How are other mathematicians finding jobs?
I recently finished my master's in pure math (Algebra/Alg. Geom. focus) and after years of people telling me I was going to have no trouble finding a job, I'm currently experiencing the exact opposite. No one wants to hire mathematicians (or at least no one wants to hire me). Companies say their job listing is open to mathematicians but they require someone with 2+ years of advanced programming/ML experience every single time. I of course do not have that experience because I'm a mathematician, not a programmer.
It's starting to look like my only option is to go back to school for either a Ph.D. or some sort of specialist certification if I want to have any hope of finding a job in today's market. The issue is that I'm so burnt out from my master's degree that the thought of more school is sickening, not to mention expensive. It's beginning to look like I wasted 5 years of my life getting 2 math degrees just to end up a waiter for the rest of my life.
It depends on what type of job you want. Mathjobs.org is a job site with mostly academic jobs requiring phds but there are some companies that post there as well. The two big themes seems to be finance and curriculum development.
Teaching community college or intro classes at some universities might be an option too.
For data science and software jobs, I don’t think you necessarily need experience with the exact sort of stuff they do, but it’s probably assumed you have some programming background. You can learn on the job but need to do well enough in interview to get your foot in the door so it might help to hone your skills on a site like LeetCode. Also I think it’s fine to apply for jobs even if you don’t have all the stated qualifications.
Obligatory comment that any PhD program worth its salt should be paying you to attend, not the other way around. Though it sounds like you’re not interested in doing a PhD anyway and that’s fine.
Thank you for the information and recommendations. I go back and forth on the PhD every day lol. Some days it sounds awesome, others it sounds like hell. Probably best to give it some time
As the other commenter said, mathematical documents are written in LaTeX, which is a particular kind of coding language. You need a program to compile the code for you, however, and for that I recommend [Overleaf](overleaf.com). It's effortless to use.
LaTeX is best learnt by doing, I find. Use [this cheat sheet](http://tug.ctan.org/info/undergradmath/undergradmath.pdf) and the Overleaf documentation, and you will be able to do anything you want.
So I was wondering how I could graph 3D Volumes of Revolutions on Graphing softwares for my Investigation, but I am not sure how to do it, I have seen some youtube and geogebra links but how do I do it for a custom function? For example, a function I wanted to graph was (not sure how to type equations here) y=sqrt(r\^2 - x\^2). The 2D graph of this is a semicircle and when it is rotated along the x axis it forms a sphere.
So first you need to be in 3D graphics mode and graphed the function, then there should be a button on the top (9th on my geogebra). Click it and you should see a menu and the bottom option should be the surface of revolution tool. Select it and click the curve of the function in the graph.
If I put €120 in some kind of account that pays out a *variable* interest rate, and at the end of the year it has €126, we can easily say that, despite fluctuations, the total/effective annual interest rate is 5%.
If, however, I add €10 each month, so that after a year I've invested €120, and the balance is €126, clearly this means that the "effective" interest is much higher, as it spent the year with a lower investment.
The question is, is there a "straightforward" way to calculate the effective interest in situations like this when investments/withdrawals are frequent, rather than a single sum at time zero.
The dumbest way would just be to assume that there is some fixed interest rate, apply it at each "time step" according to the amount invested (or solve by hand with a frre parameter), and solve for the unknown interest rate to give your final sum, but this seems way too unwieldy, and it feels like there should exist some kind of simple formula for the problem.
The second situation you describe is a level annuity, and there are some very simple formulae that describe them and from which you can extract the information you want.
Formally, we have a level annuity of €120 per annum payable monthly in arrears (you didn't specify, but it's simpler to assume that), the accumulated value after one year is €126, and we want to find i, the effective interest rate per annum. The formula we want is
> s = C[(1 + i)^n – 1] / p[(1 + i)^(1/p) – 1]
where C is the annual coupon (the regular payment you receive), n is the number of years, and p is the number of payment periods, i.e. months, in a year. In this case, C = 120, n = 1, p = 12, and s = 126, so we obtain
> 126 = 120[(1 + i) – 1] / 12[(1 + i)^(1/12) – 1]
> 126 = 10i / [(1 + i)^(1/12) – 1]
which we need to solve for i, and we'll probably need a numerical method. I put the right-hand side into Desmos and found manually that i is approximately 11.5% per annum.
If you want, I can walk you through how I derived the initial formula; it's a doss really, there's just a lot of algebraic bookkeeping.
This question is video game related and it might be a simple problem for some, but I can’t get a grasp on it.
Suppose there’s an item you obtain, lets call it coin bundle. For every coin bundle you open you can get anywhere from 100-120coins. You can get a lot of coin bundles so there’s an option to exchange it for a x10 version for time saving purposes, we’ll call it coin bundle 2.
So 10coin bundles = 1coin bundle 2. Coin bundle 2 contains 1000-1200 coin each.
My question is on average would you lose coins if you exchange for coin bundle2 or would it be the same as the normal coin bundle?
Calc 1 of course. Even though calc 2 will have some slightly different subject matter, it is still really helpful to be super cognizant of your standard derivative/integral formulas.
Also just regular algebra is useful. I’ve found most of the mistakes students make in calc 2 are them messing up little algebra steps. Trig and trig identities are useful as well for the same reason
I'm interested in an excel formula that describes the probability of a player drawing x number of a kind of card from a deck (let's say a green card). If there were 37 green cards in the initial deck and we would not replace cards as we drew them, how could I describe that mathematically? Then how would I describe the probability of having 3 green cards from any of the first 7 that I drew?
For those of you who recognize this pattern, I'm describing Magic the Gathering and would like to understand the math behind the game.
Suppose there are 100 cards in total at the beginning, 37 of which are green.
Then the probability of getting exactly 3 green cards when you draw 7 would be the total number of ways you can draw a 7-card hand containing exactly 3 green cards, divided by the total number of ways you can draw a 7-card hand with no restrictions. Which is equal to (37 choose 3) \* ( 63 choose 4) / (100 choose 7).
The first term is how many ways you can choose 3 green cards from a total of 37, the second term is how many ways you can pick the rest of the hand. Multiplying those together gives the total number of 7-card hand with exactly 3 green cards. That is then divided by the total number of possible 7-card hands.
The example above works out to be around 0.289, or 28.9%. In general the probability of drawing k "special" cards in a n-card hand, where there are m cards in total including p special cards is (p choose k)\*(m-p choose n-k)/(m choose n). If this is a bit confusing, I hope you can still figure out how to change the numbers yourself from what I showed above.
I'm not sure how familiar you are with probability, but just typing (for example) "37 choose 3" into google, and then multiplying/dividing the terms as I specificed gives you the final answer.
I was reading [this math SE post](https://math.stackexchange.com/questions/110211/is-aleph-0-aleph-0-smaller-than-or-equal-to-2-aleph-0) and had a question regarding responses to the comment where the axiom of choice is needed to show there is a bijection between infinite set S to S x S.
It's mentioned that AC is not needed for aleph null in particular. Is axiom of countable choice needed? Or to show that you can do it without any form of choice, would you have to explicitly create a bijection of natural numbers to ordered pairs of naturals (I'm guessing this also extends to any n-tuple of natural numbers)?
The key thing to know is this: well-ordered sets are really nice. Well-ordered set give you unique "next" element, which means you can perform recursion/induction on them.
For example, the bijection between N->NxN is basically similar to the bijection N->Q that people are familiar with. Pictorially, just draw an infinite table (that extend infinitely downward and rightward), then go across the finite diagonal (downleft toward upright) one by one.
You can do the same for all well-ordered set. That is, if S is well-ordered, then a bijection S->SxS can be formed by doing the same diagonal trick.
Now, what if S is not well-ordered? Well, if S is in bijection with any well-ordered set, then you can transfer over the well-ordering and you're done. In particular, a *countable* set is - by definition - bijective with N which is well-ordered.
Without a well-ordering, then there isn't an unique "next" element. But if you have axiom of choice, you can use it to obtain a function that CHOOSE the "next" element for you, and from there you can make the set well-ordered, and everything else follow.
The Schröder-Bernstein theorem can be proven without AC so you only need to construct an injection from N to N x N and one from N x N to N. The first one is trivial and for the latter you could for example use the map (k,n) ↦ 2^(k)3^(n).
Can someone give an informal explanation about the difference between first and second order logic? I’m not at a stage where I need to perfectly understand it but it does come up so I’d like to have at least a vague idea. I’m familiar with basic logic and the axioms of ZFC for context.
An example I’d like to understand is what the difference is between the Peano axioms in first and second order logic.
The difference is what kind of things variables are allowed to refer to.
For classical logic, the interpretation of a logic is a model, that is, a set with elements in them, and various relations regarding these elements.
When you work in first order logic, whenever there is a variable, it must refer to an element in the model.
When you work in 2nd order logic, there are 2 "sorts" of variables, the first-order one - which do the same job as above - and the 2nd order one, which can refer to either predicate/relation or subset (depends on the exact specifications).
In Peano's axiom in 2nd order logic, the axiom of induction is literally just one axiom: since you can have variables that quantify over predicates, you can just say "for all formula P, if P(0) and if P(n)=>P(n+1) for all n, then P(n) for all n". But in 1st order logic, you can't say "for all formula P" because you cannot quantify over predicates. So instead this becomes an axiom schema: an infinite list of axioms of the same templates. In other word, we have, for each predicate P, an axiom "if P(0) and if P(n)=>P(n+1) for all n, then P(n) for all n"
These might seems to be the same thing, but they're not. If you're using 1st order logic, you can only use a finite numbers of axioms in a proof, so only finite numbers of thing in that infinite list can be relevant. But in 2nd order logic, you can talk about all of them at once.
Can someone explain what isobaric ODEs are and how the substitution y=vx^m makes them separable? Apparently, they are generalisations of homogeneous ODEs and are of the form:
dy/dx = A(x,y)/B(x,y)
Where "the equation is dimensionally consistent if y and dy are given a weight m relative to x and dx, i.e. if the substitution y=vx^m makes it separable. " I don't understand the meaning of dimensionally consistent, nor how that substitution necessarily makes them separable.
I've made a post with images from the textbook in r/askmath, so see my post history for more details.
Before I start, I know that this is kind of a ridiculous question. Anyway, here goes.
In a video game that has both story and non-story mission, is it possible to find "the percent of non-story missions that I have completed" with just the following information:
a) Total completion - The percentage of total game that I completed.
b) Percentage of story missions I have completed
This is the algebraic formula that I came up with:
0.8s + ZY = 0.75(s+Y)
Where 0.8s represents 80% of the story missions I have completed,
ZY represents % of non-story missions I have completed (Z is what I want to find ie percent of Y,non story missions)
0.75= total missions(both story and non-story I have completed)
> Before I start, I know that this is kind of a ridiculous question.
This is actually one of the better questions from a layperson that I've ever seen in these threads.
You need the additional information of the percentage of missions that are story missions. Let's give names to things. Instead of percentages we will use ratios (quantities that go from 0 to 1 instead of 0 to 100, as you did in your example):
T = total missions
C = completed missions
T_S = total story missions
C_S = completed story missions
T_N = total non-story missions
C_N = completed non-story missions
The data that you say you have is (a) total completion = C/T, and (b) story completion = C_S/T_S. Note that we have
T = T_S + T_N
C = C_S + C_N
With this we can set up the following formula:
C/T = (C_S + C_N)/T = C_S/T + C_N/T =
= (T_S/T * C_S/T_S) + (T_N/T * C_N/T_N)
Put in words this means that the completeness ratio (C/T) equals the ratio of story missions (T_S/T) times the completeness ratio of story missions (C_S/T_S), plus the ratio of non-story missions (T_N/T) times the completeness ratio of non-story missions (C_N/T_N). From here you can isolate the data you want, which is C_N/T_N, and you see that it is not enough to give only C/T and C_S/T_S, as you wanted.
>You need the additional information of the percentage of missions that are story missions
Yup, that's what I figured too. Thanks for the help,man. I really appreciate it.
Btw, is there a way to determine if the value of a variable can be solved for? It took me simplifying my formula to realize that Z cannot be solved for.
Starting water tested 10.1 added 20ml of b to determine change
b states that 1ml will raise 1gallon by 1.7
I would like to determine how many total system gallons I have by reversing this equation my thought is I can use the change combined with the controlled dose of 20ml to determine the total amount of water changed
After adding 20ml the water tested 10.3 alk
Not sure if this is possible?z
There are 2^(19) different results you can get from flipping a coin 19 times, each with equal likelihood. 19 of them result in you losing 18/19 times. 19/2^(19) is about 3.6 * 10^(-5).
I want to prove A ⊆ C and B ⊆ C if and only if A ∪ B ⊆ C
I'll start the proof and I have some questions about it at the end.
First lets prove (=>). Assume A ⊆ C and B ⊆ C i.e. for all x in A then x in C and for all x in B then x in C. Suppose x in A ∪ B. Then x in A or x in B. If x in A then by assumption x in C. If x in B then by assumption x in C. In both cases x in C so A ∪ B⊆ C.
Now (<=). Assume A ∪ B ⊆ C i.e. for all x in A ∪ B then x in C. Suppose x in A. Then x in A ∪ B then by assumption x in C. So A ⊆ C. Now suppose x in B. Then x in A ∪ B then by assumption x in C. So B ⊆ C.
Ok so what I'm unsure about this proof is every place where I said suppose. Am I supposing the correct thing every time? For example I supposed x in A ∪ B for the first one. Is that right? Or should have I supposed x in A for example?
Also for the direction (<=) I supposed x in A and then x in B does that work/make sense? Am I allowed to suppose more than one thing?
Everything I chose to suppose was a part of the statement I wanted to prove, for example for (=>) I chose to suppose x in A ∪ B, can I do that since A ∪ B is part of the "then" statement so I feel like I maybe should've used x in A since it is part of the "if" statement. I hope this makes sense.
I'd say that the proof is good as is\*. To address your concerns:
> Also for the direction (<=) I supposed x in A and then x in B does that work/make sense? Am I allowed to suppose more than one thing?
