/sci/ - Science & Math

[Anonymous No.5673929]

Hello fellow math enthusiasts. Ask me a math question, and I will do my best to answer.

[Anonymous No.5673934]

can you explain second order differentials to me? i haven't payed any attention in class
note: i don't know first order either

[Anonymous No.5673937]

>>5673934
>paid*

[Anonymous No.5673942]

That's a hefty order. First of all, differential equations is one of my weak points, not something I know a lot about, especially the more computational/non-theoretical aspects of the subject. I don't think I'd be doing you a service by even attempting to try and tell you anything. That said, there are some fantastic resources online that may be exactly what you're looking for. For example, you should try here: http://tutorial.math.lamar.edu/Classes/DE/DE.aspx and here: https://www.khanacademy.org/math/differential-equations.. I apologize again, and best of luck!

[Anonymous No.5673943]

Why did you do math and not gods work (also known as engineering)

[Anonymous No.5673946]

What is a Hilbert's space?

[Anonymous No.5673950]

>>5673943 That is a good question, I suppose. Engineering just never interested me, and to be frank with you, I would probably not be very good at it. I tend to have difficulties translating real world problems into math in that area. That said, engineering is definitely a respectable and cool subject--just not one that interests me.

[Anonymous No.5673951]

>>5673942
ty <3 u anon

[Anonymous No.5673954]

>>5673950
that was too tame. wtf is wrong with you?

[Anonymous No.5673958]

>>5673946 It is the generalization of an inner product space to infinite dimensions. That said, when you move from the wel-behaved world of finite dimensional spaces to infinite dimensional the purely algebraic side of things gets too wild. You stop being able to say anything particularly meaningful about ann inner product space, when defined to just be a vector space V along with a positive-definite bilinear pairing. Thus, you add an analytic aspect to make the theory more ruley. Namely, you no longer just consider positve-definite bilinear pairings, but those for which the induced metric is a complete one. These spaces have huge uses throughout all of mathematics (e.g. Sobolev spaces in PDE theory), their brand of soft analysis making algebraically inclined folks like me able to pretend we are good at analysis! Hope that answers your question.

[Anonymous No.5673962]

>>5673954 Engineers aren't supposed to know we don't like them--that's a time-honored, but secret hatred. More seriously, I don't know man, what do you want from me? I just said what I thought.

[Anonymous No.5673967]

>>5673929
Explain Cauchy series to me. More specifically, the formal definition.

[Anonymous No.5673975]

>>5673929
What exactly is Lesbegue integration, and different how is it different from Riemannian integration?

[Anonymous No.5673977]

Consider
<span class="math">f(x) = \begin{Bmatrix}
1\: if \: x=\frac{1}{n}, \: n\epsilon \mathbb{N}\\
0\: otherwise
\end{Bmatrix}[/spoiler]

Find
<span class="math">\int_{0}^{1}f(x)dx[/spoiler]

[Anonymous No.5673979]

>>5673967 Cauchy "series"? Ok, I think I know what you mean. Let us first define a Cauchy sequence in a metric space (M,d). A Cauchy sequence {a_n} is one for which the terms get infinitely close together as we let n tend to infinity. Rigorously, they are sequences {a_n} such that for every e>0 there exists an N such that m,n>=N implies that d(a_n,a_m)<e--more succinctly, limsup_{m,n} d(a_m,a_n)=0. Intuitively, they are sequences that "want" to converge, but are just unlucky enough to be children of an unloving parent--the point they want to converge to isn't in the ambient space. For example, the sequence {(1+1/n)^n} is a Cauchy sequence in the rationals Q with the usual metric, but it is NOT convergent since it "wants" to converge to e which is not rational. A space for which all Cauchy sequences converge, for which all sequences that don't not converge for stupid reasons, are called "complete". For example, both R and C are complete metric spaces, as well as the p-adic rationals, and every Hilbert space (see above).

Now, a Cauchy series is merely a Cauchy sequence where the terms of the sequence are partial sums. In particular, if {a_n} is a sequence let us define S_n to be sum_{m=0}^{n}a_n. Then, the series sum_{m=0}^{n}a_m is "Cauchy" if {S_n} is a Cauchy sequence. Hope that helps!

[Anonymous No.5673993]

>>5673979
>>5673977
Integrate this from 0 to 1.

