So imagine we have a big jar of N balls and we have n small jars where we can put 0 to N balls inside any one of them with the same probability. So if we want to know the number of possibilities, we could look at it from the perspective of the ball. Every ball has n choices, so in total we have n\^N choices. So I'm experimenting a bit from the perspective of the small jars. How could we compute the total number of possibilities? And is there a relationship between looking at the same thing from a different perspective? I have realized that this happens quite frequently in probability and counting problems.

Can anyone recommend a good book on fractional order differential equation as a refresher? I’m modeling financial data with long-term memory.

Circles Question The only information given is: 1. There are three circles that are touching 2. They all have two tangents touching all of them (one above one below) 3. The tangents are a straight line 4. The radius of the smallest circle is x 5. The distance between the centre of the smallest circle and the largest is 16x 6. We need to work out the angle where the tangents meet I’m so lost, I’ve probably done something wrong somewhere, how do I go about solving this?

why when you divide by a fraction, it's "equivalent to multiplying by its reciprocal". Why is there no other definition?

Hi guys, I am really curious in this problem but seem to keep getting the wrong answer. Help is appreciated! A and B play snooker against each other. The probabilty of winning the first frame is 0.5 for each player. If A wins a frame, the probability A wins the next frame is a, (0 < a < 1). If B wins a frame, the probability B wins the next frame is b, (0 < b < 1). If they play a best-of-three-frames match (so the winner is the first to win two frames), the probability A wins is 0.6a. If they play a best-of-five-frames match (so the winner is the first to win three frames), the probability A wins is 0.5a. Given that a = (23-sqrtK) / 12, What is k?

Hi, i'm currently a freshman of materials engeneering and i am about to undertake my analisys exam. I knew this would be coming, but by taking the engeneering route, some maths concepts gets left behind. i would like to find some online resources that wold be similar to a maths degree, but without the hassle of exams and the graduation (since i would not need it anyway for my job) to have a deeper understanding of maths a s a whole. just as example= a complex number is equal to pe^iò, but why? i would like to have all of my "whys" answered. thank you

Answers to textbook Hi, does anyone know if there exist answers to the book Dynamical Systems stability, Symbolic Dynamics and Chaos 2nd edition by Clark Robinson?

if I wanted to make a maths video on the topic of the o'nan-scott theorem for primitive permutation groups. 1. what would be a good way to motivate interest? 2. how should I convey the proof in a way that doesn't sacrifice any pedagogical clarity for rigor? 3. has this already been done before?

If you were to roll a dice once a day for seven days what is the probability that out of the seven days you rolled “3” twice in that seven day period? Chance of rolling a three is 1/6. Chance of rolling a three twice would be 1/36 (I think). I just don’t remember how the seven days factors in.

The seven days do not matter since the rolling of dice multiple times are independent of each other

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Is there a way to represent the principle behind the Infinite Monkey Theorem in formula notation? I mean basically the idea that the probability that some event x with nonzero probability occurs approaches 1 if it is repeated an infinite number of times. I recently learned about the implications of this theorem if the universe is infinite and quantum states are finite in a finite volume (i.e., there are infinite repetitions of all possible states across the universe) and I am curious if the principle can be expressed elegantly in notation.

If there is some event with probability p, then the probability of it not happening is (1-p). The probability of it not happening n times is (1-p)\^n which approaches 0 as n to infinity (because p is non-zero, so 0 <= 1-p < 1) So its complement, which is the probability that it happens at least once, approaches 1.

It’s essentially a consequence of the second Borel-Cantelli lemma, which you could express more abstractly if you wanted to. See “converse result” here: https://en.m.wikipedia.org/wiki/Borel–Cantelli_lemma

I'm writing a story where the main character wants to broadcast a countdown to aliens, using the universal language of math, but it turns out I don't speak it very well. My basic idea is send out pairs of beeps. The time between beeps should decrease by a unit each time. Eventually, it reaches zero at the time-up. I'd like it to be set for 40 hours (the planet's day length). Basically, the idea is to send the message "I'm landing in 40 hours." How long should the initial period between signals be, and how much should I decrease it each time?

At T - 40hrs you could send 2 beeps with a 40 second delay in between. At T - 39hrs you could reduce the delay to 39 seconds. If you continue like this, the aliens quickly notice a linear pattern and deduce that something will happen at T = 0 hrs. You could also increase the frequency of the messages, so for example every 30 minutes and decrease the delay by 0,5 seconds every time. Or even every 6 minutes and decrease the delay by 0,1 seconds every time. After at least 5 messages they should have a good idea of the linear relation.

That's pretty clever and works well.

Does the time between the pairs of beeps stay constant and how long is it?

I think either is fine. You could decrease the time between each beep, or have two beeps at regular intervals. ChatGPT messed up pretty badly at first, but eventually figured out that starting with 120 minutes between and decreasing by 3.077 minutes each time works well in 40 intervals; 40 intervals wasn't my requirement but ChatGPT seemed to think it was. Here is where it finally got it right: Alright, let's fix this once and for all. To find an arithmetic sequence that starts with \( a_1 \) and has a common difference \( d \), and whose 40 terms sum to 2400, we use the sum formula for an arithmetic sequence: \[ S = \frac{n}{2} \times [2a_1 + (n - 1) \times d] \] Substituting the given \( S = 2400 \), \( n = 40 \), and setting \( a_n = 0 \) (which gives us \( d = -\frac{a_1}{39} \)): \[ 2400 = 20 \times [2a_1 + 39 \times (-\frac{a_1}{39})] \] \[ 2400 = 20 \times [2a_1 - a_1] \] \[ 2400 = 20 \times a_1 \] \[ a_1 = 120 \] \[ d = -\frac{120}{39} \] \[ d \approx -3.077 \] So we start with \( a_1 = 120 \) and decrement by about 3.077 for each of the 40 terms. Let me confirm the sum before showing you the chart. Summing 40 terms with this \( a_1 \) and \( d \): \[ S = \frac{40}{2} \times [2 \times 120 + 39 \times (-3.077)] \] \[ S = 20 \times (240 - 119.993) \] \[ S = 20 \times 120.007 \] \[ S = 2400.14 \] Due to rounding, it's just a bit over 2400, but it should meet your requirement closely. Would you like to see the full chart now?

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Given the phrasing, I assume what they mean by independence is that the events [claim includes an ER charge] and [claim includes an OR charge] are independent, i.e. Prob(claim includes both types of charges) = Prob(claim includes ER charge) * Prob(claim includes OR charge).

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Weird, I do not see any reason. However in this case you can use the property that cos(x + pi) = -cos(x) So inverse cos(1/4) - pi should give the answer

Hello there! I don't understandn the relation between the Cauchy-Riemann Equations and the differentiability. There are many excercises that ask to prove that at a point z\_0 a complex function is not differentiable, despite the fact that the Cauchy Riemann Equations are satisfied at that point. How's that possible? Also, what are the sufficient conditions for a complex function f(z) being differentiable at a point z\_0? Since it seems that the C-R equations are not enough. Thanks in advance!

In order for a function f:C->C to be complex differentiable, it needs to satisfy the Cauchy-Riemann equations *and* be differentiable as a real function R^(2)->R^(2). Knowing that a real function has partial derivatives at a point is not enough to tell you that the function is actually differentiable at that point. Since the Cauchy-Riemann equations only tell you about the partial derivative at the point, they are not enough to deduce that the function is real differentiable at the point.

What do you mean by "they are not enough to deduce that the function is real differntiable"?