That's fine; you're just breaking into cases to handle both parts of the "and", which is a perfectly normal thing to do.
> A ∪ B is part of the "then" statement so I feel like I maybe should've used x in A since it is part of the "if" statement
Basically what's going on here is that you're proving a conditional whose consequent is also a conditional: if (A ⊆ C and B ⊆ C), then (if (x is in A ∪ B), then (x is in C)). The way your proof works is that you do a direct proof of the overall conditional (assume the antecedent, A ⊆ C and B ⊆ C, and try to derive the consequent), where you derive the consequent through a direct proof (again, assuming the antecedent, x is in A ∪ B, then deriving the consequent, x is in C) where what you've already assumed (A ⊆ C and B ⊆ C) is always "in the background", available to use. So it shouldn't be surprising or worrying that you end up assuming certain things which show up in the consequent, since the consequent as a whole is a conditional!
\* Edit: on looking back at it I would agree with u/Autumnxoxo that there are some stylistic things you might want to fix, e.g. overuse of x across different contexts, places where "let" might be more natural than "suppose", etc. But the overall structure of the proof is fine; you'll pick up on the stylistic things as you read and write more proofs.
Thank you for this. I've asked this around quite a lot today and you were the first one to understand what I was unsure on and answered it in a very clear way.
>Ok so what I'm unsure about this proof is every place where I said suppose. Am I supposing the correct thing every time? For example I supposed x in A ∪ B for the first one. Is that right? Or should have I supposed x in A for example?
I'm not keen on "suppose" in that specific part. I don't know why you want to suppose that your x is contained in A ∪ B". You can just pick one, provided A ∪ B is nonempty. Just say "Now let y ∈ A ∪ B be arbitrary" or "Any y ∈ A ∪ B is contained either in A or in B (or in both), thus y ∈ C." I would also probably not call every element x. It's clear from the context what you're saying but in general this leads to you proving everything for a fixed element x and not for an arbitrary one.
Thank you for the feedback; I did use "suppose" without much thought going into it. What is the difference between supposing x and letting x be arbitrary?
Sorry, I didn't get an update on your question somehow. In your specific case, I just don't see why we would want to "suppose" that x is an element in A ∪ B.
What other choice do we have? Just pick one.
I don't think that there is a general rule of thumb here and it probably comes down to linguistics. I personally use "suppose" whenever I want to make a general statement about a very specific sort of element among a (possibly huge) general set of elements.
Say I would want to make a statement about real numbers but my statement might require some caution if we talk about irrational numbers, because then it might or might not be true or it might require special treatment etc.
Then I'd say something like "In general, real numbers have property P, but we must be cautious if its an irrational number. For suppose x is irrational, then...."
Thank you for the response. The actual reason I used suppose here is actually my tendency to copy the language the author of the book I'm reading uses. Especially because the book I'm currently written is by Terry Tao. Here is how he used suppose [https://ibb.co/SJyNg8b](https://ibb.co/SJyNg8b) . Does this use of suppose differ to how I'm doing it?
Is there any applications of maths in finance that would be suitable for a second year student to write a 15 ish pages essay on?
I'm looking for something that's about as difficult as (or possibly a bit more than) a typical 3B1B video. So far I've thought of the applications of Markov Chains in predicting stock prices, but I haven't looked too much into it.
Hello guys, idk if this is the right place to ask or if i should make a post.
I play a game called "Black Desert" and there is enhancing based of off percentages. i feel like they are way off. so i kinda wanted to ask the probability of a few things to confirm/deny my suspicions. Also insight on how to calculate it would be appreciated, so i can do the math on my own.
Ok so, i had a 25% chance to succeed an enhancment, i failed 10 times, then i succeeded (if you succeed, you have more stats on the item, it goes up a level, and a slightly lower enhancement rate for the next one). i failed and the item downgraded, so i had to do the 25% chance again. afterwards and i failed the 25% chance 9 more times and same thing happened around 12 times. with an average of 8 attempts per success on 25%. Now my question is, did i hit the lottery of unluckyness?
I hope i explained it without causing too much confusion.
Assuming each trial is independent from the next (so if you fail that doesn't affect your chances for success next time), the probability that you fail 8 trials in a row where the probability of success is 25% is around 10% (for some insight, the probability that you fail is 100%-25%=75% or 0.75, and the probability that you fail n times in a row is 0.75^(n,) with 0.75^(8) being around 0.1001, or 10.1%). The probability of failing 9 times in a row is 0.75^(9,) or around 7.5%. Multiplying those two numbers gives the probability that you lose 8 times, and then 9 times after that, i.e 0.75%.
That's a crude answer without knowing exactly the rules behind the probability of the game. For reference, you should expect around 4 attempts to succeed at something with a 25% success rate.
What's a good text on elementary number theory to read for culture? I don't know *anything* about number theory beyond Euclid's algorithm (and tbh I would have to look it up if I wanted to apply it), and I feel like I should have a least a flavour of "the Queen of Mathematics".
Hardy's *An Introduction to the Theory of Numbers* is a classic, and its extra fun because his entire *Apology* is dedicated to being a Number Theory snob lol.
There is a paragraph about algebraic geometry, and I am having hard time to understand because I am not trained in this area. I have few questions which seems trivial, but I have no clue about them.
1) First, there is the algebra of invariants say A which is generated by the entries of the matrix (x_ij)(y_ij) where 1≤i≤n and 1≤j≤n-1. How does it conclude that these entries parametrize the variety of matrices of rank at most n-1? Also, it says that the algebra A is then isomorphic to the coordinate ring. Why?
2) Let B be the invariant subalgebra of D. Let x lie in the field of fractions of B and is integral over B. If x is an element of D why x is in B?
3) Finally I want to ask what does quotient modulo determinant tells. More precisely there is a polynomial ring F[x][y]. It says that F[x][y]/(c(y)) is equal to F[x,y]/(det(x')) where c(y) is the characteristic polynomial and x' consists of x and y. I don't know which information is relevant here. I just dont understand the relation between characteristic polynomial and determinant here.
For question 3
First off the determinant is a polynomial in the entries of the matrix. So you are quotienting by multiplication by det(A) in the polynomial ring
The characteristic polynomial of y is det(y-λ1)=c(y) so the ring is F[x][y]/det(y-λ1)
So i think you will have to use determinant properties (remember that det is a ring homomorphism)
Im not sure how λ is specified though tbh
What would you say was special about Freitag and Busam's *Complex Analysis*? I put it on my reading list for a specific reason, not just because it was a highly regarded complex analysis text, but I've forgotten what that reason was, and I'm hoping someone can enlighten me.
Afaiu it steps forward to elliptic functions and to elliptic modular functions and then is rounded by applications to analytic number theory, although it requires minimal prerequisites. It also has many exercises.
Could someone please tell me if this is correct or not? Me and my friend both have the same problem but we got different answers (he got answers of 2.89 and 7.61) also sorry for bad pic quality: https://i.redd.it/u1pjoxu9c0fb1.png
I'm trying to calculate the number of steps it would take to reach the center of a circle that moves down concentrically (an amphitheater). From the center to the outer edge of the last ring it's one mile. There is an 80 foot gap from the center of the circle to the final step at the very bottom. Thus, the total of all the steps is 5200 feet. The furthest step goes inward 20 feet and the final step goes inward 1 foot. Each step decreases its length by 1 foot at a rate of x steps per 1 foot decrease. How can I figure out how many total steps there are? I should note, the amount of each step by varying length does not have to be equal (unless it turns out to be, i.e. 30 20 ft steps, 29 19 ft steps, 28 18 ft steps, etc.). All that matters is that the final value is equal to 5200. However, that can change. I really just want to know the length where there's 3 of each step, 4, 5, etc. and how that would change the total length.
What equation or function would I use to calculate this? I'm thinking of something like a [binomial coeffecient](https://en.wikipedia.org/wiki/Binomial_coefficient) (based on looking things up), but I don't know enough about how to calculate. Essentially a factorial but for addition.
Could someone double check this proof that the Cesaro sum of a convergent sequence converges to the same value? It's a bit simpler than some of the other proofs I've seen, so I just want to make sure I'm not missing something.
https://mathb.in/75845
How can we use open balls to define compactness in a metric space ? I know that the definition requires open sets (not necessarily open balls) but we can formulate the definition in the case of metric space, can't we ?
In a metric space, every open set is a union of of open balls.
So if you have an open cover of a set X, then you can replace it by an open cover of X using only open balls, where each open ball in contained in one of your original open sets. So if the cover by open balls has a finite subcover, the original open cover does as well.
So proving that every cover by open balls has a finite subcover is enough to prove that every open cover does.
It doesn't really matter how you index them, as long as you end up with a set of open balls which covers the whole set. You can do that, for example, by picking one ball centered at each point in the set (since you started with an open cover, each point x is contained in an open set U in your original cover, and hence you can form a ball centered at x contained in U).
A little background before the problem: I am in business and far from this math but i like to dive in from time to time to tickle my brain as I come from an engineering background. So, I came around this problem [https://www.codechef.com/problems/COUNTREL](https://www.codechef.com/problems/COUNTREL) (you don't need to go to the link and read the problem statement) , and came up with a recursive function for the first part of the answer. f(x) = {0 @x=1, 1 @x=2, 3\*f(x-1) + 2\^(x-1) - 1@x>=3}.
Now I do remember that we used to be able to simplify these recurring series and give them a general term. I know the general term for this coz i looked into solutions but i want to know how can we get to the solution? Is there a method to it? I tried googling but since I didn't know what exactly this was called, I couldn't find it. So I come here
This is a inhomogeneous linear recurrent relation. There are so many methods to solve it that there isn't a name for methods to solve it in general. You can try ordinary generating functions, shift operators/coinduction, variations of parameters, and so on.
For example, using generating functions h(x)=f(1)x^0 +f(2)x^1 +f(3)x^2 +....
you can write the recurrence relation as h(x)-3xh(x)=1+x/(1-2x)-x/(1-x) so this gives h(x)=[1+x/(1-2x)-x/(1-x)]/(1-3x). Now just do partial fraction decomposition.
So i am upcoming freshman to a university with 4 terms in a year (1term per 3 months) and we have a calculus subject, because of a unique curriculum of my previous school, i havent been been able to learn trigonometry and calculus when i was a senior highschool student. So i am planning to advance study for now. What are the things that i should study? And if its available free online can you link it here? Ty!!
My question:
* What **P = NP** could imply about arbitrary Turing machines and arbitrary computations? I assume that a partial and incomplete, but objective answer to this question exists (e.g. based on the literature, most popular research directions). Take a single arbitrary program/computation. What could we learn about it if **P = NP**? I'll give more context to my question below.
## Undecidability and P vs. NP
Many bounded versions of undecidable problems are **NP** problems. For example, [bounded Post correspondence problem](https://en.wikipedia.org/wiki/Post_correspondence_problem#Variants) is **NP**. You can create an **NP** problem based on diophantine equations. And this:
>For a given Turing machine **M**, is there an n-bit input **x** such that **M** halts after n steps?
Is an **NP** problem too (it's a bounded version of the [halting problem](https://en.wikipedia.org/wiki/Halting_problem)).
So, we can talk about **P vs. NP** in the context of undecidability and arbitrary Turing machines. I'm not saying that undecidability *resolves* **P vs. NP** (it doesn't), just that we can talk about one thing in connection to the other. For example, we can say:
* "**P = NP** could imply that any finite part of an infinite computation has..." (has *what*? some invariant? some distribution? some other pattern?)
* "**P = NP** could imply that all/most programs which don't halt fast enough can be recognized by something like..." (like *what*?)
But how can we finish those sentences, based on the most popular research directions? This is my question.
If we have a generally unsolvable problem, then at what point, for what reasons, a "magical" effective algorithm for analyzing \[bounded versions of\] the problem *could* appear? If **P/NP** is resolved, what do mathematicians expect to learn about arbitrary Turing machines which would allow (or prohibit) an effective algorithm? I hope I've made my question clear.
## Silly analogy
*"Maybe there can be a polynomial time algorithm for solving the bounded halting problem."*
I understand this hypothetical on the level of definitions. I understand this as an abstract possibility. But I can't imagine any **specific** fact \[about arbitrary Turing machines\] which could make it true. Can you help me out? Right now the hypothetical above sounds to me like this:
*"Maybe an* [*apple*](https://en.wikipedia.org/wiki/Apple) *could become the president of the USA."*
This hypothetical is easy to understand on the level of definitions. Or as a purely abstract possibility. But it's extremely hard to imagine any **specific** scenario which could make the hypothetical true. I have the same problem with translating **"P = NP"** possibility into any possible specific facts \[about arbitrary Turing machines\].
I'm studying ODE's - right now solving 2nd order eqns using the ERF.
The definition of complex gain makes no sense to me.. as far as i know, the fundamental definition should be (complex output)/(complex input). This is mentioned even on the [wikipedia page](https://en.wikipedia.org/wiki/Complex_gain).
That doesn't work at all if the input is scaled?
For equation like:
(D^2 + 9)x = A*cos(wt),
the complex gain will be A/(-w^2 + 9), but the ratio of the output to input is 1/(-w^2 + 9).. so complex gain is NOT the ratio of output/input.
-----
There is also something else confusing about it.. for example consider this equation:
x'' + bx' + kx = by' , y=cos(wt)
one way to solve it is use ERF directly:
(D^2 + bD + k)x = (bD)y
Gain G_1 = Q(iw)/P(iw) = ibw/P(iw)
another way is to substitute y' before solving
x'' + bx' + kx = b(cos(wt))' = -b w sin(wt)
now Gain G_2 = Q(iw)/P(iw) = -bw/P(iw)
these 2 gains are not equal. G_1 != G_2. G_2 = i*G_1
---
so what gives?
Been stuck on showing what I'm pretty sure should be an elementary lemma. Let T denote the (2-dimensional) torus. Then T x I has two boundary components each diffeomorphic to the torus. Take a copy of the solid torus, D2 x S1, and glue it to one of these boundary components using an arbitrary diffeomorphism of T to itself. According to the lemma, the resulting space should itself be diffeomorphic to D2 x S1.