[Anonymous No.5673989]

>>5673975 This is a good question. Lebesgue integration is born out of a desire to fix some of the "issues" with standard Riemann integration. In particular, while the standard Riemann integral is very nice it lacks one particularly nice property that we would want a nice theory of integration--it does not play well with limits. For example, let {a_n} be an enumeration of Q, and let f_n be the indicator function on the set {a_1,...,a_n}. One can show that f_n is Riemann integrable on [0,1], and in fact $\int_0^1 f_n=0$ for all $n$. Thus, it seems natural to want to make the claim that $\int_0^1 \lim f_n=0$, since, after all, $\lim \int f_n=0$.

Now, while this equality really wants to be true, it is encumbered with a technical difficulty--the limit $\lim f_n$ is no longer Riemann integrable! Namely, $\lim f_n$ is the indicator function $1_Q$ on the rationals, and since the discontinuities of this function are not measure zero, it is not Riemann integrable.

It is because of inadequacies like this that we want to pass from the standard Riemann integral to the Lebesgue integral--what it loses in intuitiveness it makes up for in technical usefulness.

I hope this helps!

[Anonymous No.5673992]

Why does 1 + 1 = 2 and is always = 2?

[Anonymous No.5673999]

>>5673993 Let $X=\{\frac{1}{n}:n\in\mathbb{N}}$. Since $f$ is zero on $[0,1]-X$ it follows that $\int_0^1 f=\int_X f$. But, it is trivial that $0<=\int_X f$ since $f$ is non-negative, but $\int_X f<= sup(f)m(X)=1\cdot 0=0$. Thus, $0<=\int_X f<=0$ so that $\int_X f=0$.

[Anonymous No.5674002]

>>5673992 I think we both know this is a question I can't answer. I don't know if you're looking for a Principia type argument (see attached picture), but this entirely based upon your axioms, definitions, etc. Sorry though!

[SmileyMiley !xlgRMYva6s No.5674005]

My question, in fact I came to make a thread about it, but I might aswell just use this one.

Why is it that when you have an ecuation with multiple unknown variables you need to have a system of ecuations of one ecuation per variable to determine them. Why can't you go to town with just one variable, then replace your results in that same ecuation and so forth? I know that always leads to 0=0. But why? Why does that happen?

[Anonymous No.5674012]

>>5674005

Hello, I apologize, I am not entirely sure what you're getting it. If you are given a system of *linear* equations, you may want to solve the equation. Most of the time though you want a unique solution to your system of equations. Intuitively such a system is going to have a unique solution if you have exactly the number of equations as you don variables, and the equations are independent. If you are able to reduce your system to something such as 0=0 you're reached a tautology.

Without further clarification, I don't know if I can be of more help. Sorry!

[Anonymous No.5674017]

Obligatory what's mathematics for you? question.

[Anonymous No.5674018]

>>5673999
>>5673993

To follow up actually, the more general fact is that if two functions f and ge agree "almost everywhere" (the set where they don't agree is measure zero) the Lebesgue integral cannot differentiate the two, and spits out the same result. Your function differs from the zero function on a measure zero (countable even) set, and thus the integral of that function coincides with the integral of the zero function.

[Anonymous No.5674015]

Is mathematics created or discovered?

[Anonymous No.5674025]

>>5674017
Also, what's a sheaf in simple terms and why they seem to be ubiquitous?

[Anonymous No.5674029]

>>5673943
6/10

[Anonymous No.5674030]

>>5674015

That is, of course, a good question. I don't know, I think it's a question where someone's answer is more likely to give an indication of that person's personal philosophies and life views, than elicit a useful response.

That said, I think that mathematics is created. If only because it makes me feel more excited about what I'm doing. I understand the opposite point of view though.

>>5674017

Also an interesting, and difficult question. For me, math is just structured logic (I don't mean this is any technical sense). It is the study of deducible consequences from sets of axioms. Whether or not these axioms or consequences have any interpretation in other areas of academia or real life is irrelevant (but not at all unimportant!).

More seriously though, math has a particular, intangible "flavor". I really hate to fall on the old Steward quote about pornography, but I know math only when I see it.

[SmileyMiley !xlgRMYva6s No.5674043]

>>5674012
I think I am not being entirely clear. Maybe I am not using the appropiate English mathematical terms, sorry.

