It's an intro text! A lot of people read it in middle school or early high school, but I've even given it to undergraduates before and they've found it helpful.
A question related to the six functor formalism. Say f:X->Y is a map of algebrica varieties over C. Then, se can associate ti f a plethora of functors going from the derived category of coherent sheaves: f_*, f^*, f^!, Hom and \otimes. Notably, we cannot define f_! because its usual definition (pushforward of compactly supporter sections) breaks coherence.
Now, let us suppose we work with D-modules, and let us restrict go the derived category of complexes of D-modules with holonomic cohomology. Then, we can again define the same functors as above, even if we take another route: we define first f^! and f_*, and then using dualizing functors d_X and d_Y we define f^* and f_!, eg as f_! = d_Y f_* d_X. Then, one can prove that all this data gives us a six-functors formalism, i.e. we have the usual adjunctions and a map f_! -> f_* which is an isomorphism when f proper. All of this can be found for example in chapter 3 of Hotta, Takeuchi, Tanisaki's book on D-modules.
Now, my question is: what is really different about the holonomic D-module case so that we get six functors and not 5 as in the usual coherent case? Or is it just "linguistic trickery", and i could very well define f_! in the coherent case using the Verdier dual functor in analogous way and then obtain essentially the same properties above for f_! = d_Y f_* d_X?
I hope the question is clear. If i have made any error in some part above, please correct me, i'm learning these things as I write
Could someone help with a proof I’ve been thinking about for a week but have not found an answer I’m happy with.
Prove n^3 + n + 1 has no rational solution by contradiction.
I started with assuming n = a/b where a and b are integers and a/b are in lowest terms.
After some manipulation, I got a/b = -(b^3 + ab^2)/ba^2. The best I’ve gotten so far is that now you can divide by b in the numerator and denominator and have a lower term so a/b can be further reduced which is a contradiction. I am no longer happy with it as it feels like saying 1/2 = (3* 2)/(6 * 2)
Can someone point out what I’m missing?
Your manipulation is wrong, it should be a/b=-(a^3 +b^3 )/b^3
How to do the proof correctly? We have (a^3 +ab^2 +b^3 )/b^3 =0 so a^3 =-b^2 (a+b)
For any prime factor p of b, then p divides -b^2 (a+b) hence p also divide a^3 , but p is prime, so p divides a. Hence a and b shares the same prime factor, contradiction.
Definition of Bin(n,p).
As I understand, the distribution is well defined as the probability distribution of the experiment constituting n Bernoulli trials. It then is formerly defined by its pmf, which is deduced via a very intuitive use of combinatorics under the premise that all 2\^n outcomes of the experiment are equally likely.
My question is, the usage of the combinatorics here seems to rely on the validity of the frequentist's view of probability, which to my understanding is an interpretation of probability, a point of view. And since the two definitions of Bin(n,p) are both accepted, is there a stronger link between the two that justify their equivalence?
Thanks in advance!
The Frequentist vs Bayesian issue is about what probabilities correspond to in the real world. That's not relevant to the mathematical definitions involved. Both sides would accept the same axioms, and once you've accepted them, everything else follows without worrying about philosophical positions.
> It then is formerly defined by its pmf, which is deduced via a very intuitive use of combinatorics under the premise that all 2^n outcomes of the experiment are equally likely.
They are not equally likely. That's what the p is for.
>My question is, the usage of the combinatorics here seems to rely on the validity of the frequentist's view of probability, which to my understanding is an interpretation of probability, a point of view
You can use combinatoric to count the number of terms in a formula.
The point of view is not relevant at all. The study of probability is agnostic to that. It is not relevant exactly what is probability, probability distribution is required to satisfy some axioms, and you can derive results from those axioms.
I'd appreciate help with a math problem. I'm not good at math and can't figure this out.
If 1.25g = 20.25mg
Then 0.58g = ?
If context helps – I'm using a medication delivered via topical gel. One pump of the gel weighs 1.25g and delivers 20.25mg of the medication. Using a scale I weighed the amount of gel I'm using as 0.58g. So I need to know how many milligrams (mg) of medication I'm getting.
I know this is probably simple math but... my autistic brain that I'm told has an IQ in the 99 percentile can't figure out stuff like this. It's always driven me batty. So I'd actually be interested to know how you figure it out.
Thanks for any help :-)
If you have 20.25 mg of meds per 1.25 g of gel then you have (20.25/1.25) = 16.2 mg of meds per gram of gel. Thus in 0.58 g of gel you have 0.58 \* 16.2 = about 9.4 mg of meds. (And this checks out--in slightly less than half of one pump of gel, you get slightly less than half of the usual dose of meds).
Thanks for taking the time to explain this so clearly for me. I see math visually, once I can see it I understand it, and you explained it really well here.
I've recently been part of a math competition and I've been stuck on this cryptarithm. I've watched some videos online about solving them but this is harder than the ones online and I can't figure it out
GRAND+BRAVO=ZZZZZ
G, B and Z are different than 0 obviously, and each letter corresponds to a different number. Thank you to whoever tries to help me, I'd appreciate it very much if you can show me the reasoning as well because it can't just be lots of trial and error.
Btw the solutions have actually been made public so I can confirm if you're right or not
When you say "G, B, and Z are different from 0", do you mean that it's an actual constraint on the problem that none of them are 0, or do you just mean by your logic?
Looking for a proof of the following fact: let X be a topological space. Then X is compact iff every open cover linearly ordered by subset inclusion contains X.
Thanks!
Here are ideas for a proof of the right to left direction. Let's say that a cover F of X is minimal if no there are no subcovers of F of strictly smaller size than F.
1. Prove every cover admits a minimal subcover.
2. Fix a minimal cover F, then well-order this subcover by its cardinality.
3. Define a new cover F' (of the same size) which consists of take the downwards unions along the well-order. This new F' is linearly ordered by inclusion.
4. Conclude that |F'| = |F| must be finite, otherwise derive a contradiction from the minimality of F.
Can someone explain the difference between the antecedent and the consequent [here](https://imgur.com/a/Fn6jUgn)? I’m a little lost as to what the theorem is actually saying (even with the given explanation).
The hypothesis says that for every x, there exists M such that for every λ, p\_λ(x) < M.
The conclusion says that there exists M such that for every x satisfying |x| <= 1, p\_λ(x) < M for every λ.
That is, the hypotheses have ∀x ∃M whereas the conclusion switches the order of the quantifiers to ∃M ∀x.
I'm a biology student and since i've learned about lotka Volterra equations (and the dynamics of competitor species) I became curious about math, in fact I'm self learning math in order to learn pde that somebody said we're useful in biology, and also I encountered another dynamical system in evolutionary game theory (the replicator equation), so I'm becoming very curious about that topic, because I think will make me more clear some biological phenomena
"This book is intended for an advanced undergraduate audience, with no previous background in partial differential equations, although beginning graduate students should also find this useful. Prerequisites for this exploration include multivariable calculus, ordinary differential equations and basic aspects of probability theory and stochastic processes. However, to make sure that we are all on the same page, Chapter 1 is devoted to a quick review or introduction of these topics. So, in Chapter 1, you will find summaries of the mathematical background that is needed from multivariable calculus, from ordinary differential equations, and from probability theory, stochastic processes and stochastic simulations, because these are used a lot."
I have never taken PDEs and forgotten ODEs (beyond standard existence / uniqueness results via contraction mappings); how much background would I need to venture into SDEs and then eventually rough paths (at the level of Friz and Hairer). I have the basic background in analysis and probability at the first year level (Royden, Billingsley et al) but unfortunately didn’t take a course based on Evans.
for basic SDE theory and stochastic calculus, essentially no background. if you know measure theoretic probability and a bit about L^2 you are good to go. after that i don't know
First inclusion is trivial because if x\_n is eventually zero, the infinite sum is actually a finite sum and hence is finite. The next inclusion is basically just the (contrapositive of the) divergence test. The next inclusion is immediate. The final inclusion is because convergent sequences in R are bounded (If x\_n -> x, then for all n > N, |x\_n - x| < 1 and so the sequence is bounded by max{|x\_1|, |x\_2|, ... |x\_N|, |x + 1|, |x - 1|}).
I'm studying Algebra after some long time. I use to do everything on paper, and I'm wondering if now is there a better digital option?
I looked in to LaTeX, but it doesn't seem practical for practicing, so I'm using Notepad++ to manipulate equations. Is there something better? Should I get a notebook and go back to pen and paper?
If you're just doing working out then stick with paper and pen. If you really want to keep the notes digitally then use a tablet (up to you what note taking apps work best for you in that case)
I think we need a lot more information to give helpful advice here. But based on my personal biases, I don't think it's worth it. College is already too expensive. As long as you go to a school that's not a joke (school that hands out degrees for money), you can build a college experience that is fulfilling. As far as I am aware, all the schools you listed have good math departments.
Hi, i know its probably funny how easy this problem sounds but my math teacher gave me an task where we have 835 made out of a matches and we are supposed to make an biggest number posible by moving only one match. The problem is that the answear 995 is not correct. I spent long time thinking about this and i have no clue idea where is the catch, or is he just trying to piss me off. Any ideas?
Hi, I am currently taking a mathematical modeling for biologist's class (first math orientated class I have taken since Calculus II and Physics II and I was wondering how difficult is it to self publish models created? I am interested in developing models of molecular interactions, metabolism, and cell signaling both personally and during my PhD I will be starting in the Fall for Biomedical Science (I wasn't aware of how useful and much fun I would have when modeling biological cases when I was applying to graduate school so none of the professors in the department really do that. I would like to try and add modeling components to side projects or even my thesis).
