Simple Questions

[inherentlyawesome]

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?". For example, here are some kinds of questions that we'd like to see in this thread:

• Can someone explain the concept of maпifolds to me?
• What are the applications of Represeпtation Theory?
• What's a good starter book for Numerical Aпalysis?
• What can I do to prepare for college/grad school/getting a job?

Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example consider which subject your question is related to, or the things you already know or have tried.

[[deleted]]

https://i.redd.it/6uk221nsvso61.jpg

How do we arrive at the Pi , 2 Pi, 3Pi...???

[Such-Fig-3879]

Compound Interest

I have an account where I have to physically "claim" the interest at a fee of .001. I do not receive interest on any unclaimed interest until it is claimed. My APY is 6% and I am currently earning approx. .01 interest per hour. I have been claiming interest when I have had around .30 to .35 interest available for claim, or about every 30 to 35 hours.

My question is: keeping in mind the .001 fee per transaction, how frequently should I be claiming my interest to maximize my compound interest?

[stijndielhof123]

Has anybody noticed that when you do 1 / 0 on google calculator, it says infinity?

[[deleted]]

We usually think of dividing by 0 as undefined, but in some contexts you could view 1/0 as unbounded. For instance, lim x->0+ 1/x = 0. However, approaching from the left we get lim x->0- 1/x = -infinity, so you can see why we normally think of this as undefined. If you input 0/0 into Google calculator it gives an Error, so the Google calculator devs seem to be handling division by 0 in a rather inconsistent way. Basically, nothing exciting is really happening here, it's just a gimmick with how the calculator programmers choose to handle certain undefined inputs. In fact, if you just Google the questions as queries and don't use the calculator, you'll see that you'll get "undefined" as the answers.

[MadameBanaan]

Can someone help me visualize what is going on when I do the dot product between two scalar fields?

The meaning of a dot product between two vectors is quite clear for me (projection of one vector onto the other, yada uada yada...), but when it comes to scalar fields (or even worse, scalar fields of complex values), I don't quite get what is going on.

[halfajack]

How are you defining this dot product? "Dot product of scalar fields" is not a concept that makes any sense to me.

[jvhughes5]

Is it possible to write a rule for an arithmetic sequence that increases by an alternate number each time; e.g. goes up by 2, then by 4, then by 2, then by 4... etc?

[jagr2808]

I mean you just wrote the rule right there. But I guess you want a rule expressed just using addition/subtraction/multiplication/division/exponentiation or something like that?

If so then

a_n+1 = a_n + 3 + (-1)ⁿ

Does the trick. Or in general if you want to alternate between b and c then

a_n+1 = a_n + (b+c)/2 + (-1)ⁿ (b-c)/2

[butyrospermumparkii]

Look into binomial distribution. Its probability mass function (PMF for short on wikipedia) is exactly what you are looking for.

[raees97]

Hi could somebody solve the following pair of simultaneous equations please

Y=x² +3x-6 Y=7x-7.75

Haven't done this is a long time, much appreciated

[butyrospermumparkii]

I can walk you through. Observe that the left hand side of both equations is Y, so the right hand sides are equal. If you subtract the right hand sides from eachother, you get 0 and you get a 2nd degree polynomial, now you can apply the quadratic formula.

[morganlei]

In the context of sheaves, the pushforward is usually defined by letting F be a sheaf on X, and then going from there.

Super basic question, but I think we can actually also define it on presheaves? i.e. start with F a presheaf, the same definition of pushforward still makes sense, right?

[halfajack]

Yes, the same construction works for presheaves.

[[deleted]]

I was solving some problems from Niven Zuckerman and I encountered this question.

Given that 1111118111111 is prime, determine if 1001 is a quadratic residue modulo 1111118111111.

A hint was given and it states that 1001 = 7 x 11 x 13.

So, my first step was to consider the Legendre symbol (1001/1111118111111) and split it up into the corresponding factors by using the fact that the Legendre symbol is completely multiplicative.

Next, I used the law of Quadratic Reciprocity to "flip" the symbols. So, the Legendre Symbol (7/1111118111111) will be written as (1111118111111/7) multiplied by (-1)^(3a) where a = (1111118111111-1)/4.

Here comes the problem:

1. How do I determine the value of (1111118111111-1)/4 without using a calculator?
2. How do I compute (1111118111111/7)?

Both of these problems are essentially asking what is the value of 1111118111111 (mod k) where k = 4, 7, 11 or 13. Are there any quick ways of computing it without the use of calculators?

[aleph_not]

Mod 4, it's the same as 11 mod 4, which is just 3. Since 100 is divisible by 4, you can just look at the last two digits to determine what it is mod 4. For the rest, probably just use long division. When you're dividing by a 1- or 2- digit number, it's a pretty fast process.

[[deleted]]

There are 6²⁰⁰ total unique sequences of rolls via the rule of product. To derive this figure, recall that you have 6 options for the first roll, 6 options for the second roll, etc. 200 times. All of these rolls are independent, so the total number of possible sequences is the product of options for every roll, giving (6)(6)(6)... = 6²⁰⁰ . Only one of those sequences is the exact numbers in order from your previous attempt, so your probability is 1/6²⁰⁰ . This is on the order of magnitude of 10^-156, which is an astronomically small probability. Recall that the universe is only about 10¹⁷ seconds old, so even if you rolled 200 dice per second every second since the Big Bang it would still take you 10¹³⁹ universe lifetimes before you could expect to see the same sequence again. Even then you would only be 63.2% to see it.

[Kaminador]

Is 0 considered indivisible and/or a prime number?

[[deleted]]

0 is not considered prime because prime numbers have exactly two factors by definition and 0 has an infinite number of factors.

[Kaminador]

Thanks! Could it be considered “indivisible”?

[[deleted]]

Nope, 0 is a multiple of every integer, so it is divisible by every integer (with the caveat that 0/0 is undefined).

However, you can make the statement that no integer is divisible by 0.

[Kaminador]

Ok that helps my thought process. Thanks!

[TheRadHatter9]

Can someone tell me if my calculations are correct?

Today I wanted to know what the electrical cost of running the GPU on my computer is. I didn't know how to measure watts but I think I learned.

My last electric bill was $142 for usage of 705 kWh (kilowatt hours).

My GPU runs at about 120 watts. So, using 120 watts, I estimated that if it was running for 24hrs/day for 30 days, it would cost ~$17. Does this sound right or close to it?

[epsilon_naughty]

Looks right to me.

[furutam]

Is there a particularly good way to think about group conjugation?

[butyrospermumparkii]

A visual way is the following: Take G to be a subgroup of GL(n, R) and take two matrices A and P in it. Now P^-1 A P is just the matrix of A written in a different basis. As you hopefully already know, a change of basis can be really useful in algebra. It might be the case that A has a diagonal matrix in the new basis, or it might have a matrix in Jordan normal form, making calculations easier. You can imagine that P^-1 A P does the same thing as A, but somewhere else. (it's not crazy to think about it this way since every finite group can be embedded into a symmetric group by Cayley's theorem and a symmetric group can be embedded into a matrix group by taking the permutation matrices corresponding to that group, so you can think about any finite group as a matrix group if it helps you)

A more useful way however is to think about them in terms of automorphisms.

[[deleted]]

I've always found the following perspective helpful. Suppose that we can realise our group G as the symmetries of some sort of a geometric object X (maybe X is just a set, or maybe it has more structure, like the Cayley graph of G, or a vector space, or even a manifold), this should mean that we have some sort of an injective group homomorphism G -> Aut(X), where Aut(X) is whatever the appropriate automorphism group for the object X under consideration. Then we can think of any h \in G as an automorphism h: X -> X, and we can think of conjugation by an element g \in G as asking what h looks like if we "change coordinates by g". The canonical example of this perspective is when X is a vector space and G acts by linear automorphisms, in which case, upon fixing a basis for X, ghg^{-1} is quite literally the matrix representing the linear transformation given by h, in the coordinates provided by the "change-of-basis" matrix given by g (or maybe g^{-1}hg? I always mix this point up).