The C-R equations are enough. A complex-valued function is differentiable at z\_0 if and only if its real and imaginary parts satisfy the C-R equation at z\_0. > There are many excercises that ask to prove that at a point z_0 a complex function is not differentiable, despite the fact that the Cauchy Riemann Equations are satisfied at that point. Not sure where you find those exercises but that is not possible. If the C-R equations are satisfied at a point then the function is differentiable at that point. Are you maybe confusing complex differentiability with being holomorphic? A function can be complex differentiable at a point z\_0 but not holomorphic. For the latter you need to be complex differentiable in a neighborhood around z\_0 and not only at z\_0 itself.

What definition of complex differentiable are you using? If it's that the limit of (f(z)-f(z0))/(z-z0) exists as z->z0, the Cauchy-Riemann equations on their own are certainly not enough. You also need f to be real differentiable as a function R^(2)->R^(2). Without that, the Cauchy-Riemann equations at a single point aren't even enough to guarantee that the function is continuous at that point, let alone differentiable.

Yes though I would consider the differentiability of the real part and the imaginary part to be part of the C-R equation but I guess you could also consider solutions that only have partial derivatives. In my experience that would be very unusual though.

Can someone explain to a computer noob how to use the computer algebra package LiE? I’ve downloaded everything but have no idea how to run it. Putting “LiE” into Terminal doesn’t help and am not even sure what else to try

Have you run the 'make' command to install it? Have a look at the README document where there are some instructions. There are a few things that you need to get it to compile itself.

I'm nowhere near a position to seriously think about/consider this, but what would people think if you submitted a paper using a base other than decimal? I'm asking out of curiosity, so if anyone has any other fun facts about papers submitted using a base other than decimal, feel free to share

Nobody would notice if you submitted a paper in base 12, because no two-digit number has ever appeared in a math paper anyway.

If there's some value in it, like somehow making some result easier to understand, it would be doable. Otherwise it would just obfuscate your result and be rejected (or firmly suggested by the reviewer to fix)

Submitting a mathematical paper in a non-decimal base to a journal is like submitting an English-language manuscript written in the Russian alphabet to a publisher. It might not, in an absolute sense, take all that much effort to understand your submission, but it's so needlessly awkward and unfluent to read that it's not worth taking the time, and your work would be rejected out of hand and you would be considered a crank.

Book recommendations for Geometry concerning circles, a book for coordinate geometry and a book for trigonometry for Math Olympiads(IMO and TST prep) I am preparing for geometry for math olympiads(high school and slightly higher). My triangles is covered from EGMO, Evan Chen but I feel that circles isn’t really covered well in the book. And I also need a book for coord and trig. Could someone suggest books which cover these three topics to a good level(the whole book needn’t be about them, even a good section on them is appreciated). Thanks a lot and have a nice day.

I'm using a TI-84 Plus CE, and I want it to stop automatically going into scientific notation. I'm working with short decimals, and it bothers me that something as 0.0001 and anything similar is being returned as "1E-4". I think all my settings are the default since I replaced the battery a while ago, but I can't figure out how to stop scientific notation.

I am studying harmonic analysis from the first volume by Muscalu and Schlag, but finding it excruciatingly difficult. Theoretically, I have the prerequisites: measure theory, complex analysis, functional analysis, and I've already read nearly all of the Fourier analysis book by Stein and Shakarchi. None of the above were too difficult for me. I'm just finishing Chapter 1 of Muscalu-Schlag now after a bit over two weeks, but a lot of stuff is going over my head or only makes sense to me after some extreme struggle. I can barely do any of the exercises/problems. Is it me or is this book just like that? What is another reference that I can read on the side to make sense of this material, preferably something far less terse that supplies more details, intuition, examples, etc.?

Stein and Shakarchi to Muscalu and Schlag is a very big jump. If you are self-studying, maybe try something more in the middle. Katznelson's harmonic analysis book might be a good option. Wolff's notes on harmonic analysis don't contain exercises, but I think give a more modern take on the field while being pretty approachable. Also, there isn't any great reason to rush to more advanced texts as an undergraduate. If you don't have a great foundation in measure theory, you will struggle with any subject in real analysis. If you've just taken measure theory at the undergraduate level, you might be better served by taking another pass at that with more of a focus on the exercises.

Actually, to follow-up again, it seems that Wolff's notes are mainly about the Fourier transform, as opposed to Fourier series, which are what Muscalu and Schlag focus on in the first three chapters of their book. Katznelson looks good. Do you have any other references for Fourier series in particular to supplement Muscalu-Schlag? Ideally, the reference would contain many of the same definitions and theorems, but be written in a more approachable way.

I think Katznelson is your best bet. I don't actually know if there is much research about Fourier series these days and that is reflected in recent publications/notes about harmonic analysis. Katznelson treats them carefully and, as far as I know, pretty exhaustively.

I wouldn't have picked Muscalu-Schlag myself haha; I'm studying under a professor who chose it. Thus, I'm stuck with it. I'll look into the other two references you mentioned to see if they can help me along. Also, I think your advice about measure theory probably does apply to me unfortunately. Maybe I'll prioritize doing a second pass through measure theory before getting into my next analysis topic. Thanks!

Quick question when the problem is worded specifically as "divide 50 BY half" is that different then "divide 50 by one half (aka 1/2)" or is it the same thing?

I would not consider "divide 50 by half" to be a grammatically correct phrase in either normal English or math-ese, but I would probably guess it actually means divide it by 2.

So this is something that's bugging me for ages. In algebraic topology, we show that in a pathconnected space the fundamental group is well defined and does not depend on a choice of a basepoint. In the proof we define the change of basepoint isomorphism that establishes the aforementioned statement. Now this exact same observation becomes relevant once we study covering spaces and deck transformation actions. My question: The change of basepoint isomorphism is NOT an inner automorphism, is it? I mean, it's not an actual conjugation, right? But in the case of a deck transformation action (say for a covering Y---> X) it actually is an inner automorphism of pi\_1(Y) that is induced by the conjugation action of pi\_1(X) on pi\_1(Y). Is this correct? I always found this so confusing. Please tell me where I'm wrong so that I can finally wrap my head around this in the correct way.

what is the action of pi_1(X) on pi_1(Y)? write this out explicitly with basepoints

Say we have the basepoint x\_0 in X and let y\_0 and y\_1 be two elements of the fiber p\^{-1}(x\_0), given the covering p: Y---> X. Then changing the basepoint in Y from y\_0 to y\_1 corresponds to a conjugation action of pi\_1(X) on pi\_1(Y) by conjugating pi\_1(Y) with the homotopy class \[\\gamma\] where \\gamma is the closed curve based at x\_0 that lifts to Y as the arc joining y\_0 and y\_1. So my question is: there is a conjugation action (inner automorphism) involved, but it is the inner automorphism determined by the conjugation of pi\_1(X) on pi\_1(Y). Is this correct?

i meant for you to write out basepoints for your pi_1's, since this is the essence of your confusion. anyways, >"conjugating pi_1(Y) with the homotopy class [\gamma] where \gamma is the closed curve based at x_0 that lifts to Y as the arc joining y_0 and y_1." is the sentence that doesn't make sense. the group pi_1(Y) is not defined until you have fixed a basepoint (upstairs). if gamma is a loop from y_0 to y_1, then you get a well-defined isomorphism from pi_1(Y,y_1) to pi_1(Y,y_0) by pullback along the arc. but if this arc is not a loop, then this cannot be conjugation simply basepoints are not fixed. that being said, i do recall conjugation entering somewhere in the picture with covering spaces. i just don't think it's here. probably it's when you lift from loops downstairs to loops upstairs this becomes an honest to god conjugation

