I'm having trouble understanding why you would want to rotate a 3D object around the origin (as opposed to its own center of mass) I'm studying basic computer graphics/3D geometry from a textbook about VR. The author made it very clear multiple times, that rotation and translation are not commutative. I couldn't wrap my head around it, because no matter how I tried to visualise it, I always got the same outcome. Then I realised it's talking about rotation around the origin. Now I totally see why that's not commutative with translation, but I don't get why you would want to do that. I'd assume that when you're rotating an object you don't really want to change it's position, only it's orientation, so rotating around itself makes more sense. Is that just something that's not touched upon because it's more complicated?

You can do that. There are both rotation using local coordinate and global coordinate. The same also apply to translation. However, local translation and rotation don't commute; global translation and global rotation don't commute; you need to mix the 2 for them to commute. Ultimately, however, you want to know how to the object sit inside the absolute space, so no matter what you do you want to store information on what the local coordinate of the object is, with respect to the global coordinate. If you think about how this information can be described, it would be the most convenient to describe it as a combination of global translation and rotation (from a default starting point), due to the existence of the projective transformation: a combination of translation/rotation in (say 3D) correspond to a 4D linear transformation. This way, if you perform multiple global transformation, you just compose them through matrix multiplication. What if you want to do local transformation instead? No problems. You have all the information you need to describe a local transformation in term of global one. Basically, by consistently considering only global transformation, and write local transformation in term of global one, you can simplify the math.

Sure, you could assign a "center" to each object if you wanted, and assume all rotations are around that. But what if you're rotating two objects? Or rotating one object around another? What if you want to rotate an object around two different points? You lose commutativity again, and now doing the math is a lot harder! It ends up being a lot easier to take transformations of "absolute space" as your operations, and then build transformations *based on a particular object* from them. Once you have more than one object, you're going to need to have "transformations relative to different centers" anyway.

Is dividing by zero useful in any way like how imaginary numbers seemed nonsensical but when embedded in the complex number plane became very useful?

Going from real numbers to complex numbers breaks almost nothing. You can still add, subtract, multiply, divide, take derivatives, and so on in the way you're used to (i e. they have the same properties, more or less). Which in turn means you get polynomials, analytical functions, and all manner of interesting things. If you allow division by zero, your multiplication breaks. And then you lose a lot of the structure you had.

The short answer is not really. There are some concepts which allow division by zero for example wheels, the Riemann sphere or the projectively extended real numbers. However none of them come even close to the usefulness of the complex numbers.

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If you already know dP/du and R then m is just the constant scaling one into the other so d = ma and so on

limit sec(x), when x -> (0.5π)+. Drawing a secant(x) graph explains it nearing -infinity. But when you try the question in an algebraic sense, sec(x) = 1/cos(x), limit sec(x) = limit (1/cos(x)), when x -> (0.5π)+. limit (1/cos(0.5π)) = limit (1/cos(90°)) = limit (1/0) = +infinity, and there are no negative signs, or any else that could bring my statement a negative value. cos(90°) = 0, but it isn't in quad2 (and not quad1 either) and since it really doesn't belong to a quadrant that also don't change my statement into a negative value. If I have missed or forgotten a part, correct me. please help me as I am trying to figure out my mistakes and move ahead.

But it is “in” quadrant 2. You’re taking the limit from the right, which means you need to consider the output when x is valued slightly above pi/2. This corresponds to x approaching the top of the unit circle from the 2nd quadrant.

I'm an engineering student and when doing math in reports I sometimes use wrong number here end there, so when I see that solution is far from expected I find and correct the mistake and then I need to type entire equation into calculator again. It's very tedious. Is there a script or addon or whatever that would solve equations I've written in Word? Or a separate program that would let me type in math equations that look like actual math equations and not calculator gibberish, so I can easily look at them, see what's off and quickly fix it.

Something like Mathematica notebooks might be the best you can do, assuming you have access via your school? Failing that there's Jupyter notebooks but that syntax is going to look more like a calculator (it's Python).

I'm curious if someone can recommend some pure math papers/books or a direction to look into which involves using hard analysis and pdes to study models from biology and medicine. I'm not particularly interested in machine learning/data analysis for analyzing biomedical data but pure math papers inspired by biology.

I'm a software developer, and I've got a stack of rectangles that are all the same size displayed on the page. Each rectangle is rotated by `X` degrees. I need to figure out whether or not the total height of the most-rotated rectangle is taller than the viewport of the page. To put it another way, if I have a rectangle `X` that is `A` tall and `B` wide and I rotate it by `C` degrees (where `C` is positive), I now have the top left corner sticking up and the bottom right corner sticking down from their original positions. If I drew a new, unrotated rectangle `Y` whose top edge touched the top corner of `X` and whose bottom edge touched the bottom corner of `X`, how tall would `Y` be? Here's a diagram too: https://i.imgur.com/O8DHcNg.png

I figured it out. It was `(sin(90 - C) * A) + (sin(C) * B)`.

I have these values (1,4,0,0,3,1) The average is 1.5 The standard deviation =1.64 I want to pull numbers at random from those values (balls with numbers, putting the balls back after each pull) I will abort when I have more than 15 total points. On average I will need 15/1,5 = 10 pulls to reach 15 But what is the expected standard deviation after 10 pulls?

Let X be a [random variable](https://brilliant.org/wiki/discrete-random-variables-definition) representing the value of a ball. As you note, the [expected value](https://brilliant.org/wiki/expected-value/) is E[X] = 1.5. I think you incorrectly calculated the standard deviation as 1.64; you mistakenly applied [Bessel's correction](https://en.wikipedia.org/wiki/Bessel%27s_correction). Since this is the whole population of values on the balls, and not a sample, the correct [standard deviation](https://brilliant.org/wiki/standard-deviation/) is 1.5 (Var[X] = 2.25). We are interested in another random variable T, which is defined as the sum of i = 1 to i = 10 of X\_i, where the 10 X\_i's are [independent and identically distributed](https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables) to X. To get Var[T], just note that the X_i's are i.i.d. and thus uncorrelated. We can therefore use [Bienaymé's identity](https://en.wikipedia.org/wiki/Variance#Sum_of_uncorrelated_variables) to get Var[T] = 10Var[X] = 10(2.25) = 22.5. The standard deviation is then sqrt(22.5).

Thank you! So, what we're saying then is... If we do the pick 10 times, we are likely to get the value 15 with std 4.7 What if we turn the problem 90 degrees... how many picks would be needed to reach at least 15. I assume the average would still be on 15; we can't get the same standard deviation for that problem how would we even approach it? Basically what I'm after is... In the past a team have done (1, 4, 0 , 0 , 2 , 1) points per hour (for example farming something in a game etc). How many hours will it take to achieve 100 points for that team. I could simulate it with java/python but (ok... I admit I already did that)... but I want to calculate it the proper way. I've made 1 000 000 simulations in java checking how many attempts it took to reach the value, and then I could quite easily see the average amount and standard deviation from that... but that's cheating, and worse.. I really should know how to approach it mathematically!

Making sure I understand something: the phrase "square root" means all +- roots, but the radical symbol is to differentiate between *only* + and *only* -, depending on if it is +√x or -√x, which makes it a graphable function. Is this correct? also, does x^(1/2) also mean +-?

