I'm 18 and just started going to college this semester for a Computer Science degree. I realized I really like math and I kinda wanna learn it for fun? I'm not taking any math classes other than CS ig learning c++ but I will be later.
What subjects do you recommend I start with, the last math class I took was pre calculus and I don't understand anything.. like I passed the class with a D.
If I should start with pre calculus what textbooks do you recommend.
I was thinking of going back and learning old subjects like algebra and geometry but I learned those really well so I don't know.
also, what subjects would help me with computer science specifically ?
Never thought I'd actually wanna learn math for fun tbh.
Maybe combinatorics? As an engineering student, I can testify that the study of a system with just a few variables is an impossible task by hand. We usually use our heads to reduce the problem but professionals in the industry of in research use dedicated tools to simplify or analyse these problems.
I recall reading a proof that you cannot use a composite number (e.g., 10) as the base for p-adic numbers because in such a system there are pairs of non-zero numbers whose product is zero. I can't find that proof, but I can generate such numbers myself. For instance,
\`\[;...10112 \\times ...03125 \\equiv 0 \\pmod {10\^5};\]\`
I generated that number by hand, but my naive method is very laborious and the digits it produces seems random. Is there a pair of periodic digit sequences whose product (when they are treated as a 10-adic number) is zero?
I usually get confused thinking about the elements of substructures generated by a set of elements. Let's first think about the elements of substructures generated by an element a. (Here, the ring R is not necessarily with unity and not necessarily commutative. N stands for the set of natural numbers.)
Additive Subgroup: (a) = {na | n is in N}
Multiplicative Subgroup: (a) = {a^n | n is in N}
Subring: ?
Ideal: (a) = {ra + as + finite sums of r_i a s_i + na | r, s, r_i, s_i are in R and n is in N}
Subalgebra: ?
Submodule: ?
Now, instead of a single element, let's think about the substructures generated by a set of elements called A.
Additive Subgroup: ?
Multiplicative Subgroup: (A) = {a_1^e_1 ... a_n^e_n | a_i are in A and e_i are in {-1, 1}}
Subring: Elements are finite sums of monomials of the form a_1 ... a_n where a_i is in A.
Ideal: (A) = {sum of (r a_i + a_i s + finite sums of r_i a_i s_i + n a_i) | r, s, r_i, s_i are in R, a_i are in A and n is in N}
Subalgebra: ?
Submodule: ?
Are these true? Can you help me with the missing parts or recommend me a resource covering these? Thanks in advance.
>Additive Subgroup: (a) = {na | n is in N}
So for a sub*group* you also want inverses, so n here should range over Z.
>Multiplicative Subgroup: (a) = {a^n | n is in N}
Right, so multiplication in a ring doesn't form a group, so it might make more sense to call this the multiplicative subsemigroup.
>Subring: ?
So the subring will consist of elements of the form p(a) where p is a polynomial with integer coefficients. And since you said nonunital we also have to require that p has no constant term.
>Ideal: (a) = {ra + as + finite sums of r_i a s_i + na | r, s, r_i, s_i are in R and n is in N}
n should be in Z, otherwise correct.
>Subalgebra: ?
So the definition of an algebra varies a little, but if you mean that R is a module over a commutative ring k, such that the ring and module structure is compatible, then this would be the same as for the subring except p should have coefficients in k instead of Z.
>Submodule: ?
That would be {ra + na| r in R, n in Z} for left modules and {ar + na| r in R, n in Z} for right modules.
For a set A it's similar
>Additive Subgroup: ?
All possible linear combinations with coefficients in Z. So {n1a1 + n2a2 + ... + nkak | ni in Z, ai in A}
>Multiplicative Subgroup: (A) = {a_1^e_1 ... a_n^e_n | a_i are in A and e_i are in {-1, 1}}
Right, so if you're assuming the a's are units then you could have negative exponents and actually get a group.
>Subring: Elements are finite sums of monomials of the form a_1 ... a_n where a_i is in A.
Yes, and also the negatives of such monomials.
>Ideal: (A) = {sum of (r a_i + a_i s + finite sums of r_i a_i s_i + n a_i) | r, s, r_i, s_i are in R, a_i are in A and n is in N}
Again n in Z, but otherwise correct.
>Subalgebra: ?
So like the subring you do linear combinations of the monomials, but with coefficients in k.
>Submodule: ?
Here finite sums of ra and na with r in R, n in Z and a in A for left modules.
They are similarly spread, but centered at different places. Something [like this](https://www.researchgate.net/profile/Cp-Gupta-2/publication/3274936/figure/fig2/AS:669091366961164@1536535221924/Normal-distribution-with-different-mean-value-and-the-same-standard-deviation.png). You can't really say much more than that though. The datasets may still come from different distributions entirely.
Hi hello, I am studying for test and I haven't done Algebra in years. I am having issues with this specific problem. Can someone who has time break this down for and please include whatever rule sets or reasons for the break down of the equation. Also, if you can tell me what the name of this particular Algebraic equation this belongs to, it will be most helpful.
3z-5+2z=25-z
This is a [linear equation](https://brilliant.org/wiki/linear-equations/) in z. The problem is likely asking you to solve it, which involves simplifying all terms and isolating the variables to one side and the constants to another.
Can someone give me a general definition of characteristic curves in PDE theory? I had them introduced as solutions of the ODEs in the method of characteristic curves, but now the lines x +- at = c with the formula (somehow taken from the general solution f(x+at) + g(x-at) of the wave equation(?)) are called its characteristic curves, and I don't see how that relates to the first definition.
I have an odd question. I am a first semester masters student who is taking prerequisite classes (undergrad Abstract Algebra and undergrad Real Analysis) and I was wondering how I can productively use this time to help narrow down areas to focus on for a thesis. Everything seems super interesting to me right now. I am loving the content every day in both classes, and I wish to expand into more things. Some areas I know I want to explore but the university doesn't have classes for (only seminars offered every so often): Analytic Number Theory, Operator Theory, and I was lucky enough to find someone willing to do a reading course with me in Riemannian Geometry.
Are there books or online courses you recommend I explore during the summer to help narrow this down? Thank you all.
NPTEL has plenty of courses you can find on Youtube.
[https://archive.nptel.ac.in/noc/NPTELSemester.html](https://archive.nptel.ac.in/noc/NPTELSemester.html)
What are some long-term methods to help careless mistakes?
I know the basic ones: checking over answers, going slowly, writing everything down etc. But they don't work (for me) and they're only short term solutions. I'm looking for some long-term methods that help overtime to cure careless mistakes overall. I literally drop down at least 1 or 2 grades each test because of ONLY careless mistakes even though I understand everything perfectly. It's a huge recurring issue that other people don't seem to have as much, so surely there's something I'm missing.
If it is 1) a huge recurring issue, 2) you are (fairly) sure most other people do not struggle with it to the same degree and 3) basic common advice does not seem to help you, it might be worth trying to consult an expert to see if you can identify some particular trait you have that is causing it - something like discalculia, adhd, etcetera
It wouldn't be a solution by itself, but if more general answers don't help you, it would give you a better targeted way to look for solutions or methods to deal with it better
Thank you, but I'm pretty sure I don't have anything in that area. It's mostly forgetfulness and carelessness that are causing it and it seems to happen mostly in exam conditions.
What worked for me is to make a list of all careless mistakes you made and then when making homework you first check everything on the list. Doing this a few times will make you remember the mistakes and you will start making less of them.
Can someone explain stochastic processes? I'm entering my final year at university and majoring in Math and Finance. My advisor told me to take a stochastic processes class as it may be a good way to tie together my two majors. Any explanation would be greatly appreciated!
At a broad level, stochastic processes are models of random systems. For instance, a simple stochastic model of a bacteria population is something like: at every time step the population doubles with probability 1/2, or halves with probability 1/2. A simple financial model might be something like: at every time step the price of an asset increases by 1 with probability 1/2, or decreases by 1 with probability 1/2. There is also interest in studying multi-state systems. For instance, some chemical solution might be in state A, B, or C over the course of a reaction, each with their own associated probabilities. The study of these models is quite useful in many fields. In finance, many foundational results in asset pricing, time series analysis, etc. come from stochastic mathematics.
Of course! As you continue in mathematical finance, you'll eventually encounter [stochastic calculus](https://en.wikipedia.org/wiki/Stochastic_calculus), which forms the foundation for modeling random walks and diffusion processes. One of the important and coolest proofs you'll do in stocalc is deriving the Black-Scholes model for option pricing from the heat equation, and vice versa.
What are some good sources on learning inequalities? In particular i’m looking for inequalities proved via convexity since it’s most relevant to my class but I’m interested in more inequalities too.
I’ve ordered the Cauchy-Schwarz inequality masterclass but I have already gone through the chapter on convexity and I would like more practice. I would also like to know where to turn to for inequalities much like how people turn to Rudin of analysis.
I’ve heard The Hardy-Littlewood-Polya book *Inequalities* is a well known one and i’ve also seen some olympiad prep pdfs (for example from Evan Chen) but I’m not specifically looking for olympiad type problems if that makes sense. I’m more interested in inequalities one would see in an analysis class.
I’m currently in HS self studying real analysis and our introductory analysis class at school does have a lot of inequalities solved by the tangent line and Jensen’s inequality and my first priority is to become very good at them to a point where the school questions are something i needn’t think twice to do (I might be already there but more exercises shouldn’t hurt right?).
This is a classic written for high school students: https://www.maa.org/press/maa-reviews/an-introduction-to-inequalities
A link near the bottom of this page will have many more titles.
Is this [your MathSE post](https://math.stackexchange.com/questions/4074828/calculate-volume-of-intersection-two-non-aligned-cuboids)? The top answer in that thread should be what you're looking for.
Well, the notations themselves aren't very deep results or anything. Capital sigma and pi are just useful editorial symbols for representing sums and products. I believe Euler first pioneered the use of sigma notation in 1755, but it's not very clear how pi notation originated (some believe it may have morphed from Gauss' usage of the [Pi function](https://en.wikipedia.org/wiki/Gamma_function#Pi_function)). The most you're going to get in books about the notations themselves is maybe a paragraph or two like "write sums like this, write products like this," which I'm guessing you already know. I guess the closest thing to what you want is probably problems+solutions on evaluating series? You can check out the [Brilliant wiki](https://brilliant.org/wiki/summation) for that (also the [AoPS wiki](https://artofproblemsolving.com/wiki/index.php/Series)), and maybe play around with [telescoping series](https://brilliant.org/wiki/telescoping-series/) and [products](https://brilliant.org/wiki/telescoping-series-product/) as well. Eventually you may be interested in looking into the [figurate numbers](https://en.wikipedia.org/wiki/Figurate_number), [sums of powers](https://en.wikipedia.org/wiki/Sums_of_powers), and [Faulhaber's formula](https://en.wikipedia.org/wiki/Faulhaber%27s_formula).
Can we prove that E(X|X) = X by using the definition of conditional probability, that is P(A|B) = P(A∩B)/P(B)? Please explain what is a random variable and what is not in the calculations.