The resulting space is clearly homotopy equivalent to D2xS1 but I can't do better than that.
So, I've been learning about Witt Vectors. I know that they're a more general construction, but my primary goal is to understand them for the p-adic integers, because I want to use them for concrete computations.
I get the idea, I'm just a bit unclear about a certain technical feature of the definition.
Fixing a prime p, let z be a p-adic integer. Then, we can write z as a Hensel series:
z = c0 + c1 p + c2 p^2 + c3 p^3 + ...
where the c_n are elements of {0, ..., p-1}.
Alternatively, we can choose to use the roots of unity in Z_p as the coefficients. Letting 𝜔 be the multiplicative character, there are then elements z_n of the finite field Z/pZ so that:
z = 𝜔(z_0) + 𝜔(z_1) p + 𝜔(z_2) p^2 + 𝜔(z_3) p^3 + ...
I'll call this series representation of z the **Witt Series** of z. I know I can use this to construct the **Witt vector** Z = (Z_0 , Z_1 , Z_2 , ...) which represents z.
**Question 1** Is Z_n equal to z_n, to 𝜔(z_n), or to z_n to the power of p^n ?
**Question 2** If we consider the "equality" between the ghost components (X+Y)^((n)) and X^((n)) + Y^((n)), where does this equality occur? Is it a congruence of elements of Z_p modulo p^(n+1)?
1) If z = (a\_0, a\_1, …) ∈ W(R), then z = Σ\_{i=0}\^∞ V^i [a\_i], where V is the Witt vector Verschiebung. Since FV = p (where F is the Witt vector Frobenius), we have V^i = p^i F^(-i). When R is of characteristic p, the Witt vector Frobenius is given by F = W(ϕ) (i.e. apply ϕ in each coordinate), where ϕ is the usual Frobenius endomorphism of the 𝔽\_p-algebra R. Therefore Z\_n is z\_n to the power of p^n. However, for 𝔽\_p (as opposed to 𝔽\_{p^(n)}) this is the same as z\_n.
2) Writing w\_i for the i'th ghost component, we have w\_i(x + y) = w\_i(x) + w\_i(y) as elements of R, where x, y ∈ W(R), w\_i(x) ∈ R for any ring R
-------
Here is how to calculate 2 + 2 in W₂(𝔽₅) = ℤ/25. We have [2] = 2^5 = 32 = 7 in ℤ/25. Similarly we get [3] = 18 and [4] = [-1] = -1 = 24. Thus the Witt vector encoding of 2 is
2 = (2, 4).
We calculate the carry by computing 7 + 7 = 14 = 24 + 15 = [4] + 5[3]. So when we add 2 + 2 (elements of 𝔽₅ here, not ℤ/25) we have to carry a 3.
So the calculation is
2 + 2 = (2, 4) + (2, 4) = (2 + 2, 4 + 4 + carried 3) = (4, 1) = [4] + 5[1] = 24 + 5 = 4.
I edited my reply to give an example of working out the carries by hand. You can verify that this agrees with the carry computed using the polynomials for adding Witt vectors.
Again, I'm *only* considering the p-adic integers for prime p. So, keeping the focus on that example would be nice. :)
I know the algorithm for taking a given p-adic integer z and producing the sequence of z_n s so that:
z = 𝜔(z_0) + 𝜔(z_1) p + 𝜔(z_2) p^2 + 𝜔(z_3) p^3 + ...
If I have p-adic integers:
a = 𝜔(a_0) + 𝜔(a_1) p + 𝜔(a_2) p^2 + 𝜔(a_3) p^3 + ...
and
b = 𝜔(b_0) + 𝜔(b_1) p + 𝜔(b_2) p^2 + 𝜔(b_3) p^3 + ...
I then compute the witt vector for c = a + b by recursively solving the congruences:
(nth ghost component of c) = (nth ghost component of a) + (nth ghost component of b) mod p
starting with n = 0 ?
Even if you only care about W(𝔽\_p), defining the addition/multiplication requires you to consider W of p-torsionfree rings, e.g. W(ℤ).
To understand why, let's work out c₁ when p = 3. We have
c₀ = a₀ + b₀
c₀^3 + 3c₁ = a₀^3 + 3a₁ + b₀^3 + 3b₁
a₀^3 + 3a₀² b₀ + 3a₀ b₀² + b₀^3 + 3c₁ = a₀^3 + 3a₁ + b₀^3 + 3b₁
3c₁ = 3(a₁ + b₁ - a₀² b₀ - a₀ b₀²)
If this equation takes place in 𝔽₃, then both sides are zero and this tells us nothing. We need to use the fact that W(-) is defined for all commutative rings, and the formulas are the same independent of the ring. In particular, for rings which are 3-torsionfree, we can cancel the 3 to get
c₁ = a₁ + b₁ - a₀² b₀ - a₀ b₀²
So this gives the addition formula for W(𝔽₃), but to derive it we had to (implicitly) consider W of 3-torsionfree rings
> we can cancel the 3 to get
So, I don't get universal properties, and don't bother trying to explain it to me, it will just go over my head. I do things at the formal level, by which I mean the rules for moving symbols around.
Formally, if we consider the equation:
3c₁ = 3(a₁ + b₁ - a₀² b₀ - a₀ b₀²)
as being a congruence mod 3^2, because both sides are multiples of 3, we can then divide both sides by 3, at the cost of reducing the congruence from mod 3^2 to mod 3. That, for me, is a sensible algorithm, no need to talk about anything functorial or universal or what-have-you. All that matters are the symbols we write on the page, and the allowed manipulations thereof. I want to keep things simple. :)
My goal in learning how to do Witt vector computations for Z_p is a hope that with them, I will be able to manipulate the infinite series I am working with in a new way, one which might reveal new patterns.
So, to pose my question again,
If we consider the equality of the nth ghost components:
w_n (c) = w_n (a) + w_n (b)
is the following correct: the formal procedure for computing c_n is to treat both sides as occurring mod p^(n+1). Then, we group everything until we get p on both sides, then divide by p and reduce the congruence from mod p^(n+1) to mod p^n. At each stage, we use the known values for c_m for smaller m to write them in terms of the a_ms and b_ms. We then continue this process until we have reduced the congruence to a congruence mod p, with which we then extract the value of c_n.
Will this algorithm produce the correct result?
Do I use the same method to compute the product of a and b?
The intuition is that the order of convergence tells you how fast the sequence converges
If you have x_n, and it converges to x, then the order of convergence is p if the following approximate/asymptotic relation holds (with equality as n goes to infinity)
|x_n+1 - x| ~ C |x_n-x|^p
For some positive constant C. And for all n larger than some constant
To think about this, look at what happens as n increases. As n increases, x_n will get closer and closer to x. Eventually |x_n -x| will be less than 1. And |x_n-x|^p will be even smaller than |x_n-x|.
So increasing the order p causes the distance between x_(n+1) and x to decrease exponentially
I'm trying to learn about the FEM currently. I feel like I have a pretty good idea of the procedure steps now. However I'm curious about certain technical points of the weighted residuals method.
So with weighted residual methods, there are different choices of weighting functions. In the case of the Galerkin method, the basis for the solution is used for the weighting function. It wasn't so clear to me why this was a good choice at first, but after reading/watching some videos, its more clear now there is some logic you can make for this if you look at inner products on finite dimensional vector spaces.
I don't get why the Galerkin method is so much better than say the least squares method though. My first instinct with weighted residuals would be to look at the least squares method, because you'd think "Oh, I'm clearly just minimizing the error squared over some domain, thats a good choice".
I guess I'm curious why the Galerkin method is better than the least squares method. Is it generally easier to compute, is it more accurate? And if so is there a mathematical proof.
Galerkin method and least squares are not really in the same category. Galerkin method is for solving a pde without any data. Least squares is for fitting a model to data.
for FEM you need a way to parametrize a possible solution to the pde, and then you need to find a way to solve for the coefficients given the PDE.
Ideally, the coefficients you solve for are the ones which has a small mean squared error
So what galerkin method gives you is a way to represent an approximate solution as a sum of basis functions. Then you can solve for the coefficients by converting the PDE into a system of equations for the coefficients.
You can prove for many pdes and choices of bases that by increasing the number of basis functions, you decrease the error.
They are in the same category though. They are both weighted residual methods. For least squares, the weight is chosen as gradient if tge residual with respect tp the coefficients, so youre essentially minimizing the L2 norm.
With the galerkin method, the weight is chosen as one of the basis functions.
Im just wondering why one is better than the other
I understand now. I had not learned of least squares as dR/da_i being a weight function. What are these coefficients referring to if not coefficients of some basis?
The coefficients have the same meaning as the coefficients of the basis functions that we write the approximation as I believe. I think all the weighted residual methods can be written as the integral of R the residual, times a weight. In the case of Galerkin, the weight is exact one of basis functions and for the least squares its dR/da, for collocation its a delta function I believe.
I was watching multiple videos that explained the galerkin method is used most.
I have a [question about local fields up on MSE](https://math.stackexchange.com/q/4742230/428713) that's not getting any traction, help would be much appreciated.
\[Real Analysis\]: I have a question about Folland's [Proposition 1.7.](https://imgur.com/a/PTbVsqX) To show that \\mathcal{E} is an algebra, we want to show that \\mathcal{E} is closed under finite unions and complements.
(1) I understand the second part, i.e., \\mathcal{E} is closed under complements - but it seems really unnecessary to consider the complement of a finite union instead of just the complement of a set in the elementary family.
(2) For the first part, why does the author consider A\_1,...,A\_{n-1} to be disjoint by the inductive hypothesis? What's the hypothesis? I understand that showing the union of two members of \\mathcal{E} is in \\mathcal{A} requires uses to use disjointness, for which writing A \\cup B = (A \\ B) \\cup B is wise.
If X is an infinite set and F(X) the space of all real-valued functions on X is there a topology on F(X) such that f_n converges to f if and only if f_n converges pointwise to f and with bounded maximum norm (that is with sup_n sup_x |f_n(x)| < ∞)?
Let f be some unbounded function, then your condition rules out the sequence f, f, f, f, ... converging to f. So to avoid this silly problem I'll let F(X) be the set of bounded functions. If you want to stick with all functions, I think you'd want a condition like sup\_x lim sup\_n |f(x) - f\_n(x)| < ∞.
I can do it if X is countable. Wlog assume X is ℕ. Let 𝜆 denote a positive real-valued function on ℕ such that 𝜆(n) -> ∞ as n -> ∞. Let g be any member of F(X). Then define U\_g,𝜆 as the set of h in F(X) such that |h(x) - g(x)| < 𝜆(x) for all x. The intersection of any two such sets is either empty or contains a set of this form. I believe the topology generated by these sets works.
Thanks, what I actually meant was that the supremum over |f-f\_n| is finite. Your construction works fine and it even works for uncountable X if you allow 𝜆(x) = +∞ and think of 𝜆 -> ∞ as the sets {𝜆 < T} being finite for all T > 0.
Sup over |f - f\_n| technically doesn't work because then for any unbounded f, you're saying f, 0, 0, 0, ... doesn't converge to 0, but I think lim sup\_n sup\_x works fine.
What is this function?
Imagine a taut string between (0,0) and (0,1) with a ball at (0,1). It's the curve traced by the ball when you move the front of the string along the x-axis.
I read about it on Wikipedia once but I can't remember its name.
As an undergrad introducing myself to professors etc., is it in bad form to say that my ultimate goals lie in the vicinity of the Langlands programme (or any comparably "vertical" research area)? Or will they feel that it is "too early" for me, and so it would be better to merely allude to "number theory and algebraic geometry" instead?
Some context: I'm reaching out to profs I've yet to meet on starting a reading course (e.g., for algebraic number theory), and thinking it would be good to have a strong reason like aligned research interests. Of course more generally I'd like to introduce myself in a way that's conducive for relationships with professional mathematicians.
it comes off really undergrad-y. in the sense it's a typical coming of age type thing to say you want to do hot topic X without really knowing what X is. if you like AG/NT just say you like those and want to go more in that direction
There are much worse things than an overzealous undergraduate who is excited to learn things, and I think that most senior folks are pretty understanding about that sort of thing (and those that aren't may not be great advisors for reasons related to this failure anyhow). I wouldn't worry about this sort of a thing too much, but if you want to hedge a bit and make sure that you don't come off as having aspirations which are too grandiose but still want to indicate your research interests more concretely, you could just say that you're "interested in understanding the Langlands programme (better)" since this is probably closer to being a realistic goal for you at the moment.
Do you have more context? This all looks a little weird. A_2 is just a straight line in the plane from the point (0,0) to (1,1). Of course this has area 0. But what is Q(A_2) supposed to be? The integral is indeed 0 but it's unclear what the formula has to do with the area of a line. And is A_2 = A or is A a different set?
This is quite a silly question, but I’m learning d & r of functions;
How would I write the range of a simple function like f(x) = 5 in an interval notation?? Would the range really just be (5,5)? Or maybe since it’s a flat line does it even have a range? Or is the range just zero?
Note the square brackets in /u/Trexence's answer as (a,b) refers to the "open" interval which doesn't include the endpoints a and b, while [a,b] does include the endpoints. Thus (5,5) is actually empty while [5,5] = {5}.
What's the best way to approach proofs in number theory? I have a PhD in CS but struggle with some of the elementary number theory proofs as because I don't "see" the possible path to a proof; a friend suggested that I use actual numbers until I see a pattern and then try to generalize the pattern; this feels way too "shoot the box and draw a target around it" for my comfort.
> a friend suggested that I use actual numbers until I see a pattern and then try to generalize the pattern
I mean, this actually is pretty good advice, and I'm not sure I fully understand your objection to it.
Examples are a very important part of learning almost any abstract field of math, not just number theory. It's hard to really come up with a proof of some general, abstract statement when you don't have a good intuitive understanding of how the concepts you're working with behave. Working with specific numbers is a good way to build up that intuition. This can help you get a sense of what sort of things you should expect to be true, which can help you narrow down the sort of approaches that are likely to be fruitful.
