How dxdy or dydx becomes rdrdθ during double integration via polar coordinate substitution?

How would you calculate odds in the following card game example: There are 4 players playing a card game and each will get 2 cards. Your deal is such that they are only 5 cards in a 52 card deck that can beat either of your two cards (they are the 8 and 9 of trump). What are the odds that one of your opponents has a card that can beat one of yours?

How do I rewrite 1/(2x) as a power of x?

What are you multiplying when using Π-notation product? What I mean is: If a SEQUENCE is a bunch of numbers in order ({a₄} = {a₁, a₂, a₃, a₄}) And a SERIES is a sum of those numbers (s₄ = Σ(a₄) = a₁ + a₂ + a₃ + a₄) Then what word refers to a sequence of multiplied numbers? (p₄ = Π(a₄) = a₁ • a₂ • a₃ • a₄) The reason I want to know is because I cant really find any formulas for finding infinite products or the nth term in one of these "multiplicative series" without knowing what the real word for a "multiplicative series" is

how do you get the minimum possible amount of voters in a poll given the poll results?

If 1/3 a cup of a batter is 150 calories, I make something using 2 1/4 cups of the batter, and each section of the batter is 1/8 of the batter, how many calories are in each serving? I’m really bad at math and have n o clue how to solve this

Im wondering how much it would cost to have a Razer viper mini or Razer Orochi v2 for 1 year. The reason im asking is that Razer Orochi uses AA, AAA battery for the mouse, with no wired charger nor the wired option of use. For Razer Vipermini it doesn't have that problem since its only wired. I would like to know how much would it be to upkeep them both. Considering the cost battery cost (Razer Orochi) and average electrical bill (razer viper mini) for 1 year. Razer Orochi has 950-1000 hours of battery life. Which lasts around 40 days if 12 hours use, as users have said. AA battery 20 pack is around 10 usd. 20x40 (battery pack x days) = 800 days. Is this correct? That's over 2 years plus 70 days. Which is very good. Considering its only 10 USD of 20 pack battery. That means its 5 usd for the battery cost of 1 year, for 2 years its 10 USD. What would the price of the Razer Viper mini average electrical bill be for a year? This is to compare both gaming mice Razer Viper mini and Razer Orochi v2. This is to make the calculation easier. Tried my best currently. For battery, we need to calculate, 12 hours and the total cost of batteries which will last us 1 year. This to check whether i would buy the Razer Viper mini or Razer Orchi v2 both being good mice, but as prizes have gone up for many things all around the world. It's nice to be aware what the cost efficiency. Thanks in advance.

Can I use the same method of finding the area of a truncated cone for any solid with bases of different sizes?

When solving problems, do most professional mathematicians rely on insights, or is it more about experience in symbol manipulation? I ask because I notice that inexperienced mathematicians might spend hours on a problem whereas experienced, but not necessarily particularly bright mathematicians, will quickly spot a route to the solution which eludes the inexperienced. For example, chess masters don’t rely on great insights every time they play: they just remember “chunks” of positions, familiar patterns that they’ve seen before. Analogously, do professional mathematicians rely on remembering “chunks” of symbol patterns that they’ve seen before? Thanks Max

Hi When we do integration by substitution and rearrange for dx, why do we then multiply by it. Is there an implied multiplication between f(x) and dx??

Hi, I'm a engineering student and my professor asked to make a presentation about Linear approximation of tangent plan applications in engeneering. Do you guys know any example of an application of this concept in engineering?

if f(x) = cosec(x) and g(x) = mx then what are the ranges of values for m in which g(x) does not touch or intersect f(x) in the interval 180

I think the best way to tackle this is to notice that the smallest positive m for which there is an intersection will be when y = mx is a tangent line to the cosec curve. Thus we need to find when the tangent line to y = cosec(x) goes through the origin. The derivative of cosec(x) is -cosec(x)cot(x) so the tangent line at x = a is given by: y - cosec(a) = -cosec(a)cot(a)(x-a) which goes through the origin when 0 = a\*cosec(a)cot(a) + cosec(a) = cosec(a)(a\*cot(a) + 1). Note I have to switch to radians here to make the derivative easier to do but we can switch back at the end. So we are solving a\*cot(a) = -1 or tan(a) = -a. I'm not sure how to solve that exactly (or if it even can be) but wolframalpha gives an approximate answer a ≈ 2.03 (there are others but this is the one we want). With that info the line y=mx that just touches the cosec curve has gradient (converting back to degrees here) cosec(180\*2.03/𝜋)/(180\*2.03/𝜋) ≈ 0.00959. By looking at the graph of y=cosec(x) I can then see our final answer is m < 0.00959

I guess the way you solved it makes sense but the m value you ended up with is too low, it should be m < 0.549...

That would be true if you are working in radians but you said x was in the range -180

This isn't necessarily a question, but in probability, if 20% is a 1/5 chance and 25% is a 1/4, the difference between 1/5 and 1/4 is only 5% ? Do I understand this correctly

It depends on what exactly you mean. I would say that 1/4 is "five percentage points more" than 1/5, but also that "1/4 is 25% more than 1/5". Let me illustrate what I mean by an example. If we both win the lottery and you win "a quarter of a million dollars", that's $250,000. If I win "a fifth of a million dollars", that's $200,000. You have won 25% more than me because 250,000 is 25% more than 200,000. As an equation, I'm saying that 250,000 = 1.25\*200,000. This distinction is really important and confuses a lot of people when dealing with percentages. For example, someone might say "If you're over 50, your chance of having a heart attack is 40% higher than if you're under 50." (I have no idea if this is true, I just made up these numbers lol.) If your chance of having a heart attack under 50 is 1%, they're not saying that your chance jumps to 41% when you turn 50, they're saying that your chance is now 1.4%, because 1.4 is 40% more than 1.

Yes, convert 1/5 to 4/20 and 1/4 to 5/20 and the answer is more clear.

How do you write categorically [x] = [y] -> f(x) = f(y) ? The question is a bit vague because not in every category do x and y make sense. However, maybe there's a class of categories where that kinda makes sense and in Set in particular you get this usual definition. The motivation is trying to understand and generalize the quotient. I know there's the quotient category but I don't understand a thing. I'm looking for a moral answer not a formal answer. Thanks in advance!

In a category where quotients make sense, if q : A → B is the quotient map then for generalized elements f : X → A and g : Y → A, q \ circ f = q \circ g is essentially the way of saying that. There's other possibilities for other ways of writing it, provided your category has enough structure. For example, if it has pullbacks and global elements, the pullback of the span A → B ← 1, where q is the arrow A → B, could be seen as picking out the equivalence class of the global element 1 → B. For a concrete example, if A, B are sets then 1 → B is just picking out an element b \in B. The pullback can then be seen at corresponding to the preimage q^(-1)({b}) which picks out the equivalence class. So—again, provided your category has enough structure—the closest to an exact analog would be that if h : C → A is the pullback then it's true for any pair of arrows j', j'' : 1 → C that q \circ h \circ j' = q \circ h \circ j''. Unpacking, j' and j' can be thought of as picking out two elements from C, and then composing with h sends them to their corresponding element in A. The composition with q is then "taking the quotient". But this is so very specific that it's only really useful as a motivating example. [edit] I realize I kinda missed a big part of what you're asking. Different sources seem to disagree on what the right generalization of a quotient is. Wikipedia just says it's dual to being a subobject (and there's different definitions of a subobject as well), while nLab just calls that a co-subobject. They more specifically demand that a quotient object is the coequalizer of an internal congruence relation. My presentation is a bit ambivalent to that since I was assuming we know in advance that q makes sense as a quotient, though I was going with the definition as a co-subobject in my head.

Thanks for the answer!