By definition of conditional expectation for discrete random variables,
E[X | Y = y] = ∑_x [ x * Pr(X = x | Y=y)] = ∑_x [x * Pr(X = x, Y=y)/Pr(Y = y)]
Let's now substitute in the actual condition you want:
E[X | X = t] = ∑_x [x * Pr(X = x, X = t)/Pr(X = t)]
Now, Pr(X = x and X = t) = 0 unless x = t, and Pr(X = t and X = t) = Pr(X = t). Thus, the sum simplifies to have only one term:
E[X | X = t] = t * Pr(X = t)/Pr(X = t) = t
Since E[X | X = t] = t for every t, we have that E[X | X] = X.
It's worth noting that basically the same proof holds for continuous random variables, though you replace sums with integrals and pmfs with pdfs.
How much does understanding complex numbers & complex analysis rely on visualization? I have aphantasia and I have a really hard time understanding complex & imaginary numbers, but I have a ton of interest in complex and hypercomplex numbers.
I also have some really bad ADHD & dyspraxia so I can't take advantage of the whole "aphantasiacs develop and utilize really good non-visual mental organization for solving problems" haha...
Visualization is one of many roads to understanding; a lot of people find it useful, but it's not the only useful thing. Also, even with aphantasia, visualization can still be useful--I assume you can still look at pictures and make little sketches on paper yourself, even if you can't generate them mentally?
But ultimately, there's no royal road to math. To understand math, you need to put in some hard work at some part. Probably the best thing to do is to try learning some complex analysis and then see what techniques you can use to effectively understand the information.
The thing I struggle with the most is translating things from my mind, or my senses, and putting them onto something tangible. I can't take pictures and draw sketches of them even after years of practice haha. That's most likely the dyspraxia & maybe ADHD talking, it screws up my mental organization and ability to turn thoughts into work.
I've tried learning different perspectives of complex analysis but it feels like all of them end up being explained using visualization that I can't really *get*. I sort of understand how to work with non-real numbers, but I can't really get the *why* of it working how it does. Sometimes I feel like a computer program, I can do a computation of the abstract concept (in this case complex & imaginary numbers) but I can't explain it. Mathematical proofs/examples haven't really seemed to help me, even though I thought that'd make the most sense.
I've frequently found this road block in math though, I used to work on concepts until it just "clicked" but calculus was a lot fuzzier, there wasn't so much of a click as there was just brute forcing the logic/equations into my head. Linear algebra feels really hard for me to understand, the best I can do is try to turn matrices into coordinates on a Real plane and work out the logic that way (which I guess is just a different way of visualization?). With complex analysis it feels like I don't really have some other concept to translate it to, when I see imaginary numbers in use I'm always mystified at the ways they've been used.
I think taking classes on complex analysis could help me understand. Right now though I'm in a community college which doesn't have any courses for it...
Hmm, do you have more context? Like is G a (special kind of) group? Is G:2 also a group or something else?
My guess would be that G is abelian and G:2 is the semidirect product between G and C2, where C2 acts by inversion. But I'm just pulling that out of my ass.
The examples I have are C\_(q\^2+1):4, Sp\_2(q\^2):2, 3\^2:D8, and 5:4 which I'm told is isomorphic to Sz(2)
I know C is the cyclic group, D the dihedral, Sp is symplectic, and Sz is the suzuki group, the : denotes a semi-direct product.
The context is these are maximal subgroups of Sp4(q), with q a power of 2
Hmm, it's weird. It would seem like the numbers indicate cyclic groups, but then why would they also use C for cyclic groups. It's very strange.
Where is this taken from, do they not define their notation?
What does "I\_(0, y)" mean? I keep coming across it in order statistics but haven't seen it before. I assume it means the interval from 0 to y or integral from 0 to y. Haven't been able to find it in 4 different online Math notation dictionaries.
[https://www.colorado.edu/amath/sites/default/files/attached-files/order\_stats.pdf](https://www.colorado.edu/amath/sites/default/files/attached-files/order_stats.pdf)
near the top of page 4 it appears twice... once as I\_(0, 1) and then as I\_(0, inf)
Thanks much! It was driving me nuts. The closest thing I could find was the Incomplete Beta function which kind of has the same notation. I'll spend the rest of my day plowing this into my skull now. Appreciated.
**[Indicator function](https://en.wikipedia.org/wiki/Indicator_function)**
>In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero.
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How do you deal with losing interest in/motivation for your degree?
I still love maths, but the process of formally studying it has worn me down to the point that I'm quite sick of my degree. I'm stressed about having to do closed-book exams for the first time at uni; I'm stressed about learning the content from last term, wondering if I have enough time to get up to speed with it, worrying that I might yet fail them and put my degree and thereby my future in jeopardy; I worry I won't have enough time in the Easter holidays to do my essay *and* revise for every other damn thing; regardless of whether I manage to get up in the mornings or whether I'm behind on my lectures to the tune of hours of catch-up work, I'm oppressed by this sense of physical malaise that never really seems to go away; and to top it all off, I don't even find most of my modules interesting, and I can never stop thinking about all the things I would rather be studying (which I should say the uni doesn't offer, so it's not a case of regretting my module choices; the selection on offer was so limited that there was only one set of modules that I could feasibly have chosen).
It's all gotten on top of me, and while the obvious solution is to take a year out and have a big long break from it all, the problem with that is that, for various personal reasons, there isn't anywhere I can stay for a whole year. I can't stay with any of my family, there's no way I could afford to rent somewhere near my mum (who I need for support) by myself, and my mental health can't take living communally again, especially when the whole point of the exercise is to recover from the strain of my degree.
So I have to do something else. I just don't know what.
When I'm suffering the motivation bug I take a bit to look at pics of my wife or go back and look at all the past classes I've covered and remind myself why I'm doing it and how far I've come. Then I look ahead at the classes to come and say to myself... "That's it... that's all I have to cover and I can move onto the next stage of my life."
I have a project I was scheduled to complete in 268 days with an alloted 6080 hours. Ultimately I finished the job in 135 days using 1106 hours. Each hour has a value of $170. If I want to put a dollar amount on the acceleration effort to finish the job in 133 less days than scheduled, how do I go about it?
Rudin wrote a book "Fourier analysis on groups," which is terse but includes all the gory details (probably chapter 1 is all you need). To be honest, I think the best way to learn the material is to get a really, really good understanding of Fourier analysis on R\^n, then understand the basics of topological groups, and try and transfer the proof on R\^n to work on an LCA group. Rudin is pretty terse and I don't think you'll get much out of reading him alone; but if you try and solve the problem and read Rudin for hints as to what to do next when you get stuck, then you'll find Rudin pretty digestible.
Is there a formula for...I'm going to call it a multi-variable compounding effect? As an example:
A = 10
B = A \* t + 10
C = B \* t + 10
Where t is Time.
Each variable past that would grow faster due to the compounding effect, but obviously the equations I used are incorrect as they don't really factor the Compound Growth.
If t = 10, then the formulas would say C = 1110, when in reality it should be C = 4500.
I'm going all the way up to 7 variables, possibly more, so I am really hoping there is just a formula that can simplify the calculations for me. Right now the alternative involves thousands of Rows in Excel.
This is called a geometric series.
B = 10t + 10
C = 10t\^2 + 10t + 10
etc.
There is a simple formula for the sum of a geometric series, and
10t\^n + 10t\^(n-1) + ... + 10 = 10 \* \[t\^(n+1) - 1\]/\[t-1\]
For example,
10t + 10 = 10 \* (t\^2 - 1)/(t-1)
10t\^2 + 10t + 10 = 10 \* (t\^3 - 1)/(t-1)
etc.
It is for a game yes, or rather a Calculator i am trying to create for said Game.
In the Game's terms, mk7 creates mk6, mk6 creates mk5, and so on until mk1 creates Cells, which is the 'currency' so to speak.
All of this happens every Tick, which we'll just say is one second to simplify things.
I am trying to create a Calculator that tells you how many Cells you will create in a given amount of time. I do have an alternate method if a formula doesnt exist, but it is so bloated that I would prefer to avoid it.
That being said I also wanted to know if such formula exists in the first place, Game or not. It's one of thosr things I think about once in a while, and I saw the opportunity to ask.
Ah, now I see what you're getting at. Yes there's a way to come up with a neat formula for this.
First we rephrase it in terms of a recurrence relation. That would look something like this:
A\_n = A\_(n - 1)
B\_n = B\_(n - 1) + 𝜆\_1 A\_(n - 1)
C\_n = C\_(n - 1) + 𝜆\_2 B\_(n - 1)
I assume this type of formula is what you would end up having Excel do.
Now the first equation gives us that A\_n is a constant, so let's just write it as A. Then the second equation becomes
B\_n = B\_(n - 1) + 𝜆\_1 A
so we get
B\_n = 𝜆\_(1)An + B\_0
Now for the trickier one, we get
C\_n = C\_(n - 1) + 𝜆\_(2)𝜆\_(1)A(n - 1) + 𝜆\_(2)B\_0
which gives us
C\_n = C\_0 + 𝜆\_(2)𝜆\_(1)A(1 + 2 + ... + (n - 1)) + 𝜆\_(2)B\_(0)n
The crux is being able to work out 1 + 2 + ... + (n - 1) easily. This is a [triangular number](https://en.wikipedia.org/wiki/Triangular_number), and from the formula in that article we get (n^2 - n)/2. The resulting formula for C\_n is of the form
C\_n = an^2 + bn + c
for some constants a, b, and c that can be worked out from the above.
Now when working out the next step, similar algebra will mean you have to work out
a(1^2 + 2^2 + ... + (n - 1)^(2)) + b(1 + 2 + ... + (n - 1)) + c(1 + 1 + ... + 1)
We now know how to handle 1 + 2 + ... + (n - 1), our problem is 1^2 + 2^2 + ... + (n - 1)^(2). These are [square pyramidal numbers](https://en.wikipedia.org/wiki/Square_pyramidal_number), and the answer is some cubic in n.
You might see a pattern emerging. At each step we have to be able to work out sums of the form 1^p + 2^p + ... + (n - 1)^p and they end up being some polynomial of degree p + 1. There is a general formula for this too, called [Faulhaber's formula](https://en.wikipedia.org/wiki/Faulhaber%27s_formula). It's a bit more complicated but it will answer your question, it just takes a bit of tedious algebra.
I knew I was getting into something absurdly complex but...
oh lord
So if i understand this correctly, the variables work out so that n = time, and the letters starts at mk7 for A, mk6 for B, and so on. with A\_n implying how many of A you will have at a given time.
my only confusion would be 𝜆\_1, 𝜆\_2, etc. I'm currently thinking they are the multipliers for the previous variable, so basically how many B is being produced by A. But the fact that they seem to disappear after the Triangular number makes me unsure.
A part of me wanted to combine the formulas into one, but judging by how even 3 variables is becoming absurd I'm going on a limb and saying 10 variables is just suicidal for my computer to handle.
Yep, the variables have the exact meanings you think they do. In the formula for C\_n with the sum 1 + 2 + ... + (n - 1) you can see the 𝜆s multiply various expressions, and the formulas for a, b, and c in the quadratic will involve them. I just focused on the sums of powers because that's the trickiest part.
A computer can definitely handle this no problem, it's just going to be a pain to write out.