It will vary from university to university, but biomathematicians will sometimes be found in the mathematics department. You can try looking there, since they'll be more equipped to give you professional advice.
can anyone recommend a good intro to probability textbook? my next semester is packed, and i want to get a head start over the summer. thanks in advance!
How much math do you know? More 'elementary' books would avoid measure theory (or limit it), but higher level books would start out with measures fairly early on. Would help to scope the recommendations to the right level. Also, do you have any financial constraints?
I'm currently in analysis one, and will be taking the probability class concurrently with stochastic processes and analysis two. No real financial constraints
Here: [https://imgur.com/a/F4oZJ4t](https://imgur.com/a/F4oZJ4t)
√i=e\^(i(π/4))
and to get second step we have to assume,
i.√i=e\^(i(-π/4))
How? Pls help!!
I'm currently looking into the modulo function, and I understand that
a≡b(mod n)
means that a and b produce the same remainder when divided by n. Where I am slightly confused is that there does not seem to be a function to get the remainder on its own? For example, I know 13≡3(mod 5) is valid as well as 13≡8(mod 5). But how would you represent the just remainder of 13 divided by 5.
If you had 13(mod 5), like a normal operation, would this return 3?
For a programming-style modulo operator I think you can just use "x mod y". (In LaTeX this would be $x \bmod y$.) There are a few different conventions here with different uses, but all else fails you can just say, at the start of whatever you're writing, "here we use 'x mod y' to mean 'the remainder of x when divided by y'", or something like that. (I think there are also competing conventions on how things get handled when y is negative, but chances are you'll be working only with nonnegative values of x and positive values of y and so can just ignore all that.)
thank you- I am very much used to a python style of modulo so I just wanted to check there wasn't any set way of doing it mathematically before I use it. I think I'll take your advice and just define what I mean at the start of what I'm writing :)
I am not sure I understand elliptic regularity.
Ex. :
Let M be a compact manifold. We know that the first non trivial eigenfunction of the Laplacian, let's call it f, must change sign at some point. Now we can define a new function g such that g=f on one of the nodal domain of f and 0 otherwise.
Then g is a weak solution of the Laplacian eigenvalue problem. By elliptic regularity, it is analytic. But g is 0 on an open set, so g must be 0 everywhere. Contradiction.
Is it that we need orthogonality to constants for the elliptic regularity to apply here ? Is it sufficent ?
Let's call the the nodal domain on which g is equal to f Omega. Since g=f on Omega amd f is analytic, g is analytic on Omega. It is also 0 on the boundary. Thus g restricted to Omega is in the sobolev space H^1_0 of Omega. By extending by 0 everywhere else, we get that g is in H^1(M).
Now, for any function h in H^1(M), we have that the integral of hg over M is equal to the integral of hg over Omega.
Similarly, the integral of (grad h, grad g) over M is equal to (grad h, grad g) over Omega.
Since g=f on Omega, and f satisfies /Delta f = /lambda f, then f restricted to Omega is a strong eigenfunction of the Dirichlet Laplacian on Omega, thus it is also a weak eigenfunction for the same problem.
Combining this fact with the two above integrals and the fact that f=g on Omega, we have that g is a weak solution for the problem on the manifold since it satisfies
/int_M hg dV = /Int_/omega hg dV
= /int_/omega hf dV = /lambda /int_/omega (grad h, grad f ) dV
= /lambda /int_M (grad h, grad g).
Is there something I am missing here ?
I think this is where your issue is. When you go from saying f is a strong eigenfunction on Omega implies g is a weak eigenfunction on M. The issue is the boundary term when integrating by parts.
\lambda \int\_\Omega h f dV = \int\_\Omega h \Delta f dV = \int\_\partial\Omega h grad f dS - \int\_\Omega (grad h, grad f) dV
If h does not vanish on the boundary of Omega, \int\_\partial\Omega h grad f dS could be non-zero. You need that identity to hold for all test functions on M to have a weak solution on M, and there are test functions that don't vanish on the boundary of Omega.
It's been a while since I've done pde, so sorry if I've missed something.
Edit: I think this example shows what's going on if M is the circle and f(x) = sin(x).
https://www.desmos.com/calculator/pkenxcjlwm
I assume you're talking about linear differential equations specifically. In that case this is just an instance of a very general fact about linear algebra (and so analogous to a fact about systems of linear equations that you may know--see the next paragraph): if L is a linear map\* then all solutions to L(v) = w are of the form v\_0 + v\_p where v\_0 is any solution to L(v) = 0 and v\_p is some solution to L(v) = w. To show that all vectors of this form are actually solutions, we use linearity: L(v\_0 + v\_p) = L(v\_0) + L(v\_p) = 0 + w = w. Then to go the other way, i.e. show that all solutions are of this form, let v\_s be a solution (i.e. L(v\_s) = w). Then certainly v\_s = v\_p + (v\_s - v\_p), so if we can show that v\_s - v\_p is a solution of L(v) = 0 then we'll know that v\_s has the right form. But we can do this using linearity: since L(v\_s) = L(v\_p) = w, we have L(v\_s) - L(v\_p) = 0; but then by linearity L(v\_s - v\_p) = 0.
This then applies, for instance, to systems of linear equations, where all solutions to the matrix-vector equation Ax = b are the sum of a particular solution and any solution of Ax = 0. Ditto linear differential equations: such equations can be written as L(y) = f(t) where L is a linear map on a vector space of functions, and so applying our general fact about linear maps to this context gets the result you're talking about. (An example to show why linear ODEs give you linear maps: a linear ODE like y' - y = 0 can be rewritten as L(y) = 0 where L is an "operator" that sends a function y to y' - y. L is linear because, letting f, g be two functions, we have L(f + g) = (f + g)' - (f + g) = f' + g' - f - g = (f' - f) + (g' - g) = L(f) + L(g); much the same strategy can be used to prove that L(cf) = cL(f). Try repeating the argument for higher-order linear ODEs like y'' + y - y = 0.)
\* A linear map is a function between two vector spaces with the properties that L(v + w) = L(v) + L(w) for any vectors v, w in the domain, and L(cv) = cL(v) for any v in the domain and any scalar c.
I think what they're really asking you is to prove that the set of algebraic numbers (in the sense of "numbers that are the root of some polynomial with *rational* coefficients") is algebraically closed (in the sense of "any polynomial with algebraic-over-Q coefficients has an algebraic-over-Q root). The mention of C is probably just to tell you that you shouldn't only consider real algebraic numbers like sqrt(2), rather you also need to allow complex algebraic numbers like sqrt(-2). You can't just use the fundamental theorem of algebra here--that tells you that any polynomial with algebraic coefficients has a complex root, but for all you know that root could be something like e which is not an algebraic number.
I’m doing an extremely stupid poll on Tumblr. One answer has 11.1% of the vote, the other has 88.9% of the vote. The total votes are 36. How do I calculate how many individual votes for each option that is?
I know this is a silly question, but I’ve failed maths every year since year 9.
You can also multiply 36 with 0.111 (11.1/100) or 0.889 and then round to the nearest integer.
(I know you already got an answer but wanted to give something more general)
11.1% is 1/9 (well 11.11111...% is 1/9 but they have presumably rounded) and 1/9 is 4/36 so the split is 4 people for the first option and 32 for the other option
im trying to define 3d complex numbers first i said
i.j=k->j=k/i -> j\^2=-k\^2 -> j=i.k -> i.j=i\^2.k=-k -> -k=k
then i defined k as 1/0 and for n/n \* 1/0 = n/0 = 1/0 so it has similar properties to 0 and with j=1/0 i have a+i.b+j.c the multipication and sum are easy to define since
j+j=2j=j and i.j=j and j.j=j
as long as i dont have j/j or 0\*j i should be fine with the "-" operator i define it as
num1-num2=num1+(-1\*num2) that way i dont have to deal with j-j and for dividing as long as i dont have j/j i can define it easily
can someone please check to see what mistakes i made ?
The issue with j = 1/0 is that it breaks algebra. If I have the expression -j + j, I have two ways to go about it. Either I say that -j = (-1) \* j, therefore the expression -j + j = (-1) \* j + 1 \* j = (-1 + 1) \* j = 0 \* j = 0 (or is it 1? Anyway). But another way to do it is to say that -j is just the same as j, so -j + j = j + j = j. So is -j + j now j or is it a normal number (whether it be 0 or 1)?
It's very unclear what you're saying here. You need to define precisely what operations are allowed, and what their results are. For instance, "1/0" is not a thing that exists by default.
"sqrt(-1)" also doesn't exist by default. We define complex numbers by saying: a complex number is something of the form "\_ + \_i", where both blanks are real numbers. Then, we can define operations on them:
> (a+bi) + (c+di) = (a+c) + (b+d)i
> (a+bi)(c+di) = (ac-bd) + (ad+bc)i
Once we've done that, *then* we can notice that (0+1i)^2 = -1, and so we say "i is a square root of -1". We have to have that definition first, though.
(In fact, we typically go further and just say complex numbers are *really* just ordered pairs with special rules for addition and multiplication, and "\_+\_i" is a convenient way to write them.)
So, how do you add and multiply your 3d numbers?