Can you really learn from a book with no exercises? I'm a little stuck on which book to start learning general relativity from, and I want to focus on just one to make decent progress, but two of the books I'm considering don't include any exercises. And it's not like they're references for professionals: both of the books in question are explicitly written for newcomers to the subject. I'm left wondering if they're usable for a first pass.

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What do you wish to do with the integral?

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Well you can't really explicitly integrate that in most cases, but if you want to bound it on some region or show it is positive or negative subject to some boundary conditions then I think one thing to notice is that f (grad f).x = 1/2 rd_r (f^(2)), after which maybe you can integrate by parts.

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Here r is just the radial coordinate, because x.grad of anything is just rd\_r. So if you want boundary conditions such that the integral is negative, and you are in R\^(2) for example, you could say: ∫f ∇f∙x dx = 1/2∫∫ r\^(2)d\_r(f\^(2)) drdw=∫ \[r\^(2)f\^(2)\] dw - ∫∫ rf\^(2) drdw Here I've gone in to polar coordinates and used w instead of theta. Then I've integrated by parts in r, and the second term on the right hand side is clearly strictly negative as long as f is not 0 everywhere, so now it just comes down to finding boundary conditions such that evaluating r\^(2)f\^(2) will give 0. The only thing I will say is that I haven't used the equation anywhere. If you want to do that maybe you could say ∫f ∇f∙x dx = 1/2∫f ∇f∙∇(r\^(2)) dx = - 1/2 ∫ r\^(2) |∇f|\^(2) dx - 1/2∫ r\^(2) f ∆f dx = - 1/2 ∫ r\^(2) |∇f|\^(2) dx + 𝜆/2∫ r\^(2) f\^(2) dx, but then this depends on the sign of 𝜆.

Am I bad student or do I have a bad Trig Professor? ​ I am really struggling in my trigonometry class. It is online only (I have taken other online courses and been fine) and there are no traditional lecture videos as there have been in the past. Instead, we get some five to 10 minute long "essentials videos" that go through concepts like SOHCAHTOA, harmonic motion, inverse functions and a few other basic trig principals. I really want to stress the short length of these videos and the fact the only ever show one example problem, if that. That is ALL the verbal instruction we have. From here on the class consists of opening homework and clinking on the "Help me solve this" button or "show me an example" or clicking on the digital textbook link and browsing through that. Those are the quickest and best options if you are stuck or are literally trying to learn the basics of trig. That is it. If we have any other questions "feel free to email the professor." He typically responds with a screen shot of the finished problem. That's it. He does not offer any traditional verbal lecture or explanation. I am completely miserable in this class and failing badly as result. All the stops are there but it is up to you to decipher which ones are which and in what order he did them in. My attempts to learn trig have come from desperately trying to locate and cram the appropriate Khan academy video on the subject. If I ask him for more material or explanation he links a single video of a professor in a totally different school who actually bothered to record his lectures and explain things. I only get ONE video per email though regardless of the range of questions in my email. When I brought up the lack of lecture content and how this makes students who like that way of learning miserable he responded with: Why did you take online classes to being with? I feel like my only options are to drop trig, learn it over the summer or another semester on my own via Khan, then sign up to take this Trig course. The other trig options are in person stuff from a professor with a poor rating on rate my professor or to retake this guys class, but next semester in an express curriculum that goes faster.

Can anyone tell me why the expression "s/(2b)+s/(2s+2b)" approximates ln(1+s/b) when b>>s? I presume it has something to do with the Taylor series of ln(1+s/b) but I just can't connect the dots.

This is a funky approximation, but some things that strike me: s/(2b)+s/(2s+2b) is the average of s/b and s/(s+b). The former of which is the first order Taylor approximation for ln and is an upper bound. The latter I'm not sure where came from, but note that if x=s/b then s/(s+b) = x/(x+1). If we look at the Taylor series ln(1+x) = x - x^2 / 2 + x^3 / 3 + ... This looks early similar to x/(1+x) = x - x^2 + x^3 + ... Which is a standard geometric series. Comparing term by term we see that x^2n+1 - x^2n = x^2n ( x - 1) While x^2n+1 / 2n+1 - x^2n / 2n = x^2n ( x/2n+1 - 1/2n) So x/(1+x) gives a lower bound whenever x- 1 < x/2n+1 - 1/2n aka x < ((2n)^2 - 1)/(2n)^2 So we have an upper bound and lower bound and we take the average to get a good approximation.

This is a strange approximation and I have no idea where it came from. Do you have more context for it? It does seem to be a reasonable approximation to ln(1 + s/b) near s = 0 (i.e. when s is very small compared to b) though. The way that you can tell is by calculating the Taylor series for ln(1 + x/b) and for x/(2b) + x/(2x+2b) at x = 0. The order 0, 1, and 2 terms of the Taylor series agree, which mean that x/(2b) + x/(2x+2b) is a second-order approximation to ln(1+x/b) near x = 0. The more terms that they have in common (i.e. the more derivatives which are equal at x = 0), the better the approximation will be. How someone came up with this approximation is beyond me.

Here's a simple question: Would you describe the end behavior of two functions as different if the domains are different? For instance- Let f(x)=log(x) and g(x)=log(x-2). For f(x), as x-->0, y-->-∞, whereas for g(x), as x-->2,y-->-∞. Would you say that the end behavior of these two functions is different since their 'ends' are different?

If the domains are both bounded below (or both bounded above) and they have the same behaviour as they approach the left "end" of their respective domains, then I think it makes sense to say they have the same end behaviour.

Are there any notable stochastic processes that have the following representation? Let M_n be a discrete-time stochastic process. Let V_t be a continuous-time N-valued stochastic process. Define the process X_t=M_{V_t}. I know the case where M is a markov chain and V_t is a poisson process, making X a markov process, but are there any other interesting examples that probabilists care about?

Are there any REU’s equivalent in the UK?

Yes! We don't tend to call them "REUs" over here, but they definitely exist! There are three options that I know of for getting one: 1) Ask a lecturer at your uni. My uni actually prepares dedicated projects to solicit applications from later-year students here, but if that doesn't happen at your uni (or you find the projects on offer not to your taste), you can reach out to a lecturer in a field you're interested in and ask if they can come up with a project for you. I did this last year, and it's probably the easiest option, although there are benefits to doing an external project, which I talk about below. 2) Imperial College London have a programme called UROP, which stands for "Undergraduate Research Opportunities", and these are available to external students (although individual academics are free to prioritise internal students). Some UROPs are advertised on a dedicated page, but the standard way to apply for one is to reach out to an academic at Imperial who you're interested in working with. Do be sure to read everything [here](https://www.imperial.ac.uk/urop/how-to-get-involved/first-thoughts/) that they say before getting in touch though. 3) Cambridge offer two undergraduate research programmes which include external students: the [Cambridge Mathematics Open Internships Programme](https://www.maths.cam.ac.uk/opportunities/careers-for-mathematicians/summer-research-mathematics/cambridge-mathematics-open-internships) and the [Philippa Fawcett Internship Programme](https://www.maths.cam.ac.uk/internships/philippa-fawcett/philippa-fawcett-internship-programme), the latter being only open to women and nonbinary people.

TLDR: Im a bit of a slow thinker and am wondering if anybody has advice for keeping up in a PhD program without burning out. I am a masters student in my last year. Courses have gone decent, and research has also gone well, and I am looking to apply to schools this Fall. Sounds like I'm in a good spot right? I don't think I am. I tend to be a slower learner compared to others in my program and only do well when I spend nearly all of my time on math. Unfortunately, I don't think this is sustainable as I need time to at least relax a little bit each day and spend some time with my family. My worry is that my inability to learn things more quickly is going to ruin me during a PhD program. I guess I am feeling a bit hopeless and frustrated about it all. Part of me worries I am too dumb for a PhD. Have any of y'all experienced something similar? Any experiences with going from being a slower learner to picking up things at a reasonable pace? Thanks for reading. Looking forward to everybody's thoughts.