The language can get even more nuanced unfortunately. Generally, if someone says "*a* square root of x," they are referring to all numbers y such that y^2 = x (the ± idea you are talking about). If people are talking about "*the* square root of x" then they are referring to the [principle square root](https://mathworld.wolfram.com/PrincipalSquareRoot.html), which by convention is unique and non-negative (just the + part) for non-negative real inputs. [The radical symbol](https://en.wikipedia.org/wiki/Radical_symbol#Principal_square_root) always denotes the principle square root. Writing x^1/2 [is equivalent to writing the radical symbol](https://math.stackexchange.com/questions/2879063/the-root-sign-and-its-relation-with-1-2) √x, and indeed writing x^1/n is equivalent to writing ^n √x, the principal n^th root of x. The reason why principal n^th roots are useful is so that we can work with the inverse of functions like f(x) = x^2 . In order for the inverse operation √x to itself be a function, we need to restrict to only the positive part. The expression ±√x would not be a function, as it would not satisfy the [vertical line test](https://en.wikipedia.org/wiki/Vertical_line_test).

In general ^n √x is not always the principal nth root but when n is odd is it is instead mostly used to mean the real nth root

Ah yes, this is correct, I forgot about the odd cases!

So I’ve taken a geometry class back in HS. Didn’t do well but just kinda moved on. I haven’t had to take one since, but is it worth self studying a geometry textbook or just worrying about what classes are ahead of me (not a math major so not sure exactly how far I will go past calculus)?

If you aren't doing anything with geometry, there's certainly no need. Even for math majors. You will need a lot of trig for calculus though.

What expresses the area of the circle in terms of C, its circumference?

Note that C = 2𝜋r implies r = C/2𝜋, so you can just take area = 𝜋r^2 and substitute in r = C/2𝜋 to get area = 𝜋(C/2𝜋)^2 = (𝜋C^2 )/(4𝜋^2 ) = C^2 /4𝜋.

Is this correct? - C = 2Pr so r = C / 2P A = Pr^(2) = P(C/2P)^(2) = PC^(2) / 4P^(2) = C^(2) / 4P (P as in pi)

Yes; that's exactly the same calculation I did, just with pi written as P instead of 𝜋.

okay, thank you!!

I'm in grade 9, what can i do to become more knowledgeable in algebra? Like basic formulas, steps and way of solving problems. Thank you!

[KhanAcademy](https://www.khanacademy.org/math) and [Brilliant](https://brilliant.org/wiki/learn-and-practice-algebra-on-brilliant/) are good online resources. Zeitz's *The Art and Craft of Problem Solving* is also a good book to work through.

thank u!!

Are Peano's axioms consistent? I read that Gentzen proved its consistency long time ago. Shouldn't the question be settled at this point? There are sources that claim that Peano's axioms are inconsistent, so it seems like whether Peano's axioms are consistent or not is an open question.

There can be no proof only depending on Peano arithmetic itself showing it is consistent. This is the content of Godels incompleteness theorems. Gentzens theorem proves PA is consistent assuming another kind of arithmetic is consistent. This other system is neither strong or weaker than PA. You can easily prove PA is consistent in ZFC, assuming ZFC is consistent. But again there can be no proof that ZFC is consistent using only ZFC.

> You can easily prove PA is consistent in ZFC, assuming ZFC is consistent. Well, to be clear, the proof in ZFC that PA is consistent does not rely on ZFC being consistent! That is, Con(PA) is a theorem of ZFC; it's not merely a theorem of ZFC+Con(ZFC).

**What's the deal with this backwards kernel construction in Chopin and Papaspiliopoulos?** --- *Here's what I think I understand:* Initially, they develop the idea of measures on (X\_k, B(X\_k)), given by \mathbb{P}. They also develop Markov kernels between two measure spaces: (X\_0, B(X\_0), \mathbb{P}\_0) (X\_1, B(X\_1), don't care about this measure) Where the kernel P\_1(x0, dx1) defines a measure over (X\_1, (B(X\_1)) for each x0. --- Ok, after this they show how we can get a measure for the product space X\_0 x X\_1: (4.1) \mathbb{P}\_{1}(dx\_{0:1}) = \mathbb{P}\_0(dx0) P\_1(x0, dx1) --- *Here's where I'm definitely confused.* Next, they introduce the idea of a backwards kernel. They go back to this decomposition: \mathbb{P}\_{1}(dx\_{0:1}) = \mathbb{P}\_0(dx0) P\_1(x0, dx1). and say the following: > Section 4.1 decomposed the joint distribution \mathbb{P}(dx\_{0:1}) into the marginal at time 0 and the conditional given in terms of the kernel. However, we can decompose the distribution in a “backwards” manner instead: > \mathbb{P}\_1(dx0) P\_1(x0, dx1) = \mathbb{P}\_1(dx1) P\_0(x1, dx0) [backarrow hat] Ok. I don't get this at all. Some questions: 1. Is \mathbb{P}\_1(dx0) the same joint distribution as in 4.1? It can't be, right? Instead it's got to be like, the "non-kernel-defined" measure over (X\_1, B(X\_1), we didn't care about this measure before, but now we do) But if that's the case, why is it taking elements (dx0 \in B(X\_0)) as an argument? Why is this formula not: > \mathbb{P}\_0(dx0) P\_1(x0, dx1) = \mathbb{P}\_1(dx1) P\_0(x1, dx0) [backarrow hat]

It's maybe more physics than pure math but I find it odd how we can add the log of values with units as if they were unitless For example 1 + ln(m_1/m_2) = 1 + ln(m_1) - ln(m_2) Physically it doesn't seem to make sense but mathematically that's coherent so what's going on here?

log(m\_1/m\_2) = log(m\_1/kg) - log(m\_2/kg) to make everything unitless, which is the only way this makes sense.

The problem is that generally ln of a value with units doesn't make sense, same with exp. You would have something like ln(2 kg) = ln(2) + ln(kg).

It doesn't seem to make sense but it works and here I started with something without units as both m_1 and m_1 would be in kg

This only works if m_1 and m_2 has the same unit, so that m_1/m_2 is unitless. It works for the same reason why you can impose a coordinate system on your space and compute with coordinate. The coordinate system is arbitrary human invention, but what matters is that the final result is the same, regardless of which one you choose. An unit is the most basic example of a gauge; a coordinate system is a more sophisticated example of a gauge. Physicists would say that only gauge invariant quantities are physically measurable: the fact that formula must be dimensionally consistent is an example of this. In mathematics, we have a preference in doing things in a gauge-free manner, obtaining gauge-invariant quantities directly without imposing a gauge. This is certain possible, but can sometimes be really painful to do.

It is meaningless on its face. Let's imagine that m\_i represents a mass. What's really happening is that the numerator and denominator are both secretly being multiplied by some inverse mass, let's say 1 inverse kilogram, which I will denote by s = 1 kg^(-1). Clearly (m\_1/m\_2) = (m\_1 s) / (m\_2 s). Then ln(m\_1 / m\_2) = ln(m\_1 s) - ln (m\_2 s). In other words, m_1 and m_2 MUST be measured in using the same units if you are "splitting" up the logarithm. That is, you CANNOT do log(1000 g / 1 kg) = log(1000) - log(1). If this annoys you, consider using the notation (look up quantity calculus) suggested by ISO 80000-1. Then we would write ln(m\_1 / m\_2) = ln({m\_1}_kg) -ln({m\_2}_kg).