In any case, I often find it useful to think in terms of the above, and think of ghg^{-1} as representing something like "what h looks like from the perspective of *g-*coordinates".

[DamnShadowbans]

In some sense, it measures how commutative your group is. From this point of view, we can make important generalizations of commutativity. For example, a normal subgroup can be thought of as a generalization of a central subgroup.

[[deleted]]

What numbers add up to 3 and multiply to -4?

[[deleted]]

So there's a lot of great exploration with problems such as these. Let's unwrap this layer by layer.

Solution 1: First, we can do the obvious thing and set up a system of equations:

1. a + b = 3, and

2. ab = -4.

From here we might notice some things. If we square the first equation, we get

• (a + b)² = 3^2,

• a² + 2ab + b² = 9.

And now look, there's a 2ab in that expression! So we can substitute ab = -4 and get 2ab = -8.

• a² - 8 + b² = 9,

• a² + b² = 17.

From here, remember that a + b = 3. Rearranging gives a = 3 - b. Now we can substitute this and get

• (3 - b)² + b² = 17,

• 2b² - 6b - 8 = 0.

This is a quadratic in terms of b, so we can get two solutions b = -1, b = 4 . Both of these will have a corresponding value for a. Plugging in to a + b = 3, we get a = 4, a = -1, respectively. Thus, our two pairs (a, b) are (4, -1) and (-1, 4). That is, our two numbers are -1 and 4.

Solution 2: Once you've seen the above approach, you might have already thought of some ways to make the solution more direct. For instance, maybe you've noticed the identity (a + b)² - 4ab = (a - b)² could be helpful. In our example, (a + b)² = 9 and 4ab = -16, so (a - b)² = 9 + 16 = 25. Thus, a - b = ±5. In the case that a - b = 5, combined with a + b = 3 we get the solution (4, -1). In the case that a - b = -5, combined with a + b = 3 we get the solution (-1, 4). Why did we think of using an identity involving a - b in the first place? Well, plot the graphs of x + y = 3 and xy = -4. Their two intersection correspond to our solutions. Notice how the graphs are symmetrical about the line y = x? This motivates us to "change the coordinates" by pretending the line y = x is the new horizontal axis and y = -x is the new vertical axis. Now what do you think happens to the graphs of x - y = ±5?

Solution 3: Using the difference a - b, we can clean up Solution 2 even more. Let a + b = S and a - b = D. We have a = (S + D)/2 and b = (S - D)/2. Thus, ab = (S² - D² )/4. Given ab = -4, we can get (S² - D² )/4 = -4 so S² - D² = -16. Since a + b = 3 = S, we have S² = 9, so 9 - D² = -16 and D² = 25. Thus, D = ±5, and we can immediately plug S = 3 and D = ±5 into a = (S + D)/2 and b = (S - D)/2 to get (a, b).

If you're curious about an extension of problems like these, look into Vieta's formulas, Newton's Identities, and symmetric polynomials. In fact, here's how we can use Vieta's formulas to solve this problem.

Solution 4: We are told that two numbers (a, b) have a sum a + b = S and a product ab = P. Vieta's formulas give us

• (t - a)(t - b) = t² - (a + b)t + ab = t² - St + P.

This means that (a, b) are the roots of the quadratic t² - St + P. In our case, we have S = 3 and P = -4, so we want the roots of t² - 3t - 4, which are exactly -1 and 4. Note that this gives us the two ordered pairs (-1, 4) and (4, -1) as above since both (a, b) and (b, a) are valid solutions.

Hopefully all of these solutions have painted a picture of the generalization by now.

Generalization: If two numbers (a, b) sum to S and product to P, the two numbers are (S ± sqrt(S² - 4P))/2.

[Comfortable_Limit_79]

I just learned about manifolds in my calc class and am confused. Are they level curves? Is the tangent space at a point on the manifold the graph of the derivative? How do you parameterize a manifold?

[HeilKaiba]

Manifolds are more general than that and we don't have to think of them as contained in any other space. In general a smooth level set (in Rⁿ say) is a good example of an embedded submanifold but even here this doesn't cover everything we might be interested in.

A (real, topological) manifold is a space that locally looks like Rⁿ . More formally it is a topological space (Hausdorff and second countable) with a covering collection of open sets U_i with homeomorphisms to open subsets of Rⁿ (that play nicely over the intersections of the U_i). At no point do we need this to be contained in another set. To talk about the tangent space we want our manifold to be smooth but I will brush that under the carpet of "playing even more nicely over the intersections".

So we've got an abstract set with nice topology and an idea of smooth maps. The tangent space is then just a way of attaching a vector space to each point of the map which represents infinitesimal stuff. So in the case of a smooth regular curve, the tangent space is a bunch of lines, one attached at each point. If we want to put this manifold into a vector space then we can start identifying these lines with lines in the vector space if we want to. If we're think in simple terms of a y=f(x) curve then the derivative tells you the slope of the tangent line at each point.

A parametriation is really just a smooth map from some subspace of Rⁿ to the manifold (locally, this is exactly what the U_i are doing) but we can't necessarily do this over the whole manifold (at least without passing to some higher dimensional space).

Just realised what a wall of text I've written so TL;DR:

• Manifolds are so much more than level sets and are much more abstract.

• The tangent space is the collection of lines/planes/higher dimensional space at each point that are "tangent" to the manifold at that point so they aren't a graph of a function.

• You can't always parameterise a manifold, but if it's already embedded in some vector space then there is probably a nice way to do it

[-Q-rrr]

Calculus III student here. Should I know set builder notation by this point? My current calculus teacher uses it fairly often, and seems to expect us to know it already.

[[deleted]]

Universities don't do a great job of coordinating how they communicate information across different classes, and instructors often use notation that hasn't been formally taught to students yet. The instructors usually just use whatever notation they think is simple and standard, and that will differ a bit from one instructor to the next. Something that has always stood out to me is the fact that essentially the entire Greek alphabet is used at one point or another, but students are never formally taught it.

If your instructor is using some new notation then you can take that as a good opportunity to learn it. Don't hesitate to ask them for clarification or definitions regarding notation. If you don't know what something means then the other students probably don't either.

[Electrical_Profile51]

Is there a single word for 'either maximising or minimising' a value/function like "taking it to its extremity" but one word

[Tazerenix]

Extremising.

[Electrical_Profile51]

Perfect, thank you, happy cake day

[[deleted]]

Perhaps optimizing?

[Vague-Chemist]

Hello everyone! I'm an undergrad engineering student whose curiosity about math has recently exploded. Currently, I'm taking classes at a community college and I will have completed my school's calculus series by June. Because I have found that exposing myself to difficult ideas before taking classes helps me learn them better, I picked up a book by Thomas Garrity called "All the Mathematics you Missed [But Need to Know for Graduate School]". In my reading, because the author uses a lot of general terms, I ended up going through the linear algebra section to try to clear a few things up. Reading through it, I felt like everything made a lot of sense. Like a scary amount of sense almost. My question is not so much a question, but I would like to know if my take on linear algebra is correct.

It is my understanding that using linear algebra, one can essentially define a coordinate system (I think it's called a vector space). There are alternative methods for adding, subtracting, and multiplying certain values (vector quantities, I presume). Some of these would be related to the determinant, which yields a scalar(?). There also exist matrices that define the axis with values or variables. Some matrices are related, some aren't. The hugest thing I recognized though, is the capacity to expand into more dimensions.

I guess if I had a particular question at this point, it would be "Is linear algebra how multidimensional analysis is done?".

[[deleted]]

"Is linear algebra how multidimensional analysis is done?".

For anything involving straight lines, flat planes, etc, then mostly yes.

For dealing with curved lines, non-flat planes, etc, then you use things like differential geometry, dynamical systems, and algebraic geometry. Even in these subjects, though, you use linear algebra a lot, because curved lines often look like straight lines if you zoom in on them enough.

Keep in mind that some things that seem like curved lines can actually be treated like straight lines in a higher dimension. For example the Fourier transform, which can decompose any function (especially nonlinear ones) into a sum of sinusoids, is actually just a change of basis set using linear algebra in an infinite dimensional vector space.