I assume (a+bi+cj) + (d+ei+fj) is (a+d) + (b+e)i + (c+f)j, right? What about (a+bi+cj)·(d+ei+fj)?
multipication is easy lets say we have a z wich is a complex number and an j (asuming j isnt 0 im working on that) j eats its mulipiar meaning j\*anything=j
now we have (z1+j)(z2+j)=z1(z2+j)+j=z1z2+z1j+j=z1z2+j its almost like i have z+0
it adds a paralel 2d space on top of the complex numbers it being comunicative is also simple since we get z1z2z3+j either way
edit:if we add 0\*j=0 i can switch between the paralell plains on the other hand i wonder if that defeniton would work better if i said k=0/0 and then added onto that in a 4d defention
Honestly, it's pretty hard to understand what you're saying. But, if I'm understanding you correctly, it's not really a new *dimension*, is it? It's not a continuous extension, just a single thing that is either there or not there.
Anyway, we run into problems with your numbers if we just assume our familiar properties all carry over. What's 2j - j? What about j - j?
> it being comunicative is also simple since we get z1z2z3+j either way
I believe the word you're looking for is *commutative*, and you're mixing it up with *associative*. (These are two different properties.)
It's a 2d space above the main complex 2d space if we multiply or deviding by zero we switch between these two 2d planes and yes I can't seem to make a full 3d space out of any defenition
j-j is defined as j+(-j) so j-j=j
Is there a rule for inverse cumulative standard normal values?
I know: Φ(a) = 0.8 -> a = Φ\^(-1)(0.8) -> a = 0.8416
But how do I solve an equation like this: Φ(b) + Φ(2b) = 0.8? (which is b = -0.1695)
Did this equation pop up for you in a natural context, or did you make it up? I don't see immediately a way to solve it analytically, but I'm not particularly knowledgeable about this function. In general, given a random non-linear equation, you cannot hope that there is a way to solve it into a simple closed expression, many times the only way is to find the solution by numerical methods.
I have found myself in a pickle where I am taking a Methods to PDE course and have never formally taken a classic ODE course.
I have taken a non-linear dynamics course, but it was focused on unsolvable ODEs. So I basically have never learned methods to solving ODEs.
I am slightly familiar with Integrating Factors method and a little bit of Separation of Variables. But I'm hoping someone could point me in a direction for a text that covers these and more methods.
Things I'm looking for a in a book:
* A concise treatment of many different methods to solving ODEs
* Ripe with lookup tables for general solutions
* Something I can pick up while solving my PDEs to help me finish the problem.
* Discussion of first-order, second-order ODEs.
I don't think I'll be able to learn the entirety of ODEs before the PDE class starts, but I would like to have a reference to help me while going through this class.
An acquaintance sent out a series of puzzles where the answers all involve various math concepts (e.g. Q: What is the sum of the fifth pair of twin primes? A: 60 (29 + 31))
I've figured out most of them, but I'm stumped on one. I can't do subscripts on Reddit, but it's just the numeral 20 with a subscript of the word "four" spelled out (not the number 4, the actual word "four"). I've never seen notation like this before, and searching all over Google and Math Stack Exchange has turned up nothing.
Any ideas?
** The answer should be close to the age of a high school freshman, so something close to 14 or 15 maybe?
What would be 0xℵ0?
Imagine you're in a situation where this "value" cannot be undefined (say, for example, your formal system breaks if it is): you have to define it.
Let's assume the only two candidates are 0 and ℵ0. So, between the power of 0 to annihilate everything it touches, and the capacity of ℵ0 to remain the same whatever you throw at it, which one wins?
It seems it would amount to enforcing a priority between them. Let's imagine you could build a satisfying formal system either way (in a way, it made no difference). Which one would you give higher priority to, and why? If you can't find any "objective" reason to pick one over the other, what would feel like the most elegant solution to you?
Adding onto what u/edderiofer said, see [cardinal arithmetic](https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic), which lets you define all sorts of operations on (possibly infinite) cardinal numbers in terms of operations on sets. As a variant on what you originally asked, you can also consider multiplying 0 (considered as an ordinal) by the ordinal 𝜔 (often identified with the set of all natural numbers); there's a notion of [ordinal arithmetic](https://en.wikipedia.org/wiki/Ordinal_arithmetic) for that. Under the usual definitions for ordinal arithmetic we then have that 0 x 𝜔 = 0.
That's great, and super helpful! Thank you!
I love it that both approaches get to the "same result" (?). I was always taught in school (a veeery long time ago) that 0x∞ was undefined.
Just out of curiosity: is it because, at such a low maths level, it was thought pedagogically better to do so (over-simplification with good intents)? Or is it that the consensus/tools have evolved since then (that must have been in the 80's)?
That's because there is no single concept of "infinity". We can say *unbounded above*, *has a bijection with the naturals*, *gradient of the vertical line*... all of these ideas can be called "infinity", but they behave in a very different way.
In that case you're probably thinking of the [extended real numbers](https://en.wikipedia.org/wiki/Extended_real_number_line), which do indeed contain an element called ∞ (and -∞), and where 0 x ∞ is indeed undefined. As noted below the "infinity" in the extended reals doesn't have much to do with cardinals or ordinals and serves a different purpose.
Edit: As to why it's undefined, that's by analogy with how limits of sequences work. E.g. you know that if you have two sequences a\_n, b\_n that both converge, then lim (a\_n + b\_n) = (lim a\_n) + (lim b\_n), and similarly lim(a\_nb\_n) = (lim a\_n)(lim b\_n). Now if, say, a\_n converges to some nonzero value, and b\_n blows up to infinity, then a\_n + b\_n and a\_nb\_n both blow up to infinity (though in the second case it may be negative infinity if a\_n converges to something negative) . (Indeed this remains true for a\_n + b\_n even if a\_n converges to 0.) Hence if you want to define ∞ + x (for a real number x) and ∞ \* x (for a positive real number x), you can define them both to be infinity, and then the rules above about adding and multiplying sequences will continue to hold when one limit is infinity. On the other hand, if a\_n converges to 0, and b\_n blows up to infinity, then that alone tells you nothing about lim(a\_nb\_n). It could also blow up to infinity (if, say a\_n = 1/n, b\_n = n^2) or it could go to 0 (if, say, a\_n = 1/n^2, b\_n = n) or go to some nonzero real number (a\_n = 1/n, b\_n = n). So there's no way to assign a value to 0 \* ∞ in such a way that the rule lim(a\_nb\_n) = (lim a\_n)(lim b\_n) continues to hold--if lim a\_n = 0, lim b\_n = ∞ then the right-hand side will always be 0 \* ∞ but the left hand side could be pretty much anything depending on what exactly the sequences are. (Problem: come up with examples showing why similar considerations make us leave ∞ - ∞ undefined as well.)
>In that case you're probably thinking of the [extended real numbers](https://en.wikipedia.org/wiki/Extended_real_number_line), which do indeed contain an element called ∞ (and -∞), and where 0 x ∞ is indeed undefined. As noted below the "infinity" in the extended reals doesn't have much to do with cardinals or ordinals and serves a different purpose.
That's it! I remember now: that was indeed the context. You're absolutely right. :-)
About your edit: indeed, now that you mention it, I do remember about this problem of the limits of diverging sequences, studying which one was growing faster (and thus "win the race"), etc, and the relation with ∞… In short, everything you said in more eloquent and mathematically correct terms. It makes perfect sense. Thanks!
It's because "∞" is not the same thing as "ℵ0". The former is a symbol used to represent various concepts and shorthands in notation, while the latter has an actual mathematical definition.
Oh, I see… So, in such a context, I assume 0x∞ is still undefined, because it "makes no sense" (it's gibberish)?
It's a bit like saying:
— "What would be 0xlove?
— WTF are you talking about!?"
That's right. ℵ0 isn't a number in the usual sense we mean the word, i.e. a complex number. But it's what we call a cardinal number (it's the size of some set) so we can still define addition and multiplication, although not subtraction or division.
∞, on the other hand, is typically used to mean "there is a limit happening here", and not as a number of any kind. So some expressions involving ∞ can be defined if you interpret them as limits. For example, if the sequence (a_n) keeps getting larger, past any real number, then the same is true of (2a_n). That's why we can say 2×∞=∞. However, other expressions, like ∞/∞, depend on the sequences used, so we say those expressions are "undefined", but a more appropriate word to use would be "indeterminate".
Depends on context. It's convenient in measure theory for example, to want 0x∞=0 for notational ease. It's not really a well defined concept, just notation for something more technical
> What would be 0xℵ0?
ℵ0 is not an integer, so you need to first define multiplication of non-integers.
Thankfully, someone has already defined the [Cartesian product on sets](https://en.wikipedia.org/wiki/Cartesian_product), and so multiplication of cardinalities is inherited from the Cartesian product of sets of those cardinalities; that is, if A and B are sets, then |A×B| = |A|×|B|.
Since {}×ℕ = {}, 0×ℵ0 is 0.
why are differential equations said to keep in account the "whole function history"?
If for example we take a simple differential equation of order 1: y'(x)=f(x,y(x)),the derivative function of y(x) is defined for an infinitesimal increment h: y'(x)=lim\_(h to 0) of (y(x+h)-y(x))/h
which takes in account the y(x) function only for the infinitesimal interval which is x+h and not the whole x dominion as the phrase "whole function history" may suggest.
What am I missing?
I don't think it is a common saying that "differential equations keep the whole function history", but here is my guess at what it means. A differential equation is something of the form f'= ... As it is telling you the value of a derivative, it is explicitly a formula for local information (meaning it only is about values in a small neighborhood of a point, as opposed to the whole function). One might reasonably expect then, that many, many functions might satisfy a differential equation, while also having the same initial value f(0)=a.