Consider any system of equations like the one in the pic. To get the values of x, y, and z we need at least 3 different independent equations using all of the variables.

If you were use only one equation to replace the values within itself, thus making a bigger equation with less variables, and you would solve the values for x, y, and z this way you would always end up in 0=0. Why does this happen?

[Anonymous No.5674047]

>>5674025
A sheaf is just an assignment of local data on a topological space. It is more of an organizational tool that allows us to formalize certain local-to-global principles on objects where this, otherwise, would be difficult.

Their ubiquity is a consequence of the above. In mathematics, in particular geometry and number theory, we always think globably but work locally, and we'd like a way to reconcile our intentions with our methods.

I don't know your background, but their is a very good reason why certain types of sheaves actually give us geometric information. For the sake of example, let us consider the category of vector bundles on a complex manifold X. One can then show that this category is equivalent to the category of locally free, locally finitely generated O_X-modules (a type of sheaf) on this space. Coherent sheaves are then just a more specialized version of this, that is a natural consequence when we start working with vector bundles, now thought of as sitting inside the ambient category of Sh(X).

The above paragraph should give you intuition about why sheaves on other objects give us serious information. It is patently true that a lot of information about a space is contained in its set of vector bundles (i.e. for a Riemann surface, the solvability of the Weierstrass problem is equivalent to the triviality of the Picard group [the group of holomorphic line bundles]). Now, we would love to consider vector bundles over spaces whose ground field is not R or C. Unfortunately though, there just isn't sufficient enough data to see exactly how to analogize vector bundles to these situations. That said, the idea of locally free, locally finitely gerated O_X-modules (where we need a ringed space now to make sense of O_X) makes total sense in this level of generality! Thus, we see that studying these sheaves is exactly the analogy of studying vector bundles [by the equivalence of categories I mentioned above] but in a more general, algebraic setting

[Anonymous No.5674052]

>>5673929
What's your main area of study?

[Anonymous No.5674054]

>>5674029
Is 6/10 always equal to 3/5? Is there any reason the lowest term isn't equal to whatever it reduced from?

[Anonymous No.5674058]

>>5674043
I apologize, but I still am not sure I understand. I believe you are asking why we don't just take a set of n-equations, with n-variables and just substitute the various equations into one equation eliminating the variables in that equation.

We can, and we do! That is precisely how we deduce the value of a given variable, by taking an equation of n-variables and using substitutions to create an equation of one variable which we can easily solve.

The fact that you get 0=0 is good. In essence it is telling you that the system you wrote down is coherent, and consistent. That two of the equations don't contradict each other (i.e. you would have ended up with something like 0=1).

I hope this helps.

[Anonymous No.5674065]

>>5674052
I'm sorry, I'm not sure what you mean. If you meant this at the "what's your major" level, then I am studying math. If you meant what type of math I am more specifically interested in, that varies often. Things that are on the roster though are arithmetic geometry and complex geometry.

I hope that answers your question!

[Anonymous No.5674074]

lets say you have a ball connected to the ceiling by a string of length L, and the ball is thrown from position Pi with an initial velocity Vi (Vi is perpendicular to the ray of the string from the ball, but when projected onto the horizontal plane may or may not be directed towards the "center" of the sphere to which it is confined). The ball is under the influence of gravity (G) and experiences no loss of energy due to air resistance

Write a function P(t) that gives the position of this ball at any given time t

[Anonymous No.5674076]

>>5674074
You're stumped me :)

[Anonymous No.5674082]

>>5674065
Ah, you're still an undergraduate, okay. I wasn't sure.

Also, arithmetic geometry is cool. I might end up studying that. Or maybe some other sort of algebraic geometry. Not sure yet.

One more question: Do you share the feelings of Kummer?

[SmileyMiley !xlgRMYva6s No.5674083]

>>5674058
No, that's not what I asked!

Following the image from ealier, if we have

2x+3y+z=11
3x-2y+3z=39
-x+4y-2z=-3

But instead of solving it using all the equations you say fuck you to the last two and do this:

2x+3y+z=11
x=(11-3y-z)/2
2((11-3y-z)/2)+3y+z=11
0=0

And you can do that too with y, or z, or two variables, or all 3. Why does this happen?