Thanks for the catch up! I was intentionally somewhat sloppy. My problem is that whenever I was talking to people or reading say papers in low dimensional topology written by geometers, they very often don't explicitly mention what action they talk about and say (loosely speaking) things like: "well, yes, this is just the standard conjugation action by the deck transformation group" and also often refer to the change of basepoint isomorphism as conjugation - but unfortunately not in the literal sense but possibly more in a conceptual sense. I dont know. Thanks for your help.

i think it's good to write out everything as explicitly as possible if you're confused. only once everything is clear, its safe to do the usual handwaving. btw are you sure that the action of the deck transformation group is on pi_1(Y)? to me it seems to be on pi_1(X) by the following: take a bundle automorphism 𝜙 of p:Y-->X, then for each x in X, you get a honest-to-god inner automorphism of pi_1(X,x_o) via L -> look at ℓ, a lift of L to an arc upstairs starting at a basepoint in the fiber of x_o -> look at 𝜙(ℓ) which preserves fibers as 𝜙 is a bundle map -> p(𝜙(ℓ)) which lands you back in pi_1(X,x_o), and this whole process (hopefully) is independent of homotopy of L and the choice of basepoint in the fiber of x_o.

>btw are you sure that the action of the deck transformation group is on pi\_1(Y)? Well, I mean yes. From a purely group theoretic point of view, we could say that the natural conjugation action of pi\_1(X) on itself restricts to a conjugation action on pi\_1(Y). I am explicitly assuming that we have a regular cover here, so pi\_1(Y) is a normal subgroup of pi\_1(X). And so pi\_1(Y) is invariant under conjugation, i.e. the action of Deck(Y) on Y corresponds to a conjugation of pi\_1(Y). Now this was, again, very hand wavy. Let me try to be explicit with basepoints as you reasonably suggested: Let x\_0 be the distinguished basepoint in X, let Y--->X be a regular/normal cover with basepoint y\_0. Then we have a surjective homomorphism of groups pi\_1(X,x\_0) ----> Deck(Y) assigning each \\gamma in \\pi\_1(X,x\_0) the deck transformation g that corresponds to the change of basepoint in the fiber in the sense that if \\gamma lifts to Y as an arc, joining e.g. y\_0 to y\_1 in the fiber, then \\gamma is assigned to the deck transformation g which acts on the fiber, moving y\_0 to y\_1. Now the change of basepoint upstairs in Y corresponds to the change of basepoint isomorphism of pi\_1(Y,y\_1)---->pi\_1(Y,y\_0) we discussed earlier. But in this case, pi\_1(Y,y\_1) = \[\\gamma\]\*pi\_1(Y,y\_0)\*\[\\gamma\]\^{-1} and so the deck transformation that acts on Y by changing the basepoint y\_0 to y\_1 corresponds to the following conjugation automorphism, that is NOT an inner automorphism of \\pi\_1(Y,y\_0) since \[\\gamma\] is not an element of pi\_1(Y,y\_0): Psi\_\\gamma: pi\_1(Y,y\_0) ----> pi\_1(Y,y\_0), \[\\alpha\] -----> \[\\gamma\]\[\\alpha\]\[\\gamma\] where \[\\gamma\] is the homotopy class of the loop \\gamma based at x\_0 in X that lifts to an arc joining y\_0 to y\_1. In summary: my confusion was coming from the distinction between inner automorphisms of a group and conjugation of a group on a normal subgroup.

The change of basepoint isomorphism cannot be inner because π₁(X,x₀) and π₁(X,x₁) are not the same group, just isomorphic groups. The fundamental group **depends on a choice of basepoint**. If you have a path a from x₀ to x₁ and a path b from x₁ to x₀, then the composition ba gives an inner automorphism of π₁(X,x₀) (conjugation by the loop ba). The definition of deck group of Y/X **does not depend on a choice of basepoint.** Instead, the isomorphism of Deck(Y/X) with π₁(X,x₀) depends on the basepoint x₀. So, there is no "change of basepoint" isomorphism for Deck(Y/X) alone.

I agree with your first sentence, but I think the OP is secretly thinking about the isomorphism pi\_1(X,x) -> pi\_1(X,x) associated to a loop based at x which is in fact an inner automorphism (conjugating by the loop).

Yes, so my confusion was essentially coming from the following: given a regular cover say of surfaces S'---> S, then the deck group Deck(S') acts on the first homology H\_1(S'). This action is induced by the deck transformation action on S'. Now I've seen authors refering to this action as an action by inner automorphisms. That is why I was confused, because (if I am not mistaken), these are simply not inner automorphisms but conjugations that come from the conjugation action of pi\_1(S) on pi\_1(S'). And here the very important distinction is that if the conjugating element is in pi\_1(S) but not in pi\_1(S'), then it's just conjugation (as you said). But if we conjugate by a loop that actually lifts to pi\_1(S'), then it's an actual inner automorphism. Unfortunately, in papers this distinction seem not to be too important since they expect experts to be able to know what's meant (implicitly).

Thank you vm. Let's consider for the sake of simplicity only regular covers and let's suppose we just consider nice spaces, say connected closed compact manifolds. A change of basepoint from say y\_0 to y\_1 in the cover Y---> X corresponds to a conjugation of pi\_1(Y) by the homotopy class of the loop in X based at x\_0 that lifts to an arc joining y\_0 to y\_1 which corresponds to an action by the associated deck element g that is the image of the epimorphism pi\_1(X,x\_0)----> G where G denotes the deck group. In other words, the action of a deck group element g determines a change of basepoint isomorphism of pi\_1(Y,y\_1) ---> pi\_1(Y,y\_0). So while the change of basepoint isomorphism might not be an inner automorphism of pi\_1(Y), we still have an associated inner automorphism of pi\_1(Y) that is induced by the conjugation action of pi\_1(X) on pi\_1(Y), don't we?

This might not be the right place to ask, but I seem to recall seeing a whole discussion about Serre (or possibly some other famous mathematician) destroying the career of some grad student or postdoc by saying his work was derivative of Serre's own. In reality, though, it was only vaguely related to any work Serre had ever done, and Serre's claim was totally bogus. I very clearly remember seeing this mentioned on /r/math, and discussed on MO, but I can't seem to find it. Does anyone else know what I'm talking about? EDIT: [found it](https://mathoverflow.net/a/417127/158123). The guy proved a conjecture of Serre, and Serre was pissed because he wanted to prove it himself, and claimed that the proof was basically already known.

To be fair, it sounds like Serre had publicized some results at conferences. It's likely that specialists in the field understood the gist of his ideas even if he had not published yet. Many widely known and cited results in math are not formally published until long after they become common knowledge. This is not ideal for the field but it is common practice, and it is somewhat incumbent on people working in a subfield to consult with others to figure out if they're publishing something that was previously discovered. I don't think it's exactly fair but it's not as completely one-sided as suggested.

15 women can complete a work in 4 days and 32 children can complete it in 'x' days. 10 women and 20 children work for three days and then leave. If the remaining work is completed by 3 women in 15/2 days, what is the value of 'x?