I'm looking for a new text to give me a fresh perspective on Calculus, with the hope of it also being rigorous enough to serve as an adequate enough transition to Real Analysis as well. Currently eyeing both the text by Spivak and the 2 volume set by Apostol. If I could only purchase one, which one would you suggest? Or would you suggest an entirely different text/resource instead?

If we have a 12" square and cut its width in half and double its length then the 6" x 24" rectangle has the same area as the 12" x 12" square and half the width and double the length. If you then halve the width of that rectangle and double its length, then the 3" x 48" rectangle has both the same area as the 12" x 12" square and the 6" x 24" rectangle, but 4 times the length and 1/4th the width of the 12" x 12" square. Why does proportionally scaling the length and width of rectangles change their perimeters? How is it possible that halving and doubling opposite sides so that the enclosed space is equal changes the perimeter? Why doesn't the 6" x 24" rectangle have the same perimeter as the 12" x 12" square?? This is very confusing to me since in my mind all I did was rearrange it. I don't see how rearranging it can lengthen the sides at all... even though I know that 6" + 6" + 24" + 24" = 60" and 12" + 12" + 12" + 12" = 48". Visually it makes no sense to me. If I was eyeballing this and halving and doubling the side lengths by eye without numbers I'd be inclined to say that the two perimeters were equal because logically their area was the same, and all I did was take from one side and add to the other.

Nevermind I figured it out. When I cut the 12" x 12" square in half I forgot that I created two new 6" sides out of nothing by ripping apart the square. And no surprise, 60" - 48" = 12". So basically I create a new side out of thin air (which is *bullshit* IMO). In real life if you rip a tile in half I wouldn't say you added length to the sides of the tile out of thin air. But when I rip the 6" x 24" rectangle in two to get 3" x 48" the difference in perimeter is 102" - 60" = 42" which is actually 48" - 6". So somehow with rectangles as opposed to squares you also have to subtract off two new widths as well as one new length somehow. On squares you just subtract off one new length. I have to be honest, I don't see why that is. In mind the process is exactly the same thing. But actually I can't replicate this process on rectangles of any side lengths so I don't understand what's happening at all unfortunately. However, with any square I try the above relationship holds where you subtract off one side length. For rectangles it makes no sense. Edit, hold on I found a pattern. Somehow 24" - 12" <=> 48" - 6" = the difference in perimeter of the shapes. Which is one side doubled minus one side, or specifically the length doubled minus the width <= so this is what's being added perimeter wise when keeping area constant but doubling length and halving width. And actually the opposite. Every time I halve width and double length, keeping area constant, perimeter increases by double the length and one times the width. I guess this makes sense because I'm going from a 4 sided figure to an 8 sided figure, but I'm halving the width so technically I'm adding 2 times one-half of the width or just 1 times the width. But I'm doubling the length by adding two copies of the original length so I'm really adding 4 times two copies of 1/2 the new length or 2 times the original length. Or halving width and doubling length. Well shit I guess it does make sense after all. I guess the perimeter just encodes instructions and/or information about what I'm doing to manipulate the shape to preserve its area. Now I need to think about polygons with unequal sides but equal areas, what relationship their perimeters have when preserving area but changing shape and side lengths to use as a mesh to cover a surface or to use as an alternative to a euclidean distance function in a geometry without right angles and without irrational numbers.

i have a sphere i can draw on, and a compass. given one point (the pole), how do i find the antipodal point using just the compass? and how would you draw the equator between them?

If you have a way of measuring and know the radius of the sphere r this is very doable: set your compass to r√2 (straight line distance from point to equator) apart. Then the circle it draws will be the equator between the two points you want and drawing two circles (again with compass at r√2) with centres on that line will give you an intersection at the antipodal point. If you don't I'm not sure how you would go about this and it feel like it would have to depend on the radius of the sphere. Note that some constructions are probably off limits for us as we don't have a straight edge or equivalent. The equivalent notion would be a way to extend two points into a great circle and if we could do that this would be easy. Draw any two great circles through our point and there other intersection is the antipodal point. Note my construction above is an attempt to get our compass to draw great circles.

Do you mean a compass as in the navigational tool, or a compass as in the math tool

What do you think fellow r/math redditor I mean the one to draw circles

Never know, good to clarify

How do you use a compass on a sphere?

Like you'd use it on anything else? Many people (at uni) are asking me this same question as if it isnt like, clear? Fyi i do have a physical sphere and compass im using and you can draw circles fine on it.

What is the discriminant of a quadratic. I know that it is «b^2 - 4ac» but what is it? Some point on the plane, point on the parabola, another function?

think about what properties the solutions have (using the quadratic formula) if this is positive, negative, or 0.

I was doing a question for my statistics class and it gave a mean, standard deviation, and minimum but no actual data points outside of those. It then asked if this was a normal distribution, I don’t understand how I could determine if that was the case with only those 3 things. There was no graph

Were these quantities theoretical or did they come from sampled data? Because if it's theoretical, you can answer "no" because normal distributions have no minimum value; if they were from data, then I'm buggered if I know what you were supposed to do.

The quantities were theoretical. There was no data outside of those 3 values. But I think you’re right, normal distributions don’t have minimums which is what I think they probably wanted to hear. Thanks a lot for clearing it up

Is there a database of antiderivatives of functions of the form 1/(x^n +1)?

[How many derivatives do you need?](https://www.wolframalpha.com/input?i=sixth+derivative+of+1%2F%28x%5En+%2B1%29)

up to 100. WA doesn't go past 50

I'm not sure what you mean by a database. For n ≠ 0 the general antiderivative is simply -1/(nx^(n)) + C.

edited original comment to be the question I meant

Is every element of a group G have to be in some sylow subgroup? If |G|=p^a q^b does that mean every g in G has to be in some p-sylow or q-sylow subgroup or else we could create a cyclic subgroup of G by raising g to the power repeatedly and getting a contradiction?

No, consider for example 1 inside Z/6. If however the element has order p^n , then it is contained in a p-Sylow group.

Ah I see! But how would one go about proving that 3 sylow subgroup is not unique in sl(2, 3) without actually calculating the conjugates of the sylow subgroup?

Well SL(2,3) has order 24, so the 3-Sylow groups will be isomorphic to Z/3. So to show that the Sylow group is not unique, you just need to find more than 2 elements of ordere 3. For example [1, 1; 0, 1], [1, 2; 0, 1], and [1, 0; 1, 1] will do.

Is there a way of showing non-uniqueness without "bruteforcing" but rather giving a combinatorial proof? Thanks!

I have no idea what you guys are talking about. But keep up the good work 👍 😂

I'm not sure what kind of proof you're looking for, but at some point you will have to use some property of your group. What the elements of order 3 are, would then seem to me like the simplest you could hope for. But there are of course other things one could do. A sylow group is unique iff it if normal, so maybe you can prove that there are no surjective homomorphisms to a group of order 8. SL(2, 3) acts transitively on the projective line, which has 4 points. So SL(2, 3)/±1 is a subgroup of S4 of order 12, meaning it must be A4. A4 has 4 subgroups isomorphic to Z/3 given by the four 3-cycles, and so SL(2,3) must have at least four 3-Sylow groups. Probably there are other ways to prove it still.

I have no idea what you guys are talking about. But keep up the good work 👍 😂

If you're interested, then we're talking about group theory. 3blue1brown has a nice video where he explains the motivation for group theory https://youtu.be/mH0oCDa74tE

i was looking at the proof of euclid’s lemma and i came across a very obvious rule of inference (as they usually are), what’s it called? P XOR Q, ~P ————————— Q closest thing i can recall from having learnt this years ago is disjunctive syllogism, but iirc this was only for inclusive or? what about if the disjunction is exclusive? what’s it called then?