[Anonymous No.5674085]

>>5674082
I don't know if this is a thinly-veiled joke that I am missing, but what feelings of Kummer? I like things that he has done in number theory, and algebra.

Was he famous for some philosophical feelings?

[Anonymous No.5674087]

>>5674083
I'm sorry, I'm not exactly sure what aspect of equation manipulation you are doing. To the naked it eye, it appears as though you just, as you said "fucked" the last two equations, and took the first and created two more equations.

What meaning do you think this has? The set of (x,y,z) which satisfies the first block of equations is entirely different than those that satisfy the second block of equations. In fact, the first block is describing a singular point in R^3 and the second block is describing an entire plane!

[Anonymous No.5674088]

>>5674030
Ok, one last philosophical question from me. Are, for you, coincidences in mathematics really coincidences? Formal systems of mathematics with very different axioms and purpose tend to converge to the same results, patterns you'd never think of before but that in hindsight seem so obvious, are we just unreasonably lucky or is there anything deeper to it?

>>5674047
Welp, that's way more than I can digest right now. I'll stick to presheaves until I understand why would you call a contravariant functor into Set a presheaf.

[Anonymous No.5674090]

>>5674083
I'm not OP, but I can explain this. What you're basically doing is this:
2x + 3y + z = 11
x = (11 - 3y - z)/2
(11 - 3y - z)/2 = (11 - 3y - z)/2
0 = 0
Of course, if you have an equation like A = B, and you substitute B for A, you'll just get B = B, which is trivial. That's why.

In other words, you can always deduce a trivial statement (like 0 = 0) from anything.

>>5674085
Just a reference to the best commutative ring theory notes ever:
http://math.uchicago.edu/~chonoles/expository-notes/courses/2013/326/notes/math326notes.pdf
Reads like a modernist novel. I wish I could have been in that class.

[Anonymous No.5674096]

>>5674088
As per your first question--that's heavy man. I know what you mean, but at the same time I don't. I believe what you're asking about, is whether or not coincidences should arise from some deeper theory. For example, the idea of moving from one subject to another, and the fact that many of the techinques involved in this, pervaded many parts of mathematics seemed coincidental. This coincidence was then made rigorous via the definition of categories and functors, and the realization that "changing subject"="applying functor" and the method were the same since they were just functorial arguments.

I am not sure whether or not this holds in general man, I don't have very satisfying answer right now. I think that coincidence is a human construct, and we can rigorize any coincidence away by defining Notion X to be the thing which makes those coincidences a coherent theory.
>5674052

As per your second question. If you parse out what it means to have a contravariant functor from Open(X)^op (the opposite category on the ordered set of open subsets of a topological space X) into a category C, this just says that to every open set U you've assigned some object F(U) (like I said, the assignment of local data) and that you can "restrict" this data--so that there is a natural map F(U)-->F(V) if V is contained in U (this is just intuitively the restriction of functions).

This is a presheaf. A sheaf is really just a presheaf that has local determinacy (it is determined locally) and has the gluing property (you can glue local data to global data--this is this desired local-to-global principle I said earlier).

[Anonymous No.5674099]

>>5674090

Yeah, Zev's notes are pretty good. It's strange though, because Kato is famous for being CRAZY--I wonder how much of the notes are doctored by him.

If those notes interest you, I highly reccomend the book on algebraic number theory by Neukirch or the book on arithmetic geomtry by Lorenzini--they are in a similar spirit.

[SmileyMiley !xlgRMYva6s No.5674121]

>>5674087
Nevermind, thanks.

>>5674090
I am not randomly changing A=B for B=B without any reason whatsoever. I am following the logic used to resolve an unknown variable but with multiple ones.

I have this
2x+3y+z=11

separate x
x=(11-3y-z)/2

in the original equation, exchange x for what I got when I isoleted it
2((11-3y-z)/2)+3y+z=11

Now solve this new equation that I have, derived from the original 2x+3y+z=11
0=0

Why then do I reach a tautology by donig this?
I am not asking how to solve a system of equations. I am only asking why does this happen every time?

[Anonymous No.5674124]

Here's a math problem and not a philosophy question.

A swimmer stands on the west bank of a river 100m wide. She wants to swim to a point on the east bank 100m north. She can walk or swim at 5m/s, and the river is flowing south at 3m/s. To reach that point the fastest, how far should she walk before she enters the river?