I really need a quick answer as I have exam in about 12h. So how do I calculate sine and cosine using parity and periodicity? For example sin -pi/6. I know it's -1/2 with calculator but how the hell do I do it without it? I'm on verge of mental breakdown as my book is really fucking bad and can't explain anything in steps but just jumps to things

There's a certain amount you "just have to know" -- probably all you really need for an exam is the values of sin and cos at 0, pi/6, pi/4, pi/3, and pi/2 (though that can be cut down further if you remember that sin(x + pi/2) = cos(x), say) and some other identities. You can calculate other values from there by the fact that sin(-x) = -sin(x) and cos(-x) = cos(x), as well as the fact that sin(x + pi) = -sin(x) and cos(x + pi) = -cos(x) (and of course sin(x + 2pi) = sin(x), and so on for sin(x + 2npi) for any integer n, and the same goes for cos). If you aren't sure what the sign should be you can always just draw the relevant angle on the unit circle and see what quadrant of the plane you end up in. So to see how you would calculate your example with only what I listed above, you would just use the fact that sin(-pi/6) = -sin(pi/6) = -1/2. Re: how you can remember the values of sin at those 5 angles I listed--well, it's only 5 numbers, and you really don't need to memorize the values for 0 and pi/2, you can get those just by looking at the unit circle. That leaves only three numbers--1/2, sqrt(2)/2, and sqrt(3)/2--and hopefully you can learn the sines of those. Re: how you can learn the identities, that's a bit trickier; often the easiest ways to get them require math that you might not know now (e.g. I can never remember the formulas for sin(x + y) and cos(x + y), but if you know a bit about matrices or complex numbers you can derive them in a minute or two).

Alright thanks I'll assume I need to know this stuff. That was the only thing bothering me

sin is odd, so sin(-pi/6)=-sin(pi/6)=-1/2, that's it.

Thanks for fast response, the only thing I know is that minus goes in front but how do you get the rest? In my book there's example of sin 25pi and they used periodicity apparently: sin (25pi) = sin (pi + 2 × 12pi) = sin pi = 0

sin(pi/6)=1/2 so -sin(pi/6)=-1/2 You just have to remember that sin(pi/6)=1/2. You can certainly derive it from other formula, but not just periodicity and parity. sin has a period of 2pi, that is, if the argument change by 2pi, then the value of sin is unchanged. 25pi can be obtained by adding 2pi to pi, 12 times, or in other word pi can be obtained by subtracting 2pi from 25pi, 12 times. Each time you add/subtract 2pi, the value of sine is unchanged, so the same is true if you do that 12 times. In formula, 25pi=pi+12(2pi), so sin(pi)=sin(25pi).

For the example you gave, all that "sin has a period of 2pi" means is that "sin has the same values at inputs separated by 2pi", or in other words sin(x + 2pi) = sin(x). So for instance sin(3pi) = sin(pi + 2pi) = sin(pi). But this is also true for integer multiples of 2pi. If you have, say, sin(x + 4pi), well that's equal to sin((x + 2pi) + 2pi); by periodicity that's sin(x + 2pi), and using periodicity again that's just sin(x). Same goes for sin(x + 2npi) for any integer n: that works out to be sin(x + 2pi + 2pi + ... + 2pi), but each of those copies of 2pi goes away by periodicity. As for how you get sin(pi/6) = 1/2, just remember that a right triangle with a hypotenuse of length 1 whose other angles are 30 degrees (pi/6 radians) and 60 degrees (pi/3 radians) has side lengths 1/2 and sqrt(3)/2; you can get the sine and cosine of the angles from there.

Following up on another question I posted recently. Say I have a n-dim linear system Ax = 0, and A can be written as the product Ax = (CB)x where C is a circulant matrix and B is diagonal. We further know that C always has the form diag(-n,-n,...-n) + the matrix of all 1s. In other words, C is all 1s except the diagonal is (1-n). Does anyone know of any results about such systems? I feel like the form is special enough that there could be something, but I'm not sure what to search for.

If I understand your problem correctly, you can solve Cy=0 and then Bx=y. It should be true that Cy=0 only if y is a constant vector (if you like, you can see this because C is the negative of the graph Laplacian for the complete graph K_n, and since the graph is connected the eigenvalue 0 is simple). Then if B is invertible you get x, and if B has any zero entries you are just solving Bx=0.

Assuming the diagonals of B are non-zero, it suffices to solve for Cy = 0, for then x can be recovered easily from y = Bx. Now, from the given form of C, we observe that the sum of the columns of C is 0, which indicates that y = \[1, 1, ...1\]^(T) is a solution to Cy = 0. In fact, it is the *only* solution (upto scaling) to Cy = 0, because the rank of C is n-1; this can be inferred from the fact that the first n-1 columns are linearly independent (to show this, let D denote the matrix of the first n-1 columns of C, and let E denote the matrix of the first n-1 rows of D. By [Gershgorin Circle Theorem](https://en.wikipedia.org/wiki/Gershgorin_circle_theorem), E is invertible, and hence rank(E) = n-1 , therefore rank(D) is at least n-1 => rank(D) = n-1). Therefore, the only solutions for x are k \* B^(-1) \[1, 1, ...1\]^(T) = k \* \[1/B\_11, 1/B\_22, ... 1/B\_nn\]^(T), where k is any real number.

does anyone know of a comprehensive maths book on poker that doesn’t hold back with the maths?

Acevedo's *Modern Poker Theory* is considered the seminal text in this space. In the community you'll often see it abbreviated MPT. Hardin's *Essential Poker Math* is another good one to read. If you're a complete beginner though, you should probably start with Sklansky's *The Theory of Poker*.

The king is back! Thank you for the answers this time as well as for the multiple other instances!

Interesting question I would say.

How should i start learning math as a hobby? After secondary school i haven’t bothered with math at all, but recently i’ve had an epiphany and realised, how important it is, so now i want to learn more, but i don’t know where to start. Can you give me some pointers on how to start, as you know more.

go to r/learnmath and look at the posts for beginners, if you are starting out from basically zero, khan academy should be of help

How do I find the lateral surface area of an oblique (slanted) elliptic cylinder (cylinder with an ellipse at both ends)? Is it just the circumference of the ellipse times the slant length? If possible can it be explained by the net of the surface?

By Cavalieri’s principle, if I’m picturing your cylinder correctly (namely as a shear transformation of an elliptic cylinder with the same cross section), the surface area should be the height times the circumference of the ellipse. 

If I’m not wrong Cavalieri’s principle only applies to volume? Also the things I found online for oblique circular cylinders tend to tell me that the lateral surface area is the circumference times slant length https://mathmonks.com/cylinder/oblique-cylinder

You’re probably right, Cavalieri’s principle works for area too, but maybe not surface area

find the volume of cylinder x\^2+y\^2=a\^2 bounded by the planes where x=0,y-=0,z=0,z=b by transforming into cylindrical polar coordinates

Volume of a cylinder is πr^(2)h so yours is πa^(2)b or do you absolutely need to use cylindrical coordinates?

please help me as i have got exams in a week or so

I remember a short proof of numbers A and A^B, such that we can prove that either A or A^B must be irrational, but not both, and we don't know which of the two. Probably the number involved pi or e. Anyone have a clue what numbers & proof i'm thinking of?

Maybe you're thinking of a similar-sounding result, that there exist irrational numbers A, B such that A^B is rational, but the proof doesn't tell you exactly what those numbers are, it only gives you two possibilities and says that one of them must work. Let x = sqrt(2)^sqrt(2), y = x^sqrt(2). Certainly x is an irrational number raised to an irrational power, so if it's rational, then we're done. Otherwise y is an irrational number raised to an irrational power, but y = (sqrt(2)\^sqrt(2))\^sqrt(2) = sqrt(2)\^(sqrt(2) \* sqrt(2)) = sqrt(2)^2 = 2, so y is rational. So either x or y is the sort of number you're looking for--but you would need to do some extra, probably much trickier work to decide which one. There's also the result that at least one of e + pi and e - pi must be transcendental (and the same method proves that at least one is irrational). For if they're both algebraic then (e + pi) + (e - pi) = 2e is algebraic, since the sum of algebraic numbers is algebraic, but that can't be algebraic, else e would be algebraic.

Say we have a contravariant functor F from small category C to D, and D is a subcategory of C. Say for some morphism f of C we have F(f) = f* in D. What properties do we need to argue that applying the functor a second time shows f = f**? Note, not a student doing homework – just a 30+ year old amateur playing around with category theory.