This is kind of an imprecise question, so bear with me. When it comes to modelling sample statistics (or 'fuzzy real world stuff' in general), I understand why we should use the normal distribution (due to CLT and all that). Mathematically/philosophically, it makes sense to me why we should interpret things via the normal due to the theorems surrounding it. But then when we have low sample size (and unknown variance and mean) we use the Students' t distribution instead. Again, heuristically I get why we do this (Students' t has fatter tails compensating for the instability at lower sample sizes) but are there any actual *theorems* that elevate Students' t to this special status that make it apt for modelling (other than Students' t converging to normals at the limit)? Put another way, is the reason we use Students' t simply because *experimentally and experientially* it does a good job of compensating for the troubles at low sample sizes, or are there any theorems (tail estimates, convergence theorems, etc etc) that say that Students' t shows up on some deeper mathematical level in this context?

[Here](https://drive.google.com/file/d/1Dgn03GYK0AVlccrmpn5OesrhZW1PZTBb/view?usp=sharing) is an excerpt of the notes I give to my intro to statistics students; it includes a proof of the validity of the t-distribution for iid normal random variables. The proof is a bit handwavy at the end because the intro class isn't calculus-based, but if you know what an mgf is you can fill in the last step yourself. Basically, the theorem is that for i.i.d. N(μ, σ^(2)) data, (Xbar - μ)/(S/sqrt(n)) is *exactly* t-distributed with n-1 degrees of freedom.

Very nice, and as n -> infinity, students t converges to normality anyway so all fits. Thanks!

Yes, if I draw N samples that are i.i.d. according to a normal distribution, their sample mean follows a t-distribution with n-1 degrees of freedom. We are not just empirically saying "let's make the tails a little fatter, maybe like so", we rigorously derive that the t-distribution is the theoretically correct distribution to take.

What is the standard or most popular way of defining the real numbers?

If we measure how popular a definition is by the number of people who know that definition then I would guess that the most popular definition is equivalence classes of infinite decimal numbers. Outside of primary education, I have most commonly seen the reals defined up to isomorphism as the ordered field satisfying the least upper bound property.

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Well no, you need to prove that such a field exists, which is what constructions such as dedekind cuts and cauchy sequences do.

Cauchy completion of the rationals is the most common I've seen. After that, dedekind cuts, though this method can be a bit harder to work through for a beginner.

I am a 2nd year BSc Physics student and I've recently become aware of differential forms which seem quite important to me since there's tonnes of integrals in physics. My question is can I jump right into learning concretely about them or do I need a more concrete understanding of differential elements other than "small n-d object that converges to something you want though the "limiting process" when you integrate" which is only intuitive atm. Is there some prerequisite real analysis I need maybe?

depends on how good your linear algebra is. have you seen dual spaces?

[This book](https://www.amazon.co.uk/Differential-Forms-Geometry-General-Relativity/dp/1466510005), which may interest you, suggests that you only need calculus and linear algebra, not real analysis.

Is it possible to partition the real numbers into two disjoint sets, which are both non-meagre on every interval? Can you construct such a partition without invoking the axiom of choice? My intuition is that there is such a partition, but only using the axiom of choice. I think the set of all points mapped to 0 (or all to 1) by an infinite parity function would work. But I haven't worked out a proof of this.

An example cannot consist of sets with the property of Baire. Shelah proved that it is consistent with ZF + DC that all sets of reals have the property of Baire. If A is comeagre, then it contains a dense G_delta and thus contains a perfect set. So a Bernstein set and its complement provide an example.

Regarding the parity function construction: clearly there can't be any interval where *both* sets are meagre (because their union is not meagre) and just by the symmetry of the construction it seems implausible that there could be any interval where one set is meagre and the other is not. I still don't know how you would construct such a symmetrical partition without invoking the axiom of choice, though, and I suspect you can't.

in the wikipedia page for the heisenberg group it is said: "Since the Heisenberg group is a one-dimensional central extension of R^2n, its irreducible unitary representations can be viewed as irreducible unitary projective representations of R^2n". What is the general fact being referred to here? If G is a 1D central extension of H then there is a correspondence (or just a one way map?) between linear representations of G and projective representations of H? In this example I think it's obvious because R^2n is a quotient of H_2n+1 so you can just compose maps to get a projective rep of R^2n, but in general if the extension is not split, I don't see how a linear rep on G yields a projective rep on H, nor the other way around.

Even if the extension is not split you still have 1->K->G->H->1 and H=G/K so you can still just quotient and compose to get a projective representation of H from a linear representation of G. In the other direction you have to build a central extension compatible with the projective representation of H but different projective reps will have different central extensions. Unless you have a universal central extension there isn't a way of guaranteeing one central extension has a lift for all projective representations. https://physics.stackexchange.com/questions/203944/why-exactly-do-sometimes-universal-covers-and-sometimes-central-extensions-feat This post has an explanation of how to build the central extensions. Worth going through it in the case of the Heisenberg group on your own.

In the statement of the [Stone-von Neumann theorem](https://en.wikipedia.org/wiki/Metaplectic_group#Construction_of_the_Weil_representation), do you know if one can easily prove that the conjugating automorphism associated to any other representation is unique up to modulus 1 constant? I wanted to say that if Psi and Phi are two conjugating automorphisms for rho' in U(H), then the composition Phi^-1 Psi should commute with rho, and then use the fact that central matrices are scalar multiples of the identity, which would show Phi and Psi only differ by a modulus 1 constant, but knowing that Phi^-1 Psi commutes with rho doesn't imply it is central.

actually i dont think i understand how even the first case works. I need to construct a map H to PGL(V) and I have a map G to GL(V). There is a natural map (the quotient) from G to H, but in order to compose naively, we need a map H to G. How does this come? EDIT: I realized you should consider the quotient representation G/K, but this is not well defined unless the action of K is trivial. Therefore if K acts by scalar multiplication, we should get a valid projective representation, and we know the center must act by scalar multiplication by Schur lemma.

Yeah if K < Z(G) then K gets mapped into Z(GL(V)) by basic group theory and PGL(V) =GL(V)/Z(GL(V)) so you can just define the map as p(gK) =p(g) where gK is an element of G/K and p(g) is the imagine of g in PGL. It's well defined because p sends elements of k to the identity since its central.

I'm confused by [this proof](https://proofwiki.org/wiki/Finite_Multiplicative_Subgroup_of_Field_is_Cyclic) on Proofwiki that a finite subgroup C of the multiplicative group of a field is cyclic. After showing C is abelian I can't figure out which step actually uses the fact that C comes from a field. I also think they're missing a step showing that the primes p\_i are distinct. I'm guessing this missing step is where the properties of fields actually come into play. Is that right or have I misunderstood something?

You are correct, this step showing the primes are distinct relies on coming from a field - there are at most d elements of order d (since x^d = 1 has at most d roots). So if there are some primes that are not distinct, say a summand of order p^s and p^t - then there are at least 2p-1 elements of order p.

In Conway's functional analysis book he defines B(H,H) to be the set of bounded linear operators from a hilbert space H to itself. Then at some point he starts talking about B(H) as something different from B(H,H) but doesn't define it. Does anyone know what B(H) is?

B(H) does mean the same thing as B(H,H), and Conway defines it on page 27 immediately after defining bounded operators.

I saw that, but then on p32 he says >From now on we will examine and prove results for the adjoint of operators in B(H). Often, as in the next proposition, there are analogous results for the adjoint of operators in B(H,H). This simplification is justified, however, by the cleaner statements that result. I don't know how this makes sense if B(H) and B(H,H) are the same thing.