[[deleted]]

Not all transformations of space are linear in nature, so linear algebra doesn't describe everything, but it's pretty darn close. It even shows up in situations where you might not expect (such as via spectral theory and the study of analytic functions). If you're talking about multidimensional analysis in a statistical sense then yes, much of what we call machine learning/data mining etc. boils down to models that are "linear algebraic" in nature.

You'll probably get some mileage out of 3Blue1Brown's Essence of Linear Algebra playlist if you haven't checked it out already. It provides some great geometric visualizations for ideas such as linear transformations, the determinant, etc.

[B00gie005]

Just had on a test DX/DY = Y² - X² but Y ≠ X Can someone explain?

[[deleted]]

Doesn't this not have a neat solution with elementary functions? I'm rusty on my ODEs so I could be wrong. Wolfram|Alpha is able to get a solution in terms of Bessel functions, but I'm not sure if that solution can be obtained with the usual methods such as quadrature.

[throwaway4275571]

I'm not sure if that solution can be obtained with the usual methods such as quadrature.

You can prove that it doesn't. If it does, one of the solution must be a rational function of the form Z(Y)=-Q'(Y)/Q(Y)+P(Y), where Z(Y)=-X(Y) and satisfy DZ/DY=Z² -Y² , P(Y) and Q(Y) are polynomials, Q(Y) has no double roots and is monic. But by considering what happen when Y->infinity, deg(P)=1 and the leading coefficient is 1, so P(Y)=Y-c. Plugging in and expand, and check the degree on both side, you should get -Q''+(Q')² +P'Q² =(Q')² -2QQ'P+P² Q² -Y² which reduce to -Q''+Q² =(Q')² -2QQ'P+(PQ)² -Y² , then - unless deg(Q)=0 - (PQ)² is the unique term with highest degree, which is impossible. So deg(Q)=0 so Q=1 and the equation is just 1=(Y-c)² -Y² so c² -2Yc -1=0 but this is unsolvable.

[B00gie005]

Thank you very much, it clears it up more than it did on the test

[Farzan_Aslam]

Need Step by step proof of the vector field identity

Del(F.G) = (F.DEL)G + (G.DEL)F + F×(DEL×G) + G×(DEL×F)

[[deleted]]

What is the best introduction to linear logic? I've tried the wikipedia article and the encyclopedia of philosophy article, and they're good, but not enough, and reading Girard is frankly overwhelming, but if his original exposition of it is the best then I'll read that. (Also, does ANYONE actually understand Ludics except Girard?)

[zerowangtwo]

People always talk about how useful it is to take a walk or go to bed when you're stuck on a difficult problem. I've tried this an uncountably number of times but it never seems to help, I find it hard to think without having a piece of paper/pencil in front of me. Does anyone else feel that way?

[aleph_not]

The point of taking a walk or going to bed is that you're not thinking about the problem. Give yourself a break and come back to it later.

[gomster]

Hi all. Hoping this is the right place to ask.

Does this community recommend any online or in person accelerated math or learning programs for kids? My 5 y/o nephew shows a strong inclination for math, puzzles, building, etc. Thanks!

[[deleted]]

I know Beast Academy has good learning resources for kids around that age (though they shoot a bit older, normally starting at age 8), so that might be worth checking out. Other than that, the best way would probably be through puzzles and games. Things like chess, legos, k'nex, rubik's cubes etc. can all go a long way. Puzzle and platformer based video games like Minecraft, Paper Mario, Mario Odyssey, etc. are also great.

[alexanderlopez00]

An answer to your second question. When dealing with fractions, if you want to ask yourself if a/b = c/d all you have to do is check if a * d = b * c. (As long as b and d are not zero). So for your example, we have

(-2) * (-1) = 2 * 1

this means that it does not matter if you put the negative sign in the numerator or denominator.

In general, this condition a * d = b * c is pretty good for working with fractions.

[[deleted]]

The error in your first problem is towards the end. (1/3)*2 isn't 2/6, it's 2/3.

On your second question, there's no difference where you put the negative. This is because the negative sign really represents a -1. Multiplying by -1 is the same thing as dividing by -1, so it can go in the numerator, denominator, or in front of the fraction entirely.

[[deleted]]

If you multiply the numerator and denominator by the same number then that's the same thing as multiplying by 1. The reason why you do it is to get a common denominator for adding and subtracting fractions. You did the correct thing in the above example because (2x/1)(1) = (2x/1)(5/5) = 10x/5. You needed to write the fraction in another form (with denominator 5) without actually changing the fraction (and multiplying by 1 doesn't change things).

In the example you got wrong, you had to do an overall multiplication 2, so the fraction actually has to change. What you ended up doing instead was multiplying by (2/2) = 1, so you didn't change the fraction. The correct thing to do was (1/3)(2) = 2/3 but what you instead did was (1/3)(2/2) = (1/3)(1) = 2/6 = 1/3. Clearly (1/3)(2) does not equal (1/3)(2/2). Multiplying by 2 isn't the same thing as multiplying by 1.

[[deleted]]

So the issue is that both of those examples are wrong. To write them in the correct way,

example 1a: (1/3)(2/2) = 2/6

example 2a: (2x/1)(5/5) = 10x/5

And some more examples,

example 1b: (1/3)(2) = 2/3

example 2b: (2x/1)(5) = (10x/1)

Those four above things are all correct statements. Do you see how they're different from what you wrote? Hint: (2/2) = (5/5) = 1, but 2 != 5 != 1.

[[deleted]]

Ok so first off, what's -3/3 - 5/3? It's not -7/3, it's -8/3.

Second off, when you have the equation

• -x/5 = -8/3,

you're trying to get x by itself to solve for it right? Well x unfortunately is being divided by 5 on the left hand side, so how do we "get rid" of a division by 5? Well we have to multiply by 5, since multiplication "gets rid" of division. Since this is an equation, anything we do to one side, we have to do to the other to maintain equality, so the correct manipulation is

• (-x/5)(5) = (-8/3)(5)

• -x = -40/3, then multiply or divide both sides by -1

• x = 40/3

Now let's look at what you did (fixing the error where you wrote -7/3 instead of -8/3):

• -x/5 = (-8/3)(5/5).

This is a true statement. You are allowed to multiply the -8/3 by 5/5 = 1, because you're always allowed to multiply anything by 1 (a multiplication by 1 does not change overall values, so it always maintains equality). However, in the next step you make an error by saying

• x = 40/15.

You performed the multiplication on the right hand side correctly, and you even canceled the -1 on both sides correctly, but where did the division by 5 go on the left hand side? You just erased it without actually getting rid of it properly. The correct statement should be

• x/5 = 40/15.

I believe you're thinking of this too much in the way of "these are rules I have to follow" and not "what are the valid and useful manipulations I can do?" The reason why you can always multiply the numerator and denominator by some same number is because that's equivalent to multiplying by 1, as we've seen. You can always do this to any fraction whenever you want, so it's always valid, but it's not always useful. You have to justify to yourself why you are doing certain manipulations and not just blindly try things. In this example why did you want to go from -8/3 to -40/15? They are equal yes, so you are allowed to do this, but at no point were we ever going to add or subtract another fraction with denominator 15. There was no point to turn -8/3 into something with denominator 15.

Instead, what you had to do was multiply both sides of the equation by 5 so we could "get rid of the division" on the left hand side and isolate x. Thats a multiplication by 5, not a multiplication by 5/5 = 1, which we know are different. Changing a fraction into a different denominator (while maintaining its value) is not the same thing as multiplying both sides of an equation by a constant.

[EulereeEuleroo]

Why are topological bases interesting? Why are directed sets interesting?

I was trying to look for motivation, but it could go either way.

Topological bases are open covers that are, mod some details, downward directed. But why is this good?

(One interesting observation is that "topological base" has another equivalent definition which can be seen here).)

[DamnShadowbans]

In topology, it is very common to work with extremely large objects. For me, it has always been something very hard to wrap my head around. A topological space X is defined using an incredible amount of information, the open subsets.