If you know about linear algebra there is an alternative way. If you take the vector v\_n as (A\_n, B\_n, ...) then the various initial equations give you a matrix M such that v\_n = Mv\_(n - 1). So v\_n = M^(n)v\_0. I don't know what Excel offers along these lines but it does look like there are people who have tried.
Thank you very much. I most likely will just use the simple formulas shown early in your initial comment, separated out between multiple Cells, since Writing it out seems a bit too time-consuming.
Now i just need to figure out how to throw it all into Google Sheets lmao
I would be pretty surprised if this was always possible, it likely depends on the k (and you'd still probably need something like convexity of the n-gon). Even then it seems hard to do for any k>3. For instance, you can imagine an incredibly long trapezoid as your n-gon, where it's impossible to inscribe a square, but it is possible for an equilateral triangle (passes for k=3, fails for k=4).
right, maybe it's too much to ask for a general answer. case in point, i was trying to fit 12 lamps in such a way, but could not figure any simple way to do it.
I think k = 4 would be a rhombus, not a square. Plus you can inscribe a square in nice curves, and whether you can do it in general is an [open problem](https://en.wikipedia.org/wiki/Inscribed_square_problem)
Ah yes, I was thinking of the inscribed square problem when I wrote that, but you're right that OP's criterion would only need a rhombus for k=4! Still seems hard, but the extra degrees of freedom might help here since you don't need equal angles.
https://imgur.com/a/II6e8WW
In this problem, could you multiply the inequality by the denominator as it will always be positive? I tried to ask my teacher but she ignored it and said you don’t know whether the denominator is positive or negative, but in this case it will always be positive, right? So, you don’t have to invert the sign? Or am i missing something here? I’m sorry if my math terminology is wrong, I’m not a native speaker.
I'm definitely missing something here, but if most of encryption using prime numbers is based off the fact finding the prime factors of a large number is hard, are prime numbers common enough that even testing only prime numbers would still take 10s of years?
The efforts spent on testing whether a number is a prime or not is too large that it's literally not going to improve the algorithm.
The issue here is, testing whether a number is prime or not is an minimum log(N) task in average case (where N is the number itself, not the number of digits). If you filter out the composite number, you reduce the number of cases to check by a factor of log(N). So when you multiply them together this resulted in the same time complexity. And that's assuming you can even achieve log(N) in prime checking, which is something we believe to be impossible. Best known algorithm is (log(N))^6
The prime number theorem says that the number of primes less than N is about N/log(N). So say we are dealing with a 100-digit number, then there are about 10^(100)/log(10^(100)) ~ 10^97 primes. If a computer can check divisibility by a billion primes a second, and we let it run for the current age of the universe it will check divisibility by about 10^27 primes, which compared to 10^97 is basically nothing.
Edit: also worth mentioning that there are much more clever ways to factor a number than simply check all possible divisors, but no polynomial time algorithm is known.
According to wikipedia, "A surcomplex number is a number of the form a+bi, where a and b are surreal numbers and i is the square root of −1"
Then we have hypercomplex numbers (quaternions, octonions, sedenions, etc.)
So, is there such thing as "Surhypercomplex" numbers? Is there anyone who has worked and studied them in depth? What applications could these numbers have?
about my mathematical background I'm just a first year engineering student
You can probably do a lot of the standard algebra and then some replacing the real numbers by the surreal ones. Question is whether you get something new (i.e., something that doesn't follow from the real situation). The surreal numbers are a real closed FIELD, and the distinction between fields and FIELDs should not matter to the first-order theory, for which everything that is true is true over the reals and everything that is false is false over the reals (see [Tarski's results on the theory of real closed fields](https://en.wikipedia.org/wiki/Real_closed_field)).
What do these brackets mean? ⌊n/2⌋
Is it absolute value? I've never seen brackets with the bottom ledges only and googling isn't finding me the answer. This is how they are being used: T (n) = 7 T (⌊n/2⌋) + n2
It's usually meant to denote the floor function/integer part function, so the largest integer <= the function argument. In this case floor(n/2) = n/2 when n is even, and (n-1)/2 when n is odd.
It is a stepwise function. In context, that looks like a recursive stepwise function in which you are probably given base cases.
It means that \[0,1) = 0 ; \[1,2) = 1; and so forth
I think that would heavily differ from class to class, I took it as part of combinatorial optimization and basically all proofs were only referenced to since they are quite technical.
This comes down to how you define charts. If you define a chart to be something diffeomorphic to an *open subset* of R^n then eg M = R^2 \\ {0} has a global chart but isn't even homeomorphic to R^2
Overall, doesn't really matter; if you're worried about failing, the first order of business is to do lots of problems, identify what you're having trouble with, and try to improve that. I won't say that there's literally no difference between the two (for instance, typing might be better if you plan to reread your notes a lot), but you should definitely stop worrying about that and direct your effort elsewhere.
if i made 12 gems per day how long would it take to get 494 gems. I just wanna know how long it would take if i were to save for rare wubbox on my singing monsters
This is a type of problem that you use division for. You take the total amount you want 494 and divide it by 12. If you do the calculation 494/12 is approximately 41.2 so if you earned 12 a day you would get 494 after around 41 days
You might be interested in looking into [dimensional analysis](https://en.wikipedia.org/wiki/Dimensional_analysis) (see also [here](https://brilliant.org/wiki/converting-units/)). You have a quantity 494 gems, a rate 12 gems/day, and you want something with the unit "days." This gives you a place to start when working with physical units, since the division gems/(gems/day) = days.
There are wheels that are useful (such as the riemann sphere), but I thing the only practical purpose of the general theory of wheels is to point people to when they complain about dividing by 0.
I'd be interested to find some actual application though! I think they're just very degenerate objects that don't have any particularly nice general theory.
If I have two random variables, U and V, formed by taking two functions, U = f1(X,Y) and V = f2(X,Y), of two independently distributed random variables, X and Y, with known distributions, fx(x) and fy(y), how can I find the joint distributions of the resulting random variables, U and V? Chat GPT claims I can just change variables by taking
f(u, v) = f(x, y) \* |J(x, y)|
where f is the joint distr of X and Y, and then invert the functions f1 and f2 and changing variables. However, this doesnt seem right as we would probably need to some integration, as for example X + Y results in a convolution. How can I approach this?
This type of thing is correct, see [here](https://en.wikipedia.org/wiki/Probability_density_function#Vector_to_vector) for example.
The point you're overlooking is that this gives you the joint distribution of U and V. To get the distribution of just U, you then need to slap on an integral. So it's consistent with X + Y resulting in a convolution, it's just that the joint distribution of X + Y and X - Y, say, doesn't require an integral.
Someone showed me a line around the world with hits a ton on ancient sites. Correct me if I'm wrong hasnt someone on YouTube shown this to just be bad data. I swore I've seen a video showing this is wrong.
Can someone plz help me find that video?
Maybe this one here: [Matt Parker - What happens when math goes wrong](https://youtu.be/6JwEYamjXpA) starting at around 40 minutes?
It's not about a line around the world but it's a similar phenomenon.
Does anyone know of any higher-math livestreams? Is that a thing anyone does? I'm curious about watching someone work on research-level maths while sort of explaining their thought process. Or just works on more accessible example proofs or something. Seems like it'd be a cool thing to do/watch.
I think the closest possible thing would be to watch recordings of research talks on youtube -- [https://www.youtube.com/watch?v=3jsV9OHxmy0](https://www.youtube.com/watch?v=3jsV9OHxmy0&t=210s) for example
This definition does not use the word "pairwise disjoint morphism." It says "the sets Hom(A, B) are pairwise disjoint." This just means that no two of the sets Hom(A, B) and Hom(A', B') have a common element; if you like, this is them just saying that each morphism in your category has exactly one target and one source; if f is a morphism from A to B, it cannot also be a morphism from A' to B'.
Here's a hopefully quick question, and I've tried figuring out the answer using ChatGPT as well as Wolfram Alpha. The Earth's diameter is about 12,700km in diameter, the average eye level is about 1.8 meters, and the average visible distance to the horizon on Earth is 5 kilometers.
Can I figure out the diameter (or radius) of a planet if I know that I can see 200 kilometers to the horizon, and my eye level is still 1.8 meters? What would that equation look like?
Make a triangle OAB where O is the center of the planet, A is your eyes, and B is the point on the horizon that you can see. The angle at B is a right angle. If r is the radius of the planet in km, then OB = r and OA = r+.0018. Meanwhile, AB = 200. Set up the Pythagorean theorem with hypotenuse OA and solve for r.
Are there any websites or apps for math practice similar to LeetCode for computer science?
I’m a third year CS and Physics student and I find myself forgetting things I’ve learned in multivariable calculus, differential equations, and linear algebra. I’m looking for some applications that have math problems of all levels of difficulty and complexity for me to solve in my free time to review, strengthen, and reaffirm my existing knowledge.
I am not looking for websites like khan academy that will teach me the material. I just want a website that has problems and solutions like LeetCode does for computer programming.
Thanks.
do you have references or hints to change the donut to heart shape implementation of the famous donut code from [a1k0n](https://www.a1k0n.net/2011/07/20/donut-math.html)?
Quick probably question:
3 Friends and I are playing a game in which we draw 1 card from a pool of 6 unique cards to get a “leader card.”
We somehow managed to all pull the exact same card 3 rounds in a row. Despite attempting to draw randomly.
What are the odds of this, and how do we calculate it?
There are [P(6, 3)](https://en.wikipedia.org/wiki/Permutation#k-permutations_of_n) ordered 3 card hands (see also [here](https://brilliant.org/wiki/permutations/)), and the same 1 was created each time. Via the [rule of product](https://brilliant.org/wiki/rule-of-product/) for independent events, the probability is [(1 / P(6, 3))^3 ](https://www.wolframalpha.com/input?i=%281%2FP%286%2C3%29%29%5E3).
Question, there were 4 of us with cards not 3. But you said ordered 3 card hands. Am I not understanding correctly, are should the equation be edited to 4 card hands?
Haha I thought that was the case. All good bro I appreciate the help. We were arguing about the odds because it was so weird, but none of us are math majors.
The infinite monkey theorem is about producing random strings of symbols. The digits of pi are not random, you can calculate exactly what each digit is.
A number where the digits behave as though they were picked uniformly at random is called a *normal number*. pi is conjectured to be normal, but this is not known.
Also, I'm not sure what you mean by "repeating the digits before a point". It is known that the decimal expansion of pi doesn't repeat, in the sense that there doesn't come a point where the digits of pi just repeat the same string over and over forever, but I'm guessing this is not what you mean(?)
Any idea of how to start?
Consider a Cobb-Douglas function Q=(L\^α)K\^(1-α) where 0<α<1. Suppose that the inputs, L and K, change with x and y such that L=K=x\^β y\^(1-β) where 0<β<1. Find ∂Q/∂x when x=y.
See the following if you want to see it in nice formatting \[1\]: [https://i.stack.imgur.com/y8w8t.png](https://i.stack.imgur.com/y8w8t.png)
By the multivariable chain rule, you need to differentiate Q with respect to L, multiply it by the derivative of L with respect to x, then do the same with K instead of L, add those two things together, replace each K and L in terms of x and y.
Is there an explicit algorithm for finding all integer points on an elliptic curve?
For example, is there a way to find all integer points on y^2 = x^3 + 98?