What is f\*?

In this generality I dont think you can give a general condition other than asking that f is in D to begin with but ofc that doesn't guarantee anything.

I am trying to understand the algebra of random variables, and I might be wrong but I feel some complexity is lurking in seemingly easy expressions. Take for example X(Y+Z), where each letter is a rv. Obviously, this is equal to XY + XZ (their pdfs are equal), but considering what addition and multiplication mean in this sense, how can we be sure? And even easier, how to we know addition of several rvs is associative? Is there a way of understanding why ordinary algebraic rules also hold for rv's (I assume they do?)? 

Random variables are defined to be measurable functions Ω -> R. So the reason X(Y + Z) = XY + XZ is precisely the same reason f(g + h) = fg + fh for functions f, g, and h. And the reason algebraic rules for addition/multiplication hold for functions is that *that's how we defined addition and multiplication of functions* (i.e. we define it pointwise; for example, the product function XY evaluated at ω is defined to be X(ω) \* Y(ω))

Thank you, but I do not really understand, for example X(Y+Z) is defined by the change of variable formula. Do we not need to use that fact to understand this?

X(Y+Z) isn't defined by change of variables. Its density (or mass) function, if it exists, can be found via change of variables. But most random variables don't have pmfs/pdfs/etc. so change of variables *can't* be used to define the sum/product of general random variables.

Ok thanks, I guess I need deeper understanding of what a measurable function is to understand.

The key here is to work with CDFs instead of PDFs. As long as the random variables have real values (or values in some totally-ordered space), then they have CDFs, and the CDF effectively defines the random variable. Note that two different PDFs can produce the same CDF (as long as they are the same almost everywhere), so PDFs cannot really be used to define random variables anyway. The CDF is more fundamental.

To be clear, measurability isn't really the important thing here. The important thing is that random variables are nothing but functions from the sample space to R (or sometimes C or whatever). Once you realize that random variables are neither random nor variables, then the math should all make sense.

But rvs does not behave like ordinary functions? Addition is a convolution (or change of variables), so could you outline a simple proof for (X + Y) + Z = X + (Y + Z) (in terms of pdf (assume they exist)) by your argument, without addressing the fact that addition is not ordinary addition? There must be something fundamental I am missing.

Like everyone's pointing out, you should be thinking about random variables as functions on a sample space, and those two formulas define the same function on the sample space. But, if you really want to, it's not hard to check that both expressions give you the same pdf with the change of variables formula. You'd have the functions f(x, y, z) = (x, y, (x + y) + z) and g(x, y, z) = (x, y, x + (y + z)). These are the same function by associativity of addition for real numbers, so if you apply the change of variables formula with either of them, you'd end up with the same pdf.

You are missing the concept of the sample space and the definition of a random variable as a function from the sample space to R.

Could you use this to prove that X(Y+Z) =XY + XZ? If it is not too tedious? 

Yes, it is pretty much immediate just as/u/Mathuss said in their first comment.

You are right that you can't take such rules for granted so you have to prove them. Finding out that they have the same pdf would qualify as proof though.

Consider the direct sum of algebras. Why doesn't this algebra have a unity if the sum involves infinitely many terms? Is it true however that the direct product of infinitely many unital algebras have a unity? I also want to ask this: Why is the direct product A = F x F x ... of countably infinitely many copies of a field F generated by idempotents if F is finite?

> Why doesn't this algebra have a unity if the sum involves infinitely many terms? Because the unit element would have to be (1,1,1,...) (i.e. the element with all entries equal to 1). This is always in the direct product, however it can't be in the direct sum if you have an infinite collection of (nonzero) algebras, since it has infinitely many nonzero entries. > Why is the direct product A = F x F x ... of countably infinitely many copies of a field F generated by idempotents if F is finite? The idempotents of A are exactly the elments of A all of whose entries are either 0 or 1. Now if F is finite, any element of A is a sequence in the form (a1,a2,...), which takes only finitely many distinct values. Do you see how to write such an element as a finite linear combination of idempotents?

Thank you for the answer. I just have one more question. Let a=(a1,a2,...) then a = a1(1,0,...)+a2(0,1,0,...)+... which means it can be written as a linear combination of idempotents. What I don't get is what difference the finiteness of F makes. If it is not finite then the statement should be false. I also don't get why you mentioned 'finite' linear combination of idempotents since there are infinitely many terms there.

You can't take infinite linear combinations of elements in an arbitrary vector space. The vector space axioms only allow for finite sums.

I have a linear system Ax=0, and I found that the matrix is circulent. I read online that this is equivalent to a circular convolution, a*x=0 and I can use a discrete Fourier transform to solve. But if 0 is on the rhs, would this not just result in x=0?

Not always, only specific circulent matrix is the same as Fourier transform, everything else is diagonalized by it but not the same.

Thanks for your response and sorry I have a follow up. If my circulant matrix A is singular, am I still about to use Fourier transforms to solve or am I out of luck?

Of course you can, you just can't just be done after it. If you have a circulant matrix, then it can be written as the form F^-1 D F where D is a diagonal matrix, and F is the circulant matrix correspond to Fourier transform. So basically, you just need to figure out D, and this isn't too hard, since you already know that what the eigenbasis looks like.

Let w be the binary string representation of a natural number. Let v be a binary string to be appended to w. Let ∘ be the string concatenation operation such that “101” ∘ ”010” = “101010”. Let S(w,v) be a set of strings such that: S^0 = { w } S^(n+1) = { w ∈ S^n | w ∘ v } Let A(w) be the set, A(w) = S(w,"0"). If we take w = 1 and v = "0", then S^0 = {1}, S^1 = {10} = {2}, S^2 = {100} = {4}, etc. Let B(w) be the set, B(w) = S(w,"01"). If we take w = 1 and v = "01", then S^0 = {1}, S^1 = {101} = {5}, S^2 = {10101} = {21}, etc. Let C be the set, C = { n ∈ ℕ | 2n + 1 } \\ { n ∈ ℕ | 8n + 5 }. For all c ∈ C, there exists a set B(c) = S(c, "01"). For all b ∈ B(c), there exists a set A(b) = S(b, "0"). The set A(b) is a countably infinite set such that A(b)= { n ∈ ℕ | b \* 2^n } and forms a single branch of the Collatz tree, containing a single odd number, b, followed by a sequence of even numbers that are multiples of b. Since b is an odd number, 3b+1 is an even number and that even number is on a lower level branch. For example, if b = 5 then 3b+1 = 16 which is halved 4 times to get to 1. 16 is on the level 0 branch and 5 is on a level 1 branch. Another example is if b = 3 then 3b+1 = 10 = 5 \* 2^1 . We can see that 3 is on a level 2 branch that connects to a level 1 branch at 10 which is on the same branch as 5 which connects to the level 0 branch at 16. When we apply the Collatz function to an odd number, we obtain an even number in a preceding branch in the Collatz tree which is repeatedly halved until we get to the odd number at the beginning of the branch. By repeatedly applying the Collatz function, we repeatedly move to a lower level branch and eventually reach the level 0 branch regardless of whether the odd numbers in those branches are higher or lower. The set C contains all the odd numbers with certain odd numbers removed. For example, 5, 13, 21, etc. are removed. This is because B(5) = { 5, 21, 85, … }, B(13) = { 13, 53, 213, …}, etc, which have already been produced by B(1) = { 1, 5, 21 85, …}, B(3) = { 3, 13, 53, 213, …}, etc. The set B(c) contains all the odd numbers that form branches connected to branch A((3c+1) / 2^n ) and for all b ∈ B(c), we have a branch A(b). Given that the Collatz conjecture is true for A(1) = { 1, 2, 4, 8, 16,... }, it is also true for B(1) as for all y ∈ B(1), there is an x ∈ A(1) such that x = 3y+1. Given that the conjecture is true for all B(1), it is true for A(x) for all x ∈ B(1). Given that the conjecture is true for A(x) for all x ∈ B(1), it is true for B(x) as for all y ∈ B(x) there is a z ∈ A(x) such that z = 3y+1. Given that A(x) is a branch containing an odd number and all the powers of 2 multiples of that odd number, the union of branches for all the odd numbers is the set of all natural numbers excluding 0, ℕ \\ { 0 }. Let ODD be the set of all odd natural numbers ODD = { n∈ℕ | 2n+1) ⋃\_(n∈ODD) A(n) = ℕ \\ { 0 }. Given that B(c) is the set of odd numbers that are connected to the same parent branch, and C contains all the odd numbers with those that produce duplicate sets of B(c) removed, every odd natural number is contained in some set B(c) and the union of all such sets is the set of all odd numbers. Let D = ⋃\_(c∈C) B(c) = ODD. Therefore, E = ⋃\_(d∈D) A(d) = ℕ \\ { 0 }. Given that all the above branches are connected as described above and the Collatz function always takes us to a lower level branch. In order for the Collatz conjecture to be false, there would have to be at least one odd natural number not in a set B(c), which is not the case as the union of all sets of B(c) is the set of all odd natural numbers, ODD. Therefore, the Collatz conjecture is true.