[Anonymous No.5674133]

>>5674121
But you are just changing A=B for B=B in a sense. Think about it this way. You're starting with the equation 2x+3y+z=11, the equality meaning that (x,y,z) is a triple of real numbers such that equation is true. You correctly solved that those tuples must look exactly like ((11-3y,z)/2,y,z), or in other words, (x,y,z) is a solution to your equation if and only if (x,y,z)=((11-3y-z)/2,y,z), or in other words, x=(11-3y-z)/2. So, then you take a point that looks like that, one of the form ((11-3y-z)/2,y,z), and plug it into your equation. But that is exactly the equation for which points of that form are a solution! So, of course you can reduce this to a tautology!

Summarizing. You took your equation, and said that you're solution set to that equation is {(x,y,z):2x+3y+z=11}. You then showed that set is exactly equal to the set {((11-3y-z)/2,y,z):y,z are real}. You then plugged a point in the second set into the equation 2x+3y+z=11, which since this second set is equal the first, is BOUND to be tautological!

[SmileyMiley !xlgRMYva6s No.5674139]

>>5674133
Thanks! It's very clear now.

[Anonymous No.5674143]

>>5674096
>I don't have very satisfying answer right now. I think that coincidence is a human construct, and we can rigorize any coincidence away by defining Notion X to be the thing which makes those coincidences a coherent theory.
That's more than satisfying for me, it's the first coherent formalist answer I got. I stand more on the platonist side, it's easier to dismiss the question because "we don't know enough mathematics to know" and sleep well at night.

>A sheaf is really just a presheaf that has local determinacy (it is determined locally) and has the gluing property (you can glue local data to global data--this is this desired local-to-global principle I said earlier).
Now it's starting to make more sense.
Do you have any book recommendations on sheaves in topos theory (or just topos theory in general) for someone with a background in logic?
Thank you.

[Anonymous No.5674150]

>>5674143
As someone that wants to learn more logic, and needs to understand topoi to some degree, I don't know if I'm the best person to answer.

I think the canonical reference would be Sheaves in Geometry and Logic by MacLane and Moerdjik. A more algebraic reference would be Bredon's book on Sheaf Theory. Probably the best place to learn it though is to force yourself to sit down and understand it from a complex analytic/geometric side of things. This is from where the theory was born, and where the true intuition lies. There are many good references for that, a technical one being Global Calculus by Ramanan.

I hope that helps!

[Anonymous No.5674189]

>>5674065
What year are you in?

[Anonymous No.5674200]

>>5674189
Senior

[Anonymous No.5674206]

How do you go about proving that a number is transcendental, besides proving that it isn't by finding a polynomial that has it as a root?

[Anonymous No.5674207]

>>5673929
Hey man, I'm a sophomore in uni and I'm currently taking complex analysis, real analysis, and this one course that is basically all about Fourier Transforms and Diff. Eq.s, even though I already took Diff Eq, doing that course for the A. Anyway, I'm signed up next semester for advanced set theory, abstract algebra, and intro to general topology, and I was wondering if you could give me some advice on where to go from there. I'm trying to become a professor, if that helps.

[Anonymous No.5674208]

>>5674206
There is an entire subject in number theory dedicated to this question! A standard approach is to show that the number can be approximated *too well* by rational numbers, which proves it's transcendental. See here: http://en.wikipedia.org/wiki/Transcendence_theory#Approximation_by_rational_numbers:_Liouville_to_Roth

[Anonymous No.5674211]

>>5673975
>>5673989
There is still the issue with Lebesgue integration though that it can't prove the Fundamental Theorem of Calculus without a few assumptions, which is where the Generalized Riemann Integral with tagged partitions comes in.

[Anonymous No.5674212]

>>5674150
Will look into them, thanks!

[Anonymous No.5674215]

>>5674207
I would love to answer some questions you have, but for the sake of time, and so that you get out of my answers exactly what you want, it would be most helpful if you could pose some questions--I would love to help then!

Thanks!

[Anonymous No.5674216]

>>5674211
Hey, could you explain what you mean precisely? Thanks!

[Anonymous No.5674223]

>>5674215
Well, I love pure mathematics and theory in general. Considering that I have a pretty good grasp on analysis, linear algebra, discrete math, calculus, number theory, and differential equations, what particular branch of mathematics, in your opinion, would you recommend I try to tackle next? My school's not good at these recommendations, so I generally end up stumbling blindly into courses.