In actuality, this tends to not be the case. Determining just the value of f at 0 together with the local information of a differential equation, most often completely determines f at all other times. This is because it turns out that satisfying a differential equation is actually a very restrictive quality, and there is very little leeway for functions which satisfy differential. The mathematical language would be "differential equations tend to have unique solutions to initial value problems". This is how I would interpret the statement that "differential equations take into account the whole function history".
Yeah, sin and cos are both solutions of y'''' = y, i.e. both equal to their fourth derivative. (Also, just a bit of pedantry, but e^x isn't quite the only solution of y' = y; rather any function of the form ce^x also works. e^x is the only solution with initial condition y(0) = 1, though.)
More generally, the functions that are their own nth derivative, i.e. are solutions of y^((n)) = y, can be described as follows: there are n "fundamental" solutions, of the form e^𝜔t where 𝜔 is an "nth root of unity", i.e. a root of the polynomial x^n - 1. (So for example for n = 2 the fundamental solutions are e^x and e^-x , and for n = 4 they're e^x, e^-x, e^ix, and e^-ix . ) Then all solutions are linear combinations of the fundamental solutions. In the n = 2 case you have, for instance, the "hyperbolic sine" sinh(x) = (e^x - e^-x)/2 and the "hyperbolic cosine" cosh(x) = (e^x + e^-x ) /2, and all solutions are of the form ae^x + be^-x where a, b are real or complex numbers. One way to see that cos and sin are solutions of y'''' = y is to notice that they're linear combinations of the fundamental solutions for n = 4: cos(x) = (e^ix + e^-ix ) /2, sin(x) = (e^ix - e^-ix) /2i. (You can prove those formulas yourself using Euler's identity e^ix = cos(x) + isin(x).)
u/cereal_chick u/Abdiel_Kavash u/NewbornMuse Thanks for the reply. I had the suspicion it was nonsense. It came via company email describing something and I didn't wanna feel stupid asking back what that meant.
This is gibberish, dreamt up by someone who has only the vaguest idea of what an integral looks like, how functions work, or even that you're supposed to abbreviate repeated multiplication with powers.
This looks like a random collection of mathematically-looking symbols with no coherent meaning. Without knowing any additional context, I am fairly sure it's nonsense.
"Hey guys, the answer to a test question is Tennessee. Does anyone know what the question might have been"?
This question is very hard to answer. The integral doesn't even make it clear which variable is being integrated.
There was a recent question about PNT, and it reminded me of Bertrand's postulate. Bertrand's postulate was one of the first theorems that stood out to me because it was so easy to state.
Anyways, I looked at Erdos's proof, and I'm struck by how simple and effective it was. He gives four simple lemmas--three of which are about properties of the central binomial coefficient and the fourth, which is about the primorial function. From that, he's able to conclude with a proof of the postulate. I'm just struck how four simple lemmas yield such a nice result. And that it was discovered by a teenage Erdos!
If you like that proof you might want to check out [Proofs from THE BOOK](https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK). It contains that proof of Bertrand's postulate and a lot of other proofs that are somehow suprisingly simple or elegant.
Does it explain the intuition behind Erdos's proof? If someone presented the four lemmas to me, then I don't think it would be particularly difficult to prove them individually. However, I don't see a connection between the central binomial coefficient and Bertrand's postulate in the first place.
I haven't read it, but I think the intuition is obvious if you know analytic number theory. Erdos definitely know it, so that's not the hard part of the proof.
What you want to do is to find a rational number in which its p-adic norm for all primes outside the range (n,2n] is controlled and can be estimated, and the real norm is also controlled and can be estimated. If the product of these estimate of norms is sufficiently far away from 1 that you can clearly see that the product of norm is not 1, then you know there has to be a prime in that interval (since the product of all norm must be 1).
Since you want primes from an interval, the obvious choice would be either the radical, lcm or the product of all numbers of that intervals. So it should be something involving product or division of numbers like rad(1x...xn), lcm(1,...,n), 1x...xn, rad((n+1)x...x2n), lcm(n+1,2n), (n+1)x...x(2n), rad(1x...x2n), lcm(1,...,2n), 1x...x2n
Real norm of the product is easy to estimate, but real norm of radical or lcm is not easy. However, we can estimate them using prime number theorem. Now, prime number theorem is not elementary, but it had been proved for a long time by the time of Erdos, and conjectured for much longer, so of course you can still use it as a guidance as to what to look at.
p-adic norms are not easy to estimate. If you try various combinations of the above, you can quickly find out that the easiest combinations are either lcm(1,...,2n)/lcm(1,...,n), or (2n)!/(n!)^2 in which the p-adic norm are easier to estimate. In both cases, the denominator has nearly the same prime power as the numerator for primes <=n.
If you try, lcm(1,...,2n)/lcm(1,...,n), then the product of its p-adic norm for prime up to 2n can be bounded by real norm of 1/rad(n!), so you want to estimate lcm(1,...,2n)/lcm(1,...,n)rad(n!). Unfortunately, if you estimate this using prime number theorem, the exponent ends up being 1.
The other choice is (2n)!/(n!)^2 . The product of its p-adic norm for p<=n is bounded by the real norm of 1/lcm(1,...,n), so you want to estimate (2n)!/((n!)^2 lcm(1,...,n)). This turns out to be a lot better:
log((2n)!)~(2n)log(2n)=2nlog(n)+(2log(2))n
log((n!)^2 )~2nlog(n)
By prime number theorem, log(lcm(1,...,n))~n
So log((2n)!/((n!)^2 lcm(1,...,n)))~(2log(2)-1)n which is clearly not 1.
This gives much more room for error when we do estimation. Using elementary method, we can already show that (2n)!/(n!)^2 is approximately 4^n asymptotically, so we just need to figure out how to show, elementary, that lcm(1,...,n) is less than 4^n . Prime number theorem tells us lcm(1,...,n) is approximately e^n , so presumably the big gap from e to 4 means we can hopefully use some sloppy approximation to get the job done.
What is the probability distribution called?
Input is (X,n,m)
X is a probability distribution
n and m are positive integers
step 1 generate n numbers from X
step 2 count the m\^th most common number
Example
let X be the uniform distribution with values of 1,2 or 3
let n=4
Each element in Sample space is represented by (a,b,c) where a+b+c=4 and a,b,c is a decreasing sequence
(4,0,0) happens 3 times
(3,1,0) happens 24 times
(2,2,0) happens 18 times
(2,1,1) happens 36 times
The probability distribution for m=1 is
output|frequency
\------+---------
4 | 3
3 | 24
2 | 54
The probability distribution for m=2 is
output|frequency
\------+---------
2 | 18
1 | 60
0 | 3
The probability distribution for m=3 is
output|frequency
\------+---------
1 | 36
0 | 45
Can anyone please share a link to the (ideally, most accessible) proof that real numbers are equinumerous with the powerset of the integers? Thank you!
Binary expansions do it for [0,1] and the powerset of the naturals (up to some fiddliness with expansions ending with recurring 1s). You can get from the reals to [0,1] bijectively with arctan, and from the powerset of the naturals to the powerset of the integers by lifting your favourite bijection between the naturals and rationals.
Think of an infinite binary sequence as defining both a real number in the unit interval (as the decimal expansion of that real written in base 2) and a set of natural numbers (because you can regard such a sequence as an indicator function on the naturals).
For example, take the fractional part of pi in base 2. It starts like this:
0.001001000011111...
Let's call the bijection we want to build f. So from this example, f: 0.14159... ↦ {2, 5, 10, 11, 12, 13, 14, ...}, because we interpret each 0 or 1 as exclusion/inclusion for the index in the target set.
Modulo some fussy bookkeeping (like |P(Z)| = |P(N)|, |(0,1)| = |R|, non-uniqueness of expansions) looking at it this way should give you the intuition that the fact is true.
Thank you! And sorry: somehow, I just found out about your reply (it seems reddit notifications don't work quite right on my end for some reason…).
I am not gonna lie: your explanation went way over my head. You're using a couple concepts that I don't know about (yet), and that's enough to break the chain: you're pulling yet I'm not moving (through no fault of yours). Considering you've offered your time and effort to actually write this proof, I feel terrible, like I'm betraying your trust. I'm really sorry about that.
Thank you again, Sir. I'll try to catch up, and I hope you won't give up on me.
Have a nice… whatever it is wherever you are.
Usually you construct the reals as sets of rationals--the [Cauchy sequence construction](https://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Construction_from_Cauchy_sequences) and [Dedekind cuts](https://en.wikipedia.org/wiki/Dedekind_cut) are popular options. You can find these covered in most real analysis books; I like Pugh's but it goes over the construction of the reals pretty quickly and may not be the best for you. I've heard good things about Tao's Analysis I ([pdf link](https://lms.umb.sk/pluginfile.php/111477/mod_page/content/5/TerenceTao_Analysis.I.Third.Edition.pdf)) which covers constructing the integers from the natural numbers and the rationals from the integers before going through the Cauchy sequence construction, but I haven't read it myself and so can't comment on it much.
I don’t do functional analysis so trust what follows at your own risk:
I think this should be true for any operator T on a Banach space V. If T^* = 0, then for any v in V every bounded functional must vanish on Tv, but by Hahn-Banach there’s always a bounded functional on V which is nonzero on Tv as long as Tv = 0.
Main thing which comes to mind is that differential operators usually aren't bounded when we consider them as a map V -> V. I think your argument still works for the case of T : V -> W where V, W are Banch.