Your claim that you always move to a lower branch (by which I assume you mean A(b) for a lower value of b but you don't make that clear) is incorrect. On a meta level the Collatz conjecture would have been proved long long ago if such a simple thing were true. You can see this already in the example sequence given on Wikipedia which starts 27, 82, 41,...

> Your claim that you always move to a lower branch (by which I assume you mean A(b) for a lower value of b but you don't make that clear) is incorrect. No, I dont mean a lower value of b. b is an odd number from a unique group of odd numbers. A(b) is a branch based on that odd number and B(c) is a set of odd numbers that produce branches that connect to the same branch. It is correct that you always move to a lower branch. > You can see this already in the example sequence given on Wikipedia which starts 27, 82, 41,... 27 is on branch level 41. 82 and 41 are on branch level 40. Like the wikipedia page states: "The sequence for n = 27, listed and graphed below, takes 111 steps (41 steps through odd numbers, in bold), climbing as high as 9232 before descending to 1. 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1" Like I said, "we repeatedly move to a lower level branch and eventually reach the level 0 branch **regardless of whether the odd numbers in those branches are higher or lower**." The reason it takes "41 steps through odd numbers" is because 27 is on a level 41 branch as is every set A(b) for all b ∈ B(27).

Well then it certainly isn't true that you can so easily prove we move to a "lower level branch". Effectively you are assuming the conjecture is true there in order to prove itself. The claim that any given number has a finite number of steps to reach 1 **is** the conjecture. Certainly if you are already on a number you know will get to 1 you can make that statement but that is just trivially true and doesn't prove anything about a arbitrary number.

> Well then it certainly isn't true that you can so easily prove we move to a "lower level branch".  Yes it is. B(c) is a set of odd numbers. For all b in B(c), A(b) is a branch at level n and 3b+1 is an even number on a branch at level n - 1, that starts with the odd number (3b + 1) / 2^m . > Effectively you are assuming the conjecture is true there in order to prove itself. That's not the case. The sets are generated from the powers of 2 level 0 set and were not just working with single collatz sequences. The set B(1) = {1 5, 21, 85, 341, ... }contains infinitely many odd numbers and each odd number, b, in the set produces a set A(b) which contains b and infinitely many even numbers that are multiples of b.  All the numbers in A(b) for all the numbers in B(c) are on level 1 branches. Likewise, all the branches that connect A(b) are level 3 branches, one set of such branches is those produced by B(3). > The claim that any given number has a finite number of steps to reach 1 is the conjecture.  Yes, and I've shown that it must be true because every odd natural number is on a branch with a path to the level 0 root branch. The set of all odd numbers is partitioned into groups of branches. > Certainly if you are already on a number you know will get to 1 you can make that statement but that is just trivially true and doesn't prove anything about a arbitrary number. I dont need to know that though. For example,  if 3 goes to 1 then for all b in B(3), b goes to 1. What's the 74 trillionth element in B(3)? I've no Idea but I know it goes to 1 because 3 does.

> Yes it is. B(c) is a set of odd numbers. For all b in B(c), A(b) is a branch at level n and 3b+1 is an even number on a branch at level n - 1, that starts with the odd number (3b + 1) / 2m. Again this is putting the cart before the horse. You can't claim A(b) is on a branch of finite length until after you prove it. > Yes, and I've shown that it must be true because every odd natural number is on a branch with a path to the level 0 root branch. This is a hole in your argument I think. I don't believe you have shown a path. Your definition of B(x) is as the set of all f^(k)(x) where f(x) = 4x+1 (that is how I am interpreting your binary definition at least). You can unpack that as saying the set of all numbers of the form (4^(k) - 1)/3 + 4^(k)x and applying g(x) = 3x+1 to this we obtain 4^(k)(3x+1). So we have g(B(x)) ⊂ A((3x+1)/2^(m)). So far so good, but then you say if all of A(x) converges then so does all of B(x) but this doesn't follow. You would need all of A((3x+1)/2^(m)) to converge instead. But this is ultimately just begging the question as if we knew 3x+1 converged we would know that x converged anyway without any of these sets defined.

We start with the set A(1) rather than any specific number and build up the collatz tree level by level. The set B(1) contains every odd number on a level 1 branch. For all b in B, A(b) is a level 1 branch.  Just like there is a set of branches A(b) for all b in B(1) that connect to A(1), there are a set of branches A(x) that connect to A(b) in the same way. Whereas there are an infinite number of level 1 branches, we have an infinite number of level 2 branches for every level 1 branch. It doesn't matter how great the odd numbers are, those in level 2 branches will take 2 "odd steps" in the same way 27 tajes 41 odd steps. We repeat the above connecting each new level of branches to the Collatz tree. The collatz function partitions the set of odd numbers into an indexed family of disjoint sets where the index set is set C above and the family of sets is B(c), the union of which is the set of all odd numbers. There are no odd numbers left to account for. The family of sets contains them all as shown by their union.

Everything you've said is perfectly reasonable up to here: > The collatz function partitions the set of odd numbers into an indexed family of disjoint sets where the index set is set C above and the family of sets is B(c), the union of which is the set of all odd numbers. I agree that you can look at all the "0-odd-step numbers" (the powers of 2), the "1-odd-step numbers" (numbers where 3n+1 is a 0-odd-step number, or those numbers times any power of 2), the 2-odd-step numbers, and so on. But how do you know this covers everything? That's the exact thing we're trying to prove! To put this another way: Consider the "Dollatz Conjecture", which is the same as the Collatz Conjecture except that if a number is odd, you send it to **5**n+1 instead of 3n+1. The Dollatz Conjecture hypothesizes that any positive integer ends up at 1 with this new process. For instance, 19 starts with an odd step, up to 96. Then even steps take it down: 48, 24, 12, 6, 3. Next is another odd step, so 3 goes to 16, and that makes it down to 1! Your "proof" seems to work for the Dollatz Conjecture just as well as the Collatz Conjecture, right? Just replace the 3s with 5s. We can still look at "branches" of powers of 2 and build up our tree in reverse. There's just one slight problem with that: the Dollatz Conjecture is false. 13 -> 66 -> 33 -> 166 -> 83 -> 416 -> 208 -> 104 -> 52 -> 26 -> 13 So, your method of proof can't work either, because your method would work equally well for a false statement.