[Anonymous No.5674224]

Open question to anyone that knows the answer:

What would be considered the essentials that one should know for differential equations (ordinary) and linear algebra? I found out the other day that I'll be working with a bunch of math shit for a lab here in a couple of weeks and I don't know either of those subjects. I took calc 3 a few years back, but my memory is hazy on it.

[Anonymous No.5674225]

I am not a mathemitician/physicist, but I am an engineering student, as such I only care about applications. Do you know what the following could be used for? Also if you could list some difficulties associated with these, that would be great.

Eigenfunctions
functions of complex variables

[Anonymous No.5674226]

>>5674223
Hmm, without more to go off of, this is somewhat difficult. I don't know what "number theory" means, but I'm going to assume that it was "elementary number theory". From that list, a logical next step would be to tackle either a topology or an algebra course. What are your options at this point?

[Anonymous No.5674228]

>>5674225
You may have gleaned this from my other responses, but applied math is NOT my forte. That said, both have HUGE applications.

Both of these topics though have a huge prescence in PDEs which is a topic of fundamental importance of a engineer. Beyond that, I am not sure what I can tell you. Sorry! :\

[Anonymous No.5674230]

What's the most efficient way to work through pure math textbooks?

[Anonymous No.5674232]

>>5674216
Whew, okay. So Lebesgue integration is all about splitting your domain into sets of various measure and using that measure as a means of determining your integral. The Generalized Riemann Integral is basically the partition approach of the ordinary Riemann integral, except that you include an additional set of points with your partition, one within each interval defined by the partition, and you create a new sum based on the value of the function at those points, as well as your infimums and supremums, which is called R, and basically you set up your standard inequality of L(f,P)<=U(f,P), but you add in R. So L(f,P)<=R(f,P,c)<=U(f,P). I'm not very good at this without a chalkboard, I'm sorry.

[Anonymous No.5674235]

I'm pretty interested in number theory, where should I start on it, any prerequisites I need? any books you recommend for a beginner?

(right now i've done a 1st year course for b.eng course with calc 1, linear algebra and complex analysis

[Anonymous No.5674236]

>>5674230
Efficient? What do you mean by this?

If you mean how does one maximize their intake of the information? If so, I recommend either writing your own set of notes, or at the very minimum, giving two sentence synopses for every proof, theorem, definition, example, etc. Putting things in your own words not only helps you absorb the information, but prepares you to start thinking independently for when you start research!

[Anonymous No.5674243]

>>5674226
Oh, yeah, it's elementary number theory. Sorry. I'm all set to take a course in advanced set theory, one in abstract algebra, and one that's an introduction to general topology, but there are also courses being offered for algebraic geometry, some higher-level combinatorics course I'm pretty vague on, and one about diaphantine equations. Those are the ones open to me right now.

[Anonymous No.5674254]

>>5674235
To learn number theory with any semblance of modernity, algebra (abstract) is a must. That said, you should never let prerequisites get in the way of your interests! There are classical approaches to number theory that require *less* algebra (none you don't have right now).

For the more modern approach, I suggest Ireland and Rosen (difficult), for the more classical less algebraic approach I suggest Niven and Zuckerman (still fairly difficult), or if those are a little fast [don't be afraid if they are--you'll get to the more advanced stuff sooner than you think!] I would try Rosen's book on number theory.

Good luck!

[Anonymous No.5674259]

>>5674243
Definitely do not take the course in algebraic geometry. I think that algebra and topology are your best bets. They will give you the broad range understanding you need to start absorbing more and more mathematics.

Once you do those two you will have the three basics (analysis, algebra, and [point-set] topology) which will allow you to move on to more difficult and interesting topics (but algebraic goemetry may still be a ways off :) )

[Anonymous No.5674263]

>>5674259
Thanks! Glad to hear I accidentally made the right choices.

[Anonymous No.5674267]

>>5674263
Glad to help :) Feel free to ask more!

[Anonymous No.5674279]

>>5674267
Well, I am working on formalizing and proving this weird idea I had and I was wondering if there were a good, continuous function that closely approximates n choose r?