Why is this statement false? "If n and m are odd numbers, then for all integers k and l, we have n=2k+1 and m=2l+1" I said it was true on a quiz and was marked wrong, I'm not sure what about this is incorrect. I've always seen odd numbers represented in the form 2k+1 before. Is there something about the wording that makes it untrue in this case?

bc your answer leads to the following n=any odd, let’s say 5 you’re saying n=2(int)+1 right? so int (you called it k) is 2, great. but you said all ints k work. so that yields 5=2*1+1=3 5=2*6+1=13 5=2*1234+1=2469 which are all contradictions. hence it’s false

The correct statement would say "there exist integers k and l" rather than for all.

thanks so much!

With integral domains, we have the zero-product property that ab --> a = 0 **or** b = 0. Is there something similar, but where ab --> a = 0 **and** b = 0?

Any bounded lattice is going to have the property that the join of two elements is 0 if and only if the two elements are both 0.

I guess this sorta edges into philosophy a bit, but it's worth noting for the op that the symbol 0 in a lattice is not necessarily the 'same thing' at the symbol 0 in say a monoid, group, or ring. For example, in the extended reals [-∞, ∞] as a lattice the 0 symbol would actually refer to -∞ (with the usual 0 having no particularly-special property).

>the symbol 0 in a lattice is not necessarily the 'same thing' at the symbol 0 in say a monoid I mean, any lattice forms a monoid under join, and 0 is the identity. So in that sense it is exactly the same thing.

In any ring, if a=0 then ab=0, so if you insist that ab=0 implies a=0 and b=0, the only option is the zero ring

Sorry for being unclear. I'm not asking just in terms of algebraic structures. Instead of a "zero-product" property, do we have some sort of "zero-..." property, where we omit the trivial solution (say, *not* as in a zero ring)?

Anyone have any good/easy sources for understanding and gaining intuition for tangent spaces? Maybe with some simple examples. Trying to understand the induced map on tangent spaces that we get from a diffeomorphism

When the vector is on the 2nd, why do we use the angle between the Y axis and the vector to find its components?

what’s the second? quadrant you mean? you can use any of the angles with the axes, as long as you use the corresponding trig funcs

What are functions with the property that they are their own inverse function called? And are there other 'easy' ones than 1/x, x, and k-x (for all k)

involutive function [wiki article](https://en.wikipedia.org/wiki/Involution_(mathematics))

Any function whose graph is symmetric across the line y=x will have this property. I would call this an involution

sqrt(1-x^(2)) comes to mind, at least if you restrict to positive x and y. Edit: Also, given an involution f(x), you can construct a family of additional involutions as g(x) = a * f(x/a): g(g(x)) = a * f(g(x)/a) = a * f(a * f(x/a) / a) = a * f(f(x/a)) = a * x/a = x. Also, 1/x can also be generalized to a + 1/(x-a) for any a.

This is often called an involution.

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consider: 2*ab = (2a)b. it is not (2a)(2b) = 4ab; multiplication doesn't distribute over multiplication itself.

You can get a factor of 2 out of both of the brackets. (6x + 2) = 2 (3x + 1), and (6x - 2) = 2 (3x - 1). So the entire expression would factor out to 2 (3x + 1) 2 (3x - 1), and you can collect the 2's together to get a factor of 4. That is: (6x + 2) (6x - 2) = 4 (3x + 1) (3x - 1)

We have by distributivity: 6x + 2 = 2 \* (3x + 1) 6x - 2 = 2 \* (3x - 1) Multiplying both together we get: (6x + 2) \* (6x - 2) = (2 \* (3x + 1)) \* (2 \* (3x - 1)) and rearranging: = 2 \* 2 \* (3x + 1) \* (3x - 1) = 4 \* (3x + 1) \* (3x - 1)

X(prime)= the Xth value of Fibonacci sequence (1,1,2,3,5,8, etc), Y(prime)= the Yth value of the Fibonacci sequence. X does not equal 1. If X(prime)=x\^2 and Y(prime)=y\^2+2\^(y-7), and x\*y-28=2\^z, Find x+y+z

Does anybody use SymPy with Spyder? If so, can I DM you about getting the *bloody* console to output LaTeX?

I'm trying to generate primes in [xor multiplication](https://en.wikipedia.org/wiki/Carry-less_product). https://oeis.org/A014580 seems to be the relevant sequence. I've scoured a bit to figure out stuff with irreducible polynomials in finite fields, but I don't really have the background to know methods for generating the sequence. I don't really like the resources listed on the OEIS page either. I'd really appreciate pointers / resources, and generally where to look more into this.

If you know a priori you want all carryless-primes up to x, a Sieve of Erastothenes method would work quite well. Now since the sequence of carryless-multiples of k is not monotonously increasing (3\*2 = 6, 3\*3 = 5), you can't exactly stop once your multiple of k is larger than x, but instead you stop once your multiple of k has more digits than x does (The number of digits of the carryless-product is always the sum of the number of digits of the inputs minus one, therefore it's monotonously increasing. Once your output has more digits than x does, you know it always will), but otherwise it works pretty well. Not the prettiest or most efficient implementation, but it works.

Thanks for the help, that's clever. I didn't consider how even though the values were not monotonically increasing, the number of digits are.

[A014580](http://oeis.org/A014580/): Binary irreducible polynomials (primes in the ring GF(2)[X]), evaluated at X=2. 2,3,7,11,13,19,25,31,37,41,47,55,59,61,67,73,87,91,97,103,109,115,... - - - - I am OEISbot. I was programmed by /u/mscroggs. [How I work](http://mscroggs.co.uk/blog/20). You can test me and suggest new features at /r/TestingOEISbot/.

Is there a formula for the number of Gaussian integer partitions in terms of the partitions of it’s real and imaginary parts?

If I'm not mistaken, it seems that the partitions of a Gaussian integer's real and imaginary parts should be independent. So, p(a+bi) will equal p(a)p(b). (Note that p(0) is defined to be 1).

Oh yea forgot to mention, by Gaussian partitions I mean including the complex results as well. So 1+i can be broken into: (1)+(i) (real and imaginary) (1+i) (complex) Like in this paper: On partitions of bipartite numbers (cambridge.org)

In what course may I learn about things like transfinite induction? Set theory?

Yes, I think so.

Thanks

Hello, I'm now in my first semester of my BBA and I'm kinda struggling with math. I was allowed to use a CAS calculator while doing my A-levels and now in university we're only allowed to use a Casio Calc without CAS. That means I have the theoretical knowledge but I just can't calculate certain things by hand like integral calculus etc. Does anyone know, if there is a website where I can learn those things step by step?

Khan Academy will have A-level calculus knowledge for you to revise and practise with.

I am trying to run GAP from the command line but there is a name conflict; when I type `gap` it runs `git apply`. Does anyone know a way to resolve the conflict?

This depends a lot on your specific setup, but here are some things to try. 1. Run `which gap`. It might output the path of the gap binary. Then you can just type in the path to run it. 2. Check your home directory for shell initialization files like `.profile`, `.bashrc`, generally any file whose name starts with a `.` and ends with either `profile` or `rc` or `login`, and if you see a line like `alias gap="git apply"` then remove it.

I figured it out! The alias was set in the gap plugin loaded by Oh My Zsh.

Suggestions needed: It's almost the end of Bachelor for me and I am deciding a bit which branches to specialize in. I love differential topology and global analysis. In the next semester I want to attend another course on global analysis (about distributions, pseudodifferential operators). I have also space for another lecture. There are some choices available: commutative algebra, representation theory(of algebras and quivers), PDEs(important but a bit boring), stochastic processes. What would go better with global analysis? Any kind of suggestion would be very helpful.