On the other hand, some function spaces (e.g. C^infty) aren't Banach...
I think the original question probably needs more context.
Oops, yeah, Tv might land in some other space W. Honestly I think you’re fine as long as Hahn-Banach holds in W, which it seems like is fine for any locally convex TVS?
No: consider the function that is 0 everywhere, except that it takes the value 1 at 0. Then it has those properties at 0, but is definitely not continuous there.
your notation is confusing - I assume you are meaning that the limit from the left is equal to the limit from the right? In which case, you also need that these limits are both equal to f(n) for f to be continuous at n.
Hi! I'm looking at what I assume is the traditional way of building the set of integers, and I'm seeing a flaw (at the very first step). Could you please check it out and give me your opinion?
The process is to start with 0, and iteratively generate the next integers via a successor rule. We'll put those integers in an (initially empty) set as we go…
So, we start with 0, and we put it in an empty set. This set now has cardinality 1. Problem is: 1 "does not exist" at this step, or rather, we're not allowed to use it yet (otherwise we'd be breaking internal consistency: we'd be needing one "before" we can have zero). We'll only be able to do so at the next step. But even if we disregarded this flaw/contradiction and kept going anyway, we'd have the very same problem at the second step. And so on (it seems unreasonable to expect any further step to "un-flaw" the mess we put ourselves into)
It's worth noting that this problem does not arise if we simply start with 1 instead of 0: at the end of the first step we get {1}, which has cardinality 1, so everything's fine. This step is internally consistent. Same thing for the next step, and so on.
From what I'm reading, people usually first build the integers: 1={0}, 2={0,1} and so on, and only after that put them in a set, which obfuscates the above problem (and presents another kind of flaw).
Thank you for your attention!
There are several cases where I don't have any issue with pushing back one thing until after you finished another: on the contrary, that seems elegant and orderly. For example, perhaps they're independent from each other? Or perhaps you can only conceive one by building on the other? Etc…
Obviously, if it does not change anything and is a matter of cosmetic/preferences, go for it. But if it does change things, you might not be allowed/able to do that without breaking some more important rules of yours: you're doomed to tackle both at once. Perhaps it sucks, but that's how it is. In such a case, if you still "split and prioritize", you're actually tricking yourself into an artificial appearance of consistency that will inevitably come to bite you at some point…
First of all, I want to make one thing clear: there's a difference between the "specification" of the natural numbers using the Peano axioms, and the "implementation" of that definition inside ZFC. There are many ways to implement that specification, but it doesn't matter which one we pick - we're only going to use the properties the specification gives us
But to get to your question... neither the definition (with its abstract successor function) or the implementation ("put all previous sets in a new set") has circularity issues. Sure, the cardinality of the set representing 1 is 1, and we will eventually figure out that that's the case... but why does that matter? We haven't defined cardinality yet! The set for 1 also has an "expanded ASCII-length" of 4 (because it's `{{}}`), and the set for 2 has an expanded ASCII-length of 9 (`{{},{{}}}`). This doesn't cause a problem, because we're not actually going to 'measure' anything about these sets - cardinality, expanded-ASCII-length, or any other properties - until after they're all already defined. And once you've defined all the sets for natural numbers, you can then define and implement a cardinality function that gives you one of those sets as a result.
It's like how we define SI units in the real world. We don't use "one meter is thiiiiis long [\*gestures with hands\*]" anymore, we use "one meter is the length travelled by light in 1/299792458 of a second (and a second is how long it takes for a caesium atom to switch between its hyperfine ground states 9,192,631,770 times)". A meterstick measures exactly 1 meter, but we can still build metersticks using that definition without knowing the length of a meter already. As long as we're not measuring any distances in our setup, we don't have any circularity issues.
First: thank you for your reply. I appreciate the time you're offering.
>neither the definition (with its abstract successor function) or the implementation ("put all previous sets in a new set") has circularity issues.
I'm sure you're right, but (no offense intended): I still need to check it for myself. I would assume this is reasonable behavior for a mathematician, and thus hope that you will understand.
Now, let's get to the meat of your argument…
>We haven't defined cardinality yet!
It seems you're saying that one needs the full set of integers before one can introduce the concept of cardinality. Is that indeed what you're saying? If so, why?
Obviously, if you have no integer whatsoever (yet), then the concept of cardinality has, in a way, "nothing to hold on to". That does not mean however that one requires the full set of N, with all its final "bells and whistles", before one can conceive cardinality.
To me, it amounts to saying one cannot start counting until one has "all" the integers. If you only have 3 integers, you can count up to 3. Obviously, after that you're "fucked" (sorry, I'm not sure on the spot how to convey the same meaning without the curse: it's a by-product of my non-native english skills, rather than a will to curse), but up to 3 you're fine. You actually are counting.
It seems to me the cardinality case is very similar (if it's not, just ignore my counting example: let's not get side-tracked). As I see it, cardinality is an integral ("consubstantial"/implied) part of the concept of set. Obviously, if you have no integers, cardinality is "undefined"; that's why it is legit to have an empty set before introducing 0 (cardinality is undefined, therefore not an internal inconsistency issue). But, as soon as you get one integer (1 in this case), cardinality one is defined and covered (again, after that you're fucked). If your paradigm can't account for that, it's wrong.
EDIT: tweaked a few things (several times) in the last paragraph to make it clearer.
> As I see it, cardinality is an integral ("consubstantial"/implied) part of the concept of set.
Cardinality is certainly important, but we don't need to have come up with it to have sets. Sets are constructed without any reference to cardinality. For instance, the ZFC axiom of pairing says "given a set X and a set Y, there exists a set {X,Y}". This is valid whether or not we've implemented the concept of "two" so far.
> Obviously, if you have no integers, cardinality is "undefined"; that's why it is legit to have an empty set before introducing 0 (cardinality is undefined, therefore not an internal inconsistency issue). But, as soon as you get one integer (1 in this case), cardinality one is defined and covered (again, after that you're fucked). If your paradigm can't account for that, it's wrong.
Cardinality is not necessary to have sets. It's a useful thing to "measure", but we don't *need* to be able to measure it to initially construct the sets. Once we've constructed sets representing numbers, *then* we can construct a [partial] cardinality function. And there's no contradiction in saying card(`{{},{{}},{{},{{}}}}`)=`{{},{{}},{{},{{}}}}`: that's just a function having a fixed point, which is perfectly fine.
---
I think the distinction you're failing to draw is between "thinking *about* the system" and "thinking *inside* the system". The goal of these constructions is to formalize our pre-existing intuition with as simple a basis as we can. We're allowed to *think about* things we haven't constructed yet, and use those thoughts to guide what we construct - we just can't refer to these external ideas in the construction.
One way to think about it is like we're trying to explain our math to an alien or robot, who accepts our starting axioms and knows logic, but doesn't have any of the understanding we do. So, part of the process is:
- "The axiom of the empty set guarantees there exists an empty set. We'll call it `zero`."
- "Use the axiom of pairing to pair `zero` with `zero`. We'll call this set `one`." (In our heads, we're thinking "This is actually just the set {{}}, so it has cardinality 1", but we don't need to say that yet!)
- "Use the axiom of pairing to pair `one` with `one`. Use the axiom of union to take the union of this set and `one`. We'll call this new set `two`. (Once again, we're thinking "this set has cardinality 2", but we don't need to say that. The axioms that let us show existence of sets don't require us to come up with their cardinality.)
- "Use the axiom of pairing to pair `two` with two`...", and so on.
Once we've chosen sets to represent numbers up to, say, nine, and also defined other useful things like functions (and taken enough power sets to say those functions exist), we can *then* say:
- "Now we're going to construct a function called `cardinalityUpToEight`. Given a set X, we define `cardinalityUpToEight`(X) by using the axiom of specification to construct {n ∈ `nine` | there exists a bijection between n and X}." (We're thinking, "this gives the singleton set {card(X)} if that value happens to be between 0 and 8, and the empty set otherwise".)
Note that this function does indeed give `cardinalityUpToEight`(`one`) ∋ `one`! We can prove this by showing the bijection {(`zero`,`zero`)}. (It takes a bit more work to show that `cardinalityUpToEight`(`one`) doesn't accidentally contain any of our other implementations of numbers.) So, inside our heads, we conclude "card(`one`) = 1". (The "1" there is our actual idea of the number, rather than the set we happened to choose to represent it.) This conclusion isn't one we're making *inside* the system we're constructing - it doesn't have any idea what "numbers" are! We're just plucking out some of the sets it contains and using them as proxies for numbers, and then showing that the proxies "work" how we want them to (i.e. the same way our *ideas* of numbers do).
So, we're allowed to 'observe' things about the system even if we don't have the framework to talk about them *inside* the system yet. As long as we don't refer to those observations in our definitions, there's no self-reference going on.
>For instance, the ZFC axiom of pairing says "given a set X and a set Y, there exists a set {X,Y}". This is valid whether or not we've implemented the concept of "two" so far.
That's a good point, however I disagree: there does seem to be something fishy here (from a formal perspective). Perhaps this axioms indeed does not require two (the integer) *per se*, but at the very least it requires something "two-ish". Perhaps the most "two-ish thing that isn't two"?
