which implies [math] |x+y| \leq 2max(|x|,|y|) [/math]

My physical intuition tells me your supposed "proof" is wrong.

Is there an expression equivalent to arccos((a*cos(x) + f)/(a + f*cos(x))) which does not use an inverse trig function or infinite series? Where a and f are constants and a > f > 0.

My physical intuition tells me "no."

My physical intuition tells me "yes".

What is the physical intuition for the axiom of infinity?

I don't see a proof of this group supposedly being "abelian" in your post.

My physical intuition tells me "maybe".

arccos((a*cos(x) + f)/(a + f*cos(x))) =

π/2 + i log((i (f + 1/2 a (e^(-i x) + e^(i x))))/(a + 1/2 f (e^(-i x) + e^(i x))) + sqrt(1 - (f + 1/2 a (e^(-i x) + e^(i x)))^2/(a + 1/2 f (e^(-i x) + e^(i x)))^2))

What is the last "how to self-learn math" pasta nowadays? I want to brush up calc 1 and Linear algebra and go from there.

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Nice meme.

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I agree! My physical intuitions actually told me this :D

I've spent the whole summer up to this point doing gen eds and have barely had anytime to do math. How screwed am I for this upcoming semester? It's only Calc III and an intro proof course so it's not too tough but I am a brainlet.

Try gaining some additional physical intuition for the math and you'll be fine. That's what I usually do lol
I don't even read any "books".

logicmatters.net/2017/01/01/teach-yourself-logic-2017
>Analytic propositions

Why does Stone's theorem require the one-parameter group to be strongly continuous? What if I only have the weak operator topology?

lol way to throw the baby out with the bathwater. You do realize you don't have to agree with everything someone says in order to get useful value out of them right? Stop being so 1 dimensional

>Analytic propositions don't require axioms...
Example?

At what point can an autodidact pick up an "Advanced Calculus" text, or even an "Analysis" text?

I've been working through Keisler's "Elementary Calculus" and have just finished learning about Integrals. I own a copy of an Advanced Calculus text. Looking through the TOC, there's quite a bit of overlap between that text and introductory calculus texts. Pic is a page from the TOC of the book in question.

I also recently picked up a copy of Apostol's "Mathematical Analysis" at an estate sale, which also seems very accessible.

I already have a foundation in logic, set theory, algebra, and trig, in addition to the calc I've already learned. Can I just dive into the "Advanced Calculus" or even "Mathematical Analysis" texts at this point?

I would encourage you to finish a computational calculus book before trying to proceed to those other two books. It will build the necessary background and mathematical maturity. I honestly might advise you to use a different book than Keisler's, he takes a pretty non-standard approach, the books by Stewart and Kline would be more traditional.