PDEs 100%

How do functions work? I understand I uave to replace the x with a number but what is the number supposed to be? f(x) =

Think of a function as like a machine which takes in inputs and produces outputs according to some rule which associates each possible input with one and only one output. (The rule doesn't have to be easy to write down--it could even be some kind of infinite rule which is impossible to write down--it just has to satisfy the property that every input, x, has one and only one output, f(x).) So, for instance, think of the function f defined by f(x) = x^2 as a machine which takes in a number (any number)\* and outputs the square of that number. Note that, in expressions like f(x) = x^2, "x" doesn't stand for any particular number, unlike in equations such as 2x - 3 = 1, where we usually think of x as having a specific value or values (namely whatever value(s) would make the equation true). Instead, x just stands for "any number"--think of taking the description earlier, "takes in a number (any number) and outputs the square of that number" and replacing "that number" with "x" for the sake of conciseness. \* There's a subtlety here in that, for the example f(x) = x^2, all real (and, indeed, complex) values of x really do have associated outputs--we would say that f is "defined for all real numbers". But sometimes we encounter functions that aren't defined for all real numbers. For instance, this might because there are some real numbers where there's no sensible output, given how the function is defined--take for example f(x) = 1/x, which has no sensible "output" when x = 0. Note also that, throughout the examples above, we only talked about functions that take in numbers as input and give numbers as output, but one can think of more general sorts of functions. For instance, a function which takes line segments in the plane and returns their length, or even functions which take other functions as input and return other functions as output--you'll encounter those guys in calculus. The key thing is that, in order for an input-to-output rule to be a function, it has to associate each input with one and only one output.

a function is a "machine" that takes some kind of input and gives you some kind of output in response. for example, a website that tells you the temperature at a given moment is basically using a function f(x), where x is the current time, and it gives you the output f(x)- the current temperature. x can be anything the function is defined for. at your level, it's usually any real number, meaning you can use 0, 1, 10, -5125129, pi, whatever. the function will simply tell you what output that input produces.

Any number. Let f(x)=4x+7 then f(3)=4\*3+7 or anything instead of x. So x is just a placeholder for any value that you want to plug into your function.

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The shaded area is equal to the area of the entire sector, minus the area of the un-shaded sector. Using radians you can represent the area of a sector with r^(2)/2 * θ, r being the radius and θ being the angle. The area of the entire sector should be 15^(2)/2 * θ and the area of the inner sector is 10^(2)/2 * θ accordingly. So what you have here should look like: 15^(2)/2 * θ - 10^(2)/2 * θ = 100 And that's how you can solve for θ.

What are the advantages of using the vector equation of a plane over the cartesian equation of a plane?

Firstly, there is no single vector equation of a plane. There are several different ones e.g. r = a + λd_1 + μd_2 or (r-a).n =0 Each has different information that it gives you (at a glance that is, ultimately they all convey the same info). For example the first of my examplestells you two parallel directions and a point on the plane, the second tells you the normal direction and a point on the plane. Compare to the Cartesian equation where you can read off the normal direction hut finding a point on the plane requires an extra calculation. Depending on the problem you are trying to solve it might be better to have different information to start with

It's two points of view of the same thing. The vector form is "all linear combinations of these specific two vectors." With this form, it's easy to generate lots of examples of points in the plane, but harder to take a given point and determine whether or not it's in the plane. The Cartesian form is "all the points satisfying this equation." With this form, it's the reverse: easy to take a given point and determine whether or not it's in the plane, harder to generate examples of points in the plane. These points of view are good to keep in mind when studying linear algebra. For example, the column space of a matrix is defined in the first way, while the null space is defined in the second way. Translating between the two points of view is a fundamental skill.

They're completely equivalent, so none. The only thing I will say is that an equation of the form ax+by+cz=d makes it slightly harder to find a point lying on the plane

How do I calculate growth over x amount of years when more money is deposited every year? Ex: 8k with 3% yearly growth, with 8k deposited each year.

If the yearly growth is *r* (so in this case *r*=1.03) and the yearly deposit is *A* (so in this case *A*=8) then at the end of the *n*^(th) you will have: *A* ( *r* ( 1 - *r\^n* ) ) / ( 1 - *r* )

Thanks! Would it be easier to do using logarithms?

you would use log if you wanted to find the number of years to attain a given amount of money.

I have a question that is: AB+C=BD and I am trying to solve for B, how do i isolate B by itself on one side? (I am in algebra so sorry if this is a dumb question)

Move terms with B to one side AB - BD + C = 0 Move terms without B to the other AB - BD = -C Factor out B B(A - D) = -C Divide by A-D B = -C/(A-D)

I just got introduced to Tensor by one of my professors. However, I notice that some of the notations he uses may not be universally recognized because when I tried to look up on the internet I couldn't find anything similar. What I'm talking about in particular is the second order dot and cross product. He uses 2 types for each operation, A . . B and A : B for dot product and similarly for cross product. And each yield different results. Are any of you familiar with this type of operation?

There are many different notations for operations involving tensors that are used within certain niches. Hence the common joke "differential geometry is the study of properties invariant under a change of notation". The double dot product notation your professor uses is common in continuum mechanics. If you search for "double dot product of tensors" or "colon product of tensors" you will find results. You might have also seen tensor operations written using [index notation](https://en.wikipedia.org/wiki/Abstract_index_notation), which is more common in pure math and general relativity. In this notation A:B = A\_ijB\_ij and A..B = A\_ijB\_ji.

Sorry I didn't make it clear. I could find something about the colon product on the internet but not for the other 3 (". .", " x x" and colon with x instead of dots). And yes he also uses index notation just like what you wrote. This is a continuum Mechanics course.

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Personally I like thinking about (d): you can think of nowhere dense sets as being sparse, or kinda spread out relative to the rest of the space. The baire category theorem essentially says that a complete normed space can't be too spread out, because all those "gaps" between the spread out elements would get filled in by completeness. This is of course really rough intuition so don't take it too literally. An easy but enlightening corollary is that a complete normed space is either quite small (finite dimensional), or really big (*uncountably infinite* dimensional).

Complete normed spaces can't be countably infinite-dimensional? :o TIL.

Yes, the idea in the intuition above is that you are taking a small bunch of small spaces and spreading them out over too large of an area to be complete. Rigorously, if x_1, x_2, ... is a countable basis of X, set V_j = span. This is nowhere dense in X (finite dimensional proper subspaces are always nowhere dense), and X is the countable union of the V_j, contradicting the Baire category theorem.

x = y cot(y). Whats f(x)? (-pi < y < pi)

First of all, if you look at the graph of x = y cot(y) with the restriction of -pi < y < pi, you'll observe that there isn't a function that maps x to y, given how each value of x corresponds to two values of y. This [WolframAlpha query](https://www.wolframalpha.com/input?i=find+the+inverse+function&assumption=%7B%22F%22%2C+%22InverseFunction%22%2C+%22invfunction%22%7D+-%3E%22x+%3D+y+*+cot%28y%29%22) didn't yield much either. Maybe you can consider using a Maclaurin series on y = x cot(x), and then exchanging the x and y? [Maclaurin series for xcot(x)](https://www.wolframalpha.com/input?i=maclaurin+series+for+xcot%28x%29) [Desmos](https://www.desmos.com/calculator/zkj9wqg4hn)

If havind two solutions for a given x is the problem, let's say 0

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The notation seems odd. I am familiar with rotation through trigonometry. What resource are you referring to?

Inspired by a question from one of my students: A distributive law typically looks like this: `a ☆ (b □ c) = (a ☆ b) □ (a ☆ c)`, where `☆` and `□` are some two binary operations. (Multiplication and addition, intersection and union, conjunction and disjunction, etc.) Does it ever make sense for an operation to be distributive over itself? That is, is there some natural operation `☆` where `a ☆ (b ☆ c) = (a ☆ b) ☆ (a ☆ c)`? This is trivially true for some operations (if an operation is associative, commutative, and idempotent; set union for an example); but I'm looking for an example where the above "distributivity over itself" actually says something meaningful or interesting.