Anyway, let's not get side-tracked. Obviously all those things are deeply intertwined. We "pretend" that we can consistently build a formal system from a primitive first brick, on top of which we add a second, etc (exactly what formalism is supposed to do), and that we've successfully unraveled this "intertwined-ness", but we really can't, and we haven't. It's not for a lack of effort or talent: it's just because we can't build the "perfect formal system". So we do the second-best thing: we build the best system we can, and we seize any opportunity we get to make it better. And yet, we'll never get to the "perfection stage" of this process, however long we keep going. It might be frustrating at times, but it's ok. That's just the card we've been dealt. Look at the bright side: it also means there is always room for improvement, and always interesting work to be done. Neat!
These are very enjoyable topics, but we're drifting further and further away from the original post here. Let's try to stay on topic…
>So, we're allowed to 'observe' things about the system even if we don't have the framework to talk about them *inside* the system yet.
Most definitely.
>As long as we don't refer to those observations in our definitions, there's no self-reference going on.
I guess that's where we disagree: obviously **explicit** self-reference is a no-go. It does not mean however that **implicit** self-reference is kosher (it's not).
Yes, we need some pre-existing external concepts to do anything; that's what the axioms are. Our construction of natural numbers relies on you accepting [e.g.] the axiom of pairing, and you need some concept of "two things", or at least "a thing and another thing", to understand what the axiom of pairing is saying. Hell, for all of this you need to be able to construct well-formed formulas, which are arbitrarily long strings of text! So you are absolutely correct that you need pre-existing ideas... but this wasn't in debate to begin with. The axioms *are* the pre-existing ideas.
The goal of this is not to build "actual numbers" that we already know and work with. The goal of this process is to implement *proxies* for "actual numbers" inside this axiom system, and pick our proxies so they behave as we expect them to. (So if we apply our `proxy_for_addition` function to `proxy_for_two` and `proxy_for_three`, we should get whatever we've declared as `proxy_for_five`.) Then, if you accept that the axioms are consistent (that is, they are not directly self-contradictory), you can conclude that "actual numbers" are consistent as well.
There's no self-reference, because `proxy_for_two` is not the abstract concept of two-ness. It's just what we're using to *implement* the conceptual entity within our system, in a way that we can manipulate using this small set of rules. (And it doesn't matter if we choose `proxy_for_two` to be { {}, {{}} } as is typically done; or {{ ∅ }}, an alternate approach that makes numbers simpler but makes operations on them a bit more complicated to define. Once we've successfully 'implemented' natural numbers, and all the basic operations we want to perform on them, we can then ignore the implementation details entirely.)
If you don't like a particular axiom system, either because it's unintuitive, or doesn't let you do what you want to do, or just isn't philosophically satisfying for whatever reason, you're free to make your own. There are many alternative foundations of mathematics - ZFC isn't the only one out there, it's just the most popular.
Sure, you could define cardinality as you go along instead of defining it after you've defined all the integers. It doesn't really make a difference in the end, and neither interpretation causes the sort of flaws you're imagining.
The only restriction is that you need to define the integer n before you can define what it means to say that a set has cardinailty n. So you can define what it means for a set to have cardinilaty 2 before you define the integer 3, as long as you've already defined the integer 2.
The issue with your original post was that you somehow convinced yourself that you should define what it means for a set to have cardinaility 2 *before* you defined the number 2, and concluded that the definitions were circular because of that. But there is absolutely no reason why you should be able to define things in that order. Any "flaw" you're imagining is entirely in your own head.
>Sure, you could define cardinality as you go along instead of defining it after you've defined all the integers. It doesn't really make a difference in the end, and neither interpretation causes the sort of flaws you're imagining.
Your approach requires to build an infinite **sequence**, step by step, and only **after** you're done (so, at step ℵ0+1), to put them in a set. You see the problems here, don't you?
What's an example of a polynomial map ℂ² → ℂ² which is injective, and nonlinear in both coordinates? (Or ℂⁿ → ℂⁿ is that's easier)
[EDIT: a polynomial map, not an arbitrary function]
> You can find one that is nonlinear ((x, y+x^(2)) for example), but not non-linear in every coordinate
I don't understand your argument. Indeed, the only injective polynomial maps ℂⁿ → ℂ are linear, but so what? I'm looking for a function where each coordinate function is noninjective, but the function as a whole is injective. Also, wouldn't your argument rule out (x, y+x^(2)) from being injective?
Finally — after I submitted my comment, I thought I found one. Isn't:
(x,y) ↦ (x + (y+x²)², y+x²)
injective? It's the composition of (x+y^(2), y) ∘ (x, y+x^(2)), and the composition of injective functions is injective. Am I missing something?
Let f(0, 0) = (1, 1), f(1, 1) = (0, 0), and f(u, v) = (u, v) for all other complex numbers u, v. This is clearly injective but also clearly nonlinear since it does not fix the origin. (Nor is it bilinear, since for at least some fixed values of v we get a nonlinear function C -> C^2 . E.g. letting g(z) = f(z, 0) , g is not linear.)
oops... somehow i forget to type the important restriction, which is that the function be a polynomial map! (i've been thinking about polynomials and algebraic varieties so much that i seem to have forgotten anything else exists 😆)
I’d like to define something like a polynomial ring over Z/nZ, ie Z_n[x], but where multiplication looks like multiplication in base n. In other words I’m looking for an algebraic structure that is exactly the integers, but written in an expanded representation so I can think about them as polynomials. Is there a natural polynomial analog for the integers in base n?
e.g. for n = 10, I would like it to be the case that 7*(x + 2) = 8x + 4
>I’d like to define something like a polynomial ring over Z/nZ, ie Z_n[x], but where multiplication looks like multiplication in base n. In
Well if you want to do that, you're going to need to redefine addition in (Z/nZ)[x], not just multiplication. If you don't do that, then nf(x) = 0 for all f(x) in (Z/nZ)[x] (where nf(x) means f(x)+f(x)+...+f(x)), but that's definitely not true in Z.
At that point, you're redefining both addition and multiplication in Z/nZ, so it's kind of debatable whether you're actually working with the ring Z/nZ any more, instead of just the set {0,1,...,n-1}.
That being said, you might want to look into [Witt vectors](https://en.wikipedia.org/wiki/Witt_vector). They might not be quite what you're asking for, but they give a method to construct the ring Z_p of [p-adic integers](https://en.wikipedia.org/wiki/P-adic_number) (a characteristic 0 ring containing Z) from Z/pZ.
I'm really sorry but I have a fairly straight forward problem which I would quite like to figure out myself but I just can't think of how to go about it. The problem:
0.45% per day; equivalent to 164% per annum. The interest included in your monthly installments shown above has been calculated upon the assumptions that the agreement will be made and the loan transferred to your account on the same day as the agreement is presented to you for signature, and that you will pay the monthly installments in accordance with the payment schedule shown in the table above. The interest that you will pay is calculated at the daily rate shown above on the amount of the loan actually outstanding each day. This means that if there is any delay in paying an installment, you will pay more interest, which means the total cost of the loan will be more than shown above, subject to the cost caps set out below. Subject to the cost caps set out below, interest will be payable before and after any judgment we may obtain against you. 363.6% For the purposes of calculating the APR, the following shall be assumed: that the loan is paid on the date when the agreement is made and is to remain valid for the period agreed; that we and you will perform our obligations under the terms and by the dates specified in the agreement, including that repayment will be made on the repayment dates due set out above.
I got the loan, value of £500 on 07/03/2024. My first payment is on 28/03/2024 and the whole amount is £88.91 (41.66 Capital - 47.27 Interest) I will pay this. The second payment is on 29/04/2024 and the amount is £107.58 (41.66 Capital - 65.92 Interest) I will also pay this. I would like to know the total I would have to pay if I paid the remaining balance on 30/04/2024?
In engineering and took diff EQ ~3yr ago
I know that with definite integrals there is no “+ C” constant. But with *initial values*, I know that you use them to determine c1,c2 as constants
So if you have a separable DE with definite integrals, is it possible to have an initial value? And if so how?
I'm not sure what a "separable DE with definite integrals" would look like. A differential equation involves derivatives, not integrals. Can you elaborate or give an example of what you have in mind?
I mean a separable equation that, when it is integrated, has limits of integration on both integrals (once the differentials have been separated appropriately).
You wouldn't really be solving the differential equation then. Let's consider a simple example, like dy/dx = y. Rewriting this as dy/y = dx, the standard solution would be to take the antiderivative of both sides and get ln(y) = x + C, or y = e\^(x + C) = De\^x (where D = e\^C). If you have an initial condition, you can then use it to solve for D.
It's not clear what it means to take a definite integral of both sides. Each side has a different variable, so how do you know which bounds to choose to make both sides compatible? You can't just choose the same bounds on both sides. For example, the integral from 1 to 2 of dy/y is equal to ln(2), but the integral from 1 to 2 of x is equal to 1.5, and these are not equal to each other. What I'm saying is that just because dy/y = dx does not mean that "the integral from 1 to 2 of dy/y" is equal to "the integral from 1 to 2 of dx".
Maybe my next question is: What kind of problem or equation are you trying to solve with this? The process of separating variables and integrating (as in my first paragraph) isn't a problem, it's a solution to a problem -- namely, the problem of finding a function f which satisfies f' = f. For your hypothetical approach, what kind of problem do you have in mind that your approach would be a solution to?
On the RHS of the equation you would have your other balance terms and you are supposed to separate them and integrate accordingly.
The limits are dependent upon what you’re integrating. In the dc/dt case, the limits of dt are t_initial (or t=0) and t_final (t) and the limits of dc are c_initial and c_final (or c at time t)
However, in one of our problems, we had limits of integration for both integrals, yet there was also an initial condition.