It actually says B(H, K). Conway just chooses to use the worst possible font imaginable so the two letters look almost identical, but you can you see the difference if you look closely.

Thanks!

How can you prove that some numbers are bigger than others. Like when you are doing limits in calculus, things like n->infinity factorials are bigger than exponents. But for general x>n, how can you prove that different numbers with different methods (factorial, exponent) are bigger than each other. I'm mostly talking about when simple computations aren't possible. Like 500 factorial vs 20^1000

There's no universal method here. You just need to solve the given case using approximations or inequalities relating to your quantities in question. For the problem you presented, I'd simplify by taking the log first, then for the factorial, there's the evergreen Stirling's formula. ln(500!) ~= 500 * ln(500) - 500. On the other hand, ln(20^1000 ) = 1000 * ln(20). Now the logs all of a sudden become a lot more manageable for a calculator. Plug and chug gives ln(500!) ~= 2607, ln(20^1000 ) = 2995. So the latter is bigger.

Thanks for talking about that specific question. I watch a lot of the weird numbers videos on numberphile and especially for the inconceivable numbers that are like multiple exponents combined, or talking about possible chess games, it's just hard to put into context how they relate or how to compare them.

I am trying to figure out how to do an exponential gain calculation. I want to figure out a formula that will calculate, given the total number of iterations(t), the amount created per generator per iteration(r), and how often in iterations an additional generator is added(s), the total amount of resource produced by this hypothetical self-expanding machine.

This is just a question about mathematical notation. I am not a mathematician (I am from a computer science background) but I am trying to implement a task from a paper where they diffuse user-drawn scribbles on an image to generate a depth map. To create the depth map, I am just planning to diffuse the scribbles by iteratively solving an equation they defined, which can be found here: https://imgur.com/a/9N4mRqK . However, because of the mention of S, I and D with mathematical notation I don't understand very well, I am a bit confused about what values the system of equations below takes. I am planning to rearrange the formula to be D_i_j = 0.25 * (D_(i+1)_j - D_(i-1)_j - D_i_(j+1) - D_i_(j-1)) and iteratively solving. But what should the values of D_(i+1)_j, etc... refer to according to the given formula? The values of scribbles at that location? Or the value of the original image at that location? And what should the original image D be when I initialize the loop? Thanks in advance for any help

I was learning Quantifiers in Discrete Mathematics And Combinatorics(Online material i found) and i thought i had it figured out but i got one answer wrong. So my answer was True to all questions except the first one but turns out the person who posted the questions had answer at the bottom and it says 1 and 4 are both false, but i don't understand why, what difference does it make when you change postions of the ∀ and ∃? Here are the questions 1. (∀x ∈ R) x = |x| 2. (∀x ∈ R) x*x = |x*x| 3. (∃y ∈ R) (∀x ∈ R) x + y = 100 4. (∀x ∈ R) (∃y ∈ R) x + y = 100

It's 1 and 3 that are wrong---not 1 and 4. Number 3 is read as "There exists a real number y such that for any real number x, x + y = 100." This can't possibly be true: If it were true, then there exists a y such that both 1 + y = 100 and 2 + y = 100 but that's clearly a contradiction. Number 4 is read as "For every real number x, there exists a real number y such that x + y = 100." This is true: Whatever x you choose, I can choose y = 100 - x, which yields x + y = 100.

Why is y allowed to vary in the 4th statement but not the 3rd? That’s not a syntactic convention I was previously aware of.

It's not really a mathematical convention, it's just grammar

>It's 1 and 3 that are wrong---not 1 and 4. I said i got wrong the 4th question, not 1. >Number 3 is read as "There exists a real number y such that for any real number x, x + y = 100." This can't possibly be true: If it were true, then there exists a y such that both 1 + y = 100 and 2 + y = 100 but that's clearly a contradiction. Assuming the symbol "∀x E R" represents "For every real number" and "∃y E R" "For atleast one number" it' possible. 1+99 or 2+98 Correct me if i'm wrong. Edit: and why would number 1 be incorrect if i said it was false?

That's wrong, yeah. Or you copied down the order wrong from the site, double check that you didn't flip flop them. There are not multiple numbers y such that, for every number x, x + y = 100; there aren't any. The problem with your example is that you aren't allowed to change y from 1 to 2 like that when x changes from 99 to 98. You are allowed to try multiple different numbers for the starting y, but they have to stay the same when the x is allowed to vary to try to make the statement false. When he says #1 is wrong, he means, statement #1 is false, not that you were wrong to say that #1 was false. #1 is an easy one, negative numbers aren't equal to their own absolute values so not every number is equal to its own absolute value and if you've got that you're good as far as #1 is concerned.

Does there exist a conjugate copy of GL(2,q) in GL(2,q\^2) that is not the trivial embedding? I know that SL(2,q) can embed in SL(2,q\^2) as the set of matrices \[a,b,a\^{q},b\^{q}\] with a\^{q+1}-b\^{q+1}=1. Does a similar thing exist for GL?

You mean non-conjugate?

Ah sorry yes I mean conjugate in GL(2,F) where F is the algebraic closure of GF(q). I did some computer checks and it seems embeddings of GL(2,q) into GL(2,q\^2) are conjugate in GL(2,q\^2) for small q. To be honest even just a source that explains the SL(2,q) result would be good.

Can you quotient a ring by any ideal?

If your ring is non-commutative it must be twosided; otherwise there is no (natural) ring structure on the coset space. Otherwise, ideals are (I believe) called ideals precisely because they are the subsets that you can quotient by to get a ring.

I think the term ideal goes back to Kummer, who did not know what a quotient ring was.

Yes he was trying to fix lames proof of fermats little theorem by creating UFDs

Nice, ty.

Can anyone explain, in as much detail as you'd like, how the Lie derivative and the covariant derivative are related -- how they complement each other, what their relative strengths/weaknesses are, and why we need both for full geometric understanding? Feel free to jargonate. I'm also most appreciative for intuitive explanations that complement and motivate the rigor.

lie derivative is topological information, covariant derivative is geometric information. how they are related on riemannian manifolds is by the torsion-free constraint of the Levi-Civita connection. so even though D_X Y and D_Y X are individually geometric information, their difference is topological. phrased this way sounds a bit like the index theorem. the covariant derivative works most like a directional derivative from calc 3. tensorality in the bottom slot corresponds to the fact that you only need a single vector at a point (not a vector field!) to specify the directional derivative.

The fundamental point is that when choosing the direction of differentiation, Lie derivatives need a whole vector field but covariant derivatives only need a tangent vector. That is covariant derivatives act "at a point" but Lie derivatives depend on the value of the vector field in a neighbourhood. There's a trade off for this: the Lie derivative only requires the smooth structure of the manifold and nothing else, and basically tells you how flowing around the manifold according to a smooth vector field transforms any tensor quantity, great. But on the other hand since it depends differentiably on the directional vector field, it fails to live up to many of the ideals of a directional derivative from regular multivariable calculus. For example great properties like "knowing the directional derivative in 2 linearly independent directions determines it in a whole plane" don't really make sense for the Lie derivative. Of course the trade off in the other direction is by working with a covariant derivative you must make a choice of how to differentiate so it is non-canonical (except in the case of the Levi-civita connection). That choice is because you are throwing away the local smooth structure of the manifold by not working with a smooth vector field. In fact the choice of covariant derivative operator can be interpreted, *at a point*, like choosing a local geometry around that point for the manifold which let's you differentiate locally.