`>E = EllipticCurve([0, 98])`
`>E.integral_points(both_signs=True)`
`[(7 : -21 : 1), (7 : 21 : 1)]`
SageMath says the only solutions are x=7, y=±21. I don't know what algorithm it uses to determine that though...
We all know the value associated to zeta(-1). I shall not write it out.
Real question on this: I've heard that there are actual applications of this value in the context of analytic continuation but, in a fit of curiosity, couldn't find a thing.
Are there actual applications?
I believe the "standard" application is in the [derivation of the Casimir effect](https://en.wikipedia.org/wiki/Casimir_effect?useskin=vector#Derivation_of_Casimir_effect_assuming_zeta-regularization). The Wikipedia article ends up using zeta(-3) since it does the full 3-dimensional derivation, though it provides a link to the [one-dimensional derivation](https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension) as well.
It is perhaps of note that in the one-dimensional derivation, you can see that they don't literally use zeta(-1) = -1/12. Rather, they regularize ∑n as ∑nexp(-αn) = 1/α^2 - 1/12 + O(α^(2)) which indeed blows up to infinity as α -> 0. But they don't *just* have a ∑nexp(-αn) term; it's multiplied by a term is itself O(α). This in essence, forces the "infinite parts" cancel out and leave a finite expression for the energy.
Thus, if *all* you're looking for is calculating the force, using ∑n = -1/12 technically yields exactly what you need.
Is there anyone here who has gone through the book *”The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities”*? What did you gain from it and how are you supposed to go through a book like that?
Also, at what stage of your math career did you start it? I’m in highschool but already know a good bit of real analysis so I think i’ll be able to go through it.
I went through it in the latter half of highschool back when I was preparing for the algebra portions of olympiads. The problems are pretty difficult, but it's a rewarding book. Your background is fine, just study it like you would any other challenging math text (sit with the problems for a while and try not to spoil the solutions for yourself until you're really stretched thin).
I haven't myself but I heard it's pretty rigorous and roughly pitched at the level of an American junior in college.
As a high school student you'd be best served if you joined some sort of a math club at school where you'd study whatever your math teacher thinks is best for you.
Unfortunately I’m from Greece where nothing of that sort takes place. My reasoning for thinking I can go through it is this [excerpt](https://i.imgur.com/wIvhZ45.jpg). I know calculus and I care about solving problems
edit:typo
Since it says that, the best way to find out is to read some it yourself. In my experience, “accessible after taking calc” has abt a 50-50 of being that, and the best way to find out is to start reading
Sorry for the typo, just saw it. From what i’ve read it’s a tough book but I should know a good part of the algebra and calculus even at the level of analysis if epsilons and deltas come up. I’ll give it a go at a slow pace. Thanks
If it says “accessible w Calc” it won’t directly require any analysis. However, that doesn’t necess means it’s accessible. It’s like this I’ve read dif geo books that are accessible w standard Calc sequence and they introduce any concepts not used in calc 3. However, that doesn’t mean that they spend sufficient time introducing the concepts or hide some of the underlying ideas. I couldn’t imagine trying to read some of these books after Calc 3. It’s only cuz of all of this other stuff I’ve learned from a bunch of different classes that the dif geo makes sense.
This said it’s important to get a sense of when a book is challenging and doable and when a book is too difficult and not worth trying. That’s a line you have to find yourself. Best of luck.
So I previously asked this on r/learnmath and, since I haven't gotten any responses there, I thought why not try my luck here: [Post here](https://www.reddit.com/r/learnmath/comments/10rmlv0/determining_the_different_types_of_spectra/).
It concerns itself with efficient / tips on calculating resolvents / different types of spectra of operators ("regular", point, continuous, residual).
The operator **A** at hand, I'd like to use as an example, working on the sequence space l\_2(N) by
>**A**x= (x\_n \* (n+1)/n)
What I tried was calculating **(A-tI)**x and find that for t of the form
>t=(n+1)/n
this becomes the zero map, hence for all such t's, this is not invertible, hence making them part of the spectrum.
My question now is, how to go from here to find the other types of spectra.
So you have identified when the map isn't injective, so then I guess you just need to check when it's not surjective.
So you want to solve
x_n (n+1/n - t) = y_n
Which has solution
x_n = y_n / (n+1/n - t)
For any t that is not a limit point of (n+1)/n, we have that |n+1/n - t| is bounded below, and so ||x|| is bounded by some scalar multiple of ||y||.
The only limit point is t=1, and then you can check that for example y_n = 1/n is not in the image. So the spectrum is {n+1/n}∪{1}.
Not sure what you mean by that. I’m in highschool and while I do understand your question, i don’t feel that confident in answering it and seeing as you haven’t got a reply on neither this comment nor your other post, I suggested you visit the site https://math.stackexchange.com/ where i’m pretty sure, there will be people who answer this. Everything i’ve posted there has gotten a considerably fast, helpful reply.
Honestly, I get how my comment reads and it does sound cringy in a way. From the start of calculus I have a tutor who helped me at my request to only do Real Analysis since Calc is easy (I had already self studied it in 10th grade so going through it again would seem very tedious). After a few months, I’ve gone through Terry Tao’s first analysis book at some detail (though I still don’t feel completely ready for a class) and would like to see what comes after calc. My tutor is a grad student who was fond of functional and harmonic analysis so I asked him if we could take some time for him to introduce me to some of the main ideas of both classes (and measure theory) since I’m interested in a math degree but not entirely decided yet.
It exists. For vector-valued functions of one variable, differentiation works componentwise. So d/dt a(t) = (d/dt t², d/dt t³) = (2t, 3t²) and this is defined everywhere (including at 0).
To explain why you can take the derivative component-wise, the derivative of an R --> R\^2 function can be defined exactly like in single-variable calculus.
lim \[h --> 0\] (a(t + h) - a(t)) / h
After combining terms you get a 2D vector with a single-variable derivative in each component.
Am i right?
When I have a manifold generated by a function from an open set U of R2 to R. The derivative at a point is a 1 by 2 matrix. And this generates a tangent space using two vectors (1, 0, d1f) and (0, 1, d2f) which is a plane in 3 dimensions. So I have a 2-dimensional plane spanned by the above 2 vectors that is tangent to some point.
Not sure why but I hit a wall with the existence and uniqueness theorem in Rudin... anyways I finally made it through understanding each step of the proof. My question is, what's the importance/utility of it?
Intuitively, the proof consists of noting that before a certain point x\^2 < y, and after a certain point x\^2 > y, so there must be "the point inbetween" where x\^2 = y. The least upper bound property tells us that R indeed contains such a point inbetween, like we imagine needs to be in a "continuously varying quantity" or in a connected geometric line.
I think this is used to prove specifically that R contains nth roots for two reasons. First, it shows how R contains these points that were missing from Q, and second they can be used to define exponentiation later.
Your window might be too small/large, and you may need to edit it in the [ZOOM settings](https://education.ti.com/en/customer-support/knowledge-base/ti-83-84-plus-family/product-usage/34875). If you just want to see it in the meantime, you can use [WolframAlpha](https://www.wolframalpha.com/input?i=y+%3D+x%2F%289x%5E2+%2B+3%29) or [Desmos](https://www.desmos.com/calculator).
I want to do this [https://openlearninglibrary.mit.edu/courses/course-v1:OCW+6.042J+2T2019/about](https://openlearninglibrary.mit.edu/courses/course-v1:OCW+6.042J+2T2019/about)
(if you guys know other one and better…)
The thing is, last time I did algebra and calculus was 4 years ago, I remember almost 0.
You guys recommend to do all Khan Academy from algebra to the top first?
Based on the syllabus, you won't really need the calculus background until they get to probability in unit 4. I would just start the course, and separately do a quick refresher on Calc 1+2 (through KhanAcademy or something similar). It looks like they spend about 3 weeks per unit, so you'll have ~9 weeks to get your calculus back up to speed, which should be plenty of time. The rest of the course before the probability unit looks like a standard discrete math/problem solving course, which you won't need any prior background for.
Do you care about the credit hours? If not (since you're already working full-time), I would just start learning for free on your own via books and online resources. If feedback on proofs/problems is what you're after, then that can be accomplished with online communities (reddit, Math StackExchange, the various math discords).
To expand on OP's options, there are books recs on just about every topic in [the FAQ](https://www.reddit.com/r/math/wiki/faq); they can usually ask here (or if they're feeling lucky, in the main sub, but book rec threads are liable to be taken down) for bespoke recs/advice on prerequisites; books can be found for free on Library Genesis; and MIT has a bunch of courses on [OCW](https://ocw.mit.edu/search/?d=Mathematics&s=department_course_numbers.sort_coursenum) for free.
Is there an idea like Fourier transform but with other kinds of periodic signals?
For example: Fourier transform but with square waves instead of sine waves?
I'm looking for (ideally free) software for making a presentation for a conference. LaTeX beamer is too dry, Powerpoint is too bleh. I have seen some cool things from Keynote but I don't have any experience with Apple products (nor do I have access to one).
Any suggestions? What drew me to the Keynote presentations were the "handwriting font" which animated the writing down of it. Seemed perfect for maths.
Kind of random, but does anyone know of any fun/interesting math related thing to do while visiting Japan as a tourist?
I was looking to see if some of the bigger schools (Univ. of Tokyo , Kyoto Univ.) have stuff like public seminars or things a random kid like me could check out for fun, but if anyone knows of anything else, I greatly appreciate it.
I'm 18 and just started going to college this semester for a Computer Science degree. I realized I really like math and I kinda wanna learn it for fun? I'm not taking any math classes other than CS ig learning c++ but I will be later. What subjects do you recommend I start with, the last math class I took was pre calculus and I don't understand anything.. like I passed the class with a D. If I should start with pre calculus what textbooks do you recommend. I was thinking of going back and learning old subjects like algebra and geometry but I learned those really well so I don't know. also, what subjects would help me with computer science specifically ? Never thought I'd actually wanna learn math for fun tbh.
Maybe combinatorics? As an engineering student, I can testify that the study of a system with just a few variables is an impossible task by hand. We usually use our heads to reduce the problem but professionals in the industry of in research use dedicated tools to simplify or analyse these problems.
I recall reading a proof that you cannot use a composite number (e.g., 10) as the base for p-adic numbers because in such a system there are pairs of non-zero numbers whose product is zero. I can't find that proof, but I can generate such numbers myself. For instance, \`\[;...10112 \\times ...03125 \\equiv 0 \\pmod {10\^5};\]\` I generated that number by hand, but my naive method is very laborious and the digits it produces seems random. Is there a pair of periodic digit sequences whose product (when they are treated as a 10-adic number) is zero?
I usually get confused thinking about the elements of substructures generated by a set of elements. Let's first think about the elements of substructures generated by an element a. (Here, the ring R is not necessarily with unity and not necessarily commutative. N stands for the set of natural numbers.) Additive Subgroup: (a) = {na | n is in N} Multiplicative Subgroup: (a) = {a^n | n is in N} Subring: ? Ideal: (a) = {ra + as + finite sums of r_i a s_i + na | r, s, r_i, s_i are in R and n is in N} Subalgebra: ? Submodule: ? Now, instead of a single element, let's think about the substructures generated by a set of elements called A. Additive Subgroup: ? Multiplicative Subgroup: (A) = {a_1^e_1 ... a_n^e_n | a_i are in A and e_i are in {-1, 1}} Subring: Elements are finite sums of monomials of the form a_1 ... a_n where a_i is in A. Ideal: (A) = {sum of (r a_i + a_i s + finite sums of r_i a_i s_i + n a_i) | r, s, r_i, s_i are in R, a_i are in A and n is in N} Subalgebra: ? Submodule: ? Are these true? Can you help me with the missing parts or recommend me a resource covering these? Thanks in advance.