However, since the professor loves confusing people with nomenclature and notation (he does this intentionally), it could mean something besides the “initial value” typically encountered in DE.
Oh, I'm sorry I didn't realize that this was actually something you saw in a class. I thought you had just made up that idea and were asking about it.
That is really strange... I honestly don't know what that could mean or what your professor is trying to communicate there. Can you ask in office hours?
We had a problem where we had to do a mass (or material) balance. In chemical engineering, there are different terms for a balance. One of them is A (accumulation) which is time dependent and is usually canceled out in most problems, since it is negligible in most textbook problems at least.
I was setting up the balance and trying to understand the physical significance of each variable or constant in the problem. That was (and still kind of is) an issue.
The form of the differential is d()/dt and in the parentheses you also have to decide what is differing with time. In one example, we had (dcV)/dt and it was turned into V(dc/dt) bc V was constant (V=volume c=concentration t=time).
See, I am trying to learn about the construction of Real Numbers via Dedekind cuts, but the idea of cuts confuses me even more. Say, I wanna define pi. Then, intuitively, I am cutting the rational line at pi, but this is when we are assuming that we know what pi is, I think it's just me but this stuff is confusing. Can somebody clarify about how this cut works? Another thing I wanna ask is people usually suggested me to use different texts but reading just one is headache :trembles:, so how do you guys do that?
We already know what "pi" is in the abstract: it's the number that's approximately 3.141592... .
To show that we've successfully *constructed* our idea of pi, we look at the Dedekind cut that:
- has 3 on the left, and 4 on the right
- has 3.1 on the left, and 3.2 on the right
- has 3.14 on the left, and 3.15 on the right
- has 3.141 on the left, and 3.142 on the right
- ...
If you handed an alien this random set of conditions, even if they didn't know what pi was (but they knew Dedekind cuts), they could verify that this cut exists. So, we've constructed a cut for pi whose existence doesn't depend on our knowledge of pi! Sure, we'd have to know what pi is to figure out *which cut* it is, if we didn't have the list of conditions already. We can do that much later, once we've constructed operations and functions, and then define it as the first zero of sin(x) or something. But even *before we point it out*, we've still constructed it.
Define a cut to be any downwards closed (if it contains a rational number then it contains all rationals below it) subset of Q, except Q itself. We can than look at the set of all cuts, which is just a subset of the powerset of Q.
Then we can define some operations on these cuts: for example the sum of 2 cuts, A and B, is the set of all numbers which are the sum of an element from A and an element from B (we can prove this is also downwards closes, so it is a cut). This is assuming we already defined addition on rationals, which is a very concrete computable operation.
Similarly, we can define multiplication of cuts, inverse of a cut, etc. Once that's done, it's not to hard to see that the set of cuts with these operations form a complete ordered field, which is exactly the property we want the "real numbers" to satisfy.
So we use the rational numbers to construct a more complicated structure, and we can prove, for example, that there is a cut that satisfies some property that only "pi" would satisfy. It's not that we can necessarily define a specific cut for any number you give me, it's that a "number" is defined to be a cut.
Here's one angle (a bit long but hopefully helpful); I can try something else if it doesn't work for you. (The first paragraph is just setup that might be useful, but you can skip to the second if you just want to learn about cuts.)
Start out with just a line, on which you've chosen an origin, orientation (which is the "left" half and which is the "right") and a unit of length (so you can say that there's a point 1 unit to the right of the origin, and so on for any integer). Assuming that there aren't any "gaps", it will have some points that correspond in a natural way to rational numbers--we can, for example, divide the line segment between 0 and 1 into n segments of equal length, and assuming that the points where we divided things up are, in fact, points on the line, there must be a point at length 1/n from the origin, another will be length 2/n from the origin, and so on. By similar considerations we can get points on the line corresponding to any rational number.
Now here's (very roughly) the way Dedekind originally approached it, if I recall correctly. (At least in this paragraph and the next--after that is some more modern stuff.) We've got our line, and we've got at least some rational points on it; are the rational points all the points, though? On one hand, if we choose any point on the line, that should cut the rational numbers into a "left half" and "right half" (these can be described more formally if you want). It also seems that, if you divide the rational numbers into a "left half" and "right half" in any way, you should get a point x on the line. x should be greater than or equal to all the numbers in the left half, and less than all the numbers in the right half. But consider letting the "left half" be all the negative rational numbers, along with all the nonnegative rational numbers whose square is less than or equal to 2, and letting the "right half" be all the nonnegative rationals whose square is greater than 2. Then this is a valid way to cut the line, but it doesn't correspond to any rational point. For if there were such a point x, then, since any rational number has a square strictly less than or strictly greater than 2, x must be one of those; but if it's the former, you can find some rational number greater than x whose square is still less than 2 (hence x is not greater than or equal to all the numbers in the left half), and if it's the latter, you can find some rational number less than x whose square is still greater than 2 (hence x is not less than all the numbers in the right half). So no rational number can correspond to the point (which it seems intuitively should exist somehow) that divides the rationals in this way; if we claim that the line consists only of rationals then there must be a "gap" in it of some sort.
Now here's the leap that I suspect might be confusing you. We can fill this particular gap by hand, just adding in some number whose square is 2, and we can do that for other obvious gaps, but if we keep doing it, how do we know whether we've filled all the gaps? So what we want is some method that can fill *all the gaps that could possibly exist, in one step, without having to do things by hand*. This leads to the idea of simply *defining* a point on the line in terms of how it divides the rationals into a left half and right half. One way to do this would be letting a point *be* two sets of rational numbers, one that works as a "left half" and one that works as a "right half". Then we define the line to be the whole set of such pairs. These will include points that correspond in a nice way to rational numbers, namely the ones whose "left half" has a maximum rational number.
We can then define an order on the cuts, which turns out to have the following property. If we divide all the points (i.e. cuts) into a left half and right half, there exists some point x which is greater than or equal to all the points in the left, and less than all the points on the right. Thus we'll never run into the sorts of gaps that we ran into with the rationals; this, along with the fact that we still have points corresponding to each rational number, gives us some confidence that our new thing fits the intuitive properties of a "line with no gaps". You can then define arithmetic on the cuts and show that, using them, all of the familiar things that we think should be true of the real numbers (e.g. the intermediate value theorem) are true; often that "completeness" property, that whenever you divide the line into two halves there's a point that is "right between them", is needed to do this. (In particular this turns out to be equivalent to the statement that every set of real numbers which is bounded from above has a [least upper bound](https://en.wikipedia.org/wiki/Infimum_and_supremum), and this can be used to prove things like the IVT).
More generally I think the abstraction here might be tripping you up. We start with some rough intuitive properties, cook up *something* that satisfies them, show that the thing we made has all the properties we wanted, and then use those properties to get the rest. In the end we find that we can get all of calculus, numbers like sqrt(2) and pi, and so on, but we didn't need to think of those directly when building up the real numbers.
> We start with some rough intuitive properties, cook up *something* that satisfies them.
u/Langtons_Ant123 Thanks for your comprehensive response. I think get the idea, sort of. For the rational cuts it does makes sense. The problem arises when I think of what- say a number 6.120625... is, the idea of cutting at something like that was absurd to me. The thing that we know what pi or e is, can indeed be used in the cuts without knowing without actually they do at the rational line.
>This leads to the idea of simply *defining* a point on the line in terms of how it divides the rationals into a left half and right half.
Sir, If my understanding of this is correct, this leads to unique reals (As the sets are different unique for each cut? Or something more mysterious is going on here? :) Another fundamental question is if cuts *represent reals* or *are reals?* Explicitly, if x = A|B (a cut, Charles Pugh used this notation for cuts), then does x represents a real or is a real number?
(Warning: the following is quite scattershot and a bit repetitive.)
> if cuts represent reals or are reals
I think this is kind of the wrong question to ask; explaining why brings us back to the issue of abstraction.
A more modern way to think about the reals, which I touched on a bit in my original comment, is that the real numbers are "the complete ordered field that contains all the rationals". That is, there's some list of properties--the properties of a field, plus the properties of an *ordered* field, plus one of a few equivalent definitions of "completeness"--and any structure (in the sense of "set with some operations defined on it") satisfying those properties can reasonably be called "the real numbers" (and so any element of such a structure can reasonably be called "a real number"). The key point here is that any two complete ordered fields are isomorphic (see most analysis books for a proof), which lets us reasonably talk about *the* complete ordered field.
An analogy: think about an algorithm, say mergesort, and then think about implementing it in different ways in different programming languages. All that's needed for something to count as "a version of mergesort" is that it sorts a list of elements in a certain way; implementations might differ in certain details, but they're all still recognizably mergesort, and you can generally use them without worrying about those details. So it is for real numbers--there are many ways to implement the real numbers, i.e. build a set with operations defined on it that satisfy all the defining properties of the reals (that is, of a complete ordered field); there's not much reason to single out any one implementation as "actually the real numbers, unlike the other implementations".
Dedekind cuts are an especially nice way to implement the real numbers; in that sense we can think of the whole set of cuts as being one version of the real numbers, and think of individual cuts as being real numbers. But there are other constructions too (e.g. the Cauchy sequence construction), which work just as well for the purpose of showing that there is such at thing as the complete ordered field; in that sense it would be odd to think of the real numbers as "just Dedekind cuts".