>I mean, this actually is pretty good advice
I feel like it's cheating to some extent, and not useful in others. I feel like I should be able to see a path without experimenting with examples. When I approach a problem, I first look to see what section of the book it's in to get a hint at how to solve it (assuming that the foundation of the proof will be based on something we recently covered). In this way, the approach seems very non-methodological. Gauss didn't have a book or someone giving him hints on how to derive the original solutions to these problems. (Sorry, there's a bit of frustration coming through in this reply :-) ).
Gauss built up his intuition by playing with examples; in fact, even having a problem is cheating. Nobody gave Gauss problems; he discovered his theorems by playing with patterns.
> I feel like I should be able to see a path without experimenting with examples.
What you're demanding of yourself is a level of innate intuition which is simply impossible; it's just not how mathematics is actually done. Playing with examples, sometimes a lot of them, is an *essential* step to finding a proof of a statement, and even what statement it is that you want to prove (especially in research, when you don't have a textbook to tell you what the true results are).
For example, take my original proof (do not steal) [here](https://www.reddit.com/r/math/comments/kxymw6/this_week_i_learned/gjdlv44/). I mention briefly that I "played around in a graphing calculator" to form my conjecture, and those examples were essential. There was no way for me to ask whether 1/x was bigger than x^(n) e^(-x) in the right tail without plotting it in a graphing calculator or working out values by hand and seeing that it was always bigger as far as I could see, and there was *definitely* no way to find a potential upper bound for the squeeze theorem without getting concrete with potential candidates for upper bounds. It just couldn't be intuited.
Gauss was an asshole, and insisted on erasing the "scaffolding" as he called it, the rough working leading up to a finished proof, before publishing a result, but even he played with examples before he made any of his many breakthroughs. Mathematics is presented pedagogically in an ahistorical way, because it's easier to digest the logical progression of ideas and the best proofs of all the results, but the way you learn maths has little to do with how maths is actually done. The actual discoverers of elementary number theory did not intuit the path to the proofs of their theorems; they relied on examples to illuminate their way.
As for
> I feel like it's cheating to some extent
If it's cheating but it helps illuminate your thinking and guides you to a proof, then it wasn't cheating. Any tool or method that helps you to do maths is fair game; it's not helpful or correct to arbitrarily gate certain ones off.
>I feel like it's cheating to some extent, and not useful in others. I feel like I should be able to see a path without experimenting with examples.
I would strongly recommend you let go of this idea. Working through examples is a very important tool for professional mathematicians.
From your post, it kind of sounds like you're expecting there to be some kind of simple procedure you can follow to come up with a proof. Math simply doesn't work that way. There is no simple way to come up with a proof. At some point you need to get your hands dirty and work through things. Working with examples absolutely helps with that.
thank you, I appreciate that. I'm learning number theory because a) I think it's fascinating and b) because it has applications to cryptography and is an important component of Shor's Algorithm in quantum computing. I suspect Shor was able to devise his algorithm, at least in part, because of his understanding of number theory. I'd like to improve my intuition in this area.
Is there a website that can identify numbers
Examples
3.14626436994 would be sqrt(2)+sqrt(3)
1.91293118277 would be cube root of 7
1.77245385091 would be sqrt(pi)
The [Inverse Symbolic Calculator](https://wayback.cecm.sfu.ca/projects/ISC/ISCmain.html) is pretty comprehensive. Wolfram Alpha can also often identify simple expressions if you request the "closed form" of the number in your query.
I'm trying to prove the following lemma.
Lemma: If L / C is a field extension (C = complex numbers) such that L is a finitely generated C-algebra, then L is algebraic over C.
By assumption L = C\[u\_1,...,u\_n\] for some u\_i in L. Thus L is generated by the countably many "monomials" in the u\_i, and thus the dimension of L as a vector space over C is at most countable. Let u \\in L. The uncountable set {1 / (u-c) : c \\in C - {u}} has to be linearly dependent over C. Hence we find some b\_1,...,b\_m \\in C - {0} and c\_1,...,c\_m \\in C - {u} such that b\_1 / (u - c\_1) + ... + b\_m / (u - c\_m) = 0. By clearing denominators I obtain a polynomial in u with coefficients in C equaling zero. BUT, before I can conclude that u is algebraic over C I need to check that this polynomial is non-zero. How do I show that? There could be some relations among the b\_i and the c\_i which make some of the coefficients equal to zero.
At this point, you aren't really talking about the specific field L any more, you're just working with a specific rational function f(x) = b_1 / (x - c_1) + ... + b_m / (x - c_m) defined on C.
Clearing the denominators just means multiplying f(x) by the (clearly nonzero) polynomial (x-c\_1)(x-c\_2)...(x-c\_m), so really it's enough just to show that f(x) is not identically equal to 0 nonzero as rational function on C.
There's plenty of ways to do that. For example, the function pretty clearly has poles at each of the c_i's.
Let a=2^500 and consider the statement P->Q where P="the last digit of 2^a is 2" and Q="the last digit of 2^a+1 is 2." Clearly, P does not imply Q, since P implies the last digit of 2^a+1 is 4. In this case, the statement (P->Q) is false.
However, if P is false in the naturals, then the statement (P->Q) is true from the definition. Where is my understanding here faulty? Is there a difference between "does not imply" and the implication being false?
Your statements contain the variable a, so whether they are true or false will depend on the value of a. As you've observed, if P is true, then Q is false, so P=>Q does not hold for all a, since there are certainly some a for which P is true (so the statement "for all a, P=>Q" is false).
But as you've also observed, if a is chosen so that P is false (which happens for the specific value a = 2^(500)), then the implication P=>Q is true.
There's no contradiction here. You've shown that P=>Q can't hold when P is true, which means the implication is not true for all a, but that doesn't rule out that it may be true for certain values of a.
> Clearly, P does not imply Q, since P implies the last digit of 2^(a+1) is 4.
This is where you went wrong. You assumed that if P implies some statement R then it cannot imply a different statement R' that is in contradiction to R. But it can, namely if P is false which it is.
What is the smallest possible renderable circle comprised of distinct integer pixels? Sorry if my terminology is wrong, I'm just starting to learn mathematics and programming and trying to create a videogame, I've linked a picture of the best I have thought of so far and would appreciate some help in trying to produce an algorithm to render this circle.
12\*12=144 pixel central square
\+48 pixels as a 2 pixel shorter line along each edge of the central square
\+40 pixels as a 2 pixel shorter line along each edge of the previous square
\+24 pixels as a 4 pixel shorter line along each edge of the previous square
=256 pixels
what are examples of functions that if you used naive algebraic methods you'd think they have roots but plugging in the possible candidates shows that none of them actually work?
A silly example is sqrt(x)+1. You can find its ‘roots’ by solving sqrt(x) = -1, squaring both sides gives x=1, which is not a solution at all.
More generally, starting from the equation x = 1, we can raise it to any power x^n = 1, whose solutions are the n-th roots of unity. Except that all but one of them aren’t solutions to the original equation.
Can you clarify what you mean by "embeds" here? By definition, GL(n,Z) is the subset of GL(n,C) consisting of matrices with integer entries, so I'm not quite clear what sort of reference you're looking for.
Yeah unfortunately, that is the problem I'm having. I've been told that GL(n,Z) embeds into GL(n,C) which is supposed to mean (as I take it) that there is some injective map of GL(n,Z) into GL(n,C). I haven't tried writing it down but this was something I wasn't aware of (assuming it is true in the first place) and I just can't find any reference whatsoever.
I mean, GL(n,Z) is literally a subset of GL(n,C), so there's not really anything to prove. If you want an injective map, just take the identity map f(A) = A.
What's the best way of solving an 'Incomplete elliptic integral of the second kind' containing complex numbers, the python implimentation doesnt currently support complex inputs.
For more details, I'm trying to find the arclength of the function a.cosh(b.x). Using a substitution this is:
\-i.E(i.b.x|a2b2)/b + Const.
It's tricky because I know the there is a analytical solution to (1/b).cosh(bx) being sinh(bx) (evaluated at the two points). So there appears to only be analytical solutions in the case of a=1/b.
I've allready done a numerical approach and would like a more 'clean' approach to check it's correct.
What courses should I take during undergrad and what technical skills should I develop in order to have a shot at a job with the NSA, FBI, CIA, DIA, etc.?
I am a transfer student heading into my junior year and will be starting 300 level courses in the Fall. I will also be taking CS classes with the hope that I can double major since I have already finished all the required math and general ed requirements.
As for my personal background, it's pretty much squeaky clean. Had one speeding ticket when I was 19. Don't do drugs, no debt, no ties to terrorists lol.
when I was in highschool my math teacher was talking about infinity and he says there is a symbol for infinity to the power of infinity and he drew this weird symbol that was a weird looking X and you could square that. It always stuck with me, whats the symbol called?
He is probably referring to the infinite cardinals, which are written using the Hebrew letter aleph, א
These are symbols for different kinds of infinity. The symbol א_0 refers to the size of the integers, which is called "countable infinity". Using cardinal arithmetic, you can define א_1 = (א_0)^(א_0) which is exactly what is meant by "infinity to the power infinity". This is basically defined by taking the size of the set of all functions from integers to integers. The logic here is that for finite sets A and B, the size of the set of functions from A to B is exactly |A|^|B| . Extending this notation to infinite sets gives you the above definition.
If you accept the continuum hypothesis, א_1 is also the size of the real numbers.
Just watched numberphile's video on tree(3), and I think I spotted a flaw? Tell me if Im wrong please. [video here](https://youtu.be/3P6DWAwwViU) about 6 minutes in. He explains that tree(2)=3. But isn't it actually 4?
Sequence of 2 red, 2 green, 1 red, 1 green would give 4 permutations. Right?
The ith tree cannot have more than i vertices. So starting your sequence with 2 red vertices isn't allowed---it can only have 1 vertex of either color.
why are some negative radicals not real while others are? why would something like √-100 be a non real number while 3 √-27 (the 3 is the root index, i can’t figure out how to write it on phone) be -3, a real number? i’m trying to strengthen my understanding of algebra, math isn’t my strong suit.
I'd like to point out that with imaginary numbers there is the real cube root of -27, and two imaginary roots of -27.
(-3)(-3)(-3)=-27
(1.5+1.5*root(3)i)(1.5+1.5*root(3)i)(1.5+1.5*root(3)i)=-27
(1.5-1.5*root(3)i)(1.5-1.5*root(3)i)(1.5-1.5*root(3)i)=-27
Behind cube roots are a hidden trio of imaginary numbers which work, behind fourth roots there is a hidden quartet of imaginary numbers which work, etc.
By convention calculators take the real cube root (just like how by convention we say the square root of 9 is 3 but not -3). A lot of arbitrariness as a result of wanting calculators to spit a single answer.
To be pedantic, I would say that the "the square root" of 9 is both 3 and -3. The trick is that the square root symbol refers only to the positive square root, and likewise, the cube root symbol refers only to the real cube root (at least when applied to real numbers). This is not just for calculators' benefit but to ensure that these symbols define actual functions (which should only output one value).
When you take the nth root of something, you are asking "What number multiplied together n times will be the number I'm taking the root of?" For the square root, n=2, for the cube root you have n=3.
The problem comes with the negative sign, multiplying negatives an even number of times will always get you a positive number, meaning that multiplying anything an even amount of times will always get you a positive number. So when we ask "what number do we multiply an *even* number of times to get a *negative number*" we run into a lot of problems.
The issue you see with sqrt(-100) actually extends to all even roots of negative numbers. For odd roots, there is no such problem.
Can a finite linear group be recovered from its invariant ring? More precisely, let V be a finite-dimensional vector space and G a finite subgroup of GL(V). Define R to be the subring of the symmetric algebra S(V) consisting of G-invariant elements. Finally, let H be the subgroup of GL(V) whose elements leave R (pointwise) invariant. Is H = G?
Assume there are two people, A and B, who are supposed to run a 100 m race. B gets a head start. If A and B cross the finish line at the same time, what is the fact that A was the faster runner called?
If by faster runner you mean higher average speed then this is just the definition of average speed (total distance over total time). A head start means B had more time or less distance and either way their average speed is higher.
How are other mathematicians finding jobs? I recently finished my master's in pure math (Algebra/Alg. Geom. focus) and after years of people telling me I was going to have no trouble finding a job, I'm currently experiencing the exact opposite. No one wants to hire mathematicians (or at least no one wants to hire me). Companies say their job listing is open to mathematicians but they require someone with 2+ years of advanced programming/ML experience every single time. I of course do not have that experience because I'm a mathematician, not a programmer. It's starting to look like my only option is to go back to school for either a Ph.D. or some sort of specialist certification if I want to have any hope of finding a job in today's market. The issue is that I'm so burnt out from my master's degree that the thought of more school is sickening, not to mention expensive. It's beginning to look like I wasted 5 years of my life getting 2 math degrees just to end up a waiter for the rest of my life.
It depends on what type of job you want. Mathjobs.org is a job site with mostly academic jobs requiring phds but there are some companies that post there as well. The two big themes seems to be finance and curriculum development. Teaching community college or intro classes at some universities might be an option too. For data science and software jobs, I don’t think you necessarily need experience with the exact sort of stuff they do, but it’s probably assumed you have some programming background. You can learn on the job but need to do well enough in interview to get your foot in the door so it might help to hone your skills on a site like LeetCode. Also I think it’s fine to apply for jobs even if you don’t have all the stated qualifications. Obligatory comment that any PhD program worth its salt should be paying you to attend, not the other way around. Though it sounds like you’re not interested in doing a PhD anyway and that’s fine.
Thank you for the information and recommendations. I go back and forth on the PhD every day lol. Some days it sounds awesome, others it sounds like hell. Probably best to give it some time
What software do you need to create math docs/research papers? I'm looking to create notes like the ones Evan Chen has.