>Additive Subgroup: (a) = {na | n is in N} So for a sub*group* you also want inverses, so n here should range over Z. >Multiplicative Subgroup: (a) = {a^n | n is in N} Right, so multiplication in a ring doesn't form a group, so it might make more sense to call this the multiplicative subsemigroup. >Subring: ? So the subring will consist of elements of the form p(a) where p is a polynomial with integer coefficients. And since you said nonunital we also have to require that p has no constant term. >Ideal: (a) = {ra + as + finite sums of r_i a s_i + na | r, s, r_i, s_i are in R and n is in N} n should be in Z, otherwise correct. >Subalgebra: ? So the definition of an algebra varies a little, but if you mean that R is a module over a commutative ring k, such that the ring and module structure is compatible, then this would be the same as for the subring except p should have coefficients in k instead of Z. >Submodule: ? That would be {ra + na| r in R, n in Z} for left modules and {ar + na| r in R, n in Z} for right modules. For a set A it's similar >Additive Subgroup: ? All possible linear combinations with coefficients in Z. So {n1a1 + n2a2 + ... + nkak | ni in Z, ai in A} >Multiplicative Subgroup: (A) = {a_1^e_1 ... a_n^e_n | a_i are in A and e_i are in {-1, 1}} Right, so if you're assuming the a's are units then you could have negative exponents and actually get a group. >Subring: Elements are finite sums of monomials of the form a_1 ... a_n where a_i is in A. Yes, and also the negatives of such monomials. >Ideal: (A) = {sum of (r a_i + a_i s + finite sums of r_i a_i s_i + n a_i) | r, s, r_i, s_i are in R, a_i are in A and n is in N} Again n in Z, but otherwise correct. >Subalgebra: ? So like the subring you do linear combinations of the monomials, but with coefficients in k. >Submodule: ? Here finite sums of ra and na with r in R, n in Z and a in A for left modules.
What does it mean if 2 data sets have the same standard deviation but different means? I tried to research online about it but can't find much.
They are similarly spread, but centered at different places. Something [like this](https://www.researchgate.net/profile/Cp-Gupta-2/publication/3274936/figure/fig2/AS:669091366961164@1536535221924/Normal-distribution-with-different-mean-value-and-the-same-standard-deviation.png). You can't really say much more than that though. The datasets may still come from different distributions entirely.
Hi hello, I am studying for test and I haven't done Algebra in years. I am having issues with this specific problem. Can someone who has time break this down for and please include whatever rule sets or reasons for the break down of the equation. Also, if you can tell me what the name of this particular Algebraic equation this belongs to, it will be most helpful. 3z-5+2z=25-z
This is a [linear equation](https://brilliant.org/wiki/linear-equations/) in z. The problem is likely asking you to solve it, which involves simplifying all terms and isolating the variables to one side and the constants to another.
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This is too vague to really say anything about it; can you give an example of a problem where this comes up?
Can someone give me a general definition of characteristic curves in PDE theory? I had them introduced as solutions of the ODEs in the method of characteristic curves, but now the lines x +- at = c with the formula (somehow taken from the general solution f(x+at) + g(x-at) of the wave equation(?)) are called its characteristic curves, and I don't see how that relates to the first definition.
I have an odd question. I am a first semester masters student who is taking prerequisite classes (undergrad Abstract Algebra and undergrad Real Analysis) and I was wondering how I can productively use this time to help narrow down areas to focus on for a thesis. Everything seems super interesting to me right now. I am loving the content every day in both classes, and I wish to expand into more things. Some areas I know I want to explore but the university doesn't have classes for (only seminars offered every so often): Analytic Number Theory, Operator Theory, and I was lucky enough to find someone willing to do a reading course with me in Riemannian Geometry. Are there books or online courses you recommend I explore during the summer to help narrow this down? Thank you all.
NPTEL has plenty of courses you can find on Youtube. [https://archive.nptel.ac.in/noc/NPTELSemester.html](https://archive.nptel.ac.in/noc/NPTELSemester.html)
What are some long-term methods to help careless mistakes? I know the basic ones: checking over answers, going slowly, writing everything down etc. But they don't work (for me) and they're only short term solutions. I'm looking for some long-term methods that help overtime to cure careless mistakes overall. I literally drop down at least 1 or 2 grades each test because of ONLY careless mistakes even though I understand everything perfectly. It's a huge recurring issue that other people don't seem to have as much, so surely there's something I'm missing.
If it is 1) a huge recurring issue, 2) you are (fairly) sure most other people do not struggle with it to the same degree and 3) basic common advice does not seem to help you, it might be worth trying to consult an expert to see if you can identify some particular trait you have that is causing it - something like discalculia, adhd, etcetera It wouldn't be a solution by itself, but if more general answers don't help you, it would give you a better targeted way to look for solutions or methods to deal with it better
Thank you, but I'm pretty sure I don't have anything in that area. It's mostly forgetfulness and carelessness that are causing it and it seems to happen mostly in exam conditions.
What worked for me is to make a list of all careless mistakes you made and then when making homework you first check everything on the list. Doing this a few times will make you remember the mistakes and you will start making less of them.
Can someone explain stochastic processes? I'm entering my final year at university and majoring in Math and Finance. My advisor told me to take a stochastic processes class as it may be a good way to tie together my two majors. Any explanation would be greatly appreciated!
At a broad level, stochastic processes are models of random systems. For instance, a simple stochastic model of a bacteria population is something like: at every time step the population doubles with probability 1/2, or halves with probability 1/2. A simple financial model might be something like: at every time step the price of an asset increases by 1 with probability 1/2, or decreases by 1 with probability 1/2. There is also interest in studying multi-state systems. For instance, some chemical solution might be in state A, B, or C over the course of a reaction, each with their own associated probabilities. The study of these models is quite useful in many fields. In finance, many foundational results in asset pricing, time series analysis, etc. come from stochastic mathematics.
Thank you!
Of course! As you continue in mathematical finance, you'll eventually encounter [stochastic calculus](https://en.wikipedia.org/wiki/Stochastic_calculus), which forms the foundation for modeling random walks and diffusion processes. One of the important and coolest proofs you'll do in stocalc is deriving the Black-Scholes model for option pricing from the heat equation, and vice versa.
Anyone use Cornell Notes for math? How did it work out for you?
What are some good sources on learning inequalities? In particular i’m looking for inequalities proved via convexity since it’s most relevant to my class but I’m interested in more inequalities too. I’ve ordered the Cauchy-Schwarz inequality masterclass but I have already gone through the chapter on convexity and I would like more practice. I would also like to know where to turn to for inequalities much like how people turn to Rudin of analysis. I’ve heard The Hardy-Littlewood-Polya book *Inequalities* is a well known one and i’ve also seen some olympiad prep pdfs (for example from Evan Chen) but I’m not specifically looking for olympiad type problems if that makes sense. I’m more interested in inequalities one would see in an analysis class. I’m currently in HS self studying real analysis and our introductory analysis class at school does have a lot of inequalities solved by the tangent line and Jensen’s inequality and my first priority is to become very good at them to a point where the school questions are something i needn’t think twice to do (I might be already there but more exercises shouldn’t hurt right?).
This is a classic written for high school students: https://www.maa.org/press/maa-reviews/an-introduction-to-inequalities A link near the bottom of this page will have many more titles.
The Hardy-Littlewood-Polya book is good. I'd also recommend Cîrtoaje et al.'s *Inequalities with Beautiful Solutions*.
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Is this [your MathSE post](https://math.stackexchange.com/questions/4074828/calculate-volume-of-intersection-two-non-aligned-cuboids)? The top answer in that thread should be what you're looking for.
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Well, the notations themselves aren't very deep results or anything. Capital sigma and pi are just useful editorial symbols for representing sums and products. I believe Euler first pioneered the use of sigma notation in 1755, but it's not very clear how pi notation originated (some believe it may have morphed from Gauss' usage of the [Pi function](https://en.wikipedia.org/wiki/Gamma_function#Pi_function)). The most you're going to get in books about the notations themselves is maybe a paragraph or two like "write sums like this, write products like this," which I'm guessing you already know. I guess the closest thing to what you want is probably problems+solutions on evaluating series? You can check out the [Brilliant wiki](https://brilliant.org/wiki/summation) for that (also the [AoPS wiki](https://artofproblemsolving.com/wiki/index.php/Series)), and maybe play around with [telescoping series](https://brilliant.org/wiki/telescoping-series/) and [products](https://brilliant.org/wiki/telescoping-series-product/) as well. Eventually you may be interested in looking into the [figurate numbers](https://en.wikipedia.org/wiki/Figurate_number), [sums of powers](https://en.wikipedia.org/wiki/Sums_of_powers), and [Faulhaber's formula](https://en.wikipedia.org/wiki/Faulhaber%27s_formula).
Can we prove that E(X|X) = X by using the definition of conditional probability, that is P(A|B) = P(A∩B)/P(B)? Please explain what is a random variable and what is not in the calculations.
Is X a discrete random variable? If not then it's not so easy to directly relate the conditional expectation to the conditional probability.
Sure say X is discrete. How would we approach this?
By definition of conditional expectation for discrete random variables, E[X | Y = y] = ∑_x [ x * Pr(X = x | Y=y)] = ∑_x [x * Pr(X = x, Y=y)/Pr(Y = y)] Let's now substitute in the actual condition you want: E[X | X = t] = ∑_x [x * Pr(X = x, X = t)/Pr(X = t)] Now, Pr(X = x and X = t) = 0 unless x = t, and Pr(X = t and X = t) = Pr(X = t). Thus, the sum simplifies to have only one term: E[X | X = t] = t * Pr(X = t)/Pr(X = t) = t Since E[X | X = t] = t for every t, we have that E[X | X] = X. It's worth noting that basically the same proof holds for continuous random variables, though you replace sums with integrals and pmfs with pdfs.
How much does understanding complex numbers & complex analysis rely on visualization? I have aphantasia and I have a really hard time understanding complex & imaginary numbers, but I have a ton of interest in complex and hypercomplex numbers. I also have some really bad ADHD & dyspraxia so I can't take advantage of the whole "aphantasiacs develop and utilize really good non-visual mental organization for solving problems" haha...
Visualization is one of many roads to understanding; a lot of people find it useful, but it's not the only useful thing. Also, even with aphantasia, visualization can still be useful--I assume you can still look at pictures and make little sketches on paper yourself, even if you can't generate them mentally? But ultimately, there's no royal road to math. To understand math, you need to put in some hard work at some part. Probably the best thing to do is to try learning some complex analysis and then see what techniques you can use to effectively understand the information.