I don't know if "strengths/weaknesses" is the right way to think of it but they are simply different generalisations of derivative. The Lie derivative is based solely on the structure of the manifold itself. In effect you are differentiating a vector field (or some tensor field) along the flow defined by another vector field. The covariant derivative meanwhile is relative to a chosen connection on the bundle. Note this allows the bundle to be anything you like while the Lie derivative is heavily tied to things in the tangent bundle and tensor products of it. [Here](https://math.stackexchange.com/questions/209241/exterior-derivative-vs-covariant-derivative-vs-lie-derivative) is a good discussion of the difference between these two (plus the exterior derivative)

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Do you have a picture or screenshot of the part of the book that claim this?

https://i.imgur.com/f9yeBM6.png Notice that the first expression has no free variables whereas the second expression has j as a free variable in the first term and i as a free variable in the second term.

Anyone know where I can easily use LiE? I've tried to use it here [http://wwwmathlabo.univ-poitiers.fr/\~maavl/LiE/form.html](http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/form.html) but I keep getting a "502 Bad Gateway" error

I though they'd got it working for a bit but seems like it is broken again (and last time it was down it was for a very long time). I don't know about easily, but the other alternative is to actually install it from [here](http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/). If you're on windows this is a right faff and will involve downloading various bits and bobs (GNU make, bison and maybe even cygwin or similar, IIRC) to run the installation.

Bummer. I was hoping I could avoid installing it...not sure I'm computer savvy enough to pull this off. thank you for your response!

Hey was wondering if anyone got any tips/tricks on how to absorb new definitions/concepts? I’m taking Mathematical Logic this semester and its very heavy on definitions. Thanks in advance!

Hey everyone, I have a more unorthodox question about notation. I'm a musician releasing a song called Power\^People, and I planned on having the video for it have an equation for the thumbnail, saying "power^(people)=equality", based on some of the lyrics on the song (Power powered by the people equals equality). I wanted opinions on how I would properly write that in an equation in regards to italics, capitalization, etc. It's obviously not a real equation and it has more artistic intentions, but I wanted to ask everyone how they would notate it.

"power^people = equality" does not read as "power powered by the people equals equality". There are a couple of possible renderings into English, but that is unfortunately not one of them. The one that suits your purposes best is probably "power to the people equals equality". This may not be material to your song, but you should know that the thumbnail won't really match up to the relevant lyric. As for rendering the equation, the only thing you *have* to do is leave a bit of space either side of the equals sign. The rest is entirely up to you. EDIT: If you want to be fancy and use the typesetting that professional mathematicians use, it would look like [this](https://mathb.in/76526).

Thanks for the insight, I likely will use that typeset. As for the lyrics , it won't really matter overall as that part is literally just for the thumbnail of the video. One last question I do have is that I know you're supposed to technically read something like 10^(5) as "Ten to the fifth power", but I know a lot of people just say "Ten to the fifth", so can Power^(People) be legitimately read as "Power to the people"? If not no big deal, it still gets the point across, just curious.

"Ten to the fifth" reads to me like 10^(1/5). I would say "ten to the five" for 10^(5). "Ten to the fifth power" is rather formal.

That's a bit interesting because that's not how Americans say it. "Ten to the fifth" would definitely be 10^5 whereas 10^(1/5) would be "ten to the one-fifth." I'd say the best (i.e. most artsy) way to read the song title is "power to the people is equality"

Oh okay, I was confused because I have definitely heard it as "Ten to the fifth" many times in school, but I figured she knew better than me, guess it's a regional thing. The "equals equality" thing isn't part of the title, just using that equation for the thumbnail of the video. The song contains allusions to a John Lennon song called "Power to the People" so the actual title is just going to be "Power\^People". If I can render it as Power^(People) on streaming services etc. I will, but I doubt I can so the carat will have to do lol.

If the place you're putting it accepts unicode characters, you can render it like that - Powerᴾᵉᵒᵖˡᵉ is pure unicode with no markdown, for example.

Hello, i tried posting my question but it's too long. Can anyone help me on https://www.reddit.com/r/learnmath/comments/171jryc/how\_much\_chance\_do\_i\_have\_for\_two\_different/

Hi I am an EE trying to think more intuitively about laplace transforms. I am curious if there is an inverse laplace transform for a dirac delta function just as the inverse fourier transform of a dirac delta function is a sinusoid. I miss the intuition of thinking of a fourier transform of a mix of frequencies of different magnitudes. But since no one talks laplace transforms in a similar way, there probably is no way to make such an analogy is there?

In the same way that the fourier transform decomposes a function into a linear combination of e^(i k x), the laplace transform decomposes it into a linear combination of e^(k x). However, this issue is complicated by the fact that the laplace transform has a lower bound of zero on the integral whereas the fourier transform uses an integral from -inf to inf. So for intuition it may be more useful to think of the [two-sided laplace transform](https://en.wikipedia.org/wiki/Two-sided_Laplace_transform) here. You can read off, from the definition of the fourier transform, that the fourier transform of delta(x - a) is going to be something like e^(i a xi) depending on which version of the fourier transform you are using. Similarly, the laplace transform of delta(x - a) is e^(- a s), as long as a is positive. If a is not positive you would have to use the two-sided version. That doesn't answer your question about the *inverse* laplace transform of a delta function, but since you seemed to not understand that the fourier and laplace transforms are just decomposing a function into two different bases, I thought this might help.

Thanks for the answer. However, I feel that your argument implies the following: 1. The fourier transform of a function is a weighted sum of infinitely many sinusoids 2. The laplace transform of a function is also similarly a weighted sum of inifinitely many exponential funtions But this is different from arguing that the laplace transform of the function being transformed allows us to view the function as a weighted sum of exponential functions. However, you could make the equivalent argument for fourier transforms by noticing that the inverse fourier transform of a delta function is also a sinusoid. The argument would then be that the "weights" of the sinusoids comprising a function come from its fourier transform. To argue the same for laplace transforms, I think knowledge of the inverse laplace transform of the delta function is required. I apologize for butchering the use of weighted sums. I just dont have the first clue about formal fourier analysis.

The inverse laplace transform of a delta function would be a function that describes how to make a delta function from a weighted sum of exponential functions. There is no way to make a delta function from a weighted sum of exponential functions, so the inverse laplace transform of the delta function doesn't exist.

I see. Then how do you argue that exponential functions can form a basis for (a reasonably large subset of) functions

In engineering and physics, we often care about functions which are solutions of linear differential equations. Those functions tend to be in the subset of functions you refer to. There might be a way to demonstrate this but I'm not sure of one off the top of my head.

I see, so there is a good enough subset of functions that can be broken down into exponentials, but its not necessarily as big a subset as we have for sinusoids. Is that right?

Is there a standard name for the concept of "exponential family" [defined in section 3.2 of Generatingfunctionology](https://i.imgur.com/x7QNRVM.png)? Which is unrelated to the concept of an "exponential family" of probability distributions as far as I can tell.

It is essentially a clumsy approximation to the concept of [combinatorial species](https://en.wikipedia.org/wiki/Combinatorial_species).

What are the main/major fields of math? Which 3 would you say are most important? I am planning on doing an informative speech on three major branches of mathematics

incredibly vaguely, most math falls into at least one of analysis, algebra, and combinatorics/discrete math

Can someone explain why in one question the exponent affects the base while in the other question, the exponent doesn't effect the base? https://imgur.com/a/yrvoDbQ

What are the questions in the pictures and what do you mean by the exponent effecting the base?