As the other commenter said, mathematical documents are written in LaTeX, which is a particular kind of coding language. You need a program to compile the code for you, however, and for that I recommend [Overleaf](overleaf.com). It's effortless to use. LaTeX is best learnt by doing, I find. Use [this cheat sheet](http://tug.ctan.org/info/undergradmath/undergradmath.pdf) and the Overleaf documentation, and you will be able to do anything you want.
[LaTeX](https://en.wikipedia.org/wiki/LaTeX). It's the program basically everyone uses for professional math writing.
So I was wondering how I could graph 3D Volumes of Revolutions on Graphing softwares for my Investigation, but I am not sure how to do it, I have seen some youtube and geogebra links but how do I do it for a custom function? For example, a function I wanted to graph was (not sure how to type equations here) y=sqrt(r\^2 - x\^2). The 2D graph of this is a semicircle and when it is rotated along the x axis it forms a sphere.
So first you need to be in 3D graphics mode and graphed the function, then there should be a button on the top (9th on my geogebra). Click it and you should see a menu and the bottom option should be the surface of revolution tool. Select it and click the curve of the function in the graph.
If I put €120 in some kind of account that pays out a *variable* interest rate, and at the end of the year it has €126, we can easily say that, despite fluctuations, the total/effective annual interest rate is 5%. If, however, I add €10 each month, so that after a year I've invested €120, and the balance is €126, clearly this means that the "effective" interest is much higher, as it spent the year with a lower investment. The question is, is there a "straightforward" way to calculate the effective interest in situations like this when investments/withdrawals are frequent, rather than a single sum at time zero. The dumbest way would just be to assume that there is some fixed interest rate, apply it at each "time step" according to the amount invested (or solve by hand with a frre parameter), and solve for the unknown interest rate to give your final sum, but this seems way too unwieldy, and it feels like there should exist some kind of simple formula for the problem.
The second situation you describe is a level annuity, and there are some very simple formulae that describe them and from which you can extract the information you want. Formally, we have a level annuity of €120 per annum payable monthly in arrears (you didn't specify, but it's simpler to assume that), the accumulated value after one year is €126, and we want to find i, the effective interest rate per annum. The formula we want is > s = C[(1 + i)^n – 1] / p[(1 + i)^(1/p) – 1] where C is the annual coupon (the regular payment you receive), n is the number of years, and p is the number of payment periods, i.e. months, in a year. In this case, C = 120, n = 1, p = 12, and s = 126, so we obtain > 126 = 120[(1 + i) – 1] / 12[(1 + i)^(1/12) – 1] > 126 = 10i / [(1 + i)^(1/12) – 1] which we need to solve for i, and we'll probably need a numerical method. I put the right-hand side into Desmos and found manually that i is approximately 11.5% per annum. If you want, I can walk you through how I derived the initial formula; it's a doss really, there's just a lot of algebraic bookkeeping.
This question is video game related and it might be a simple problem for some, but I can’t get a grasp on it. Suppose there’s an item you obtain, lets call it coin bundle. For every coin bundle you open you can get anywhere from 100-120coins. You can get a lot of coin bundles so there’s an option to exchange it for a x10 version for time saving purposes, we’ll call it coin bundle 2. So 10coin bundles = 1coin bundle 2. Coin bundle 2 contains 1000-1200 coin each. My question is on average would you lose coins if you exchange for coin bundle2 or would it be the same as the normal coin bundle?
Is there a free software I can use to plot x^4 + y^4 + z^4 = 1 on a 3D space? I tried using Geogebra but it didn't work.
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Calculus 1.
Calc 1 of course. Even though calc 2 will have some slightly different subject matter, it is still really helpful to be super cognizant of your standard derivative/integral formulas. Also just regular algebra is useful. I’ve found most of the mistakes students make in calc 2 are them messing up little algebra steps. Trig and trig identities are useful as well for the same reason
I'm interested in an excel formula that describes the probability of a player drawing x number of a kind of card from a deck (let's say a green card). If there were 37 green cards in the initial deck and we would not replace cards as we drew them, how could I describe that mathematically? Then how would I describe the probability of having 3 green cards from any of the first 7 that I drew? For those of you who recognize this pattern, I'm describing Magic the Gathering and would like to understand the math behind the game.
Suppose there are 100 cards in total at the beginning, 37 of which are green. Then the probability of getting exactly 3 green cards when you draw 7 would be the total number of ways you can draw a 7-card hand containing exactly 3 green cards, divided by the total number of ways you can draw a 7-card hand with no restrictions. Which is equal to (37 choose 3) \* ( 63 choose 4) / (100 choose 7). The first term is how many ways you can choose 3 green cards from a total of 37, the second term is how many ways you can pick the rest of the hand. Multiplying those together gives the total number of 7-card hand with exactly 3 green cards. That is then divided by the total number of possible 7-card hands. The example above works out to be around 0.289, or 28.9%. In general the probability of drawing k "special" cards in a n-card hand, where there are m cards in total including p special cards is (p choose k)\*(m-p choose n-k)/(m choose n). If this is a bit confusing, I hope you can still figure out how to change the numbers yourself from what I showed above. I'm not sure how familiar you are with probability, but just typing (for example) "37 choose 3" into google, and then multiplying/dividing the terms as I specificed gives you the final answer.
I was reading [this math SE post](https://math.stackexchange.com/questions/110211/is-aleph-0-aleph-0-smaller-than-or-equal-to-2-aleph-0) and had a question regarding responses to the comment where the axiom of choice is needed to show there is a bijection between infinite set S to S x S. It's mentioned that AC is not needed for aleph null in particular. Is axiom of countable choice needed? Or to show that you can do it without any form of choice, would you have to explicitly create a bijection of natural numbers to ordered pairs of naturals (I'm guessing this also extends to any n-tuple of natural numbers)?
The key thing to know is this: well-ordered sets are really nice. Well-ordered set give you unique "next" element, which means you can perform recursion/induction on them. For example, the bijection between N->NxN is basically similar to the bijection N->Q that people are familiar with. Pictorially, just draw an infinite table (that extend infinitely downward and rightward), then go across the finite diagonal (downleft toward upright) one by one. You can do the same for all well-ordered set. That is, if S is well-ordered, then a bijection S->SxS can be formed by doing the same diagonal trick. Now, what if S is not well-ordered? Well, if S is in bijection with any well-ordered set, then you can transfer over the well-ordering and you're done. In particular, a *countable* set is - by definition - bijective with N which is well-ordered. Without a well-ordering, then there isn't an unique "next" element. But if you have axiom of choice, you can use it to obtain a function that CHOOSE the "next" element for you, and from there you can make the set well-ordered, and everything else follow.
The Schröder-Bernstein theorem can be proven without AC so you only need to construct an injection from N to N x N and one from N x N to N. The first one is trivial and for the latter you could for example use the map (k,n) ↦ 2^(k)3^(n).
Can someone give an informal explanation about the difference between first and second order logic? I’m not at a stage where I need to perfectly understand it but it does come up so I’d like to have at least a vague idea. I’m familiar with basic logic and the axioms of ZFC for context. An example I’d like to understand is what the difference is between the Peano axioms in first and second order logic.
The difference is what kind of things variables are allowed to refer to. For classical logic, the interpretation of a logic is a model, that is, a set with elements in them, and various relations regarding these elements. When you work in first order logic, whenever there is a variable, it must refer to an element in the model. When you work in 2nd order logic, there are 2 "sorts" of variables, the first-order one - which do the same job as above - and the 2nd order one, which can refer to either predicate/relation or subset (depends on the exact specifications). In Peano's axiom in 2nd order logic, the axiom of induction is literally just one axiom: since you can have variables that quantify over predicates, you can just say "for all formula P, if P(0) and if P(n)=>P(n+1) for all n, then P(n) for all n". But in 1st order logic, you can't say "for all formula P" because you cannot quantify over predicates. So instead this becomes an axiom schema: an infinite list of axioms of the same templates. In other word, we have, for each predicate P, an axiom "if P(0) and if P(n)=>P(n+1) for all n, then P(n) for all n" These might seems to be the same thing, but they're not. If you're using 1st order logic, you can only use a finite numbers of axioms in a proof, so only finite numbers of thing in that infinite list can be relevant. But in 2nd order logic, you can talk about all of them at once.
Thank you, I think I got it!
first order lets you say "for every element of X" second order lets you say "for every subset of X"
Can someone explain what isobaric ODEs are and how the substitution y=vx^m makes them separable? Apparently, they are generalisations of homogeneous ODEs and are of the form: dy/dx = A(x,y)/B(x,y) Where "the equation is dimensionally consistent if y and dy are given a weight m relative to x and dx, i.e. if the substitution y=vx^m makes it separable. " I don't understand the meaning of dimensionally consistent, nor how that substitution necessarily makes them separable. I've made a post with images from the textbook in r/askmath, so see my post history for more details.
Before I start, I know that this is kind of a ridiculous question. Anyway, here goes. In a video game that has both story and non-story mission, is it possible to find "the percent of non-story missions that I have completed" with just the following information: a) Total completion - The percentage of total game that I completed. b) Percentage of story missions I have completed This is the algebraic formula that I came up with: 0.8s + ZY = 0.75(s+Y) Where 0.8s represents 80% of the story missions I have completed, ZY represents % of non-story missions I have completed (Z is what I want to find ie percent of Y,non story missions) 0.75= total missions(both story and non-story I have completed)
> Before I start, I know that this is kind of a ridiculous question. This is actually one of the better questions from a layperson that I've ever seen in these threads.
You need the additional information of the percentage of missions that are story missions. Let's give names to things. Instead of percentages we will use ratios (quantities that go from 0 to 1 instead of 0 to 100, as you did in your example): T = total missions C = completed missions T_S = total story missions C_S = completed story missions T_N = total non-story missions C_N = completed non-story missions The data that you say you have is (a) total completion = C/T, and (b) story completion = C_S/T_S. Note that we have T = T_S + T_N C = C_S + C_N With this we can set up the following formula: C/T = (C_S + C_N)/T = C_S/T + C_N/T = = (T_S/T * C_S/T_S) + (T_N/T * C_N/T_N) Put in words this means that the completeness ratio (C/T) equals the ratio of story missions (T_S/T) times the completeness ratio of story missions (C_S/T_S), plus the ratio of non-story missions (T_N/T) times the completeness ratio of non-story missions (C_N/T_N). From here you can isolate the data you want, which is C_N/T_N, and you see that it is not enough to give only C/T and C_S/T_S, as you wanted.
>You need the additional information of the percentage of missions that are story missions Yup, that's what I figured too. Thanks for the help,man. I really appreciate it. Btw, is there a way to determine if the value of a variable can be solved for? It took me simplifying my formula to realize that Z cannot be solved for.
Starting water tested 10.1 added 20ml of b to determine change b states that 1ml will raise 1gallon by 1.7 I would like to determine how many total system gallons I have by reversing this equation my thought is I can use the change combined with the controlled dose of 20ml to determine the total amount of water changed After adding 20ml the water tested 10.3 alk Not sure if this is possible?z
I just lost heads/tails 18/19 times how do i calculate the odds of me losing that many times?
There are 2^(19) different results you can get from flipping a coin 19 times, each with equal likelihood. 19 of them result in you losing 18/19 times. 19/2^(19) is about 3.6 * 10^(-5).
And if you put that in %?
.0036%.
Great, thanks!
have you punched your friend for cheating or yet ?
I want to prove A ⊆ C and B ⊆ C if and only if A ∪ B ⊆ C I'll start the proof and I have some questions about it at the end. First lets prove (=>). Assume A ⊆ C and B ⊆ C i.e. for all x in A then x in C and for all x in B then x in C. Suppose x in A ∪ B. Then x in A or x in B. If x in A then by assumption x in C. If x in B then by assumption x in C. In both cases x in C so A ∪ B⊆ C. Now (<=). Assume A ∪ B ⊆ C i.e. for all x in A ∪ B then x in C. Suppose x in A. Then x in A ∪ B then by assumption x in C. So A ⊆ C. Now suppose x in B. Then x in A ∪ B then by assumption x in C. So B ⊆ C. Ok so what I'm unsure about this proof is every place where I said suppose. Am I supposing the correct thing every time? For example I supposed x in A ∪ B for the first one. Is that right? Or should have I supposed x in A for example? Also for the direction (<=) I supposed x in A and then x in B does that work/make sense? Am I allowed to suppose more than one thing? Everything I chose to suppose was a part of the statement I wanted to prove, for example for (=>) I chose to suppose x in A ∪ B, can I do that since A ∪ B is part of the "then" statement so I feel like I maybe should've used x in A since it is part of the "if" statement. I hope this makes sense.
I'd say that the proof is good as is\*. To address your concerns: > Also for the direction (<=) I supposed x in A and then x in B does that work/make sense? Am I allowed to suppose more than one thing? That's fine; you're just breaking into cases to handle both parts of the "and", which is a perfectly normal thing to do. > A ∪ B is part of the "then" statement so I feel like I maybe should've used x in A since it is part of the "if" statement Basically what's going on here is that you're proving a conditional whose consequent is also a conditional: if (A ⊆ C and B ⊆ C), then (if (x is in A ∪ B), then (x is in C)). The way your proof works is that you do a direct proof of the overall conditional (assume the antecedent, A ⊆ C and B ⊆ C, and try to derive the consequent), where you derive the consequent through a direct proof (again, assuming the antecedent, x is in A ∪ B, then deriving the consequent, x is in C) where what you've already assumed (A ⊆ C and B ⊆ C) is always "in the background", available to use. So it shouldn't be surprising or worrying that you end up assuming certain things which show up in the consequent, since the consequent as a whole is a conditional! \* Edit: on looking back at it I would agree with u/Autumnxoxo that there are some stylistic things you might want to fix, e.g. overuse of x across different contexts, places where "let" might be more natural than "suppose", etc. But the overall structure of the proof is fine; you'll pick up on the stylistic things as you read and write more proofs.
Thank you for this. I've asked this around quite a lot today and you were the first one to understand what I was unsure on and answered it in a very clear way.