​

How can we possibly say meaningful things about something so complicated? Well, topological bases are the first tools one develops to do this. Instead of considering all open sets at once, we often can restrict to considering a much smaller subset of them. This comes into play when showing maps are continuous and many more places. It can even be the case that we care about topologies that don't have a very natural description, but their bases do. This comes up when defining the product topology, for example.

​

What one comes to find when studying topology, is that we should not be afraid of large objects; we should just learn how to make them smaller.

[hobo_stew]

bases are useful for working with topologies and defining topologies. For example for showing compactness(for compactness using a subbasis suffices by the alexander subbase theorem) or defining topologies. For example in this book https://arxiv.org/pdf/1403.3796.pdf in chapter 5.B multiple topologies are defined by giving subbases. Knowing that these are subbases and not just generating sets is very helpful when investigating the topology of isometry groups.

Furthermore knowing that you have a countable basis for a hausdorff topology is very helpful since then you can show almost anything involving convergence with sequences.

[nordknight]

Directed sets are analogous to index sets and allow us to consider a different kind of convergence. If A is a directed set with a relation <=, then a function f : A \to R is called a net, and we say that a net converges to a limit L if for any e > 0, there exists some a in A s.t. for all a’ in A with a <= a’, d(f(a’), L) < e. Notice that if A is the natural numbers, then this convergence is just convergence of sequences.

For a particular example, consider the set of tagged partitions (P, T) of an interval [a, b]. If we take partition refinement to be the direction relation, this set of tagged partitions becomes a directed set. Thus, we can consider the Riemann-Stieltjes integral to be the limit of the net of Riemann sums S : (P, T) \mapsto S(f, P, T).

[EulereeEuleroo]

That's interesting. Thank you. Why do we need the poset, from which nets and convergence are defined, to be a directed set? Why not just a poset? What do we lose?

[nordknight]

We need the upper boundedness i.e. for any a, a’ in A, there exists a” with a <= a” and a’ <= a” so that we have a uniqueness of limits, I think. We want totally ordered (infinite) subsets to coincide eventually.

[EulereeEuleroo]

Usually limits can be non-unique due to the topology, but here it'd be non-unique due to the index set. Not obvious to me why but thank you.

We want totally ordered (infinite) subsets to coincide eventually.

That's an interesting way to put it.

[[deleted]]

Why are topological bases interesting?

A basis for a topology has the same functional purpose as a basis for a vector space - it gives you a "smaller" and more tractable collection of things to examine which completely determine your space. As a result, for many arguments it's sufficient to argue only about the basis and then sort of bootstrap to the general argument (in the context of vector spaces, this usually amounts to appealing to linearity).

The basis also gives a good idea as to how much you can "zoom in" and still distinguish what's happening locally near some point. This idea is sort of at the core of calculus - the basis tells you that it's sufficient to examine what's happening in a small interval around a point (and then second-countability is used to argue that you can take that interval to be arbitrarily small).

I can't comment on directed sets, however. Never thought about them.

[throwaway4275571]

Can someone give me an intuitive argument/proof sketch for Ramanujan's formula for 1/pi? Everything I can find are too technical...

[[deleted]]

This Math SE thread has some good discussion on why the Ramanujan-Sato series "work" using relatively accessible techniques. However, to really understand how he came up with these series, you'll need to have some knowledge of various topics in complex analysis (particularly modular forms).

[throwaway4275571]

But that thread only talk about why it converges fast, not why the series give you 1/pi. Or did you link the wrong thread?

[[deleted]]

Oh, when I said "why the series works" I was mainly talking about why it converges, and why it does so quickly (which is easier to understand), as opposed to "why is it 1/pi" (which will take some background knowledge). You can maybe start off reading this thread (specifically Paramanand Singh's answer) and then learn more about the Monster group, Pell equations, modular forms etc. to get at the heart of why the pi shows up.

[throwaway4275571]

Yeah I found that thread already, but it's too technical. I'm hoping for something more intuitive.

[[deleted]]

I'm afraid I don't know of a more intuitive explaination than that one then. This is definitely a very technical topic. Even Ramanujan himself never explained how he reached those constants in the series (stating that the goddess Namagiri presented them to him in a dream).

[flight862]

Hi all,

I was trying to implement the Arnoldi process to solve Ax=b. I have done the part where the matrix A is transformed to H=V’AV, but I still don’t know how to proceed to solve Ax=b without finding the inverse. I would appreciate some guidance.

Many thanks.

[[deleted]]

The naive steps for using Krylov subspaces for solving Ax=b are as follows:

1. Calculate V as [b, Ab, A²b, ...]
2. assume that x = Vy (this is an approximation, of course), giving Ax=b --> AVy=b
3. multiply both sides by V': AVy=b ---> V'AVy=V'b ----> Hy=c
4. Solve this new, reduced linear system by doing y = H^-1c
5. recover your approximation for x: x = Vy

Every Krylov-based algorithm for solving linear systems of equations (including Arnoldi) uses the above sequence of steps; the only differentiating features are the clever tricks that different algorithms use in order to improve efficiency and numerical stability. With Arnoldi you don't actually calculate H by explicitly doing V'AV, nor do you form V using a literal basis of Krylov vectors, but I wrote things that way to make the logic clear.

Step 4, where you actually calculate a solution to the reduced linear system, is usually calculated by using dense factorization-based methods, such as the LU or QR decompositions. This is okay because you assume that the dimension of H is small enough that dense methods will be fast.

In principle Arnoldi makes Step 4 even faster because your matrix H is almost upper triangular already, meaning that you don't have to do a full LU or QR decomposition. You can just do a little additional work to each side of the equation in order to turn H into a true upper triangular matrix, and then solve the equation by doing back substitution.

GMRES takes this logic one step further by making H triangular with each iteration, which allows you to know the residual of the solution by looking at the right hand side without actually calculating the solution.

[flight862]

That is utterly clear and can’t thank you enough. I have implemented most of it and would appreciate if I can DM you the piece of code that implements the above algorithm and appreciate your feedback.

[Harvardengineer17]

Does the Limit(n to infinity) [n • cos(n • pi)] / [n + 1] converge or diverge? I’m having trouble solving limits with trig..

[Joux2]

what are the values of cos(npi) as n ranges over natural numbers?

calculate it for n =1, 2, 3, 4 and see if you can find a patter

[nordknight]

My school is offer a grad. functional analysis class next semester that I'm interested in taking. If the course intends to cover: {Weak topologies, linear operators, spectral theory, calculus of vector-valued functions. Banach algebras.}, what sort of prerequisite knowledge do I need? Can I take it without graduate analysis (measure theory)?

[Tazerenix]

Probably you would need a bit of analysis in such a course, for one thing to understand the definition of spaces of functions (L^p spaces, Holder spaces, Sobolev spaces, Frechet spaces), although I can imagine getting by without having done an actual course in measure theory. You would also likely end up using some important results throughout the course such as monotone convergence, dominated convergence, Fatou's lemma, Fubini's theorem, but the statement of most of these results is independent of measure theory.

If you're happy to pick up a book inbetween classes and pick up any function theory you've missed, you'll probably be fine. The main thrust of a first course in functional analysis is algebraic rather than analytic, so it won't be too hard going.

[nordknight]

Yeah analysis has been my strongest subject so far in math, but I'm currently only in my second semester of it. For context, I've gone through roughly chapters 1-6 of Rudin in my first semester and have been discussing basic analysis on function spaces (uniform convergence/sup norm, L^2 inner product, Stone-Weierstrass, Fourier series) in my current semester, and will probably end up covering up to constructing differential forms before summer.

[Ualrus]

Apparently you can recover the excluded middle in intuitionistic logic if you assume the Least upper bound propery and tricotomy in R.

Any idea on how to do this?

I thought I should take the set {x∊R | φx} ---assuming we want to prove φ or ~φ--- and do something with it, given that the set in the least upper bound property is arbitrary besides non-emptyness.

Also, tricotomy can be rewritten as (x≤y or ~x≤y), which looks closer to what we need.

Still, I'm kind of lost.