[Anonymous No.5674282]

Hey OP, how do you study for a maths exam? (go through notes, read books, practice..) I find myself most of the time simply applying theories to questions until I understand why I'm doing it. I don't think this is the right way.

Also, what maths do you recommend for a physicist?

[Anonymous No.5674285]

>>5674279
I'd prefer if it were uniformly continuous, but any kind would be wonderful.

[Anonymous No.5674296]

>>5674285

Approximation is a bit of a tricky word. What exactly do you mean? The choose function takes in discrete values?

There is a continuous (holomorphic!) function which restricts to the choose function (almost!) on the integers. Check this out: https://en.wikipedia.org/wiki/Beta_function

[Anonymous No.5674300]

>>5674282
Hey. Look above where I answer how to read books efficiently. Same idea applies :) Ask again if this does not answer your question!

Barely knowing mechanics, I sadly cannot say. I'm sorry :( The one thing that seems to be prevasive in all the sciences though is Lie groups and Lie algebras!

[Anonymous No.5674315]

>>5674296
Oh, I just meant a function continuous over R2 that's a function of n and r that could give the value of the choose function at integer-valued points is all. I looked at the Beta function but I don't quite understand it; I'll keep looking at it, I guess.

[Anonymous No.5674316]

What makes math interesting?

[Anonymous No.5674320]

>>5674315
Look at the bottom of properties on the wiki page. It's almost exactly what you want! There is another, simpler, example of such an interpolation

[Anonymous No.5674323]

>>5674300
I was talking more about after you've read those books, and you need to quickly look over for revision. Do you simply just flip through notes or do you do practice questions? Also, slightly irrelevant, but how much sleep do you get OP? I'm trying to find the right amount to have just enough energy to study all day, but also maximizing my awake time.

[Anonymous No.5674325]

>>5674320
Oh wow, thanks again! I just saw it. This is going to be incredibly helpful.

[Anonymous No.5674326]

>>5674323
Practice questions I guess, or see if I can explain it to a friend without a hiccup.

I sleep less than I wish I did. Probably about 4-5 hours a night.

[Anonymous No.5674329]

>>5674325
You're welcome :) Be careful though that while the beta function takes in continuous parameters like you want, that other interpolation doesn't, unless you replace the factorial by the Gamma function and the product by...something, I don't know I'm tried. Best of luck though!

[Anonymous No.5674339]

>>5673929
What is an easy way for me to memorize the 2nd fundamental theorem of calculus?

[Anonymous No.5674346]

>>5674339
That, in a sense, the association of a function to its antiderivative, and the association of a function to its derivative, are inverse functions. Does that at all help?

[Anonymous No.5674354]

>>5674346
So the 1st is going forward through integration and the 2nd is going backward?

[Anonymous No.5674359]

>>5674354
Yes, in a sense.

[Anonymous No.5674367]

<span class="math">Suppose we have $M$ movies, $N$ users, and integer rating values from 1 to $K$. Let $R_{ij}$ represent the rating of user $i$ for movie $j$ , $U \in R^{D×N}$
and $V \in R^{D×M}$ be latent user and movie feature matrices, with column vectors $U_i$
and $V_j$ representing user-specific and movie-specific latent feature vectors respectively.

After some tweaks, I have this objective function:

$$E = \frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{M} I_{ij}(R_{ij} - U_{i}^{T}V_{j})^{2} + \frac{\lambda_{u}}{2}\sum_{i=1}^{N}\left \| U_{i} \right \|_{Fro}^{2} + \frac{\lambda_{v}}{2}\sum_{i=1}^{N}\left \| V_{j} \right \|_{Fro}^{2}$$

where $\lambda_{u} = \frac{\sigma }{\sigma_u}$ and $\lambda_{v} = \frac{\sigma }{\sigma_v}$ and $\left \|. \right \|_{Fro}^{2}$ denotes the Frobenius norm.

and I would like to do gradient descent to update $U$ and $V$.[?math]

What the fuck is the derivative of E wrt to U_ab ?[/spoiler]

[Anonymous No.5674371]

>>5673929

A simple one for you:
How do I prove that every even integer greater than 2 is the sum of two prime numbers?

[Anonymous No.5674376]

>>5674371
Don't come in here with that Goldbach shit you smug pseudo-intellectual asshole, people are being helpful.