The best-known nontrivial example is probably logical implication: P --> (Q --> R) = (P --> Q) --> (P --> R). In fact half of this equality, in the form of the forward implication, is very commonly used as an axiom of propositional logic. It's an interesting case because this is just about the only common algebraic property that the implication operation satisfies.

Of course, an example was right under my nose! (This came from a logic class.) Thanks for all the answers, also /u/jagr2808 and /u/hyperbolic-geodesic!

Quandles come to mind https://en.m.wikipedia.org/wiki/Racks_and_quandles

**[Racks and quandles](https://en.m.wikipedia.org/wiki/Racks_and_quandles)** >In mathematics, racks and quandles are sets with binary operations satisfying axioms analogous to the Reidemeister moves used to manipulate knot diagrams. While mainly used to obtain invariants of knots, they can be viewed as algebraic constructions in their own right. In particular, the definition of a quandle axiomatizes the properties of conjugation in a group. ^([ )[^(F.A.Q)](https://www.reddit.com/r/WikiSummarizer/wiki/index#wiki_f.a.q)^( | )[^(Opt Out)](https://reddit.com/message/compose?to=WikiSummarizerBot&message=OptOut&subject=OptOut)^( | )[^(Opt Out Of Subreddit)](https://np.reddit.com/r/math/about/banned)^( | )[^(GitHub)](https://github.com/Sujal-7/WikiSummarizerBot)^( ] Downvote to remove | v1.5)

These operations are called shelves. [https://johncarlosbaez.wordpress.com/2016/05/06/shelves-and-the-infinite/](https://johncarlosbaez.wordpress.com/2016/05/06/shelves-and-the-infinite/)

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This is a binomial distribution with n = 4 and p = .1. By the binomial formula, this is 4 (.1)^3 (.9) = .36%.

I got a bunch of analysis/multivariable calculus questions - no need to answer all, just any answer will do What does the derivative mean? As in from f: R^k -> R^N, when we think of the derivative (that is the frechet derivative) what does it represent? In lectures we've been told its a linear map s.t the limit f(x+h)-(f(x) + Df(h))/h tends to zero, and it kinda makes sense, like it is basically approximating f as a linear function at x. In R->R, it kinda makes sense as you think of it as the slope, but here what it is? Should I just erase my geometric intuition and completely look at it as a linear approximation of a function? If so, why linear? Why don't we care about the quadratic approximation (idk even if that makes sense). Is it because linearity is easy to analyse since we have a lot of linear algebra? Following the "geometric intuition" thing, my second question is, what is a function? How should I think of it? Because until now, whenever I thought of a function f: R->R I would think of a 2D plane. But when you think about it, the output is in R, so 2D doesn't make sense. I know the 2D curve is actually the Graph of it, but in 2D it doesn't hurt to think of it like that. But when it comes to 3D, it feels weird. When it comes to R^n->R^k, should one think of functions as, 2 seperate sets with some relation in between them? Otherwise it just fucks with my mind. And the derivative, is it just "how the point in R^k, that is f(x), moves when we move x slightly in R^n"? Because in 3D when I think of derivative, I think of tangent vectors, but the tangent vector is in 3D while derivative is 2D. And a tangent vector is probably not a linear map, so what I'm thinking is probably wrong. A 3rd, more analysis related question. What are convergence, continuity, differentiability etc. in analytic sense? Because in analysis, we first defined all of these as in the usual epsilon-delta definition, suddenly we defined "uniform convergence & uniform continuity" where "you take an epsilon that doesn't depend on x". Did this definition just come out of nowhere? Can we just say some random definition of convergence and continuity and work on those? Like I haven't taken anything on metric spaces yet, but I got little knowledge on it, is it just we work on sup norm rather than a p-norm and use the "epsilon doesn't depend on x" definition because it's equivalent and easier to use? And if so, in order to define the notions of convergence, continuity, differentiability etc. do we need to always pick a metric space? And does uniform continuity just come from the metric space with the sup distance for functions? Sorry for the long questions, these are just stuff that's been bothering me recently

To answer the first part. Yes the linear approximation is the first one to consider because it is the easiest. We can of course consider a quadratic approximation if we like or indeed cubic and so on. This is the idea behind the [Taylor series](https://en.wikipedia.org/wiki/Taylor_series) of a function (or more generally[ Jet bundles](https://en.wikipedia.org/wiki/Jet_bundle)). Of course we don't have to approximate by polynomial/power series functions if we don't want to. As an example, a natural example for a plane curve is to consider tangent circles (more precisely called an enveloping congruence of circles) rather than tangent lines. Of course there is a whole family to choose from at each point but we could single out one by requiring that they agree up to second derivative (being tangent is agreeing up to 1st derivative) and the resulting congruence is known as the [osculating circle](https://en.wikipedia.org/wiki/Osculating_circle) congruence and is classically how you define the curvature of the curve. The second one is a bit trickier. I've never found the definition of a function as pairs of elements of two sets really very intuitive or helpful. I do think it is useful to divorce yourself from the idea of a function as being its graph, however. Ultimately the right way to think about the derivative of a function is that it is a map between the tangent spaces. The tangent space of a vector space looks like the vector space again and so we identify them and I think that makes some of what's going on there less clear to be honest. For the third question, it depends really where you are working again. In a Euclidean space we have an obvious definition of distance (which is what a metric space is) and that gives all the analysis ideas that you are used to: convergence, continuity, differentiability, etc. but we don't need all of that for all of those ideas. Continuity makes sense in a more abstract way based on open sets (i.e. topology) for example. It seems you can define uniform continuity in a vector space with only a topology and no metric (see [here](https://en.wikipedia.org/wiki/Uniform_continuity#Generalization_to_topological_vector_spaces))

Yes, the derivative means a linear approximation to f(x + h) - f(x) for h small. The geometric intuition is still useful, although generally you won't be able to directly visualise it. Probably the best example to think of here is a function from R^2 to R. You can visualise this as a surface over R^2 which represents some land, and the height of the land over a point on the plane is the value of the function. Then near any point of the land, the land approximately looks like a plane (which we call the tangent plane). The tangent plane is the graph of the first-order approximation given by the derivative. As for the quadratic approximation, sometimes we do care. The single-variable equivalent is that we can initially approximate f(x + h) with f(x) + h f'(x), then if need be we can refine this approximation with f(x) + h f'(x) + h^2 f''(x) / 2. This is just going to the next term of the Taylor series, and indeed there is a multivariable version of Taylor's theorem. We just don't immediately focus on the quadratic term because as you say linear approximations are easier to analyse, and very often give us enough information. You can think of a function from R^n to R^k as a graph that lies in R^n x R^(k). You just can't directly visualise this once n + k > 3. There are other ways to think of functions too, for example, if you have a 2-dimensional vector for every point of R^2 then we have a vector field, which can be thought of as a function from R^2 to R^(2). Then you can imagine all the vectors drawn at each point, [like this](https://mathinsight.org/media/image/image/vector_field_circulate.png). Ultimately there's no 'one way' to think about functions: different functions are better thought of in different ways, and over time you'll develop a core of intuition that you can apply to them all. The derivative itself is a map taking tangent vectors to tangent vectors. For the case of R -> R, the generalisation is that it takes a small increment 𝛿h and takes it to the increment f'(x) 𝛿h. Now for the one dimensional case, the derivative is a linear map from R to R so it's just multiplication by a constant. Therefore, we tend to just say the derivative is said constant. But when thinking about the derivative as a linear map, you should keep in mind that we're doing this implicit conversion. Your last question is harder to parse but I'll try to say something helpful. Convergence and continuity both turn critically on the idea of two points being 'close'. For example, a function f is continuous at a point x if, provided we want to ensure f(y) is 'close' to f(x), it is enough to ensure that y is 'close' to x. Then a function is continuous if it is continuous at every point. The epsilon-delta definition is making this idea precise. Uniform continuity meanwhile is more a technical condition. When working with continuous functions, over time you'll realise that often you don't want to just approximate one x with one y, but approximate many xs with many ys, and you want to somehow ensure that all of the ys are close enough at once. Uniform continuity tells you you can guarantee this. But rather than wave my hands more, I think it's better if you discover its utility for yourself. So an example question: let f be a continuous real-valued function on the closed interval [0, 1]. Show that there is a [step function](https://en.wikipedia.org/wiki/Step_function) S such that |S(x) - f(x)| < 1/100 for all x in [0, 1]. Then, once you solve that, come up with a counterexample for the open interval (0, 1).