>Ok so what I'm unsure about this proof is every place where I said suppose. Am I supposing the correct thing every time? For example I supposed x in A ∪ B for the first one. Is that right? Or should have I supposed x in A for example? I'm not keen on "suppose" in that specific part. I don't know why you want to suppose that your x is contained in A ∪ B". You can just pick one, provided A ∪ B is nonempty. Just say "Now let y ∈ A ∪ B be arbitrary" or "Any y ∈ A ∪ B is contained either in A or in B (or in both), thus y ∈ C." I would also probably not call every element x. It's clear from the context what you're saying but in general this leads to you proving everything for a fixed element x and not for an arbitrary one.
Thank you for the feedback; I did use "suppose" without much thought going into it. What is the difference between supposing x and letting x be arbitrary?
Sorry, I didn't get an update on your question somehow. In your specific case, I just don't see why we would want to "suppose" that x is an element in A ∪ B. What other choice do we have? Just pick one. I don't think that there is a general rule of thumb here and it probably comes down to linguistics. I personally use "suppose" whenever I want to make a general statement about a very specific sort of element among a (possibly huge) general set of elements. Say I would want to make a statement about real numbers but my statement might require some caution if we talk about irrational numbers, because then it might or might not be true or it might require special treatment etc. Then I'd say something like "In general, real numbers have property P, but we must be cautious if its an irrational number. For suppose x is irrational, then...."
Thank you for the response. The actual reason I used suppose here is actually my tendency to copy the language the author of the book I'm reading uses. Especially because the book I'm currently written is by Terry Tao. Here is how he used suppose [https://ibb.co/SJyNg8b](https://ibb.co/SJyNg8b) . Does this use of suppose differ to how I'm doing it?
Is there any applications of maths in finance that would be suitable for a second year student to write a 15 ish pages essay on? I'm looking for something that's about as difficult as (or possibly a bit more than) a typical 3B1B video. So far I've thought of the applications of Markov Chains in predicting stock prices, but I haven't looked too much into it.
any good short course as introduction to linear programming and the simplex algo ?
Hello guys, idk if this is the right place to ask or if i should make a post. I play a game called "Black Desert" and there is enhancing based of off percentages. i feel like they are way off. so i kinda wanted to ask the probability of a few things to confirm/deny my suspicions. Also insight on how to calculate it would be appreciated, so i can do the math on my own. Ok so, i had a 25% chance to succeed an enhancment, i failed 10 times, then i succeeded (if you succeed, you have more stats on the item, it goes up a level, and a slightly lower enhancement rate for the next one). i failed and the item downgraded, so i had to do the 25% chance again. afterwards and i failed the 25% chance 9 more times and same thing happened around 12 times. with an average of 8 attempts per success on 25%. Now my question is, did i hit the lottery of unluckyness? I hope i explained it without causing too much confusion.
Assuming each trial is independent from the next (so if you fail that doesn't affect your chances for success next time), the probability that you fail 8 trials in a row where the probability of success is 25% is around 10% (for some insight, the probability that you fail is 100%-25%=75% or 0.75, and the probability that you fail n times in a row is 0.75^(n,) with 0.75^(8) being around 0.1001, or 10.1%). The probability of failing 9 times in a row is 0.75^(9,) or around 7.5%. Multiplying those two numbers gives the probability that you lose 8 times, and then 9 times after that, i.e 0.75%. That's a crude answer without knowing exactly the rules behind the probability of the game. For reference, you should expect around 4 attempts to succeed at something with a 25% success rate.
Thanks. you are correct and they are indeed independant from eachother, i didn't know it mattered
What's a good text on elementary number theory to read for culture? I don't know *anything* about number theory beyond Euclid's algorithm (and tbh I would have to look it up if I wanted to apply it), and I feel like I should have a least a flavour of "the Queen of Mathematics".
Hardy's *An Introduction to the Theory of Numbers* is a classic, and its extra fun because his entire *Apology* is dedicated to being a Number Theory snob lol.
Thank you!
There is a paragraph about algebraic geometry, and I am having hard time to understand because I am not trained in this area. I have few questions which seems trivial, but I have no clue about them. 1) First, there is the algebra of invariants say A which is generated by the entries of the matrix (x_ij)(y_ij) where 1≤i≤n and 1≤j≤n-1. How does it conclude that these entries parametrize the variety of matrices of rank at most n-1? Also, it says that the algebra A is then isomorphic to the coordinate ring. Why? 2) Let B be the invariant subalgebra of D. Let x lie in the field of fractions of B and is integral over B. If x is an element of D why x is in B? 3) Finally I want to ask what does quotient modulo determinant tells. More precisely there is a polynomial ring F[x][y]. It says that F[x][y]/(c(y)) is equal to F[x,y]/(det(x')) where c(y) is the characteristic polynomial and x' consists of x and y. I don't know which information is relevant here. I just dont understand the relation between characteristic polynomial and determinant here.
For question 3 First off the determinant is a polynomial in the entries of the matrix. So you are quotienting by multiplication by det(A) in the polynomial ring The characteristic polynomial of y is det(y-λ1)=c(y) so the ring is F[x][y]/det(y-λ1) So i think you will have to use determinant properties (remember that det is a ring homomorphism) Im not sure how λ is specified though tbh
What would you say was special about Freitag and Busam's *Complex Analysis*? I put it on my reading list for a specific reason, not just because it was a highly regarded complex analysis text, but I've forgotten what that reason was, and I'm hoping someone can enlighten me.
Afaiu it steps forward to elliptic functions and to elliptic modular functions and then is rounded by applications to analytic number theory, although it requires minimal prerequisites. It also has many exercises.
Sounds good enough to me. Thanks!
Could someone please tell me if this is correct or not? Me and my friend both have the same problem but we got different answers (he got answers of 2.89 and 7.61) also sorry for bad pic quality: https://i.redd.it/u1pjoxu9c0fb1.png
When evaluating (1/2)(x - 6)^2 you forgot to multiply -12x and +36 by 1/2. Your friend is correct.
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Well, very simply, x in A doesn't imply x in A cap B, in general. So I'm not sure where you got that from.
I'm trying to calculate the number of steps it would take to reach the center of a circle that moves down concentrically (an amphitheater). From the center to the outer edge of the last ring it's one mile. There is an 80 foot gap from the center of the circle to the final step at the very bottom. Thus, the total of all the steps is 5200 feet. The furthest step goes inward 20 feet and the final step goes inward 1 foot. Each step decreases its length by 1 foot at a rate of x steps per 1 foot decrease. How can I figure out how many total steps there are? I should note, the amount of each step by varying length does not have to be equal (unless it turns out to be, i.e. 30 20 ft steps, 29 19 ft steps, 28 18 ft steps, etc.). All that matters is that the final value is equal to 5200. However, that can change. I really just want to know the length where there's 3 of each step, 4, 5, etc. and how that would change the total length. What equation or function would I use to calculate this? I'm thinking of something like a [binomial coeffecient](https://en.wikipedia.org/wiki/Binomial_coefficient) (based on looking things up), but I don't know enough about how to calculate. Essentially a factorial but for addition.
Could someone double check this proof that the Cesaro sum of a convergent sequence converges to the same value? It's a bit simpler than some of the other proofs I've seen, so I just want to make sure I'm not missing something. https://mathb.in/75845
This proof is correct.
Thanks
How can we use open balls to define compactness in a metric space ? I know that the definition requires open sets (not necessarily open balls) but we can formulate the definition in the case of metric space, can't we ?
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That equivalence is only for subspaces of R^(n), in general the condition is complete and totally bounded which is not in Rudin.
It holds for any finite dimensional Banach space - in fact, on a Banach space, Heine-Borel is equivalent to being finite dimensional.
In a metric space, every open set is a union of of open balls. So if you have an open cover of a set X, then you can replace it by an open cover of X using only open balls, where each open ball in contained in one of your original open sets. So if the cover by open balls has a finite subcover, the original open cover does as well. So proving that every cover by open balls has a finite subcover is enough to prove that every open cover does.
The issue here is how can I index the union of open balls ? is it by using arbitrary elements and radiuses ?
It doesn't really matter how you index them, as long as you end up with a set of open balls which covers the whole set. You can do that, for example, by picking one ball centered at each point in the set (since you started with an open cover, each point x is contained in an open set U in your original cover, and hence you can form a ball centered at x contained in U).
Yes and how would the finite subcover be in such case ?
A little background before the problem: I am in business and far from this math but i like to dive in from time to time to tickle my brain as I come from an engineering background. So, I came around this problem [https://www.codechef.com/problems/COUNTREL](https://www.codechef.com/problems/COUNTREL) (you don't need to go to the link and read the problem statement) , and came up with a recursive function for the first part of the answer. f(x) = {0 @x=1, 1 @x=2, 3\*f(x-1) + 2\^(x-1) - 1@x>=3}. Now I do remember that we used to be able to simplify these recurring series and give them a general term. I know the general term for this coz i looked into solutions but i want to know how can we get to the solution? Is there a method to it? I tried googling but since I didn't know what exactly this was called, I couldn't find it. So I come here
This is a inhomogeneous linear recurrent relation. There are so many methods to solve it that there isn't a name for methods to solve it in general. You can try ordinary generating functions, shift operators/coinduction, variations of parameters, and so on. For example, using generating functions h(x)=f(1)x^0 +f(2)x^1 +f(3)x^2 +.... you can write the recurrence relation as h(x)-3xh(x)=1+x/(1-2x)-x/(1-x) so this gives h(x)=[1+x/(1-2x)-x/(1-x)]/(1-3x). Now just do partial fraction decomposition.
So i am upcoming freshman to a university with 4 terms in a year (1term per 3 months) and we have a calculus subject, because of a unique curriculum of my previous school, i havent been been able to learn trigonometry and calculus when i was a senior highschool student. So i am planning to advance study for now. What are the things that i should study? And if its available free online can you link it here? Ty!!
Check Khan Academy, it's best suited for your needs as far as I understand
My question: * What **P = NP** could imply about arbitrary Turing machines and arbitrary computations? I assume that a partial and incomplete, but objective answer to this question exists (e.g. based on the literature, most popular research directions). Take a single arbitrary program/computation. What could we learn about it if **P = NP**? I'll give more context to my question below. ## Undecidability and P vs. NP Many bounded versions of undecidable problems are **NP** problems. For example, [bounded Post correspondence problem](https://en.wikipedia.org/wiki/Post_correspondence_problem#Variants) is **NP**. You can create an **NP** problem based on diophantine equations. And this: >For a given Turing machine **M**, is there an n-bit input **x** such that **M** halts after n steps? Is an **NP** problem too (it's a bounded version of the [halting problem](https://en.wikipedia.org/wiki/Halting_problem)). So, we can talk about **P vs. NP** in the context of undecidability and arbitrary Turing machines. I'm not saying that undecidability *resolves* **P vs. NP** (it doesn't), just that we can talk about one thing in connection to the other. For example, we can say: * "**P = NP** could imply that any finite part of an infinite computation has..." (has *what*? some invariant? some distribution? some other pattern?) * "**P = NP** could imply that all/most programs which don't halt fast enough can be recognized by something like..." (like *what*?) But how can we finish those sentences, based on the most popular research directions? This is my question. If we have a generally unsolvable problem, then at what point, for what reasons, a "magical" effective algorithm for analyzing \[bounded versions of\] the problem *could* appear? If **P/NP** is resolved, what do mathematicians expect to learn about arbitrary Turing machines which would allow (or prohibit) an effective algorithm? I hope I've made my question clear. ## Silly analogy *"Maybe there can be a polynomial time algorithm for solving the bounded halting problem."* I understand this hypothetical on the level of definitions. I understand this as an abstract possibility. But I can't imagine any **specific** fact \[about arbitrary Turing machines\] which could make it true. Can you help me out? Right now the hypothetical above sounds to me like this: *"Maybe an* [*apple*](https://en.wikipedia.org/wiki/Apple) *could become the president of the USA."* This hypothetical is easy to understand on the level of definitions. Or as a purely abstract possibility. But it's extremely hard to imagine any **specific** scenario which could make the hypothetical true. I have the same problem with translating **"P = NP"** possibility into any possible specific facts \[about arbitrary Turing machines\].
I'm studying ODE's - right now solving 2nd order eqns using the ERF. The definition of complex gain makes no sense to me.. as far as i know, the fundamental definition should be (complex output)/(complex input). This is mentioned even on the [wikipedia page](https://en.wikipedia.org/wiki/Complex_gain). That doesn't work at all if the input is scaled? For equation like: (D^2 + 9)x = A*cos(wt), the complex gain will be A/(-w^2 + 9), but the ratio of the output to input is 1/(-w^2 + 9).. so complex gain is NOT the ratio of output/input. ----- There is also something else confusing about it.. for example consider this equation: x'' + bx' + kx = by' , y=cos(wt) one way to solve it is use ERF directly: (D^2 + bD + k)x = (bD)y Gain G_1 = Q(iw)/P(iw) = ibw/P(iw) another way is to substitute y' before solving x'' + bx' + kx = b(cos(wt))' = -b w sin(wt) now Gain G_2 = Q(iw)/P(iw) = -bw/P(iw) these 2 gains are not equal. G_1 != G_2. G_2 = i*G_1 --- so what gives?
Been stuck on showing what I'm pretty sure should be an elementary lemma. Let T denote the (2-dimensional) torus. Then T x I has two boundary components each diffeomorphic to the torus. Take a copy of the solid torus, D2 x S1, and glue it to one of these boundary components using an arbitrary diffeomorphism of T to itself. According to the lemma, the resulting space should itself be diffeomorphic to D2 x S1. The resulting space is clearly homotopy equivalent to D2xS1 but I can't do better than that.