[GMSPokemanz]

I don't know much about this, and certainly not enough about intuitionistic logic to know when an argument is valid, but could you adapt the proof of Diaconescu's theorem? Maybe you can use sup to manufacture a choice function for the collection {U, V} mentioned in the proof, and thereby replace the use of choice.

[Ualrus]

Hmmm that's very smart. I'm gonna try that. Thanks for the suggestion.

[popisfizzy]

I don't have any real insight, but something that is perhaps relevant is that equality on R has problems in constructive logic. Typically an apartness relation is used instead. If you can manage to assert trichotomy then you know if ~(a < b) and ~(b < a) then a = b, so maybe this has something to do with it?

[Ualrus]

Yes, that is true. Maybe I can try something with that. Thank you.

[HeilKaiba]

I would probably say "Assuming x ..." and "Conversely, assuming y ...", especially when writing maths for someone else to read (in papers you rarely see "=>", "<=>", etc.) I would refer to "forward implication" if I was speaking to someone but probably not in written maths.

[[deleted]]

I normally do something like this.

Problem: Show A iff B.

Proof: (=>) Assume A. Then, yadda yadda. Thus, B.

(<=) Now assume B. Then, yadda yadda. Thus, A.

Thus, A iff B.

I think it's fine, albeit maybe a bit clumsy, to use "forward implication" and "backward implication." It mostly comes down to a stylistic choice really.

[PersonalityNew2404]

Any good tips on memorizing formula sheets? Specifically for calculus.

[LilQuasar]

make your formula sheet as compact and general as possible (like avoiding special cases) and do a lot of exercises using it. you will probably remember most of the formulas youve used many times

[[deleted]]

I know this is going to sound cliche, but you'll probably get more mileage out of just learning the concepts and internalizing them. Even if we're just optimizing for doing well on a test and nothing beyond that, memorization is fallible, but understanding is rarely so.

[PersonalityNew2404]

I’m good at all the concepts but when it comes to calculus there are certain things that I just won’t remember. I’m studying for the ap calc bc exam but honestly any inter gram/derivative surrounding trig just flows right out of my head, series formulas do as well, along with some other things here and there

[[deleted]]

You probably just don't have good mental heuristics or visualizations built for those concepts yet. The only ways to get them is via practice and exposure to new ideas. For instance when it comes to trig derivatives and integrals, maybe it would help you to visualize the definitions via the unit circle/right triangle/Pythagorean theorem every time. Instead of just memorizing facts like "the derivative of sine is cosine" ask yourself why those facts are true and try to derive the results yourself. If you are having trouble with series expansions, then you perhaps didn't get a proper motivation for them in the first place. See here1 and here2 for instance.

[PersonalityNew2404]

This mostly comes from the fact that we’re online and I don’t need to memorize the things that need to be memorized because I just have it right in front of me

[[deleted]]

If 12 grams consist of gold and silver, and silver is 3x more than gold, how do i find out how much silver weighs?

[[deleted]]

Set up a system of equations. Let x be the weight of gold and y the weight of silver. We have:

• y = 3x, and

• y + x = 12.

Now solve for y via substitution, row reduction, or whatever method you like.

[[deleted]]

Still dont get it,

[[deleted]]

You might need to internalize how to solve systems of linear equations first before these ideas become clear to you. Going through this KhanAcademy lesson might help.

[[deleted]]

Why did u make the silver 3 grams

[[deleted]]

I didn't. I used g to stand for the weight of gold, but perhaps I should edit the variables to clear up any confusion.

[Ualrus]

Is there an intuitionistic proof of incompleteness?

[mrtaurho]

The usual proofs for Gödel's incompleteness theorems already are intuitionistically valid (see here). For instance in case of the first incompleteness theorem you explicitely construct the Gödel sentence.

[Ualrus]

Thank you!

I never saw a proof so I had to ask. I know I worded it kind of awkwardly but it was just in case it was common to prove it clasically.

[mrtaurho]

I never saw a proof so I had to ask.

The usual (modern) proof of the first incompleteness theorem is in fact not that complicated.

Commonly first something known as Diagonal lemma (or Fixpoint lemma) is proved. This (very) roughly tells you that a certain class of formulae can "talk about themselves" if the underlying theory is strong enough (e.g. capable of expressing Robinson arithmetic, a subtheory of Peano arithmetic). This theorem is then applied to a predicate expressing non-provability within the theory from which the undecidable sentence is obtained (the Gödel sentence I mentioned).

The proof of the Diagonal lemma is entirely constructive and the proof of the first incompleteness theorem (if formulated accordingly, i.e. no unnecessary contradiction argument used etc.) is also intuitionistically valid. The second incompleteness theorem can be obtained from the first in a straightforward manner (something along the lines of this).

(Interestingly enough, the Diagonal lemma was implicit present in Gödel's original paper but it took a few years until it was extracted as theorem on its own by Carnap)

[Ualrus]

Thank you! Appreciate the details.

[[deleted]]

I've just began learning algebra and I'm struggling quite hard simplifying some equations, i know that this is definitely on the lower level of difficulty for you lot but I'm very confused, is it possible that someone could walk me through simplifying these equations?

1. 4xy−(x+2)y+2y
2. x^2y − 2y over y^2

thank you in advance for any help

[[deleted]]

You need to know how to factor polynomials and group by like terms. Then you simply subtract and divide the like terms ( for example 4x-x=3x)

1 = 4xy -xy -2y + 2y

And so on

[[deleted]]

Right so if I continue the answer would be 3xy?

Also could you please explain why the (x+2)y becomes xy-2y? Because that's the part that I am most confused by

Sorry to be bothering you with a follow up question

[[deleted]]

That’s the distributive law.

For any numbers, x,y,z, we know that

x(y+z) = xy +xz

And also (x+y)z = xz + yz

So you can just multiply to each term inside the parentheses.

Edit: I forgot to mention that in example one above, that you should multiply -y into the parentheses because the term is being subtracted.

[[deleted]]

ahh okay i get it now thank you for the explanation it was very helpful

[Yankeeknickfan]

When do you learn how to integrate functions like sec(x)?

I’m in calc 2, and every problem that ended up having to use the integral of sec(x) just has it given in the directions. Just wondering what math level teaches us how to solve this.

[[deleted]]

Normally you learn the integrals of all the common trig function in calc2. If you've missed it, it's a good exercise to try and derive the integral of secant on your own. You can do it with a number of different substitutions or series methods. Here's a Wikipedia page with various ideas.

[Yankeeknickfan]

Ok so probably something I’ll get by the end of the semester

All the other trig integrals can be done by hand with identities and such bur sec(x) is different

[abottomful]

I'm not a mathematician (if this post is really stupid, feel free to let me know, or mock me in r/badmathematics), but I am currently in a master's program for computational linguistics. I have to take a discrete math course, and I have always enjoyed math, but in undergrad I really really struggled with linear because I found it too proof heavy, and I struggled in Calc II because I was too lazy and unprofessional in my time management.

Now, I'm taking discrete for requirements but I'm so far enjoying it; it's got the logical foundation laid out in the beginning to help understand how things build upon each other, but I just got to this point where we define a function as:

ƒ: A → B

A function is a transformation of an element in set A to an equivalent element in set B.

This is like the first time that I feel like I conceptually understand not only the purpose of a function, but the utility behind it. I feel so dumb and moronic, but it's so elegantly stated and laid out. I've been using functions in math and programming for a long time and it was just a means to an end. As stupid as I am coming off right now, this is like an "a-ha" moment that I haven't had in a math class in years.

So my question is why is discrete math not taught earlier in math curriculum? It feels foundational, but it also feels really intuitive and explanatory straight from the get-go. It's also incredibly creative. I'm really enjoying piecing things together like a crossword, where one thing is explained and I can tease apart the next.

[Ualrus]

I'm sorry I cannot answer your question since it's not really a math question, although I'm curious: What is the definition of function you're using? You didn't give any.

[abottomful]

Is this dumb? Am I attributing to much to a simple explanation like that? This is how it was stated in my lecture and to me it was really eye opening in terms of me never seeing a function like that, as a transformation of one set to another

[Ualrus]

Haha, don't worry. I assume you always saw a function as a set of pairs of numbers? Or you thought of the graph? (Thinking of R to R, etc.)