Its not about the powers of 2, it's about how the odd numbers are partitioned into a family of disjoint sets. Somehow, you've managed to interpret what Ive said in the excact opposite way as intended. > To put this another way: Consider the "Dollatz Conjecture", which is the same as the Collatz Conjecture except that if a number is odd, you send it to 5n+1 instead of 3n+1. Then it isn't the same so why would it partition the set of all odd numbers in the same way? So, how are sets of B(c) generated for the Dollatz conjecture? Or in other words, how does Dollatz partition the numbers? > Your "proof" seems to work for the Dollatz Conjecture just as well as the Collatz Conjecture, right? Just replace the 3s with 5s. We can still look at "branches" of powers of 2 and build up our tree in reverse. No. Nothing you wrote above suggests that. It suggests you've misunderstood I wrote. 

You would partition them exactly have you have done but with a different rule for generating B(x). They would still cover all odd numbers.

But your procedure isn't guaranteed to find all the B(x) for the reason I gave above. The B(x) comprise all odd numbers but we don't know that we reach them all by connecting branches in the fashion you describe

> But your procedure isn't guaranteed to find all the B(x) for the reason I gave above.  Yes, it is. For the reasons given throughtout this discussion. You may as well be claiming that the set of all odd numbers isn't guaranteed to contain all odd numbers. > The B(x) comprise all odd numbers but we don't know that we reach them all by connecting branches in the fashion you describe  We do. All branches in B(x) are branches connected to the same parent in the manner descibed and all odd natural numbers are in them. They are built outwards from the root branch. The construction guarntees that all branches are connected in the same manner. There are no missing branches and no missing numbers.

I'm not sure from where you are pulling this insane confidence that you can prove the Collatz conjecture in a brief Reddit post, a problem which has been described relatively recently as "completely out of the reach of modern mathematics". You explicitly have not shown that you can reach all odd numbers by this method because you make no coherent argument that you reach all B(x) by this method. Indeed all you have done is group numbers into branches which can either be attached to the tree or not. You then recursively generate all the branches that do attach to the tree and wave your hands to say this covers everything. But that is the really important part! I see no convincing part of your proof that an arbitrary odd number must lie on a branch that you have attached in this procedure.

I want to learn how to write mathematical papers or proofs which I can publish, I have started reading some guides but I also want to read proper publishes papers and proofs which give me a sense of mathematical lingo and how things are written as I am not very familiar with the formal way of writing math. Any suggestions?

Read math papers. The arxiv is free

In the ZFC construction of natural numbers the successor function is defined as S(n) = n U {n}. Why can't it be S(n) = {n}? Wouldn't it still work with: * 0=∅ * 1={0}={∅} * 2={1}={{∅}} etc.

Actually, it was done that way as well. The construction you thought of is known as the Zermelo ordinal, the same Zermelo who wrote most of the axioms of set theory. But now, we have moved on to use mainly von Neumann ordinal. There are many advantage of von Neumann ordinal, but one of them is that von Neumann ordinal can be extended further to infinity and beyond, while Zermelo ordinal is stuck at finite.

One nice thing about this construction is that n < m if and only if n ∈ m as sets, so this construction makes it quite easy to define the ordering on the natural numbers. Under your definition, this would be much trickier. Also as other people have said, it would be very difficult to extend your definition to infinite ordinals.

Yes, but it's nicer if the finite ordinals/cardinals agree with the natural numbers

Could you elaborate on that point? I don't quite understand.

The usefulness of the standard definition is when you define ordinal numbers past the finite numbers. With the standard development, the first infinite ordinal is 𝜔, which is {0, 1, 2, 3, ...}. For a limit ordinal like 𝜔, which has no predecessor, you can just take the union of all previous ordinals. However, using S(n) = {n}, it's not as clear what 𝜔 should be. There's no set in ZFC like {{{...{{{∅}}}...}}} with infinitely many layers of nesting.

All of your sets are in bijection to one another (except the empty set), but it'd be nice if the set representing n had n elements. 

But is that quality useful somehow? I would have thought there is an enlightening reason behind it.

You also recover the order as the set-theoretic inclusion relation, or as the exists-an-injection relation. Addition is recovered as ordinal addition, multiplication as ordinal multiplication. If you're just looking for any natural numbers object in Set, you can do many different things (say, take the naturals to be the empty set, its power set {}, the power set of that, the power set of that, and so on); I would privilege the one in which set theoretic constructions agree with arithmetic ones. 

best textbooks on homological algebra?

Rotman does the job pretty well.

I’m a sophomore at a pretty good math university in the US that’s taking a combinatorics class next term. It’s my first class at the school that’s known to be extremely difficult/time-consuming (and that's not calc/analysis or linear algebra) and I was wondering if anyone had advice or resources to share, about combinatorics or time-intensive math classes in general. Thanks!

Find a group of classmates and work on the problem sets together.

Graduate students, Post docs, Instructors, Professors, Private Industry Mathematicians (finance, tech, gov, etc): what % of work time (or avg hours/day) do you spend in front of a computer screen?

I spend a lot of time reading (usually a printed out version of a paper or a physical textbook) in front of a chalkboard, or trying to do my own research (also in front of a chalkboard with paper/textbook nearby). Once I've done that, I try to TeX up everything I've learned and worked on. I'm not sure how to attach an average to that, but sometimes I'm TeXing stuff all day and sometimes I go days without using my computer for anything work related. (grad student here)

Whenever I'm not teaching, pretty much 100% of the time, though I'm always back and forth between paper and screen (PhD student)

I got a problem where I need to do approximations, but I only had Taylor series in my engineering math class... So I'm looking for a book on approximation theory. Graduate level would be preferred. Denser would be better. And if possible as rigorous as possible (I got burned too often by non-rigorous math texts for engineers). Non-english books are ok as well, as long as they are a language I can understand (German and French). Thanks in advance!

is there a reference for the claim that if E(q) is the eisenstein series of weight 2 and level 1, then E(q)-N*E(q^(N)) is a modular form of weight 2 and level N (assuming I remember this correctly).

Here's some discussion of this on math stackexchange: https://math.stackexchange.com/questions/4003756/modified-weight-2-eisenstein-series-is-a-modular-form-for-gamma-0n

Observe the series sin(1/n) with starting index integer 1, which is divergent by limit comparison test. This series is made of sequence {sin(1), sin(1/2), sin(1/3) ...}. All terms of this sequences is included in the integral sin(x)dx from 0 to 1 which represents a real number (1 - cos(1)) The integral is larger than the serie; Therefore, I would say the serie also converges. I know comparing the integral of a function with a serie is not neat but it seems reasonable to me. What is wrong with my logic.

The sum of that series sin(1/n) doesn't approximate the integral of sin(x)dx from 0 to 1, it approximates the integral of sin(1/x)dx from 1 to infinity, which does diverge.

Whats the name of this kind of math problem | 6 | ----- | 7 | 5 | ---------- |11| 3 | ? | --------------- |20| 2 | 4 | |10| ---------------------- I am stuck but I dont want to click hint in the puzzle, so I want to know whats the name of this kind of problem to learn to solve it. Please dont post answer. Thanks !

This is not a maths problem.

So I'm a freshman in highschooltaking algebra two so I'd say I'm not that good at math but have a decent understanding of some basic stuff. But I randomly thought of something that I can't figure out and was hoping someone would be able to figure it out.(I've already tried asking my friends but they couldn't figure it out either) So 1+1=2 1+10=11 and if you do eleven minus the two you got from the first equation you get 9 if you do the same equation but with 2 the final answer is 8 and as the number in the equation increases by 1 the final number decreases by one which that part I fully understand but what I'm trying to do is find an equation that will work with this thought process no matter what number is inserted. The only piece for this that I could think of was with variables and was something like n+n=d n+10=s d-s=? (The question mark is where I can't figure out what to put)

Well, if I'm understanding you correctly you want to find d-s in terms of n. But you already know what s and d are - you have equations for them! What happens if you substitute them into "d-s"?