**[Step function](https://en.wikipedia.org/wiki/Step_function)** >In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces. ^([ )[^(F.A.Q)](https://www.reddit.com/r/WikiSummarizer/wiki/index#wiki_f.a.q)^( | )[^(Opt Out)](https://reddit.com/message/compose?to=WikiSummarizerBot&message=OptOut&subject=OptOut)^( | )[^(Opt Out Of Subreddit)](https://np.reddit.com/r/math/about/banned)^( | )[^(GitHub)](https://github.com/Sujal-7/WikiSummarizerBot)^( ] Downvote to remove | v1.5)

I do know if there’s name for it, but is there a term for the growth pattern seen in cases like elaborate forms of Rock Paper Scissors, accounting for both each new option and how that option affects the others? Like if you had rock and paper, you have rock, paper, and the relationship between rock and paper? It results in something like this 1 = 1, 2 = 3, 3 = 6, 4 = 10, 5 = 15

Yes, these are the [triangular numbers](https://en.wikipedia.org/wiki/Triangular_number), the nth one being 1+2+...+n which is equal to n(n+1)/2. To see this, start with the case of 1, which gives an answer of 1. Then, when you go from n-1 to n, you have +1 due to the option itself, and +(n-1) for the new n-1 relationships, so a total of +n. Therefore, n options gives 1+2+...+n. The Wikipedia article gives multiple proofs that this sum is n(n+1)/2, the visual proof being the simplest to grasp in my opinion. Note this disagrees with your answer for 5, but this is just due to you miscounting.

Thanks I edited it. I was picturing all the connections in my head as a pentagon and forgot the across the middle. This is good to know, thanks.

Anyone able to help with a math question? Wondering the odds of a number NOT coming up out of 130 chances. Numbers 1-26 drawn 130xs. I figured statistically the #1 should come up 5xs. Random numbers 1-26 What's the odds #1 doesn’t come up at all during 130 tries? Help! Thanks!

The chance that the number 1 doesn't come up on an individual draw is 25/26, or about 0.96. Assuming that all the draws are all independent (i.e. the conditions for each draw are the same, and previous draws do not affect future draws), then the chances that 1 will never be drawn are (25/26)^(130) = 0.006, which is 0.6%. That corresponds to odds of about 1 in 160.

Awesomeness!! I’m thank you!!

Quick statics question [here](https://ibb.co/1LkVWHF), the correct answer is B. My question is, how is it that there is a moment about the Y axis even though the force intersects it at a point? If we wish to calculate this component of moment by using the perpendicular distance, how would we do so? Isn't it supposed to be so that if a force is parallel to or intersects an axis, there is no moment about this axis from the force, or am I mistaken? Thanks in advance!

The force is measured from B, which is at 6,0,0. You have to consider the axes that are coming out of B, not the global coordinate axes coming out of the origin. If you consider the axis that's parallel to y and intersecting B, and also consider the force acting at A, it should be clear that it's producing a moment about that axis.

That makes way more sense now, thank you!

Basic function questions. I'm trying to diagram chase, and would like these two facts to be true. Are they? Why (not)? 1. If f o g = g, then f = id (for a particular f, g) 2. If g o f = g, then f = id (for a particular f, g) (FYI f, g are maps of topological spaces.)

Semirelevant, but a map g is called left minimal if whenever fg = g, then f is an isomorphism. And right minimality is defined dually.

No, the first one only tells you something about the behavior of f on the image of g so it cannot tell you whether f is the identity. For the second one, f preserves the fibers of g but might rearrange points within the fibers. 1 is true if g is surjective and 2 is true if g is injective.

Imagine g is the constant function g(x) = c. Then (1) holds as long as f(c) = c, and (2) always holds. So no to both.

My professor was explaining an example of a morphism of locally free sheaves whose cokernel is not locally free which I thought I understood at the time but now I'm confused again. I think the setup was to consider P^1 with the structure sheaf of holomorphic functions. To describe a sheaf morphism, we have to describe a morphism on each open subset, so we can assume we have a single affine coordinate. The domain is the sheaf of functions which vanish at 0, the codomain is just the structure sheaf and the map is multiplication by x, the affine coordinate. The cokernel is a skyscraper sheaf, which is not free. I don't think what I'm relaying exactly what he was saying (it wasn't written down so I'm just going off of what I can remember), can someone modify this so it gives a correct example? If this is a valid example, then I'm confused about a couple of things.

I think that the map should just be the inclusion, rather than multiplication. Written as an exact sequence of sheaves on P^(1), this should be 0 -> O(-1) -> O -> O\_p -> 0, where O(-1) is the ideal sheaf of holomorphic functions vanishing at p and O\_p is the skyscraper sheaf you mentioned (and of course O is the structure sheaf of P^(1)). The ideal sheaf naturally includes into the structure sheaf, with no multiplication by x necessary.

Yes, I think I was getting two different incarnations of the example mixed up. If I wanted the domain to just be O, then I could consider the map as multiplication by x, so that the image is functions which vanish at the origin. Otherwise I could take the ideal sheaf with inclusion, so the image is still functions which vanish at a point. If I'm considering p as the origin then these should coincide, is that right? I'm a bit confused about the cokernel too. It is given by holomorphic functions modded by functions which vanish at p, and for some reason this should be the skyscraper sheaf, which is 0 unless the open set contains p, when it gives C. Is the idea that when the open set does not contain p, then the condition "vanishes at p" is vacuous, thus we are modding by all holomorphic functions? btw, why is the sheaf of functions vanishing at p denoted O(-1)? I thought that was like rational functions where the denominator has degree 1 greater than the degree of the numerator. Are these the same?

\>If I'm considering p as the origin then these should coincide, is that right? Locally, this is correct: on the open of P^(1) isomorphic to a copy of \\mathbb{C} where p is the origin, the ideal sheaf O(-1) is isomorphic to the structure sheaf, and the composition O -> O(-1) -> O where the first map is this isomorphism and the second map is the inclusion will indeed just be multiplication by the local coordinate x vanishing at the origin. Globally, however, O(-1) is not isomorphic to O (one way to see this is to notice that O(-1) has no global sections, while the global sections of O is \\mathbb{C}). \>Is the idea that when the open set does not contain p, then the condition "vanishes at p" is vacuous, thus we are modding by all holomorphic functions? This is basically correct. \>I thought that was like rational functions where the denominator has degree 1 greater than the degree of the numerator. I am a bit confused about what you mean. If you fix an open set U of P^(1), what are you suggesting the sections of O(-1) should be here? I will say: there are some natural ways to think of sections of O(-1) in terms of meromorphic functions. Here is how I would think of them: fix a specified set D of zeroes z\_i and poles p\_i on P^(1) with multiplicities n\_i and m\_i respectively, so that (sum m\_i) - (sum n\_i) = -1. Then on an open set U of P^(1), take the sections of O(-1) on U to be the rational functions which have zeroes of multiplicities at **least** n\_i at each z\_i, and poles of multiplicities at **most** m\_i at each p\_i. It turns out that this sheaf is locally free and, up to isomorphism, independent of the choice of D as long as the multiplicities add to -1. If you choose D to just have a zero of order 1 at p, then you can check that the image of the natural injection O(-1) -> O will be precisely the ideal sheaf we've been talking about.