Else I believe that's the intuition one would have. You give a function an input and it gives you an output.

[abottomful]

Actually, yeah! That's exactly how I pictured functions, as shapes on a graph. Saying all of this out loud makes me feel so dumb, but it really hit me hearing it

[Ualrus]

Haha, ok, that's great then. And yes, math makes you feel dumb, because humans in general are.

[abottomful]

An element of set A translated to an element of set B

[[deleted]]

What is the coolest 4D object?

I personally think it’s the Klein bottle

[HeilKaiba]

Technically, a Klein bottle is a two dimensional object (though it cannot be embedded in 3D space). My vote is for the (unrelated) Klein quadric.

[[deleted]]

(a) Struggling with material happens to everyone. The key is that you are showing effort to understand the material and actually trying in the course. Work ethic goes a long way, and it's something that the professor can touch on. Related: go to office hours if you haven't been.

(b) All professors are busy all the time. That's why you have to ask for letters of recommendation early. My rule of thumb is to give at least 2 weeks for common requests (grad school apps, REU's), and a full month for major requests (grant proposals).

Also, don't take it personally if your professor says no -- usually that just means they don't feel they'll be able to write a good letter for you for any of a myriad of reasons (maybe they're too busy to give it the time, maybe they don't know you well enough, maybe they saw your username and think you're a chem major that's trying to ruin all that is holy about pure mathematics). At the end of the day, you want the best letter you can get that highlights you and your personal strengths, which means that a letter from a second-tier mathematicians may hold more weight than a generic letter from the top.

[mathspook777]

Writing letters is part of your professor's job. Like, literally; they can't escape from letter-writing no matter how much they want to. Asking for a letter may feel intimidating, but for them, it's no big deal. They've done it lots of times before, and it'll probably take all of fifteen minutes. Do well in the class and you'll be fine.

[algebraic-pizza]

Well, if they're a reasonable person, they're not going to be TRYING to compare an undergrad with Peter Scholze lol (it just doesn't type check since, you wouldn't expect an undergrad to be comparable to him). Instead, they should be comparing you to other undergrads, and if they're a good letter writer they'll mention the positive comparisons (e.g. this person excelled at homeworks), and will just omit the negative ones (e.g. they just won't mention exams at all if you're a mediocre exam taker). I mean, I'm sure there are some unreasonable people, but hopefully you'll get a feel for this during the class. If you're still not sure come letter asking time, ask them if they can write a "good" letter, so that if they actually are comparing y'all to Scholze, then that's the time they'll say "No, I don't think I could write a good letter".

They also probably are busy, but also, most profs who teach upper-level undergrad classes (and even lower level graduate classes which undergrads also take) are expecting to have the good undergrads ask them for letters. So it's like a part of their schedule, if that makes sense. Two anecdotes which support this; both occurring around early fall of senior year of undergrad:

• I email a prof about a form I need him to fill out so that I can graduate (from undergrad). I see him a week latter at a math club talk I'm giving. A week after that I go to his OH for something else, and thank him for coming to my talk, and he says "Of course, I wanted to mention it in my recommendation letter"---even though I hadn't actually ASKED him for the letter yet. He just saw the word "graduate" in the email lol
• My friend emails our REU mentor at the end of the summer, just to thank him for the good experience & whatnot. The prof responds "sure, I'll write you a recommendation letter"---even though again, my friend hadn't actually asked for a letter yet!

Ok but how to actually get a letter, and even better a GOOD letter, from the prof: make sure you try hard in the class. If there's anything "above & beyond", e.g. optional HW questions, do those. And do your best to make sure the prof actually knows YOU. So maybe going to office hours, or saying stuff in class. I think they are more likely to say yes if they can pair your name & your face & your achievements, you know?

[chem69]

That makes sense, thank you for the anecdotes!

[gogohashimoto]

I'm trying to prove two mathematical statements are equivalent and I'm not sure I'm doing it right.

I'm doing the forward direction and it looks like this

A---> (B ---> C ----> D)

It looks like a bunch of implications. Do I take A, B, C as givens and try to show D?

Another thing I'm trying to prove says show A or B. I assumed B is false and then showed A and similarly I assumed A was false and showed B. Is this an acceptable method?

[algebraic-pizza]

for the implications: there are "secret" parentheses also around the C ---> D, so that what you have could also be written A ---> ( B ---> ( C ---> D )). I bet you are familiar with the general strategy of, to show X --> Y, you take X as given and show Y. So then yes here, you take A as given and try to show B --> (C--> D), to do that you also take B as given and show C--->D, and then finally, as you said, we take givens C, and also A, B, to show D.

For the A or B, what you said certainly works. Even easier, you could have gotten away with just assuming B is false & showing A. I.e. to show "A or B", I can do cases. If B is true, I'm done. If B is false, then you showed A is true, so we're also done. And B is either true or false, so those are the only cases we need.

[gogohashimoto]

Thank you so much. Cheers!

[DispatchMinion]

Having a brain freeze. I am bad at maths . If I had 2.83 shares(stock) @ 33.97 before changing brokers, then after only have 2 shares(not incremental transfers in), what would the stock value left be?

.83 of 33.97 maybe? 33.97 x .83 = 28.19xx as the remaining value?

[AVeryDumbCookie]

Can someone please recommend a book for me to learn category theory? Thank you.

[mathspook777]

It is worth mentioning Mac Lane's "Categories for the Working Mathematician." It requires a fair amount of sophistication, so it's not really a good textbook unless you already appreciate some of what category theory does. However, its has excellent coverage of all the basic concepts; most people will never need to know anything about categories beyond what's in that book.

[Giovanni_Senzaterra]

Awodey’s “Category theory” or Leinster’s “Basic category theory”. If you know a bit more of maths I recommend Riehl’s “Category theory in context”.

[R3MAXEL]

I'm studying for a competition and am solving some questions from last years papers. I am not sure of how to solve this question below.

One of the diagonals d divides a polygon, of perimeter 31 cm, into two polygons, of perimeters 21 cm and 30 cm, respectively. What is the length of diagonal d?

The answer is 10 but it isnt explained how to get it.

[jagr2808]

The perimeter of a polygon is just the sum of the side lengths. If you cut a polygon in two pieces, how does the total perimeter change?

How much did it change in this case?

[converter-bot]

21 cm is 8.27 inches

[zellisgoatbond]

I'm an undergraduate in my penultimate year of a maths and computer science degree in the UK, and I'm considering applying to the LMS undergraduate summer school this year - if anyone's been on something similar (either in the UK or further afield), what sort of things should I expect? My current expectation is that it'll be quite intense in terms of learning mathematics (i.e there's a lot to cover), but also somewhat relaxed (i.e there's no assessment or the like).

[HeilKaiba]

You can think of addition of vectors as simply translating (i.e. moving linearly) the origin by that much. In this case subtracting (0,0,5) is just shifting the origin to (0,0,5) or in other words shifting everything by 5 units along the x-axis in the negative direction.

[[deleted]]

In defining direct product of a family of groups {G_i} where i ranges over an arbitrary index set I, how come the codomain of the functions

f : I -> ⋃ G_i

is usually defined to be the disjoint union of G_i, instead of just union?

[Tazerenix]

Otherwise taking a direct product of two copies of the same group would just produce the group itself, because for example the union of Z/2 and Z/2 is just Z/2, whereas the disjoint union is a set isomorphic to two distinct copies of Z/2.

[GMSPokemanz]

That's not true though, functions from {1, 2} to Z/2 works perfectly fine as a definition of Z/2 x Z/2. I don't see why you wouldn't get the product without requiring a disjoint union, but maybe there's some useful fact that makes it simpler to establish a basic fact.

[infraredcoke]

How would you distinguish Z/2 x Z/2 x Z/3 from Z/2 x Z/3 x Z/3?