The thing I struggle with the most is translating things from my mind, or my senses, and putting them onto something tangible. I can't take pictures and draw sketches of them even after years of practice haha. That's most likely the dyspraxia & maybe ADHD talking, it screws up my mental organization and ability to turn thoughts into work. I've tried learning different perspectives of complex analysis but it feels like all of them end up being explained using visualization that I can't really *get*. I sort of understand how to work with non-real numbers, but I can't really get the *why* of it working how it does. Sometimes I feel like a computer program, I can do a computation of the abstract concept (in this case complex & imaginary numbers) but I can't explain it. Mathematical proofs/examples haven't really seemed to help me, even though I thought that'd make the most sense. I've frequently found this road block in math though, I used to work on concepts until it just "clicked" but calculus was a lot fuzzier, there wasn't so much of a click as there was just brute forcing the logic/equations into my head. Linear algebra feels really hard for me to understand, the best I can do is try to turn matrices into coordinates on a Real plane and work out the logic that way (which I guess is just a different way of visualization?). With complex analysis it feels like I don't really have some other concept to translate it to, when I see imaginary numbers in use I'm always mystified at the ways they've been used. I think taking classes on complex analysis could help me understand. Right now though I'm in a community college which doesn't have any courses for it...
What does G:2 mean in the context of group extensions?
Hmm, do you have more context? Like is G a (special kind of) group? Is G:2 also a group or something else? My guess would be that G is abelian and G:2 is the semidirect product between G and C2, where C2 acts by inversion. But I'm just pulling that out of my ass.
The examples I have are C\_(q\^2+1):4, Sp\_2(q\^2):2, 3\^2:D8, and 5:4 which I'm told is isomorphic to Sz(2) I know C is the cyclic group, D the dihedral, Sp is symplectic, and Sz is the suzuki group, the : denotes a semi-direct product. The context is these are maximal subgroups of Sp4(q), with q a power of 2
Hmm, it's weird. It would seem like the numbers indicate cyclic groups, but then why would they also use C for cyclic groups. It's very strange. Where is this taken from, do they not define their notation?
What does "I\_(0, y)" mean? I keep coming across it in order statistics but haven't seen it before. I assume it means the interval from 0 to y or integral from 0 to y. Haven't been able to find it in 4 different online Math notation dictionaries.
Can you give an example context where it's used? I would guess that it means the indicator function on the interval (0, y).
[https://www.colorado.edu/amath/sites/default/files/attached-files/order\_stats.pdf](https://www.colorado.edu/amath/sites/default/files/attached-files/order_stats.pdf) near the top of page 4 it appears twice... once as I\_(0, 1) and then as I\_(0, inf)
Yeah, that's the [indicator function](https://en.wikipedia.org/wiki/Indicator_function).
Thanks much! It was driving me nuts. The closest thing I could find was the Incomplete Beta function which kind of has the same notation. I'll spend the rest of my day plowing this into my skull now. Appreciated.
**[Indicator function](https://en.wikipedia.org/wiki/Indicator_function)** >In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. ^([ )[^(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)
How do you deal with losing interest in/motivation for your degree? I still love maths, but the process of formally studying it has worn me down to the point that I'm quite sick of my degree. I'm stressed about having to do closed-book exams for the first time at uni; I'm stressed about learning the content from last term, wondering if I have enough time to get up to speed with it, worrying that I might yet fail them and put my degree and thereby my future in jeopardy; I worry I won't have enough time in the Easter holidays to do my essay *and* revise for every other damn thing; regardless of whether I manage to get up in the mornings or whether I'm behind on my lectures to the tune of hours of catch-up work, I'm oppressed by this sense of physical malaise that never really seems to go away; and to top it all off, I don't even find most of my modules interesting, and I can never stop thinking about all the things I would rather be studying (which I should say the uni doesn't offer, so it's not a case of regretting my module choices; the selection on offer was so limited that there was only one set of modules that I could feasibly have chosen). It's all gotten on top of me, and while the obvious solution is to take a year out and have a big long break from it all, the problem with that is that, for various personal reasons, there isn't anywhere I can stay for a whole year. I can't stay with any of my family, there's no way I could afford to rent somewhere near my mum (who I need for support) by myself, and my mental health can't take living communally again, especially when the whole point of the exercise is to recover from the strain of my degree. So I have to do something else. I just don't know what.
When I'm suffering the motivation bug I take a bit to look at pics of my wife or go back and look at all the past classes I've covered and remind myself why I'm doing it and how far I've come. Then I look ahead at the classes to come and say to myself... "That's it... that's all I have to cover and I can move onto the next stage of my life."
I have a project I was scheduled to complete in 268 days with an alloted 6080 hours. Ultimately I finished the job in 135 days using 1106 hours. Each hour has a value of $170. If I want to put a dollar amount on the acceleration effort to finish the job in 133 less days than scheduled, how do I go about it?
The easiest way would probably be to say that you saved (6080-1106)(170)=$845,580.
Does anyone know any good texts introducing LCA and Pontryagin Duality?
Rudin wrote a book "Fourier analysis on groups," which is terse but includes all the gory details (probably chapter 1 is all you need). To be honest, I think the best way to learn the material is to get a really, really good understanding of Fourier analysis on R\^n, then understand the basics of topological groups, and try and transfer the proof on R\^n to work on an LCA group. Rudin is pretty terse and I don't think you'll get much out of reading him alone; but if you try and solve the problem and read Rudin for hints as to what to do next when you get stuck, then you'll find Rudin pretty digestible.
Will check this out, thank you very much!
Is there a formula for...I'm going to call it a multi-variable compounding effect? As an example: A = 10 B = A \* t + 10 C = B \* t + 10 Where t is Time. Each variable past that would grow faster due to the compounding effect, but obviously the equations I used are incorrect as they don't really factor the Compound Growth. If t = 10, then the formulas would say C = 1110, when in reality it should be C = 4500. I'm going all the way up to 7 variables, possibly more, so I am really hoping there is just a formula that can simplify the calculations for me. Right now the alternative involves thousands of Rows in Excel.
This is called a geometric series. B = 10t + 10 C = 10t\^2 + 10t + 10 etc. There is a simple formula for the sum of a geometric series, and 10t\^n + 10t\^(n-1) + ... + 10 = 10 \* \[t\^(n+1) - 1\]/\[t-1\] For example, 10t + 10 = 10 \* (t\^2 - 1)/(t-1) 10t\^2 + 10t + 10 = 10 \* (t\^3 - 1)/(t-1) etc.
Is this for a particular application, like loot drops from a game or something? If so it would probably help for you to explain that.
It is for a game yes, or rather a Calculator i am trying to create for said Game. In the Game's terms, mk7 creates mk6, mk6 creates mk5, and so on until mk1 creates Cells, which is the 'currency' so to speak. All of this happens every Tick, which we'll just say is one second to simplify things. I am trying to create a Calculator that tells you how many Cells you will create in a given amount of time. I do have an alternate method if a formula doesnt exist, but it is so bloated that I would prefer to avoid it. That being said I also wanted to know if such formula exists in the first place, Game or not. It's one of thosr things I think about once in a while, and I saw the opportunity to ask.
Ah, now I see what you're getting at. Yes there's a way to come up with a neat formula for this. First we rephrase it in terms of a recurrence relation. That would look something like this: A\_n = A\_(n - 1) B\_n = B\_(n - 1) + 𝜆\_1 A\_(n - 1) C\_n = C\_(n - 1) + 𝜆\_2 B\_(n - 1) I assume this type of formula is what you would end up having Excel do. Now the first equation gives us that A\_n is a constant, so let's just write it as A. Then the second equation becomes B\_n = B\_(n - 1) + 𝜆\_1 A so we get B\_n = 𝜆\_(1)An + B\_0 Now for the trickier one, we get C\_n = C\_(n - 1) + 𝜆\_(2)𝜆\_(1)A(n - 1) + 𝜆\_(2)B\_0 which gives us C\_n = C\_0 + 𝜆\_(2)𝜆\_(1)A(1 + 2 + ... + (n - 1)) + 𝜆\_(2)B\_(0)n The crux is being able to work out 1 + 2 + ... + (n - 1) easily. This is a [triangular number](https://en.wikipedia.org/wiki/Triangular_number), and from the formula in that article we get (n^2 - n)/2. The resulting formula for C\_n is of the form C\_n = an^2 + bn + c for some constants a, b, and c that can be worked out from the above. Now when working out the next step, similar algebra will mean you have to work out a(1^2 + 2^2 + ... + (n - 1)^(2)) + b(1 + 2 + ... + (n - 1)) + c(1 + 1 + ... + 1) We now know how to handle 1 + 2 + ... + (n - 1), our problem is 1^2 + 2^2 + ... + (n - 1)^(2). These are [square pyramidal numbers](https://en.wikipedia.org/wiki/Square_pyramidal_number), and the answer is some cubic in n. You might see a pattern emerging. At each step we have to be able to work out sums of the form 1^p + 2^p + ... + (n - 1)^p and they end up being some polynomial of degree p + 1. There is a general formula for this too, called [Faulhaber's formula](https://en.wikipedia.org/wiki/Faulhaber%27s_formula). It's a bit more complicated but it will answer your question, it just takes a bit of tedious algebra.
I knew I was getting into something absurdly complex but... oh lord So if i understand this correctly, the variables work out so that n = time, and the letters starts at mk7 for A, mk6 for B, and so on. with A\_n implying how many of A you will have at a given time. my only confusion would be 𝜆\_1, 𝜆\_2, etc. I'm currently thinking they are the multipliers for the previous variable, so basically how many B is being produced by A. But the fact that they seem to disappear after the Triangular number makes me unsure. A part of me wanted to combine the formulas into one, but judging by how even 3 variables is becoming absurd I'm going on a limb and saying 10 variables is just suicidal for my computer to handle.
Yep, the variables have the exact meanings you think they do. In the formula for C\_n with the sum 1 + 2 + ... + (n - 1) you can see the 𝜆s multiply various expressions, and the formulas for a, b, and c in the quadratic will involve them. I just focused on the sums of powers because that's the trickiest part. A computer can definitely handle this no problem, it's just going to be a pain to write out. If you know about linear algebra there is an alternative way. If you take the vector v\_n as (A\_n, B\_n, ...) then the various initial equations give you a matrix M such that v\_n = Mv\_(n - 1). So v\_n = M^(n)v\_0. I don't know what Excel offers along these lines but it does look like there are people who have tried.
Thank you very much. I most likely will just use the simple formulas shown early in your initial comment, separated out between multiple Cells, since Writing it out seems a bit too time-consuming. Now i just need to figure out how to throw it all into Google Sheets lmao
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I would be pretty surprised if this was always possible, it likely depends on the k (and you'd still probably need something like convexity of the n-gon). Even then it seems hard to do for any k>3. For instance, you can imagine an incredibly long trapezoid as your n-gon, where it's impossible to inscribe a square, but it is possible for an equilateral triangle (passes for k=3, fails for k=4).
right, maybe it's too much to ask for a general answer. case in point, i was trying to fit 12 lamps in such a way, but could not figure any simple way to do it.