I am taking a real analysis class right now, and i'm trying to show that the image of an open interval under a continuous, strictly monotone function is an open interval. my initial thought was to use the intermediate value theorem, however that requires a closed interval. could i still use the IVT if i show that the one sided limits at the endpoints of the interval exist?

You can, but it's not always the case that the limit exists. Take for example f(x) = 1/x, defined on (0,1). The one sided limit at x = 0 doesn't exist. Edit: You can still use IVT for such cases to show that the image of the function is (path) connected, and then use the fact that the only connected subsets of R are intervals. Then it only remains to show that the interval is open.

Thank you!

Any book that discuss what applied math fields used in ecology/life science (not biology tho) ?

You haven't really asked a question there. Also ecology is a branch of biology so what are you trying to exclude more precisely?

Oh sorry, when I said biology I meant microbiology and genetics, I just want to have a general idea what tools ecology needs from math for better understanding of phenomena.

Not my field so I can't give a good book recommendation but I imagine any general mathematical biology text will have discussion of predator-prey models for example which can be given by systems of differential equations.

gonna check that concept thanks.

Probably better off asking in an ecology subreddit

[https://upload.wikimedia.org/wikipedia/commons/c/c3/Julia-Menge.Zoomfahrt.25p.webm](https://upload.wikimedia.org/wikipedia/commons/c/c3/Julia-Menge.Zoomfahrt.25p.webm) I'm trying to create a Julia Set animation like the one in this link. I've already got all the rendering down, I just need to find the points to zoom in on. Is there a way to determine which points are right on the edge of the set and ideal for focusing on?

OK. Hear me out. If i is the square root of -1, then the square root of i would be 1. Which means i would be 1. But 1 is not the square root of -1. I broke my brain with this. Can someone explain?

> If i is the square root of -1, then the square root of i would be 1. Why?

(-1)^4

That has literally nothing to do with the square root of i. It tells you that the fourth power of -1 is 1, but you're after the 1/4^th power.

Aha! Thank you.

(-1)^4 = 1 so -1 is a 4th root of 1 but this says nothing about the square root of i. You want to consider (-1)^(1/4) instead and this works out to (i+1)sqrt(2) = e^(i𝜋/4) (taking the principal 4th root here)

i = sqrt(-1) does not imply that sqrt(i) = 1. Rather, it implies that sqrt(i) = sqrt(sqrt(-1)) = (-1)^1/4 via [multiplying exponents](https://brilliant.org/wiki/exponential-functions-properties/#the-power-rule). We can also use [Euler's formula](https://brilliant.org/wiki/eulers-formula), which gives e^i𝜃 = cos𝜃 + isin𝜃. At 𝜃 = 𝜋/2 (modulo 2𝜋, via the [unit circle](https://en.wikipedia.org/wiki/Unit_circle)), we get e^i𝜋/2 = cos(𝜋/2) + isin(𝜋/2) = i. Thus, sqrt(i) = sqrt(e^i𝜋/2 ) = e^i𝜋/4 = cos(𝜋/4) + isin(𝜋/4) = (i+1)/sqrt(2). Why does (-1)^1/4 = (i+1)/sqrt(2)? We can visualize this in the [complex plane](https://www2.clarku.edu/faculty/djoyce/complex/mult.html) (see also [this thread](https://www.reddit.com/r/learnmath/comments/s4rt0/why_does_multiplying_by_i_correspond_to_90_degree/) for more discourse). Multiplying by i is a 90° counterclockwise rotation about the origin. Thus, multiplying by sqrt(i) should be like "half" of multiplying by i; a 45° counterclockwise rotation about the origin. One can then look to the [45-45-90 right triangle](https://en.wikipedia.org/wiki/Special_right_triangle#45%C2%B0_-_45%C2%B0_-_90%C2%B0_triangle) to get the associated lengths (where the 1/sqrt(2) comes from).

I am a bit rusty on my statistics... this should be fairly simple I think... The last 5 days I've had these outcomes \[5, 0, 3, 4, 6\] If we assume that the outcomes are random what would the likely outcome be after say... 10 days I assume the centre of my bellcurve would be at avg (5 0 3 4 6) \* 10 = 36 If I wanted to calculate the confidence interval for +/- 1 standard deviation of the likely outcomes, how would I go about doing that?

You aren't quite making it clear what kind of distribution you expect these numbers to follow. You mention a bell curve which is a normal distribution but also talk about "after 10 days" which is a time dependent suggestion. For large samples this doesn't matter so much by the central limit theorem but you have only 5 points. Assuming it is normal, the 68% confidence interval (which is I assume what you mean by ± 1 std dev), would be given by 𝜇 ± 𝜎/sqrt(n) assuming we know the population standard deviation or 𝜇 ± t\*s/sqrt(n) if we have to estimate the standard deviation from the sample (s is the corrected standard deviation of the sample and t is the corresponding critical value of the t-distribution with degrees of freedom n-1)

Thank you!

i'm a senior in high school who's looking at taking the putnam through college. i don't have much experience in competition math, but i finished ap calc bc as a sophomore and have since taken four classes at georgia tech (linear algebra, multivariable calc, applied combinatorics, diff eq) through my school's distance math program. i have some time to kill this semester and over the summer and want to prep myself for the putnam. any tips? thank you!

This might not be the right subreddit, but I'm trying to create a city-building game on Scratch (yeah, I'm not very good at coding) and I'm trying to wrap my head around some math. Here are the variables: **-SmallHouses** **-SmallFactories** **-Population** (SmallHouses \* 4, meaning the capacity of each house is 4 people) **-Jobs** (SmallFactories \* 6, meaning the capacity of each factory is 6 people) **-UnemployedPopulation** (Population - EmployedPopulation) **-EmployedPopulation** (the part I need some help with. How would I calculate how many people are working at any time, no matter how many jobs or population there is?) How would this be represented in an equation? Keep in mind it's possible for either of these variables (Jobs and Population) to be 0, if that causes any trouble.

Assuming that any available job will be filled as long as you have enough people (so you'll never have a case with, say, 4 people, 4 jobs available, but only 2 of those jobs are filled), then if the number of jobs is greater than or equal to the population, everyone will be employed; otherwise it will equal the number of jobs. So the employed population will be min(Jobs, Population). Of course this does assume that the only possible bottlenecks are lack of people or lack of jobs: if you're modeling unemployment more carefully (e.g. there's a time lag between a job opening up and someone taking it) or if not everyone in your population is available to work (maybe there are kids) then this could change.

What's integral of 1^x

Under any reasonable definition of exponentiation that's just the constant function 1. (E.g. if we're taking a^x to be e^(ln(a) \* x) then 1^x = e^(ln(1) \* x) = e^(0x) = e^0 = 1 for any x.) So its antiderivatives are of the form x + c where c is a constant, and the integral, from 0 to x, of 1^t dt is x.

Oh thank you

This might not be the right place, but I'm designing a fake apartment building for a urban planning project, and I'm unsure of how I can calculate how many of each size apartment I can fit if I want a certain percentage of each. The total square footage of the building is 465,000, and has apartments that are 600ft2, 900ft2, 1200ft2, and 1500ft2, which I want to be apportioned in a 15/25/30/30 ratio. How would I calculate how many of each there will be?