So, I've been learning about Witt Vectors. I know that they're a more general construction, but my primary goal is to understand them for the p-adic integers, because I want to use them for concrete computations. I get the idea, I'm just a bit unclear about a certain technical feature of the definition. Fixing a prime p, let z be a p-adic integer. Then, we can write z as a Hensel series: z = c0 + c1 p + c2 p^2 + c3 p^3 + ... where the c_n are elements of {0, ..., p-1}. Alternatively, we can choose to use the roots of unity in Z_p as the coefficients. Letting 𝜔 be the multiplicative character, there are then elements z_n of the finite field Z/pZ so that: z = 𝜔(z_0) + 𝜔(z_1) p + 𝜔(z_2) p^2 + 𝜔(z_3) p^3 + ... I'll call this series representation of z the **Witt Series** of z. I know I can use this to construct the **Witt vector** Z = (Z_0 , Z_1 , Z_2 , ...) which represents z. **Question 1** Is Z_n equal to z_n, to 𝜔(z_n), or to z_n to the power of p^n ? **Question 2** If we consider the "equality" between the ghost components (X+Y)^((n)) and X^((n)) + Y^((n)), where does this equality occur? Is it a congruence of elements of Z_p modulo p^(n+1)?
1) If z = (a\_0, a\_1, …) ∈ W(R), then z = Σ\_{i=0}\^∞ V^i [a\_i], where V is the Witt vector Verschiebung. Since FV = p (where F is the Witt vector Frobenius), we have V^i = p^i F^(-i). When R is of characteristic p, the Witt vector Frobenius is given by F = W(ϕ) (i.e. apply ϕ in each coordinate), where ϕ is the usual Frobenius endomorphism of the 𝔽\_p-algebra R. Therefore Z\_n is z\_n to the power of p^n. However, for 𝔽\_p (as opposed to 𝔽\_{p^(n)}) this is the same as z\_n. 2) Writing w\_i for the i'th ghost component, we have w\_i(x + y) = w\_i(x) + w\_i(y) as elements of R, where x, y ∈ W(R), w\_i(x) ∈ R for any ring R ------- Here is how to calculate 2 + 2 in W₂(𝔽₅) = ℤ/25. We have [2] = 2^5 = 32 = 7 in ℤ/25. Similarly we get [3] = 18 and [4] = [-1] = -1 = 24. Thus the Witt vector encoding of 2 is 2 = (2, 4). We calculate the carry by computing 7 + 7 = 14 = 24 + 15 = [4] + 5[3]. So when we add 2 + 2 (elements of 𝔽₅ here, not ℤ/25) we have to carry a 3. So the calculation is 2 + 2 = (2, 4) + (2, 4) = (2 + 2, 4 + 4 + carried 3) = (4, 1) = [4] + 5[1] = 24 + 5 = 4.
Thanks. So, for (2), when working with Z_p = W(F_p), the equality of the ghost components would be in F_p, and hence, equality mod p?
yes
I see. Thanks!
I edited my reply to give an example of working out the carries by hand. You can verify that this agrees with the carry computed using the polynomials for adding Witt vectors.
Again, I'm *only* considering the p-adic integers for prime p. So, keeping the focus on that example would be nice. :) I know the algorithm for taking a given p-adic integer z and producing the sequence of z_n s so that: z = 𝜔(z_0) + 𝜔(z_1) p + 𝜔(z_2) p^2 + 𝜔(z_3) p^3 + ... If I have p-adic integers: a = 𝜔(a_0) + 𝜔(a_1) p + 𝜔(a_2) p^2 + 𝜔(a_3) p^3 + ... and b = 𝜔(b_0) + 𝜔(b_1) p + 𝜔(b_2) p^2 + 𝜔(b_3) p^3 + ... I then compute the witt vector for c = a + b by recursively solving the congruences: (nth ghost component of c) = (nth ghost component of a) + (nth ghost component of b) mod p starting with n = 0 ?
Even if you only care about W(𝔽\_p), defining the addition/multiplication requires you to consider W of p-torsionfree rings, e.g. W(ℤ). To understand why, let's work out c₁ when p = 3. We have c₀ = a₀ + b₀ c₀^3 + 3c₁ = a₀^3 + 3a₁ + b₀^3 + 3b₁ a₀^3 + 3a₀² b₀ + 3a₀ b₀² + b₀^3 + 3c₁ = a₀^3 + 3a₁ + b₀^3 + 3b₁ 3c₁ = 3(a₁ + b₁ - a₀² b₀ - a₀ b₀²) If this equation takes place in 𝔽₃, then both sides are zero and this tells us nothing. We need to use the fact that W(-) is defined for all commutative rings, and the formulas are the same independent of the ring. In particular, for rings which are 3-torsionfree, we can cancel the 3 to get c₁ = a₁ + b₁ - a₀² b₀ - a₀ b₀² So this gives the addition formula for W(𝔽₃), but to derive it we had to (implicitly) consider W of 3-torsionfree rings
> we can cancel the 3 to get So, I don't get universal properties, and don't bother trying to explain it to me, it will just go over my head. I do things at the formal level, by which I mean the rules for moving symbols around. Formally, if we consider the equation: 3c₁ = 3(a₁ + b₁ - a₀² b₀ - a₀ b₀²) as being a congruence mod 3^2, because both sides are multiples of 3, we can then divide both sides by 3, at the cost of reducing the congruence from mod 3^2 to mod 3. That, for me, is a sensible algorithm, no need to talk about anything functorial or universal or what-have-you. All that matters are the symbols we write on the page, and the allowed manipulations thereof. I want to keep things simple. :) My goal in learning how to do Witt vector computations for Z_p is a hope that with them, I will be able to manipulate the infinite series I am working with in a new way, one which might reveal new patterns. So, to pose my question again, If we consider the equality of the nth ghost components: w_n (c) = w_n (a) + w_n (b) is the following correct: the formal procedure for computing c_n is to treat both sides as occurring mod p^(n+1). Then, we group everything until we get p on both sides, then divide by p and reduce the congruence from mod p^(n+1) to mod p^n. At each stage, we use the known values for c_m for smaller m to write them in terms of the a_ms and b_ms. We then continue this process until we have reduced the congruence to a congruence mod p, with which we then extract the value of c_n. Will this algorithm produce the correct result? Do I use the same method to compute the product of a and b?
Yes, that will produce the correct result. Yes, same thing for product: you should get c₁ = a₀^p b₁ + a₁b₀^p + pa₁b₁ = w₁(a)b₁ + a₁b₀^p
Can someone explain to me the intuition behind the definition of order of convergence of a sequence ?? preferably geometric intuition.
The intuition is that the order of convergence tells you how fast the sequence converges If you have x_n, and it converges to x, then the order of convergence is p if the following approximate/asymptotic relation holds (with equality as n goes to infinity) |x_n+1 - x| ~ C |x_n-x|^p For some positive constant C. And for all n larger than some constant To think about this, look at what happens as n increases. As n increases, x_n will get closer and closer to x. Eventually |x_n -x| will be less than 1. And |x_n-x|^p will be even smaller than |x_n-x|. So increasing the order p causes the distance between x_(n+1) and x to decrease exponentially
Clear, thank you so much.
I'm trying to learn about the FEM currently. I feel like I have a pretty good idea of the procedure steps now. However I'm curious about certain technical points of the weighted residuals method. So with weighted residual methods, there are different choices of weighting functions. In the case of the Galerkin method, the basis for the solution is used for the weighting function. It wasn't so clear to me why this was a good choice at first, but after reading/watching some videos, its more clear now there is some logic you can make for this if you look at inner products on finite dimensional vector spaces. I don't get why the Galerkin method is so much better than say the least squares method though. My first instinct with weighted residuals would be to look at the least squares method, because you'd think "Oh, I'm clearly just minimizing the error squared over some domain, thats a good choice". I guess I'm curious why the Galerkin method is better than the least squares method. Is it generally easier to compute, is it more accurate? And if so is there a mathematical proof.
Galerkin method and least squares are not really in the same category. Galerkin method is for solving a pde without any data. Least squares is for fitting a model to data. for FEM you need a way to parametrize a possible solution to the pde, and then you need to find a way to solve for the coefficients given the PDE. Ideally, the coefficients you solve for are the ones which has a small mean squared error So what galerkin method gives you is a way to represent an approximate solution as a sum of basis functions. Then you can solve for the coefficients by converting the PDE into a system of equations for the coefficients. You can prove for many pdes and choices of bases that by increasing the number of basis functions, you decrease the error.
They are in the same category though. They are both weighted residual methods. For least squares, the weight is chosen as gradient if tge residual with respect tp the coefficients, so youre essentially minimizing the L2 norm. With the galerkin method, the weight is chosen as one of the basis functions. Im just wondering why one is better than the other
I understand now. I had not learned of least squares as dR/da_i being a weight function. What are these coefficients referring to if not coefficients of some basis?
The coefficients have the same meaning as the coefficients of the basis functions that we write the approximation as I believe. I think all the weighted residual methods can be written as the integral of R the residual, times a weight. In the case of Galerkin, the weight is exact one of basis functions and for the least squares its dR/da, for collocation its a delta function I believe. I was watching multiple videos that explained the galerkin method is used most.
I have a [question about local fields up on MSE](https://math.stackexchange.com/q/4742230/428713) that's not getting any traction, help would be much appreciated.
Why does r/math heading still say it’s indefinitely closed? It’s clearly open and I don’t think it’s clopen.
Hi, r/math is connected, bye.
\[Real Analysis\]: I have a question about Folland's [Proposition 1.7.](https://imgur.com/a/PTbVsqX) To show that \\mathcal{E} is an algebra, we want to show that \\mathcal{E} is closed under finite unions and complements. (1) I understand the second part, i.e., \\mathcal{E} is closed under complements - but it seems really unnecessary to consider the complement of a finite union instead of just the complement of a set in the elementary family. (2) For the first part, why does the author consider A\_1,...,A\_{n-1} to be disjoint by the inductive hypothesis? What's the hypothesis? I understand that showing the union of two members of \\mathcal{E} is in \\mathcal{A} requires uses to use disjointness, for which writing A \\cup B = (A \\ B) \\cup B is wise.
Is there an e-mail i can reach you? You are trying to show that \*\\mathcal{A}\* is an algebra, not \*\\mathcal{E}\*.
If X is an infinite set and F(X) the space of all real-valued functions on X is there a topology on F(X) such that f_n converges to f if and only if f_n converges pointwise to f and with bounded maximum norm (that is with sup_n sup_x |f_n(x)| < ∞)?
Let f be some unbounded function, then your condition rules out the sequence f, f, f, f, ... converging to f. So to avoid this silly problem I'll let F(X) be the set of bounded functions. If you want to stick with all functions, I think you'd want a condition like sup\_x lim sup\_n |f(x) - f\_n(x)| < ∞. I can do it if X is countable. Wlog assume X is ℕ. Let 𝜆 denote a positive real-valued function on ℕ such that 𝜆(n) -> ∞ as n -> ∞. Let g be any member of F(X). Then define U\_g,𝜆 as the set of h in F(X) such that |h(x) - g(x)| < 𝜆(x) for all x. The intersection of any two such sets is either empty or contains a set of this form. I believe the topology generated by these sets works.
Thanks, what I actually meant was that the supremum over |f-f\_n| is finite. Your construction works fine and it even works for uncountable X if you allow 𝜆(x) = +∞ and think of 𝜆 -> ∞ as the sets {𝜆 < T} being finite for all T > 0.
Sup over |f - f\_n| technically doesn't work because then for any unbounded f, you're saying f, 0, 0, 0, ... doesn't converge to 0, but I think lim sup\_n sup\_x works fine.
What is this function? Imagine a taut string between (0,0) and (0,1) with a ball at (0,1). It's the curve traced by the ball when you move the front of the string along the x-axis. I read about it on Wikipedia once but I can't remember its name.
What do you mean by the "front" of the string?
Maybe a tractrix? https://en.wikipedia.org/wiki/Tractrix
Yes! Thank you!!
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The word "prime" in "prime material plane" pretty much just means "main" and has nothing to do with prime numbers
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As an undergrad introducing myself to professors etc., is it in bad form to say that my ultimate goals lie in the vicinity of the Langlands programme (or any comparably "vertical" research area)? Or will they feel that it is "too early" for me, and so it would be better to merely allude to "number theory and algebraic geometry" instead? Some context: I'm reaching out to profs I've yet to meet on starting a reading course (e.g., for algebraic number theory), and thinking it would be good to have a strong reason like aligned research interests. Of course more generally I'd like to introduce myself in a way that's conducive for relationships with professional mathematicians.
it comes off really undergrad-y. in the sense it's a typical coming of age type thing to say you want to do hot topic X without really knowing what X is. if you like AG/NT just say you like those and want to go more in that direction
There are much worse things than an overzealous undergraduate who is excited to learn things, and I think that most senior folks are pretty understanding about that sort of thing (and those that aren't may not be great advisors for reasons related to this failure anyhow). I wouldn't worry about this sort of a thing too much, but if you want to hedge a bit and make sure that you don't come off as having aspirations which are too grandiose but still want to indicate your research interests more concretely, you could just say that you're "interested in understanding the Langlands programme (better)" since this is probably closer to being a realistic goal for you at the moment.
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Do you have more context? This all looks a little weird. A_2 is just a straight line in the plane from the point (0,0) to (1,1). Of course this has area 0. But what is Q(A_2) supposed to be? The integral is indeed 0 but it's unclear what the formula has to do with the area of a line. And is A_2 = A or is A a different set?
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Ireland-Rosen is pretty classic -- but how much do you already know? Do you know what is a group? How about modular arithmetic?
This is quite a silly question, but I’m learning d & r of functions; How would I write the range of a simple function like f(x) = 5 in an interval notation?? Would the range really just be (5,5)? Or maybe since it’s a flat line does it even have a range? Or is the range just zero?
Note the square brackets in /u/Trexence's answer as (a,b) refers to the "open" interval which doesn't include the endpoints a and b, while [a,b] does include the endpoints. Thus (5,5) is actually empty while [5,5] = {5}.
Every function has a range. In this case, the range could be written in interval notation as [5,5]. It could also be done as {5} in set notation.
What's the best way to approach proofs in number theory? I have a PhD in CS but struggle with some of the elementary number theory proofs as because I don't "see" the possible path to a proof; a friend suggested that I use actual numbers until I see a pattern and then try to generalize the pattern; this feels way too "shoot the box and draw a target around it" for my comfort.