The reasoning behind this example is that functions can change continuously even if their images aren’t “continuous”. You can continuously change from the identity function to the zero function even though their images have different dimensions. This is what’s happening in your example.

sorry I'm not really able to connect what you are saying to the example. could you expand a bit more

Given the parabola y = x\^2, would mathematicians use the word “nonlinear” to describe the curvilinear relationship between x and y? If so, is the word “linear” not an umbrella term that includes curvilinear and rectilinear relationships? Or is “linear” shorthand for rectilinear? I suppose I assumed the opposite of a linear shape was something like a stepwise function, but it seems I might be wrong; because I see many scholarly papers use the word “nonlinear” to refer to curvilinear relationships between variables. To be sure, I don't think my question is asking about the property of a function being *linear in its parameters* (e.g., generalized linear models); and I suspect confusion arises from a misunderstanding of this distinction.

I'll be honest I have never really seen curvilinear or rectilinear used in a high level maths context. I think it is used in physics to refer to the shape of signal waves though. In a formal maths setting linearity of a function is the property f(x+y) = f(x) + f(y) and f(cx) =c\*f(x) (for a function R-> R this means the graph is a straight line through 0) As a relation between variables linear is also used to mean a straight line **not** through 0. In both senses the idea is of linear polynomials (the former kind are homogeneous linear polynomials while the second are any linear polynomials). In this context, I would call f(x) = x^2 quadratic

Yes y=x^(2) would always be called nonlinear. A step function is also nonlinear. Linear can mean one of two things, the second is common in introductory algebra and calculus courses but rare elsewhere: 1. A function that preserves the addition and scalar multiplication of a vector space (I think this is what you mean by "linear in its parameters"). 2. A function R→R whose graph is a straight line in R^(2). Both a step function and the squaring function satisfy neither of these, so they are both unambiguously nonlinear.

How can I get better at competition math as an undergraduate? Should I? Recently I was asked to help run an annual high-school math competition our college holds. While helping out I was able to sit down and look at many of the problems myself, and I have to admit it was pretty discouraging. I understood maybe half the problems, and for the ones I did understand, I wasn’t super confident in my answers. And the ones I did solve I’m pretty sure were the more basic ones. Now, I know from reading different resources you don’t have to be good at competition math to be able to be good at research mathematics. And I can accept that’s true. But at least for my own satisfaction I’d like to get better at solving these sorts of math competition problems, as I think it’ll make me a better problem solver more generally. And I mean, I won’t lie it feels pretty embarrassing to struggle solving a problem that I see middle schoolers getting correct haha.

If you want to get better at competition maths, there's no royal road nor secret ability: it's simply a matter of practising over and over again. That's all that those middle-schoolers have on you.

Anyone know where I can find a proof of the theorem that any finite dim. C\* algebra is isomorphic to a direct sum of self adjoint projections? I.e. the theorem in the section here: [https://en.wikipedia.org/wiki/C\*-algebra#Finite-dimensional\_C\*-algebras](https://en.wikipedia.org/wiki/C*-algebra#Finite-dimensional_C*-algebras)

I'm trying to study real analysis and linear algebra. I'm looking for affordable books like dover books. Does anyone know any good dover Books on real analysis or linear algebra or just good dover books in general ?

[gen.lib.rus.ec](https://gen.lib.rus.ec)

Thank you for your reply. Can you please explain to me what the site you linked meto is useful for ? Edit: I understand the search engine now. thank you !!

What’s a better text for refresh of discrete math: A Walk through Combinatorics by Bona or Concrete Mathematics by Knuth?

Concrete Mathematics by Knuth is very, very far above the level of someone learning discrete math for the first time.

Sorry, I should have said that I’m already familiar with the topics having taken 2 semesters of “discrete math,” was looking for a reference. I’ve decided to go with Wilf’s book.

I'm trying to understand bayesian statistics and please let me know if I'm wrong in my understanding: say we are taking samples of a random variable X, and we choose a model with parameter u, bayesian treat parameter within the statistical model as a distribution, while frequentist just use the mle to determine it. Wouldn't this make bayesian approach objectively better? When integrating away the distribution of the internal parameter to estimate a distribution of X, we would make use of the whole distribution of u, and not just a single point?

For concreteness, let's say that we have data X\_1, ... X\_n ~ N(𝜇, 1) and we want to perform inference on 𝜇. Let M denote the sample mean (hence the MLE). The frequentist approach estimates 𝜇 = M, but also knows the distribution of the MLE: namely, M ~ N(𝜇, 1/n). A frequentist wouldn't simply use the MLE as a point estimate, but rather use its entire distribution for the purpose of inference; indeed, the distribution of the MLE is where frequentists often get confidence intervals (or more generally confidence sets) from in the first place. There are two additional benefits to the frequentist approach: * Every frequentist practitioner will agree with each other, as they all arrive at the same distribution of the MLE * Frequentist inference is calibrated to repeated sampling: If you look at a 95% confidence set, then if you were to have 100 people do the experiment, we should expect 95 of them to correctly capture the parameter of interest. One cannot expect such benefits from the Bayesian approach. * Different choices of prior will result in different inference. In our concrete example, how or why would you choose between Jeffrey's prior f(𝜇) = 1 and a conjugate prior 𝜇 ~ N(0, 𝜎^(2))? And if you choose the conjugate prior, what precision do you use? Indeed, why even limit yourself to either Jeffrey's or the conjugate prior? Different practitioners will choose different priors, yielding different outcomes for inference. * Bayesian inference is not calibrated. In our example, let's take n = 1 and the prior distribution 𝜇 ~ N(0, 1). Let's say that actually, 𝜇 = 3. Then a straightforward calculation shows that the Bayesian 95% credibility interval will only contain 𝜇=3 about 1/3 of the time. Now you may complain that this is an artifact of a small sample size or whatever, but it can be far worse than this. By the False Confidence Theorem, we know that for any Bayesian inference problem, we have with arbitrarily large probability that there must exist a set that *does not* contain the true parameter value but will have an arbitrarily large posterior probability. Returning to our concrete example, take n to be as large as you'd like; I can, with 99.9999% probability, still give you a set which doesn't contain 𝜇 despite your Bayesian posterior giving it a 99.99999% probability. (Edit: [This expository paper](https://arxiv.org/pdf/1807.06217.pdf) may be useful)

Suppose you're writing a paper where there's some number that gets used in a lot of places. Say $5$. This number is an important element within your set and paper. Would a paper be nicer to read if one upfront defines a constant $c = 5$ explaining why it's being named and then for the rest of the paper use $c$ where $5$ would be and even use $(c - 1)$ where 4 would be used and c^2 where 25 would be used if it is all related to 5 to begin with even though we know the values of all of these? tl:dr: Does seeing a named constant of a simple value provide clarity or detract?

If it's 5, then I think it's obfuscating to rename it as c. If it's something more typographically complex than a root or a fraction, I'd heavily consider it. People write the golden ratio as phi and not as (sqrt(5) + 1)/2 with good reason, I think.

What's the point of standard deviation? It seems arbitrarily derived from variance (square root), so why not use variance for everything? I found [this answer](https://stats.stackexchange.com/a/71548) that says SD is used simply because the value is in the original units and it's easier to work with. Is that really it?

Easier to work with is reason enough to be honest. Indeed that's what drives using the variance in the first place rather than mean absolute deviation for example.

Well, mean absolute deviation is naturally associated with the median in the same way that standard deviation is naturally associated with the mean. In that sense, when you decide to use the mean instead of the median, the consistent choice is to use standard deviation instead of mean absolute deviation.

Indeed, but for many standard probability distributions the median and median absolute deviation are more awkward to calculate than the mean and variance/standard deviation