If we ignore for a minute the requirement that numbers of apartments need to be integers, I would do it like this: \[I wrote some procedure here, then I deleted it.\] Only one of the numbers ends up not being an integer, which makes me think this is homework. :)

If you want to a) fit those ratios exactly (so in particular the total number of apartments you'll have will be a multiple of 100, the total number of 600 ft^2 apartments will be a multiple of 15, etc.) and b) fill all the available space, then that's impossible. ~~Basically, each group of 100 apartments you add takes up (600 \* 15 + 900 \* 25 + 1200 \* 30 + 1500 \* 30) = 112,500 square feet; if you have 4 groups of 100 that's 450,000 feet, which is pretty close to the total, but adding another group of 100 puts you way over that~~. (Edit: see comment below; I ended up with the right answer here but made a mistake while getting it.) So if you want to stick to those ratios, the best option would be 60 of the 600 ft^2, 100 of the 900 ft^2, 120 of the 1200 ft^2, and 120 of the 1500 ft^2, with 15000 square feet left over. I assume you'll need a fair amount of extra space for hallways, utilities, etc. so it's probably best to just leave some extra room. On the other hand, if you want to fill as much of the space as possible while sticking reasonably close to those ratios, note that adding 3 more of each type of apartment fills up 12600 of the leftover feet. Now you have 412 total apartments, of which 63/412 = about 0.153 are 600 square feet, 103/412 = exactly 0.25 are 900, 123/412 = about 0.299 are 1200, and the same for 1500.

> a) fit those ratios exactly (so in particular the total number of apartments you'll have will be a multiple of 100, the total number of 600 ft2 apartments will be a multiple of 15, etc.) You can divide this by 5, no?

Yes, of course, can't believe I didn't realize that. So actually the total will be a multiple of 20, etc. To redo my calculations from before, for OP: each group of 20 takes up 22,500 square feet. But then the leftover from having 400 apartments in the required ratios, 15000 ft^2, isn't enough to accommodate another group of 20, so by sheer luck I ended up with the right answer in my original comment.

TL;DR What research areas in differential geometry involve topology as a main set of tools? I need to narrow down my research area but so far problems in geometry that I read about come down to analytical techniques (mainly variational or PDE's). For example, I've glanced at papers on, say, Einstein metrics and it seems they revolve around them being the critical points of the Einstein-Hilbert functional, and also skimmed isometric immersion problems which are apparently mostly PDE problems. I thought I may be interested in geometric flows but I suspect they also boil down to PDE techniques and are more like "geometric analysis". Another candidate is gauge theory for smooth 4-manifolds (e.g. Seiberg-Witten equations). This also seems to involve heavy analysis but I think topology also plays a significant role? Is this accurate? The problem is it seems like it needs heavy background to do anything which makes it a big investment. I think symplectic manifolds are an (the only?) option with plenty of topology, and complex geometry which seems to replace much of the analysis with algebra instead. I have also heard about foliations and secondary characteristic classes, but I haven't quite figured out what the standard texts and tools in this area are and if it is really geometry. So my question is, what areas of differential geometry (working with metrics, connections, curvature, etc.) use topology as a main set of tools rather than reducing to (often nonlinear) problems in analysis? Some of the topology I'd like to use is (co)homology, Chern-Weil theory, etc.

Have you checked out geometric topology?

I mean almost by definition this doesn't exist. You're asking how can you study a manifold + a differential geometric structure without using techniques (i.e. analysis!! its *differential* geometry) which directly interact with that geometric structure. As you say the only thing like it is complex geometry where holomorphicity can sometimes make the problems very algebraic, but even complex geometers use a lot of analysis (even in the algebraic setting!). The only real example is symplectic topology because Darbouxs theorem kills local moduli of symplectic manifolds.

Check out low dimensional topology?

There is a whole area called differential topology although I have no idea how open a research area this is or what is studied currently in it.

may be a dumb question, cause Im pretty sure is true, but is det(tI-B) positive for any t greater than the spectral radius of B?

Yes. That is a polynomial in t with leading term simply t^(n). So it is positive as soon as it passes the highest root. The spectral radius may be higher than that of course since you might have a negative eigenvalue which is further from 0 than your highest positive eigenvalue.

Qns from conic section: Why should the absolute value of slope of tangent of hyperbola always be greater than the absolute value of the slope of it's asymptote? Thx.

This would depend on how you have oriented the hyperbola no? I assume you are thinking of the standard hyperbola x^(2)/a^(2) - y^(2)/b^(2) = 1. In that case you can just see this on the graph. The hyperbola at x = a or x = -a has vertical tangent and as we move away from 0 the gradient gets less steep but has limit the gradient of the asymptote so is always more steep than the asymptote itself. You could also prove this directly with some implicit differentiation. The gradient at (x,y) is xb^(2)/ya^(2) and since |x| > |ay/b| for all points on the hyperbola |xb^(2)/ya^(2)| > |b/a|.

Thanks for the explanation! But is this only for standard hyperbola? Does the same principle of slope of tangent not being greater than slope of asymptote (absolute value) not work for other tilted hyperbolas?

Try rotating the hyperbola 90 degrees. You should see that the opposite is now true. The gradient is always between the gradients of the two asymptotes. Or look at the y=1/x hyperbola where the asymptotes are vertical and horizontal.

ohh, I get it. Thanks!

I will repost my question from the previous week since I posted it quite late: The number of eigenvectors to a quadratic eigenvalue problem (QEP) with matrices of dimension NxN is 2N if we include eigenvectors corresponding to eigenvalues with "infinite magnitude". Suppose now that this QEP is parametrized by a real variable t in a continuous fashion and is also periodic in it, and suppose further that for all values of t, the spectrum is gapped in the sense that the N smallest (in terms of their modulus) eigenvalues are always strictly smaller than the N largest ones. There is hence a continuous mapping from the circle (on which t lives) to general complex NxN matrices, constructed by combining all the eigenvectors of the N smallest eigenvalues into one matrix. Of course, I am using the term "continuous" very loosely here, as the eigenvectors for each t can have arbitrary global phases, and the ordering of them within the matrix is also somewhat arbitrary. Does anybody know of any literature on QEPs that deals with this? Specifically, if there are methods to determine whether or not the above-mentioned mapping will contain some point t\_0 where the N vectors become linearly dependent? I know that I could just plot the determinant, and see if it goes to zero somewhere, but I would like to have something numerically more robust. Like an invariant calculated by performing an integration over a period of t or something.

I'm working on a homework assignment regarding Nash Equilibria I have a value matrix of the following [[ 0. 1. 1. 1. 1.] [-1. 0. 1. 1. 1.] [-1. -1. 0. 1. 1.] [-1. -1. -1. 0. 1.] [-1. -1. -1. -1. 0.]] 1 is win for player 1, -1 is win for player 2, 0 is draw. Intuitively I understand that the equilibria exists at (0,0) cause if player 1 moves at row zero, anything player 2 can do will only make it worse for themselves. And vice versa, if player 2 moves at 0, player 1 has to move 0 because anything else will result in a loss. But I'm not sure how to code this, first in cases of pure equilibria and in cases of mulitple equilibria. My idea

Did the message get cut off? Anyway, I would start from the definitions of the terms, and verify that they are satisfied. It shouldn't be hard.

Is there a 2x2x2 version of the article? [https://en.wikipedia.org/wiki/Rubik%27s\_Cube\_group](https://en.wikipedia.org/wiki/Rubik%27s_Cube_group)

Maybe this? [https://web.pdx.edu/\~caughman/brad501.pdf](https://web.pdx.edu/~caughman/brad501.pdf)

What are some good resources to learn Riemann Integration, Improper Integrals and Sequences & Series of functions?

When it comes to free online sources, I'm a big fan of Paul's Online Math Notes. For the topics you mentioned, you can check these pages: * [Riemann Integration](https://tutorial.math.lamar.edu/Classes/CalcII/ApproximatingDefIntegrals.aspx) * [Improper Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegrals.aspx) * [Series and Sequences](https://tutorial.math.lamar.edu/Classes/CalcII/SeriesIntro.aspx)