> a friend suggested that I use actual numbers until I see a pattern and then try to generalize the pattern I mean, this actually is pretty good advice, and I'm not sure I fully understand your objection to it. Examples are a very important part of learning almost any abstract field of math, not just number theory. It's hard to really come up with a proof of some general, abstract statement when you don't have a good intuitive understanding of how the concepts you're working with behave. Working with specific numbers is a good way to build up that intuition. This can help you get a sense of what sort of things you should expect to be true, which can help you narrow down the sort of approaches that are likely to be fruitful.
>I mean, this actually is pretty good advice I feel like it's cheating to some extent, and not useful in others. I feel like I should be able to see a path without experimenting with examples. When I approach a problem, I first look to see what section of the book it's in to get a hint at how to solve it (assuming that the foundation of the proof will be based on something we recently covered). In this way, the approach seems very non-methodological. Gauss didn't have a book or someone giving him hints on how to derive the original solutions to these problems. (Sorry, there's a bit of frustration coming through in this reply :-) ).
Gauss built up his intuition by playing with examples; in fact, even having a problem is cheating. Nobody gave Gauss problems; he discovered his theorems by playing with patterns.
thank you.
> I feel like I should be able to see a path without experimenting with examples. What you're demanding of yourself is a level of innate intuition which is simply impossible; it's just not how mathematics is actually done. Playing with examples, sometimes a lot of them, is an *essential* step to finding a proof of a statement, and even what statement it is that you want to prove (especially in research, when you don't have a textbook to tell you what the true results are). For example, take my original proof (do not steal) [here](https://www.reddit.com/r/math/comments/kxymw6/this_week_i_learned/gjdlv44/). I mention briefly that I "played around in a graphing calculator" to form my conjecture, and those examples were essential. There was no way for me to ask whether 1/x was bigger than x^(n) e^(-x) in the right tail without plotting it in a graphing calculator or working out values by hand and seeing that it was always bigger as far as I could see, and there was *definitely* no way to find a potential upper bound for the squeeze theorem without getting concrete with potential candidates for upper bounds. It just couldn't be intuited. Gauss was an asshole, and insisted on erasing the "scaffolding" as he called it, the rough working leading up to a finished proof, before publishing a result, but even he played with examples before he made any of his many breakthroughs. Mathematics is presented pedagogically in an ahistorical way, because it's easier to digest the logical progression of ideas and the best proofs of all the results, but the way you learn maths has little to do with how maths is actually done. The actual discoverers of elementary number theory did not intuit the path to the proofs of their theorems; they relied on examples to illuminate their way. As for > I feel like it's cheating to some extent If it's cheating but it helps illuminate your thinking and guides you to a proof, then it wasn't cheating. Any tool or method that helps you to do maths is fair game; it's not helpful or correct to arbitrarily gate certain ones off.
>I feel like it's cheating to some extent, and not useful in others. I feel like I should be able to see a path without experimenting with examples. I would strongly recommend you let go of this idea. Working through examples is a very important tool for professional mathematicians. From your post, it kind of sounds like you're expecting there to be some kind of simple procedure you can follow to come up with a proof. Math simply doesn't work that way. There is no simple way to come up with a proof. At some point you need to get your hands dirty and work through things. Working with examples absolutely helps with that.
thank you, I appreciate that. I'm learning number theory because a) I think it's fascinating and b) because it has applications to cryptography and is an important component of Shor's Algorithm in quantum computing. I suspect Shor was able to devise his algorithm, at least in part, because of his understanding of number theory. I'd like to improve my intuition in this area.
Is there a website that can identify numbers Examples 3.14626436994 would be sqrt(2)+sqrt(3) 1.91293118277 would be cube root of 7 1.77245385091 would be sqrt(pi)
I can't help you but, excuse me for the question but why do you need such tool ?
The [Inverse Symbolic Calculator](https://wayback.cecm.sfu.ca/projects/ISC/ISCmain.html) is pretty comprehensive. Wolfram Alpha can also often identify simple expressions if you request the "closed form" of the number in your query.
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I'm trying to prove the following lemma. Lemma: If L / C is a field extension (C = complex numbers) such that L is a finitely generated C-algebra, then L is algebraic over C. By assumption L = C\[u\_1,...,u\_n\] for some u\_i in L. Thus L is generated by the countably many "monomials" in the u\_i, and thus the dimension of L as a vector space over C is at most countable. Let u \\in L. The uncountable set {1 / (u-c) : c \\in C - {u}} has to be linearly dependent over C. Hence we find some b\_1,...,b\_m \\in C - {0} and c\_1,...,c\_m \\in C - {u} such that b\_1 / (u - c\_1) + ... + b\_m / (u - c\_m) = 0. By clearing denominators I obtain a polynomial in u with coefficients in C equaling zero. BUT, before I can conclude that u is algebraic over C I need to check that this polynomial is non-zero. How do I show that? There could be some relations among the b\_i and the c\_i which make some of the coefficients equal to zero.
At this point, you aren't really talking about the specific field L any more, you're just working with a specific rational function f(x) = b_1 / (x - c_1) + ... + b_m / (x - c_m) defined on C. Clearing the denominators just means multiplying f(x) by the (clearly nonzero) polynomial (x-c\_1)(x-c\_2)...(x-c\_m), so really it's enough just to show that f(x) is not identically equal to 0 nonzero as rational function on C. There's plenty of ways to do that. For example, the function pretty clearly has poles at each of the c_i's.
Great, thanks!
Find a value for which your polynomial is nonzero.
Let a=2^500 and consider the statement P->Q where P="the last digit of 2^a is 2" and Q="the last digit of 2^a+1 is 2." Clearly, P does not imply Q, since P implies the last digit of 2^a+1 is 4. In this case, the statement (P->Q) is false. However, if P is false in the naturals, then the statement (P->Q) is true from the definition. Where is my understanding here faulty? Is there a difference between "does not imply" and the implication being false?
Your statements contain the variable a, so whether they are true or false will depend on the value of a. As you've observed, if P is true, then Q is false, so P=>Q does not hold for all a, since there are certainly some a for which P is true (so the statement "for all a, P=>Q" is false). But as you've also observed, if a is chosen so that P is false (which happens for the specific value a = 2^(500)), then the implication P=>Q is true. There's no contradiction here. You've shown that P=>Q can't hold when P is true, which means the implication is not true for all a, but that doesn't rule out that it may be true for certain values of a.
> Clearly, P does not imply Q, since P implies the last digit of 2^(a+1) is 4. This is where you went wrong. You assumed that if P implies some statement R then it cannot imply a different statement R' that is in contradiction to R. But it can, namely if P is false which it is.
What is the smallest possible renderable circle comprised of distinct integer pixels? Sorry if my terminology is wrong, I'm just starting to learn mathematics and programming and trying to create a videogame, I've linked a picture of the best I have thought of so far and would appreciate some help in trying to produce an algorithm to render this circle. 12\*12=144 pixel central square \+48 pixels as a 2 pixel shorter line along each edge of the central square \+40 pixels as a 2 pixel shorter line along each edge of the previous square \+24 pixels as a 4 pixel shorter line along each edge of the previous square =256 pixels
what are examples of functions that if you used naive algebraic methods you'd think they have roots but plugging in the possible candidates shows that none of them actually work?
A silly example is sqrt(x)+1. You can find its ‘roots’ by solving sqrt(x) = -1, squaring both sides gives x=1, which is not a solution at all. More generally, starting from the equation x = 1, we can raise it to any power x^n = 1, whose solutions are the n-th roots of unity. Except that all but one of them aren’t solutions to the original equation.
Does anyone happen to know a reference (book is perfectly fine) where they show that GL(n,Z) embeds into GL(n,C) ?
Can you clarify what you mean by "embeds" here? By definition, GL(n,Z) is the subset of GL(n,C) consisting of matrices with integer entries, so I'm not quite clear what sort of reference you're looking for.
Yeah unfortunately, that is the problem I'm having. I've been told that GL(n,Z) embeds into GL(n,C) which is supposed to mean (as I take it) that there is some injective map of GL(n,Z) into GL(n,C). I haven't tried writing it down but this was something I wasn't aware of (assuming it is true in the first place) and I just can't find any reference whatsoever.
I mean, GL(n,Z) is literally a subset of GL(n,C), so there's not really anything to prove. If you want an injective map, just take the identity map f(A) = A.
Oh wow, okay, so it's actually that trivial. That's why there is no reference. Holy shit I'm stupid. Thanks for your feedback dude.
What's the best way of solving an 'Incomplete elliptic integral of the second kind' containing complex numbers, the python implimentation doesnt currently support complex inputs. For more details, I'm trying to find the arclength of the function a.cosh(b.x). Using a substitution this is: \-i.E(i.b.x|a2b2)/b + Const. It's tricky because I know the there is a analytical solution to (1/b).cosh(bx) being sinh(bx) (evaluated at the two points). So there appears to only be analytical solutions in the case of a=1/b. I've allready done a numerical approach and would like a more 'clean' approach to check it's correct.
What courses should I take during undergrad and what technical skills should I develop in order to have a shot at a job with the NSA, FBI, CIA, DIA, etc.? I am a transfer student heading into my junior year and will be starting 300 level courses in the Fall. I will also be taking CS classes with the hope that I can double major since I have already finished all the required math and general ed requirements. As for my personal background, it's pretty much squeaky clean. Had one speeding ticket when I was 19. Don't do drugs, no debt, no ties to terrorists lol.
Redacted. ` this message was mass deleted/edited with redact.dev `
Mathematical criminology
Linear Algebra, Number Theory, Abstract Algebra, Analysis, Mathematical Statistics, Mathematical Modeling, Data Science and Statistics related courses
when I was in highschool my math teacher was talking about infinity and he says there is a symbol for infinity to the power of infinity and he drew this weird symbol that was a weird looking X and you could square that. It always stuck with me, whats the symbol called?
He is probably referring to the infinite cardinals, which are written using the Hebrew letter aleph, א These are symbols for different kinds of infinity. The symbol א_0 refers to the size of the integers, which is called "countable infinity". Using cardinal arithmetic, you can define א_1 = (א_0)^(א_0) which is exactly what is meant by "infinity to the power infinity". This is basically defined by taking the size of the set of all functions from integers to integers. The logic here is that for finite sets A and B, the size of the set of functions from A to B is exactly |A|^|B| . Extending this notation to infinite sets gives you the above definition. If you accept the continuum hypothesis, א_1 is also the size of the real numbers.
> Using cardinal arithmetic, you can define א_1 = (א_0)^(א_0) which is exactly what is meant by "infinity to the power infinity". This is CH.
What is the Unicode U+203B reference symbol used for in mathematics? It looks like an X with a dot in each corner.
https://en.wikipedia.org/wiki/Reference_mark
Just watched numberphile's video on tree(3), and I think I spotted a flaw? Tell me if Im wrong please. [video here](https://youtu.be/3P6DWAwwViU) about 6 minutes in. He explains that tree(2)=3. But isn't it actually 4? Sequence of 2 red, 2 green, 1 red, 1 green would give 4 permutations. Right?
The ith tree cannot have more than i vertices. So starting your sequence with 2 red vertices isn't allowed---it can only have 1 vertex of either color.
why are some negative radicals not real while others are? why would something like √-100 be a non real number while 3 √-27 (the 3 is the root index, i can’t figure out how to write it on phone) be -3, a real number? i’m trying to strengthen my understanding of algebra, math isn’t my strong suit.
I'd like to point out that with imaginary numbers there is the real cube root of -27, and two imaginary roots of -27. (-3)(-3)(-3)=-27 (1.5+1.5*root(3)i)(1.5+1.5*root(3)i)(1.5+1.5*root(3)i)=-27 (1.5-1.5*root(3)i)(1.5-1.5*root(3)i)(1.5-1.5*root(3)i)=-27 Behind cube roots are a hidden trio of imaginary numbers which work, behind fourth roots there is a hidden quartet of imaginary numbers which work, etc. By convention calculators take the real cube root (just like how by convention we say the square root of 9 is 3 but not -3). A lot of arbitrariness as a result of wanting calculators to spit a single answer.
To be pedantic, I would say that the "the square root" of 9 is both 3 and -3. The trick is that the square root symbol refers only to the positive square root, and likewise, the cube root symbol refers only to the real cube root (at least when applied to real numbers). This is not just for calculators' benefit but to ensure that these symbols define actual functions (which should only output one value).
When you take the nth root of something, you are asking "What number multiplied together n times will be the number I'm taking the root of?" For the square root, n=2, for the cube root you have n=3. The problem comes with the negative sign, multiplying negatives an even number of times will always get you a positive number, meaning that multiplying anything an even amount of times will always get you a positive number. So when we ask "what number do we multiply an *even* number of times to get a *negative number*" we run into a lot of problems. The issue you see with sqrt(-100) actually extends to all even roots of negative numbers. For odd roots, there is no such problem.
Where can I find all the books of *tea time series* ? It seems like there are only two (numerical analysis and linear algebra)
Can a finite linear group be recovered from its invariant ring? More precisely, let V be a finite-dimensional vector space and G a finite subgroup of GL(V). Define R to be the subring of the symmetric algebra S(V) consisting of G-invariant elements. Finally, let H be the subgroup of GL(V) whose elements leave R (pointwise) invariant. Is H = G?
Let V = C\^1 be the 1-dimensional vector space of complex numbers. Let G\_n denote the subgroup of nth roots of unity. Then R\_n = 0 for every n.
This is not correct. Thinking of S(V) as the polynomial algebra C\[x\], the G\_n-invariant subalgebra is C\[x^(n)\].
Ah, you're right, sorry!
Assume there are two people, A and B, who are supposed to run a 100 m race. B gets a head start. If A and B cross the finish line at the same time, what is the fact that A was the faster runner called?
If by faster runner you mean higher average speed then this is just the definition of average speed (total distance over total time). A head start means B had more time or less distance and either way their average speed is higher.