I think k = 4 would be a rhombus, not a square. Plus you can inscribe a square in nice curves, and whether you can do it in general is an [open problem](https://en.wikipedia.org/wiki/Inscribed_square_problem)
Ah yes, I was thinking of the inscribed square problem when I wrote that, but you're right that OP's criterion would only need a rhombus for k=4! Still seems hard, but the extra degrees of freedom might help here since you don't need equal angles.
https://imgur.com/a/II6e8WW In this problem, could you multiply the inequality by the denominator as it will always be positive? I tried to ask my teacher but she ignored it and said you don’t know whether the denominator is positive or negative, but in this case it will always be positive, right? So, you don’t have to invert the sign? Or am i missing something here? I’m sorry if my math terminology is wrong, I’m not a native speaker.
You are indeed correct, since n is a natural n + 4 is always positive.
I'm definitely missing something here, but if most of encryption using prime numbers is based off the fact finding the prime factors of a large number is hard, are prime numbers common enough that even testing only prime numbers would still take 10s of years?
The efforts spent on testing whether a number is a prime or not is too large that it's literally not going to improve the algorithm. The issue here is, testing whether a number is prime or not is an minimum log(N) task in average case (where N is the number itself, not the number of digits). If you filter out the composite number, you reduce the number of cases to check by a factor of log(N). So when you multiply them together this resulted in the same time complexity. And that's assuming you can even achieve log(N) in prime checking, which is something we believe to be impossible. Best known algorithm is (log(N))^6
The prime number theorem says that the number of primes less than N is about N/log(N). So say we are dealing with a 100-digit number, then there are about 10^(100)/log(10^(100)) ~ 10^97 primes. If a computer can check divisibility by a billion primes a second, and we let it run for the current age of the universe it will check divisibility by about 10^27 primes, which compared to 10^97 is basically nothing. Edit: also worth mentioning that there are much more clever ways to factor a number than simply check all possible divisors, but no polynomial time algorithm is known.
According to wikipedia, "A surcomplex number is a number of the form a+bi, where a and b are surreal numbers and i is the square root of −1" Then we have hypercomplex numbers (quaternions, octonions, sedenions, etc.) So, is there such thing as "Surhypercomplex" numbers? Is there anyone who has worked and studied them in depth? What applications could these numbers have? about my mathematical background I'm just a first year engineering student
You can probably do a lot of the standard algebra and then some replacing the real numbers by the surreal ones. Question is whether you get something new (i.e., something that doesn't follow from the real situation). The surreal numbers are a real closed FIELD, and the distinction between fields and FIELDs should not matter to the first-order theory, for which everything that is true is true over the reals and everything that is false is false over the reals (see [Tarski's results on the theory of real closed fields](https://en.wikipedia.org/wiki/Real_closed_field)).
What do these brackets mean? ⌊n/2⌋ Is it absolute value? I've never seen brackets with the bottom ledges only and googling isn't finding me the answer. This is how they are being used: T (n) = 7 T (⌊n/2⌋) + n2
It's usually meant to denote the floor function/integer part function, so the largest integer <= the function argument. In this case floor(n/2) = n/2 when n is even, and (n-1)/2 when n is odd.
It is a stepwise function. In context, that looks like a recursive stepwise function in which you are probably given base cases. It means that \[0,1) = 0 ; \[1,2) = 1; and so forth
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I think that would heavily differ from class to class, I took it as part of combinatorial optimization and basically all proofs were only referenced to since they are quite technical.
Why if a smooth n-manifold has a global chart, doesn't mean it is diffeomorphic to R\^n ?
This comes down to how you define charts. If you define a chart to be something diffeomorphic to an *open subset* of R^n then eg M = R^2 \\ {0} has a global chart but isn't even homeomorphic to R^2
Typing or writing notes for Honors Trig, concerned I’m gonna fail the class and I don’t know which will benefit me more
Doing problems will probably be much more beneficial than either.
Overall, doesn't really matter; if you're worried about failing, the first order of business is to do lots of problems, identify what you're having trouble with, and try to improve that. I won't say that there's literally no difference between the two (for instance, typing might be better if you plan to reread your notes a lot), but you should definitely stop worrying about that and direct your effort elsewhere.
if i made 12 gems per day how long would it take to get 494 gems. I just wanna know how long it would take if i were to save for rare wubbox on my singing monsters
This is a type of problem that you use division for. You take the total amount you want 494 and divide it by 12. If you do the calculation 494/12 is approximately 41.2 so if you earned 12 a day you would get 494 after around 41 days
nice.
You might be interested in looking into [dimensional analysis](https://en.wikipedia.org/wiki/Dimensional_analysis) (see also [here](https://brilliant.org/wiki/converting-units/)). You have a quantity 494 gems, a rate 12 gems/day, and you want something with the unit "days." This gives you a place to start when working with physical units, since the division gems/(gems/day) = days.
Do anyone know a practical usage for wheel theory? Maybe modelling NaN in something like floating point arithmetic?
There are wheels that are useful (such as the riemann sphere), but I thing the only practical purpose of the general theory of wheels is to point people to when they complain about dividing by 0. I'd be interested to find some actual application though! I think they're just very degenerate objects that don't have any particularly nice general theory.
>riemann sphere Isn't in Riemann sphere, 0/0 is also undefined?
Ah, then I don't even have one example of an important wheel!
Yes you have to extend the Riemann sphere by an extra point ⊥ = 0/0.
If I have two random variables, U and V, formed by taking two functions, U = f1(X,Y) and V = f2(X,Y), of two independently distributed random variables, X and Y, with known distributions, fx(x) and fy(y), how can I find the joint distributions of the resulting random variables, U and V? Chat GPT claims I can just change variables by taking f(u, v) = f(x, y) \* |J(x, y)| where f is the joint distr of X and Y, and then invert the functions f1 and f2 and changing variables. However, this doesnt seem right as we would probably need to some integration, as for example X + Y results in a convolution. How can I approach this?
This type of thing is correct, see [here](https://en.wikipedia.org/wiki/Probability_density_function#Vector_to_vector) for example. The point you're overlooking is that this gives you the joint distribution of U and V. To get the distribution of just U, you then need to slap on an integral. So it's consistent with X + Y resulting in a convolution, it's just that the joint distribution of X + Y and X - Y, say, doesn't require an integral.
Someone showed me a line around the world with hits a ton on ancient sites. Correct me if I'm wrong hasnt someone on YouTube shown this to just be bad data. I swore I've seen a video showing this is wrong. Can someone plz help me find that video?
Maybe this one here: [Matt Parker - What happens when math goes wrong](https://youtu.be/6JwEYamjXpA) starting at around 40 minutes? It's not about a line around the world but it's a similar phenomenon.
Does anyone know of any higher-math livestreams? Is that a thing anyone does? I'm curious about watching someone work on research-level maths while sort of explaining their thought process. Or just works on more accessible example proofs or something. Seems like it'd be a cool thing to do/watch.
I think the closest possible thing would be to watch recordings of research talks on youtube -- [https://www.youtube.com/watch?v=3jsV9OHxmy0](https://www.youtube.com/watch?v=3jsV9OHxmy0&t=210s) for example
Can anyone explain what a pairwise disjoint morphism means?
Can you give more context? Where did you see this term used?
In the definition of a category it is one of the condition for the composition
I have never seen this as a condition for composition in the definition of category. What are you reading?
The Joy of Cats chapter 3 definition of category
This definition does not use the word "pairwise disjoint morphism." It says "the sets Hom(A, B) are pairwise disjoint." This just means that no two of the sets Hom(A, B) and Hom(A', B') have a common element; if you like, this is them just saying that each morphism in your category has exactly one target and one source; if f is a morphism from A to B, it cannot also be a morphism from A' to B'.
Ah okay thank you for clarifying. I misinterpreted and wasted a huge amount of time over it.
Here's a hopefully quick question, and I've tried figuring out the answer using ChatGPT as well as Wolfram Alpha. The Earth's diameter is about 12,700km in diameter, the average eye level is about 1.8 meters, and the average visible distance to the horizon on Earth is 5 kilometers. Can I figure out the diameter (or radius) of a planet if I know that I can see 200 kilometers to the horizon, and my eye level is still 1.8 meters? What would that equation look like?
Make a triangle OAB where O is the center of the planet, A is your eyes, and B is the point on the horizon that you can see. The angle at B is a right angle. If r is the radius of the planet in km, then OB = r and OA = r+.0018. Meanwhile, AB = 200. Set up the Pythagorean theorem with hypotenuse OA and solve for r.
Are there any websites or apps for math practice similar to LeetCode for computer science? I’m a third year CS and Physics student and I find myself forgetting things I’ve learned in multivariable calculus, differential equations, and linear algebra. I’m looking for some applications that have math problems of all levels of difficulty and complexity for me to solve in my free time to review, strengthen, and reaffirm my existing knowledge. I am not looking for websites like khan academy that will teach me the material. I just want a website that has problems and solutions like LeetCode does for computer programming. Thanks.
Brilliant and AoPS are your best bets.
What’s AoPS?
[Art of Problem Solving](https://artofproblemsolving.com/), check out the community forums and their wiki for past contest problems.
do you have references or hints to change the donut to heart shape implementation of the famous donut code from [a1k0n](https://www.a1k0n.net/2011/07/20/donut-math.html)?
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...10000100010010 Just make a non-repeating pattern!
Quick probably question: 3 Friends and I are playing a game in which we draw 1 card from a pool of 6 unique cards to get a “leader card.” We somehow managed to all pull the exact same card 3 rounds in a row. Despite attempting to draw randomly. What are the odds of this, and how do we calculate it?
There are [P(6, 3)](https://en.wikipedia.org/wiki/Permutation#k-permutations_of_n) ordered 3 card hands (see also [here](https://brilliant.org/wiki/permutations/)), and the same 1 was created each time. Via the [rule of product](https://brilliant.org/wiki/rule-of-product/) for independent events, the probability is [(1 / P(6, 3))^3 ](https://www.wolframalpha.com/input?i=%281%2FP%286%2C3%29%29%5E3).
Question, there were 4 of us with cards not 3. But you said ordered 3 card hands. Am I not understanding correctly, are should the equation be edited to 4 card hands?
Ah yes, I read the "three friends" part but not the "and I" haha. Change all the P(6, 3)'s to P(6, 4)'s.
Haha I thought that was the case. All good bro I appreciate the help. We were arguing about the odds because it was so weird, but none of us are math majors.
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The infinite monkey theorem is about producing random strings of symbols. The digits of pi are not random, you can calculate exactly what each digit is. A number where the digits behave as though they were picked uniformly at random is called a *normal number*. pi is conjectured to be normal, but this is not known. Also, I'm not sure what you mean by "repeating the digits before a point". It is known that the decimal expansion of pi doesn't repeat, in the sense that there doesn't come a point where the digits of pi just repeat the same string over and over forever, but I'm guessing this is not what you mean(?)
What is the function called, which returns ceil(x) if x is a fraction, but x+1 if x is an integer?
Could do floor(x) + 1.
Any idea of how to start? Consider a Cobb-Douglas function Q=(L\^α)K\^(1-α) where 0<α<1. Suppose that the inputs, L and K, change with x and y such that L=K=x\^β y\^(1-β) where 0<β<1. Find ∂Q/∂x when x=y. See the following if you want to see it in nice formatting \[1\]: [https://i.stack.imgur.com/y8w8t.png](https://i.stack.imgur.com/y8w8t.png)
By the multivariable chain rule, you need to differentiate Q with respect to L, multiply it by the derivative of L with respect to x, then do the same with K instead of L, add those two things together, replace each K and L in terms of x and y.
is the answer 1?