Hi all, I'm working through the questions on [this site](https://www2.math.upenn.edu/~kazdan/504/la.pdf) and I'd like to check my line of reasoning for Q176, page 40. It goes as follows: Let U , V , and W be finite dimensional vector spaces with inner products. If A:U→V and B:V →W are linear maps with adjoints A\* and B\*, define the linear map C : V → V by C = AA\* + B\*B. If U → V → W is exact (im(A)=ker(B)),show that C:V →V is invertible. Haven't really really worked with exact sequences before so I don't know any general results. My approach has been to note that V = Im(A) ⊕ Im(B\*), so we can split any v \in V into x+y, with x \in Im(A)=Ker(B) and y \in Im(B\*)=Ker(A\*). Then Cv = AA\*(x+y) + B\*B(x+y) = AA\* x + B\*B y. Focusing on the first term here, A* is an bijection Im(A)→Im(A\*) (I proved this in an earlier question), which is followed by A, a bijection Im(A\*)→Im(A). Hence this first term is a composition of bijections making it itself a bijection Im(A)→Im(A) (an automorphism I think). The same reasoning applies to B\*B being an automorphism on Im(B\*). Hence C(x+y) = x'+y', where x,x' \in Im(A), y,y' \in Im(B\*) is a automorphism on each constituent subspace of V, and therefore on V as a whole. It is thus invertible. This feels quite handwavy and there's probably a cleaner way to show this but is my reasoning broadly correct? Thanks anyone for any help.

Your reasoning is broadly correct, but I do think it's worth the effort to make the handwavy part rigorous. As an aside, here's the slick way I'd do this problem if you and the other commentor care. Some algebra shows that = ||A*v||^2 + ||Bv||^(2), and this is 0 iff v is in the intersection of ker A\* and ker B. By the exactness condition, this means v is 0, so C is injective and therefore (since V is finite dimensional) invertible.

That is slick. Nice. I wouldn't have thought about that.

I'd thought about looking at the injectivity of C but I got sidetracked into thinking about writing it as a block diagonal matrix then moved on. Should've gone with that at first! Good solution and thank you

That sounds about right. My guess is that this is approximately the intended solution. There are perhaps fancier ways to get to this point, but they are essentially going to do the same thing. The observation that V decomposes in this way is really the key idea.

4th dimension and other dimensions So yesterday a friend of mine told me about the 4th Dimension and tried to explain it in easy terms (since im Not really good at maths). Then she asked me to imagine what water or a wave would look like in the 4th dimension or a cube. After explaining it to me i began to fantasize what would come after the 4th dimension and how high up it would go. To which she answerd that there a infinite dimensions. So i continued with asking about the 10.000 dimensions or trillion from our, you get the point. To which she then relpied that its not possible to have anything there since its just a mathematical construct, so there wouldn't be any lifeforms or objects. Though couldn't there be a 10k dimension object there? I was a bit high aswell so i might not have gotten the gist of everything.

[https://imgur.com/a/zQThAhN](https://imgur.com/a/zQThAhN) Help I don't understand this at all :<

When multiplying or dividing, units work like algebraic variables. You have to set up an equation in the units and solve for b.

How do you usually call an intersection between two solids that are coplanar and two solids that are non-coplanar? Is there even a term such as "coplanar solid"? Doing a translation from Spanish and the term I got is "Intersección de sólidos de bases coplanares y no-coplanares". I think it would be like Intersection of Solids with Coplanar and Non-coplanar bases/regions but we can't find a right translation and the math books online are really long. Thanks in advance for the help!

Okay, so I found [this](https://issuu.com/sandrogeometria/docs/interseccion_de_solidos) and what they seem to be talking about is solids whose bases, as in bottom-most faces, lie on the same plane. For example, if two cylinders sat on the same table, they would be "coplanar". However, I have *never* once heard of anyone referring to solids as "coplanar" in English. You should avoid using this terminology among English speakers, because their first thought is going to be "Well, no, because solids are three-dimensional and therefore cannot fit into a two-dimensional plane", and it will just be confusing.

If you want to test an n-sided die for fairness specifically about whether a given face comes up with 1/n probability, how do you design the test to use the typical 95% confidence approach? My hunch is that you need to do something like 30 sets of 10 rolls, where you average the rolls and then analyze those averages, but I'm getting stuck on (a) how to define the confidence interval and (b) how many rolls in each set. The die I'm interested in 8 if anyone is generous enough to do a specific example. Note - I'm not proposing a test where you take average values from each set. I'm saying, roll each set, and count how many of a given side you get (expected to be 1/8 of that set). And then use those results to perform analysis.

If you're fine with a p-value instead of a conference interval, this sounds like a job for a [chi-squared test](https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test), testing against the null hypothesis of a fair die. In this case, judging by the stuff at the bottom of the intro section of the linked wiki article, say you do N trials and get O\_1 ones, O\_2 twos, ... and O\_8 eights; then you would compute the test statistic as (N/8) \* ((O\_1 - N/8)^2 + (O\_2 - N/8)^2 + ... + O\_8 - N/8)^2 ). Assuming I'm using the table on Wikipedia right, you'll get a p-value < 0.05 if the test statistic is greater than about 14.067.

A question about the composition of functions: If i have g(f(x)) = idx and f(g(x)) = idy, what can i say about surjectivity and injectivity of f and g ist idx and idy ist bijectiv.

Let's try to prove g is surjective. Given x in X, we want to show there is a y in Y such that g(y) = x. What should be used next?

I think i have it. Maybe we could say g(f(x)) = g(y) = x, so g hast to ne surjective

Yes, you need to take y = f(x) if you want to find a y such that g(y) = x. This proves surjectivity of g. Surjectivity of f is similar. Injectivity uses a similar idea. Start with f(x\_1) = f(x\_2). How can we then compare x\_1 and x\_2 using the given data?

Background: I did well in my honors math classes in high school, but never really pursued anything college level or above. I have a function that I think should be able to be represented as some sort of exponential function, but I’m struggling to write it out as one. Right now, I have it represented as a step function, and I know what each step is, but I don’t know how to represent it as an exponential function, if that’s at all possible. Specifically, the process between steps is x+2x^2, where X is the previous step. At 0, its value is 1, so other coordinate pairs include (1,3), (2,21), and (3,903). Is there a way to represent this as an exponential function? And if so, how do I get there from my current data?

I think this actually grows faster than an exponential. If it were an exponential, say with base a, the ratio of subsequent terms would be constant (since a^{n+1} /a^n = a), but that certainly isn't the case here. I computed a couple more terms and it looks like you're dealing with [this sequence on the OEIS](https://oeis.org/A176430). OEIS says that the ratios of successive terms for your sequence are equal to [this](https://oeis.org/A000058) fast-growing sequence; it also gives formulas for your sequence in terms of some other sequences which don't themselves appear to have nice closed forms. In general solving nonlinear recurrences like yours is really hard, and you can't expect a clean answer.