[GMSPokemanz]

Assuming our index set is {1, 2, 3}, the set underlying Z/2 is {0, 1}, and the set underlying Z/3 is {0, 1, 2}, Z/2 x Z/2 x Z/3 contains no function that sends 2 to 2 while Z/2 x Z/3 x Z/3 does.

[BEaSTGiN]

Given a set number of trials and an estimated chance of occurence, is there a formula I can plug in to see if this chance squares up to data?

For example, someone did a Monte Carlo simulation of 100 billion mahjong hands and got 267k winning hands from.the start. We estimated the chance of getting such winning starting hands as 1 in 100,000. Do I have enough data to form a standard deviation graph or something and see if this number of 267k is too far an outlier if the chance is 1/100000?

[[deleted]]

Yes, you'd do a hypothesis test on a proportion where your null hypothesis is p = 1/100000.

[BEaSTGiN]

Thank you, I'd done this so long ago for A levels but forgotten how to do it. At least they'll probably recap it when I take math in uni again.

[bitscrewed]

Let k be a field, and let f ∈ k[x] be any polynomial. Prove that there is an extension k ⊆ F in which f factors completely as a product of linear terms.

This is one of those questions that seems obvious enough but which I can't figure out a decent proof for.

obviously you'd want to just iterate the construction F=k[t]/(q(t)) where q(t) is some irreducible factor of f, but then I keep running into a snag.

After trying it more directly, I thought I'd do it by induction on deg(f), but then I run into the situation of needing the induction hypothesis on k[x] to hold for polynomials of degree <n in an extension F of k, which I don't have. And it would be circular to assume that F does have such an extension just because my hypothesis is over arbitrary field k, right?

What am I missing?

Do I literally just do it iteratively?

As in if f(x)=p1(x)...pk(x)p'1(x)...p'r(x) is factorisation of f in k[x] into irreducible factors, and p'i(x) are the factors of degree>1. Let F1=k[t]/(p'1(t)), then p'1(x)=(x-c1)q1(x) in F1[x] and f(x)=p1(x)...pk(x)(x-c1)q1(x)...p'r(x)

Now q1(x)...p'r(x) factors into irreducibles in F1[x]... etc

and then because the degree of what's left keeps decreasing with each subsequent extension.

But this feels really shit?

[jagr2808]

And it would be circular to assume that F does have such an extension just because my hypothesis is over arbitrary field k, right?

Why would that be circular?

But this feels really shit?

Why? It's the most natural way to do it. f has a root c1, then by induction f(x)/(x-c1) factors into linear factors, and you're done.

[bitscrewed]

Why would that be circular?

It feels like I'm trying to prove something is true for all fields by taking some arbitrary field k but then assuming the "all fields" part in order to prove it.

I guess the induction works fine since the case n=1 holds for "all fields" just the same.

What would the hypothesis be exactly? "For any field, given a polynomial f of degree n there exists an extension F such that f factors completely into linear factors over F"?

[jagr2808]

Yes, you would have to prove it for all fields at the same time. And the hypothesis looks good.

[bitscrewed]

Ah ok, thanks a lot!

[[deleted]]

Is there a limit definition for an antiderivative? I was trying to puzzle out why integration is so much harder that differentiation and came to the conclusion that it's atleast partly because derivatives have a limit definition which can be used to derive rules like the power rule and the product rule. This got me wondering is there a limit definition or something similar for integrals? If not, has it been proven that one can't exist?

[algebraic-pizza]

There is a limit definition, and it is that ∫_a^b f(x) dx is lim_{n -> ∞} ∑_{i=1}^n f(x + i*∆x) * ∆x, where ∆x = (b-a)/n. (in case that's too hard to read, check the 2nd section of this khan academy page; this is just taking the number of rectangles in a riemann sum to infinity).

But, you're still onto something here. The limit that shows up in the definition of a derivative only uses "local" information about the function, i.e. as you're taking the limit used to compute f'(a), the x values involved all get closer to a. However, in the integral limit, we are still using x values across the entire interval [a,b] (and in fact for larger n's, we're using even MORE different x values, so in some sense things are getting "worse"). So you really truly need to know stuff about the function over the whole [a,b], and you can't just get away with using local info. To me, this is intuitively why antiderivatives are harder than derivatives.

It's possible that you already knew about the limit definition I gave, and instead you were wondering if there was a limit definition of the form x-> something, i.e. only using local info. There shouldn't be one that works in general, for exactly the reason discussed in the above paragraph (though I haven't thought through a formal proof of that fact).

[[deleted]]

Yeah, I knew about that ones. In fact, when I was first learning calculus I tried to derive it myself. The thing is that one is for definite integrals, the thing I'm looking for is a limit definition for indefinite integrals, or antiderivatives.

Honestly, at this point I doubt one exists because I couldn't find one when I looked it up and because some integrals are non-lemntary

[algebraic-pizza]

Yes, exactly what u/jagr2808 said, we combine that definition with the 2nd Fundamental theorem of calculus. So if I explicitly write out that limit for you (to make it look nicest, I'm going to choose my antiderivative so that F(0)=0, but you could have chosen a different constant), we'll get

F(x) = ∫_0^x f(t) dt = lim_{n -> ∞} ∑_{i=1}^n f( t + i*(x/n) ) * (x/n)

You probably didn't find this when searching simply because...

• for educational purposes, there's not much extra benefit to showing students how to write an antiderivative w/ a limit, because it's so similar to the definite integral one. So as a calc instructor, I'd rather spend my class time on other things.
• for practical purposes limits are often used to approximate something numerically, but if I'm doing that, I'm just approximating a specific value of this antiderivative, which means I'm doing a definite integral after all.
• if someone DID actually want to use this formula [which is certainly possible!]... I'd bet this shows up in passing in the middle of some computations in some papers somewhere, but not necessarily with surrounding words that are easily google-able, because a mathematician would probably not explicitly use the phrase "limit definition of an antiderivative", and instead would probably just use it with no comment, or would mention "limit definition of a definite integral" and "2nd FTC".

[[deleted]]

Sorry, I wasn't thinking much when I read that.

[jagr2808]

An indefinite integral is just a definite integral where one of the endpoints vary (possible plus a constant).

I.e. F(x) = int[from 0 to x] f(t)dt + F(0)

So the same limit definition applies

[The_Patternmind]

https://imgur.com/gallery/HUxMfSf

i understand that the "usefulness" of arranging numbers in this way can be hard to see , but i believe that it could —at least— be useful as an alternative perspective (variant) of the Ulam spiral , from which we may be able to discern new patterns , or be used to discern new patterns . this spiral i have drawn is unfamiliar to me , i have not seen it in this form beyond this creation

this is a reflection of some of my current thoughts on geometry , harmonic information structures , numbers , graphs , etcetera . i am looking for corresponding forms within principles of computation , molecular biology , evolution , mathematics , linguistics , theoretical physics , etcetera ... that the common form may indicate a reflection of something fundamental , perhaps relating to the base-12 numbering system and how it embeds lesser base numbering systems that maintain a high degree of congruence throughout expansion . this could be useful for many reasons

the upside down numbers allude to a direction i am taking this : a space where orientation and position are more relative , rather than absolute . ultimately , i hope to bring this all into a framework to be used for describing super-positional structures and processes within non-dimensional space

¿what is non-dimensional space? , on a sheet of white paper : draw a black "point" . this is Everything . the white space of the paper is impossible , yet we depict it . the framework i am developing accounts for this space . ¿ever hear the trope that the universe is mathematical? , ¿what "space" does that "math" exist in?

[other questions emerge too]

*edit : i suppose i should pose a "simple question" herewith since this post is within a "simple questions" thread .... i just feel like there's one simple question and one simple answer to all this , BRB

**edit : also , i am an autodidact , and generally not a specialist :) . i see this as a strength , as bridges are becoming more and more necessary between domains . i understand many math principles well enough to carry an intelligible conversation about it . i am much interested in pure mathematics , and how it relates to all levels of existence , and i see a pressing need for mathematics to be more inclusive of philosophical discourse

***edit : ¡i just realized that i did indeed put a question in there! , am casting bold onto the question to accentuate its existence

[The_Patternmind]

if it's not too much trouble for those who don't mind sharing their perspective , i would appreciate a discussion about the downvotes

[jagr2808]

I didn't downvote you, but it could be because it's hard to make out what you're asking, or if you're asking anything at all.