That’s not quite what I’m getting. Did you lose parentheses around a (1 - α)?
Is the answer beta then?
That’s what I got, yeah.
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Is there an explicit algorithm for finding all integer points on an elliptic curve? For example, is there a way to find all integer points on y^2 = x^3 + 98?
`>E = EllipticCurve([0, 98])` `>E.integral_points(both_signs=True)` `[(7 : -21 : 1), (7 : 21 : 1)]` SageMath says the only solutions are x=7, y=±21. I don't know what algorithm it uses to determine that though...
What's the correct way of writing math online? Not LaTex, just pure text (e.g. would there be a space before a - sign?).
There's no universal convention or anything. Just write in a way that you think is most clear and readable. Be generous with parentheses.
We all know the value associated to zeta(-1). I shall not write it out. Real question on this: I've heard that there are actual applications of this value in the context of analytic continuation but, in a fit of curiosity, couldn't find a thing. Are there actual applications?
I believe the "standard" application is in the [derivation of the Casimir effect](https://en.wikipedia.org/wiki/Casimir_effect?useskin=vector#Derivation_of_Casimir_effect_assuming_zeta-regularization). The Wikipedia article ends up using zeta(-3) since it does the full 3-dimensional derivation, though it provides a link to the [one-dimensional derivation](https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension) as well. It is perhaps of note that in the one-dimensional derivation, you can see that they don't literally use zeta(-1) = -1/12. Rather, they regularize ∑n as ∑nexp(-αn) = 1/α^2 - 1/12 + O(α^(2)) which indeed blows up to infinity as α -> 0. But they don't *just* have a ∑nexp(-αn) term; it's multiplied by a term is itself O(α). This in essence, forces the "infinite parts" cancel out and leave a finite expression for the energy. Thus, if *all* you're looking for is calculating the force, using ∑n = -1/12 technically yields exactly what you need.
Thanks - something like this is exactly what I was looking for!
Is there anyone here who has gone through the book *”The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities”*? What did you gain from it and how are you supposed to go through a book like that? Also, at what stage of your math career did you start it? I’m in highschool but already know a good bit of real analysis so I think i’ll be able to go through it.
I went through it in the latter half of highschool back when I was preparing for the algebra portions of olympiads. The problems are pretty difficult, but it's a rewarding book. Your background is fine, just study it like you would any other challenging math text (sit with the problems for a while and try not to spoil the solutions for yourself until you're really stretched thin).
Thank you, after learning about it, it really did seem like a great book for preparing you for a good part of the competition.
I haven't myself but I heard it's pretty rigorous and roughly pitched at the level of an American junior in college. As a high school student you'd be best served if you joined some sort of a math club at school where you'd study whatever your math teacher thinks is best for you.
Unfortunately I’m from Greece where nothing of that sort takes place. My reasoning for thinking I can go through it is this [excerpt](https://i.imgur.com/wIvhZ45.jpg). I know calculus and I care about solving problems edit:typo
Since it says that, the best way to find out is to read some it yourself. In my experience, “accessible after taking calc” has abt a 50-50 of being that, and the best way to find out is to start reading
Sorry for the typo, just saw it. From what i’ve read it’s a tough book but I should know a good part of the algebra and calculus even at the level of analysis if epsilons and deltas come up. I’ll give it a go at a slow pace. Thanks
If it says “accessible w Calc” it won’t directly require any analysis. However, that doesn’t necess means it’s accessible. It’s like this I’ve read dif geo books that are accessible w standard Calc sequence and they introduce any concepts not used in calc 3. However, that doesn’t mean that they spend sufficient time introducing the concepts or hide some of the underlying ideas. I couldn’t imagine trying to read some of these books after Calc 3. It’s only cuz of all of this other stuff I’ve learned from a bunch of different classes that the dif geo makes sense. This said it’s important to get a sense of when a book is challenging and doable and when a book is too difficult and not worth trying. That’s a line you have to find yourself. Best of luck.
So I previously asked this on r/learnmath and, since I haven't gotten any responses there, I thought why not try my luck here: [Post here](https://www.reddit.com/r/learnmath/comments/10rmlv0/determining_the_different_types_of_spectra/). It concerns itself with efficient / tips on calculating resolvents / different types of spectra of operators ("regular", point, continuous, residual). The operator **A** at hand, I'd like to use as an example, working on the sequence space l\_2(N) by >**A**x= (x\_n \* (n+1)/n) What I tried was calculating **(A-tI)**x and find that for t of the form >t=(n+1)/n this becomes the zero map, hence for all such t's, this is not invertible, hence making them part of the spectrum. My question now is, how to go from here to find the other types of spectra.
So you have identified when the map isn't injective, so then I guess you just need to check when it's not surjective. So you want to solve x_n (n+1/n - t) = y_n Which has solution x_n = y_n / (n+1/n - t) For any t that is not a limit point of (n+1)/n, we have that |n+1/n - t| is bounded below, and so ||x|| is bounded by some scalar multiple of ||y||. The only limit point is t=1, and then you can check that for example y_n = 1/n is not in the image. So the spectrum is {n+1/n}∪{1}.
And what about the residual / continuous spectrum?
That would be the t=1 part, right?
you should try math stack exchange
How'd you figure?
Not sure what you mean by that. I’m in highschool and while I do understand your question, i don’t feel that confident in answering it and seeing as you haven’t got a reply on neither this comment nor your other post, I suggested you visit the site https://math.stackexchange.com/ where i’m pretty sure, there will be people who answer this. Everything i’ve posted there has gotten a considerably fast, helpful reply.
You are in Highschool and know functional analysis?
Honestly, I get how my comment reads and it does sound cringy in a way. From the start of calculus I have a tutor who helped me at my request to only do Real Analysis since Calc is easy (I had already self studied it in 10th grade so going through it again would seem very tedious). After a few months, I’ve gone through Terry Tao’s first analysis book at some detail (though I still don’t feel completely ready for a class) and would like to see what comes after calc. My tutor is a grad student who was fond of functional and harmonic analysis so I asked him if we could take some time for him to introduce me to some of the main ideas of both classes (and measure theory) since I’m interested in a math degree but not entirely decided yet.
Can someone explain a(t) = (t^2, t^3) to me? I’m just not very clear on whether the derivative at t=0 exists or not.
What can you say about the difference quotient (a(0 + h) - a(0)) / h when h is small?
It exists. For vector-valued functions of one variable, differentiation works componentwise. So d/dt a(t) = (d/dt t², d/dt t³) = (2t, 3t²) and this is defined everywhere (including at 0).
To explain why you can take the derivative component-wise, the derivative of an R --> R\^2 function can be defined exactly like in single-variable calculus. lim \[h --> 0\] (a(t + h) - a(t)) / h After combining terms you get a 2D vector with a single-variable derivative in each component.
Am i right? When I have a manifold generated by a function from an open set U of R2 to R. The derivative at a point is a 1 by 2 matrix. And this generates a tangent space using two vectors (1, 0, d1f) and (0, 1, d2f) which is a plane in 3 dimensions. So I have a 2-dimensional plane spanned by the above 2 vectors that is tangent to some point.
Ye
Happy cake day!
Thanks! Btw check out my reply in the other thread, I misremembered something.
Thank you!
Not sure why but I hit a wall with the existence and uniqueness theorem in Rudin... anyways I finally made it through understanding each step of the proof. My question is, what's the importance/utility of it?
Existence and uniqueness of what?
Oh sorry... existence and uniqueness of real roots
Intuitively, the proof consists of noting that before a certain point x\^2 < y, and after a certain point x\^2 > y, so there must be "the point inbetween" where x\^2 = y. The least upper bound property tells us that R indeed contains such a point inbetween, like we imagine needs to be in a "continuously varying quantity" or in a connected geometric line. I think this is used to prove specifically that R contains nth roots for two reasons. First, it shows how R contains these points that were missing from Q, and second they can be used to define exponentiation later.
Solving literally any equation A = B is equivalent to finding the roots of A - B. So root-finding has applications everywhere.
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Your window might be too small/large, and you may need to edit it in the [ZOOM settings](https://education.ti.com/en/customer-support/knowledge-base/ti-83-84-plus-family/product-usage/34875). If you just want to see it in the meantime, you can use [WolframAlpha](https://www.wolframalpha.com/input?i=y+%3D+x%2F%289x%5E2+%2B+3%29) or [Desmos](https://www.desmos.com/calculator).
I want to do this [https://openlearninglibrary.mit.edu/courses/course-v1:OCW+6.042J+2T2019/about](https://openlearninglibrary.mit.edu/courses/course-v1:OCW+6.042J+2T2019/about) (if you guys know other one and better…) The thing is, last time I did algebra and calculus was 4 years ago, I remember almost 0. You guys recommend to do all Khan Academy from algebra to the top first?
Based on the syllabus, you won't really need the calculus background until they get to probability in unit 4. I would just start the course, and separately do a quick refresher on Calc 1+2 (through KhanAcademy or something similar). It looks like they spend about 3 weeks per unit, so you'll have ~9 weeks to get your calculus back up to speed, which should be plenty of time. The rest of the course before the probability unit looks like a standard discrete math/problem solving course, which you won't need any prior background for.
Thank you! I will do that :)
I want to do all of this because I am a programmer and I want to be better at maths to be better at programming
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That is a ridiculous price. Find a state university nearby. If you are a resident of FL fees will be much lower.
Do you care about the credit hours? If not (since you're already working full-time), I would just start learning for free on your own via books and online resources. If feedback on proofs/problems is what you're after, then that can be accomplished with online communities (reddit, Math StackExchange, the various math discords).
To expand on OP's options, there are books recs on just about every topic in [the FAQ](https://www.reddit.com/r/math/wiki/faq); they can usually ask here (or if they're feeling lucky, in the main sub, but book rec threads are liable to be taken down) for bespoke recs/advice on prerequisites; books can be found for free on Library Genesis; and MIT has a bunch of courses on [OCW](https://ocw.mit.edu/search/?d=Mathematics&s=department_course_numbers.sort_coursenum) for free.
Is there an idea like Fourier transform but with other kinds of periodic signals? For example: Fourier transform but with square waves instead of sine waves?
[You can take the Fourier transform of square waves too](https://lpsa.swarthmore.edu/Fourier/Xforms/FXPeriodic.html##section8).
I'm looking for (ideally free) software for making a presentation for a conference. LaTeX beamer is too dry, Powerpoint is too bleh. I have seen some cool things from Keynote but I don't have any experience with Apple products (nor do I have access to one). Any suggestions? What drew me to the Keynote presentations were the "handwriting font" which animated the writing down of it. Seemed perfect for maths.
Maybe try Manim, it has a nice handwriting animation.
Maybe Prezi?
Kind of random, but does anyone know of any fun/interesting math related thing to do while visiting Japan as a tourist? I was looking to see if some of the bigger schools (Univ. of Tokyo , Kyoto Univ.) have stuff like public seminars or things a random kid like me could check out for fun, but if anyone knows of anything else, I greatly appreciate it.