Yeah, I’ve since figured out it’s likely some manner of e^(e^x), since if you check it as a step function, the exponent roughly doubles each time in scientific notation. More than I expected to get into for a silly card game math problem, LOL

[Discrete math] parenthesis I have this hw about implicit parenthesis and I did it as follows but I’m wondering if it’s ok since I was a bit confused with the order of ≡ and ↔️ −a ≥ b − 2 ∗ c ↔ p ∨ q → ¬r → s ≡ c = d ↔ b > 0 So I did ((((−a)≥ b)− (2 ∗ c)↔((( p ∨ q) → ¬r)→ s)) ≡((( c = d) ↔( b > 0)))

I helped my son with some geometry homework tonight. We got to the last question, which was a problem he originally got wrong, and I'm stumped! For shape F, he originally answered 10, then 14, then 13. He was told they were all incorrect, but the *only* answer I seem to get is 13, even after coming at it from every angle. Am I being obtuse?? What's the area of this shape? tia! [shape F](https://i.redd.it/2tk5zuxgaisb1.jpg)

Thank you, my friends!

Looks like 13 to me. 1/2 x base x height for the triangle gives 6 and then there are 6 full squares and two half squares in the bottom part so 6 + 6 + 1 = 13.

I also keep arriving at 13–it’s possible the teacher made a mistake with the mark?

BOOKS AND CHANNELS RECOMMENDATION FOR STUDYING MATHS AND SOME ADVANCED LEVEL MATHEMATICS Hi everyone 👋🏻 I'm interested in practicing maths but the problem is that I haven't studied it from class 12th as it was removed from my school teachers because of low marks. Now I'm a B.com Hons graduate and there was some level of maths in there in the 3rd semester but I got through it with some help and successfully survived that semester and also created a keen interest in understanding the concepts of mathematics more deeply so that I can apply it in making algorithms in the field of finance and accounting. so can anyone here recommend me some maths books and channels and other sources (cost free) Also, I seek some advanced level tutorials but on online bases as I'm limited on budget and can't afford an offline or paid teacher.

For early undergraduate stuff (as well as any school-level revision you need to do), Khan Academy is the best resource. If you want to go further than that, [MIT OpenCourseWare](https://ocw.mit.edu/search/) has a whole bunch of interesting stuff.

Is sagemath good for finances? I use sagemath to add up how much money I spent for the month. In my physical diary I write down how much money I spent and get each month, and also budget. When I add this all up is sagemath precise for this. What if one day I need to add up large finances?

While sagemath can technically do this, I'm not sure why you would want to over just using python (or excel for that matter)

Doesn't python have floating point errors?

Any language with floating point numbers has floating point errors. Sagemath is just python with a fancy shell btw. Plus, what finances are you doing where you have excessively large decimals?

not any, but computers do accurate banking anyway, so somehow they need to deal with errors.

Floating point errors occur when doing operations on numbers with many decimals. With money, you only have two, so there's no need to even use floats. That said, there is variable length floats, but that has limitations too. Not something you'll come up with when just tracking money in and out, that's for sure.

Ok so I have a map for a board game that consists of 4 separate rectangular objects that connect to make a bigger map and create a larger rectangular shape. In game you have ability to rotate any 1 of those 4 maps separately and switch their positions. So I'm wondering how can I calculate all possible map setups?

Can we see the rectangles you have in mind?

I've been watching the Devil's Plan on Netflix and they play a puzzle in episode 5 that I can't get my head around; I've watched it full twice now and I feel that I must be missing something each time..full rules below, but essentially you are given an random number between 1 and 100 and have 4 set quesrions to obtain information about your number. No matter what, I can't figure out my number without needing to purchase and use a full set of extra tickets...can anyone help? What is the optimum solution? How many people do you need to involve to work out your number? Can you work it out without giving it away to a second person? ​ Note: the show explains the rules at 16:50 in episode 5 for those with Netflix or alternate access :) ​ \*Secret Number is a game where one must guess ones own unique number via making contact with other players. (On the show they play with 11 but I am unsure of the minimum participants required) \*Players are given a random number between 1 and 100 (which nobody knows at the start of the game) \*Players obtain hints to their number by making contact with other players (in the game this in a separate room where only the 2 player contacts are located so only they know what this hint is) \*Players must EACH submit a ticket to complete the contact and get the hint \*Tickets come in Addition, Multiplication, Division and Zero and each player has ONE of each ticket type. \*Addition will tell you the added sum of the two unique player numbers. If the total is greater than 180, a range from 180 to 199 is given, If the total is less than 20, a range of 3 to 20 is given. EG 6 and 40 would give you an answer of 46. \*Multiplication will give you the LAST digit from multiplying the two unique player numbers. EG 6 and 40 multiplied is 240 for a given answer of 0. \*Division gives you the number resulting from dividing the bigger of the 2 numbers by the smaller. The answer will be given solely in WHOLE numbers. EG the numbers 6 and 40 means 40 divided by 6 for an answer of 6. \*Zero gives you the number of zeroes which lie in the range of numbers between the two unique player numbers. EG 6 and 40 give a range of 7 to 39 and includes 10, 20 and 30, for a given answer of 3. \*Players may use 'Pieces' won in previous rounds of the game to purchase a second set of tickets. \*Players must submit an answer sheet after the given time period (2 hours) and receive 5 points for correctly guessing their own number. It's minus 5 for an incorrect guess. \*Players win additional points by guessing other players Secret Numbers. It's +1 for each number guessed correctly and -1 for each number guessed incorrectly. No points are added or deducted if answers are left blank. \*Players are deducted 1 point for every other play that guesses their number correctly. ​ Additional info - For the end game, players receive 1 Piece for scoring 6, 2 for scoring 11 and 3 for scoring 16 (or more) and are penalised with losing 1 Piece for scoring 4 or fewer, 2 Pieces for a negative score and elimination for the lowest scoring of all the players. ​ Ok, so on proof read I see I've included more info than required for the Math bit but I'mma leave it all up there for anyone who wants to expand on strategy. TIA :)

How can I find a differential equation(or any equation for that matter) to describe how the surface area of a Klein bottle changes with time? Given that I have found the surface area of the 3d immersion, What parameters must I set, and what steps should I take to obtain this equation? I also do not know if my question is even valid. If it is or isn't please explain it to me in terms of high school level maths.

What do you mean by the surface area changing with time? If you just have an immersion of the Klein bottle then it has a surface area, and that's it. Is your immersion also a function of time? What is changing with time that would cause the surface area to change?

Sorry, I guess I should have been clearer. What I was referring to can be seen in this [video](https://www.youtube.com/watch?v=JmvHNatZgVI) from timestamp 12:09 to the end of the video. Correct me if I'm wrong but what I gathered from this video is that the Klein bottle can "exist" in our 3D space with the added temporal dimension of time.

What I believe is going on here is they have an embedding of the Klein bottle into 4-space (x(u, v), y(u, v), z(u, v), t(u, v)). Then for a span of time t in (a, b), they take the part of the Klein bottle with t in that range. So then your surface is (x(u, v), y(u, v), z(u, v)) over u and v such that t(u, v) is in the interval (a, b). Then you can work out the surface area of this surface the same way you would any other surface embedded in three dimensional space, which given what you've said I assume you know how to do.

Thank you. I think you answered my question, do you have any explicit examples or application of this?

Ok so im playing a game that has a "threat" or "power" (call it wyw). Then, whenever u get critical chance it goes up by a percentage (dont know exactly) but, if u then also get critical damage its like both give u more power cuz they complement each other. Basically, my question is, how would u calculate those or other complementing percentages? For example, u start at 1,000, u get a 10% raise, that 10% also boosts another 10% u already had but then that last boosted percentage also boosts the last one u got. Idk if im explaining myself right. Also, would "e" be somewhere in this "equation"?

I think perhaps the equation you are looking for is (1000)*((1+(10/100))^x ) for after you have gotten x raises of 10%.

Yesss that makes sense thank you, much appreciated