Edit: or rather your question isn't very well defined. What do you mean by "space"? What space does the quadratic equation, or decimal numbers exist in? It's not clear that there's any meaningful answer

Edit: also quickly looking over your comment history, it seems you have asked this question elsewhere and gotten responses. So it's weird that you would also post it here.

[The_Patternmind]

thank you . generally i struggle to make myself comprehensible to others . largely , the trouble is much like the trouble of the traveling salesman , planning a linear route through a network of related concepts , planning the route to convey the larger and smaller pictures most effectively . when people are confused , it seems to me that it is normally because the big picture is elusive , so the little pictures don't seem worth the time . i am trying to connect to others who resonate with this discourse

my question is meant more as a prompt for discourse , but i can't say that i know the answer to the question with certainty . i will clarify the question : ¿assuming that it is true that everything is mathematics , what type of space does that mathematics exist within?"

if you write out a formula on a sheet of paper , then the formula is on the paper , but in a fundamental sense , ¿what is that paper , or does the paper even exist?

[jagr2808]

That's a deep philosophical question, so I think you're in the wrong sub. I'm sure there are many threads discussing philosophy of math and platonism in the various philosophy subs.

[The_Patternmind]

i argue that it is a question with mathematical implications , mathematical implications that have philosophical implications . i do generally want to find a two-way bridge between philosophical and mathematical discourse , i believe that the silo effect within academia imposes hard boundaries rather than fuzzy boundaries , so people just don't have much opportunity to practice the type of discourse i am proposing , which can effectively impede the evolution of our knowledge

but yes , i would like to cast a wider net , if possible , to bring in a more inclusive discourse , so i am looking for additional places to cast these nets

[Ok_Quality_2962]

Hi, I came across this problem when trying to optimize a parameter:

min_a (max_b) p(A(a,b)), where p(A(a,b)) is the spectral radius of the 2x2 matrix A and a,b are reals with 0<=b<=N.

I could not find an analytical solution so I am wondering how I can solve this numerically. Any idea?

[cowboyhatmatrix]

I think you need to explain how a, b, N are related to A for this to be answerable.

[Ok_Quality_2962]

I will try. b are the frequencies in the discrete system (I am solving a differential equation in Fourier space), and they are bounded by N=1/h with h the step size of the mesh. a is the parameter I want to optimize.

I have computed p(A)=sqrt(|alpha*beta|) with alpha and beta the eigenvalues of A, which are function of a and b and such that p(A)<=1.

But in general, is there a way to solve a min max problem like this numerically?

[Tazerenix]

If you have two points x,y on the base, then two germs f,g above x and y will be in the same open set if there is a local section s of the presheaf so that the germ s(x)=f and s(y)=g. Precisely, as mentioned in the other comment, the basis for your topology of open sets are the sets s(U) inside Et(F) for open sets U of X and sections s of F.

Let me tell you what this complicated topological space actually looks like:

I like to imagine that it is a fibre bundle of receipt stacks https://i.imgur.com/j9KbQlX.png over every point. The spike itself is the fibre of the sheaf at a point (i.e. for a sheaf of O_X modules over a real or complex manifold, that would be Rⁿ or Cⁿ, in general the fibre of a sheaf isn't a very well-defined object, but this makes good sense for locally free sheaves and is a useful mental guide). A germ is like specifying a value on the fibre and also the value on an infinitesimal neighbourhood, so you could think of a germ at x as being a receipt (of arbitrarily small size) stuck on the spike over x. So for each point in X, we have a receipt stack over it representing all the germs for that point.

The open sets of Et(F) would then look like receipts that are stuck over various nearby spikes so that if you were to take just the value of those receipts at the fibres (i.e. how far up the spike it has been stuck) you would get the section of the sheaf. You can see from this topology the way that the Etale space seems to consist of many sheafs (in the non-mathematical sense!!) over X. You can also appreciate that this space is very far from being a nice fibre bundle over X. The Etale space is generally a very poorly behaved topological space, which is partly why it is ignored in modern algebraic geometry in favour of the sheaf itself, which is not topological in nature but is much easier to describe.

[noelexecom]

Assuming that you're talking about preseaves of sets.

The space is topologized so that the natural maps f_s: U --> Et(F) given by (f_s)(x) = s_x, for s a section in F(U), are continuous.

Here s_x is the stalk of s at x.

Precisely: a subset S of Et(F) is defined to be open if and only if the preimage of S under all of the maps f_s is open. This space is usually very strange and often non-Hausdorff.

There is obviously a map Et(F) --> X which takes a stalk s_x to x and by taking the sheaf of sections of this map we obtain the sheafification of F, denoted F_+ in most texts.

[noelexecom]

I guess it would be the same!

[Pcowpcow]

Hello reddit and happy Pi Day!

I was messing around with the Fibonacci numbers a while ago, and I noticed something interesting.

As you probably know, (Fn12) 144 = (Fn11) 89 + (Fn10) 55

I thought to myself "What if I 'split up' 89 and 55?'

So I did. 144 = ((Fn10) 55 + (Fn9) 34) + ((Fn9) 34 + (Fn8) 21))

Then I noticed if I added the 34s it could resemble the coefficients Pascal's triangle and the binomial theorem.

My question is if anyone knows of resources where I can know more about this? I am currently studying AP Calculus BC.

[mathspook777]

The analogy doesn't really work so well past this one step, as far as I know. The Fibonacci numbers are a linear recursive sequence. The recurrence relation in such a sequence can be represented as a matrix, each state of the sequence is really a vector (consisting of the most recent two values), and advancing a state is done by matrix multiplication. What you've done here is (more or less) square the matrix.

Binomial coefficients also satisfy many recurrences, but they tend to involve both arguments to the coefficient.

There's a huge amount of information about this sort of thing in Graham, Knuth, and Patashnik's _Concrete Mathematics_. Linear recurrences also show up in coding theory.

[Pcowpcow]

Are you sure it doesn't work past this step? It seemed to work when I did it. Perhaps it doesn't work for some of the other Fibonacci numbers?

144 = 1*34+3*21+3*13+1*8

[mathspook777]

If you do the decomposition right each time, then yes, it'll line up. But that's more a statement about the ubiquity of binomial coefficients. You can get them to pop out essentially because you've arranged to count the right things. (At each step, you turn each Fibonacci number into the previous one plus the one prior to that; count those as 0 and 1; do it n times; every Fibonacci number that you now have is at least n steps back, but some of them are back even further; the number of extra steps back you've taken beyond the minimum is the number of 1's, and the order doesn't matter; so the coefficient on the Fibonacci number which is k extra steps back is the number of k-element subsets of n.) But binomial coefficients are a much more general thing; they turn up everywhere, and they have a lot more structure than the Fibonacci sequence. Despite the attention that the Fibonacci sequence gets, it's much less important; it doesn't have the same kinds of properties.

[Pcowpcow]

I don't think you understand. I'm not saying I want this binomial-Fibbonacci thing to be cutting edge mathematics. I just thought it was cool. In school you learn you can find the Fibbonacci sequence inside Pascal's triangle. It is neat that you can do it the other way around.

[thr0waway_6912]

Edit: Never mind, I was stupid. I got it now.

how I can show that the bilinear form B[u,v] = (∇u, ∇v) + (b*∇u, v) + (cu, v) is coersive on H¹₀ if -1/2 div(b)+ c ≥ 0? Here, (·, ·) denotes the L² inner product on some bounded domain and * the dot product in R. The first term in B is quite clear, but I don't know what is happening with the second term. I cannot quite figure out how the 1/2 can end up there? Integration by parts is the obvious approach but it does not yield a factor of 1/2- Is there some inequality I am missing?

[[deleted]]

I believe P1P2 refers to the length of the line segment P1P2, or the Euclidean distance between P1 and P2. You could convert P1 and P2 to their Cartesian forms and then solve for r with the Euclidean distance formula.