[Anonymous No.5674384]

Y=x^x^x^x^x^x... And continues to be raised to x infinitely, and y = 2, how can I solve for x?

[Anonymous No.5674390]

>>5674384
x=sqrt(2)

I'm working on these kinds of problems for a paper, I'll post it on here if I ever get a breakthrough.

[Anonymous No.5674398]

>>5673929
Suppose we have <span class="math">M[/spoiler] movies, <span class="math">N[/spoiler] users, and integer rating values from 1 to <span class="math">K[/spoiler]. Let <span class="math">R_{ij}[/spoiler] represent the rating of user <span class="math">i[/spoiler] for movie <span class="math">j[/spoiler] , <span class="math">U \in R^{D×N}[/spoiler]
and <span class="math">V \in R^{D×M}[/spoiler] be latent user and movie feature matrices, with column vectors <span class="math">U_i[/spoiler]
and <span class="math">V_j[/spoiler] representing user-specific and movie-specific latent feature vectors respectively.

After some tweaks, I have this objective function:

<span class="math">E = \frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{M} I_{ij}(R_{ij} - U_{i}^{T}V_{j})^{2} + \frac{\lambda_{u}}{2}\sum_{i=1}^{N}\left \| U_{i} \right \|_{Fro}^{2} + \frac{\lambda_{v}}{2}\sum_{i=1}^{N}\left \| V_{j} \right \|_{Fro}^{2}[/spoiler]

where <span class="math">\lambda_{u} = \frac{\sigma }{\sigma_u}[/spoiler] and <span class="math">\lambda_{v} = \frac{\sigma }{\sigma_v}[/spoiler] and <span class="math">\left \|. \right \|_{Fro}^{2}[/spoiler] denotes the Frobenius norm.

and I would like to do gradient descent to update <span class="math">U[/spoiler] and <span class="math">V[/spoiler].

[Anonymous No.5674407]

>>5674384

I love you

[Anonymous No.5674453]

>>5673929
To OP (or anyone else willing to answer):

What kind of literature should I go to in order to practice converting recurrence relations into explicit formulas?

Furthermore, any recommended reading on other somewhat high-level math related or applicable to computer science? (eg comparative growth rates, binomial expansions, etc)

[Anonymous No.5675164]

>>5674371
This of course, is a question, as pointed out to by others, which is clearly sarcastic. If I could solve Goldbach's conjecture, I would not be spending my time on 4chan answering peoples questions!

That said, I can say one small thing about it. I saw Terry Tao give a talk once where he discussed analytic number theory historically, and its goals in the future. He, and people like him, are developing techniques which, to a analytic number theorist laymen like myself, seem to be limit approaching a solution. And, while I am not a gigantic fan of the work he does (in the sense that his methods don't interest me--it's not my type of math) Terry Tao is an absolute genius. I have been told by people in entirely different fields, winners of the Cole prize in algebra, that Terry Tao is the greatest living mathematician. If it is to be solved in our lifetimes, I would make a safe bet that Terry Tao will have a large part in its solution.

[Anonymous No.5675167]

>>5674398
I apologize, I am not entirely sure what you're asking for. While I understand all of the words and symbols your using, I don't exactly understand what your objective is. This sounds like an in-depth question, one which is not answerable in a short amount of time. If you are really seriously interested in finding a solution, I would ask one of the many excellent math forums on the web.

Best of luck!

[Anonymous No.5675623]

>>5675164
>Terry Tao is the greatest living mathematician
Greatest <span class="math">active[/spoiler] mathematician, maybe. But greatest <span class="math">living[/spoiler] mathematician? To my knowledge, Grothendieck has only died in the Erdős sense of the word.

[Anonymous No.5675660]

>>5675623
I mean, it's certainly a well-said statement that Grothendieck is one of the most brilliant theory building mathematicians of the 20th century (and perhaps ever) but I really think that many people would argue that in terms of sheer Will Huntingesque brilliance, Terry Tao is superior.

This is coming from someone who is much more on the Grothendieck side of mathematics than the Tao side, mind you.

[Anonymous No.5675701]

Bamp

[Anonymous No.5676652]

Final bamping

[Magnelec No.5678544]

>>5675164
Explain the method.

[Anonymous No.5678571]

>>5675164
>Terrence Tao
>not Grigori Perelman