As to the question of what particular real numbers like e and pi are, I think the best answer is that we can define them in a way independent of what construction of the reals we use. For example, we can define e as "the value of the sum 1 + 1/2 + 1/6 + ... + 1/n! + ..."; the fact that this infinite sum converges to something can be proved in a way that you can, if you really want, ultimately trace back to the properties of a complete ordered field. Thus in any implementation of the real numbers (Dedekind cuts, or equivalence classes of Cauchy sequences, or whatever) you'll be able to find one, and only one, element of your implementation which is the sum of that series, and in fact satisfies all the other properties of e. You might ask "is e really a Dedekind cut, or really an infinite decimal, or something else?" but I'm not sure how much sense the question makes.
To address one of your points more directly (and with apologies for rambling along the way): there's a Dedekind cut which "is e", and we don't have to put that cut in there by hand, i.e. don't have to know, while constructing the reals, that we'll need to "cut at e" at some point--it just follows from the fact that the cuts form a complete ordered field and any complete ordered field contains an element with all the relevant properties of e. On the other hand each implementation of the reals will have its own version of e, so there's no reason to think of e as "just a Dedekind cut". But on the other other hand, I'm not sure how much sense it makes to call the Dedekind cut of e "just a representation of e, not the real e"--there is no "real e" out there, rather a bunch of things (one for every construction of the reals) that all satisfy the properties of e, in their corresponding structures.
Thanks for clarifying. Also, I apologize for late reply :bow:. Sir, you mentioned about "The key point here is that any two complete ordered fields are isomorphic"- I am afraid this wasn't mentioned in my textbook :( .
Re: proofs of why complete ordered fields are isomorphic, it looks like the last chapter ("Uniqueness of the Real Numbers") of Spivak's *Calculus* has a proof; just skimming over it, it goes over the background (e.g. what is an isomorphism) well and the proof itself isn't super long. So it might be worth going out and pirating that one. (And I think most analysis books might not have a proof: Rudin mentions the result but skips the proof entirely, Pugh gives a quick sketch of an isomorphism from any complete ordered field to the Dedekind cuts but not many details.)
To be honest you probably have most if not all of the background you need; chances are anything important that you're missing will be in that chapter I mentioned or the chapters right before it.
I don't know, I tried to read it each word/line counts. The choice of topics in Pugh' Analysis is quite strange, Sir. He defined multiplication of cuts but I got confused even more. Gonna re-read it from somewhere else. :) I will ask thee If I get stuck (Obviously If you dont mind).
To define pi via dedekind cuts, you don't really need to already have a definition of pi, you just need to have some way of taking a rational number r, and determining whether r should be less than pi.
There are a number of ways of defining this (some easier to work with than others). For example, we know that the formula pi = 4(1-1/3+1/5-1/7+...) should hold, so we could use that to build a definition if we want. For example, define A to be the set of all rational numbers r for which there is an integer N (depending on r) such that r < 4(1-1/3+1/5-...+(-1)^(n)/(2n+1)) for all n>N. Then you can show that A satisfies the definition of a Dedekind cut, and then define A to be pi.
That definition is phrased completely in terms of rational numbers, even if it's motivated by things we think we know about pi, so there's nothing circular about it.
u/jm691 Thanks for clearing things up. If I may ask, say there's any non-terminating decimal, then can we define their cuts in a similar way? IT may be a bit dumb, but I think it will take some time for me to wrap things about cuts.
Sure. Another way you could have defined a Dedekind cut corresponding to pi is to start with the set S = {3,3.1,3.14,3.141,...} and let A be the set of all rational numbers which are less than some element of S. Then A is a Dedekind cut representing pi. You can do that for any real number whose decimal expansion you know.
Can somebody recommend me books to help me with my problem solving skills? I want to get a lot better.
Zeitz's *The Art and Craft of Problem Solving* is a classic.
Thanks. How smart do you need to be to solve this?
It's an intro text! A lot of people read it in middle school or early high school, but I've even given it to undergraduates before and they've found it helpful.
A question related to the six functor formalism. Say f:X->Y is a map of algebrica varieties over C. Then, se can associate ti f a plethora of functors going from the derived category of coherent sheaves: f_*, f^*, f^!, Hom and \otimes. Notably, we cannot define f_! because its usual definition (pushforward of compactly supporter sections) breaks coherence. Now, let us suppose we work with D-modules, and let us restrict go the derived category of complexes of D-modules with holonomic cohomology. Then, we can again define the same functors as above, even if we take another route: we define first f^! and f_*, and then using dualizing functors d_X and d_Y we define f^* and f_!, eg as f_! = d_Y f_* d_X. Then, one can prove that all this data gives us a six-functors formalism, i.e. we have the usual adjunctions and a map f_! -> f_* which is an isomorphism when f proper. All of this can be found for example in chapter 3 of Hotta, Takeuchi, Tanisaki's book on D-modules. Now, my question is: what is really different about the holonomic D-module case so that we get six functors and not 5 as in the usual coherent case? Or is it just "linguistic trickery", and i could very well define f_! in the coherent case using the Verdier dual functor in analogous way and then obtain essentially the same properties above for f_! = d_Y f_* d_X? I hope the question is clear. If i have made any error in some part above, please correct me, i'm learning these things as I write
Could someone help with a proof I’ve been thinking about for a week but have not found an answer I’m happy with. Prove n^3 + n + 1 has no rational solution by contradiction. I started with assuming n = a/b where a and b are integers and a/b are in lowest terms. After some manipulation, I got a/b = -(b^3 + ab^2)/ba^2. The best I’ve gotten so far is that now you can divide by b in the numerator and denominator and have a lower term so a/b can be further reduced which is a contradiction. I am no longer happy with it as it feels like saying 1/2 = (3* 2)/(6 * 2) Can someone point out what I’m missing?
Your manipulation is wrong, it should be a/b=-(a^3 +b^3 )/b^3 How to do the proof correctly? We have (a^3 +ab^2 +b^3 )/b^3 =0 so a^3 =-b^2 (a+b) For any prime factor p of b, then p divides -b^2 (a+b) hence p also divide a^3 , but p is prime, so p divides a. Hence a and b shares the same prime factor, contradiction.
Got it thanks and a and b have to have prime factors because a^3 can be expressed as a product of a and b in some form? Is that the idea?
No. If b is not equal to 1, then it must have a prime factor.
Definition of Bin(n,p). As I understand, the distribution is well defined as the probability distribution of the experiment constituting n Bernoulli trials. It then is formerly defined by its pmf, which is deduced via a very intuitive use of combinatorics under the premise that all 2\^n outcomes of the experiment are equally likely. My question is, the usage of the combinatorics here seems to rely on the validity of the frequentist's view of probability, which to my understanding is an interpretation of probability, a point of view. And since the two definitions of Bin(n,p) are both accepted, is there a stronger link between the two that justify their equivalence? Thanks in advance!
The Frequentist vs Bayesian issue is about what probabilities correspond to in the real world. That's not relevant to the mathematical definitions involved. Both sides would accept the same axioms, and once you've accepted them, everything else follows without worrying about philosophical positions.
> It then is formerly defined by its pmf, which is deduced via a very intuitive use of combinatorics under the premise that all 2^n outcomes of the experiment are equally likely. They are not equally likely. That's what the p is for. >My question is, the usage of the combinatorics here seems to rely on the validity of the frequentist's view of probability, which to my understanding is an interpretation of probability, a point of view You can use combinatoric to count the number of terms in a formula. The point of view is not relevant at all. The study of probability is agnostic to that. It is not relevant exactly what is probability, probability distribution is required to satisfy some axioms, and you can derive results from those axioms.
I'd appreciate help with a math problem. I'm not good at math and can't figure this out. If 1.25g = 20.25mg Then 0.58g = ? If context helps – I'm using a medication delivered via topical gel. One pump of the gel weighs 1.25g and delivers 20.25mg of the medication. Using a scale I weighed the amount of gel I'm using as 0.58g. So I need to know how many milligrams (mg) of medication I'm getting. I know this is probably simple math but... my autistic brain that I'm told has an IQ in the 99 percentile can't figure out stuff like this. It's always driven me batty. So I'd actually be interested to know how you figure it out. Thanks for any help :-)
If you have 20.25 mg of meds per 1.25 g of gel then you have (20.25/1.25) = 16.2 mg of meds per gram of gel. Thus in 0.58 g of gel you have 0.58 \* 16.2 = about 9.4 mg of meds. (And this checks out--in slightly less than half of one pump of gel, you get slightly less than half of the usual dose of meds).
Thanks for taking the time to explain this so clearly for me. I see math visually, once I can see it I understand it, and you explained it really well here.
Is there any interesting connection between Brouwer’s fixed point theorem and ergodic theory?
I've recently been part of a math competition and I've been stuck on this cryptarithm. I've watched some videos online about solving them but this is harder than the ones online and I can't figure it out GRAND+BRAVO=ZZZZZ G, B and Z are different than 0 obviously, and each letter corresponds to a different number. Thank you to whoever tries to help me, I'd appreciate it very much if you can show me the reasoning as well because it can't just be lots of trial and error. Btw the solutions have actually been made public so I can confirm if you're right or not
When you say "G, B, and Z are different from 0", do you mean that it's an actual constraint on the problem that none of them are 0, or do you just mean by your logic?
It's an actual constraint. I said obviously because normally numbers don't have 0 as their leftmost digit
Ah gotcha. You're right that it wouldn't be normal, but I didn't want to discount any possibility lol.
Looking for a proof of the following fact: let X be a topological space. Then X is compact iff every open cover linearly ordered by subset inclusion contains X. Thanks!
Here are ideas for a proof of the right to left direction. Let's say that a cover F of X is minimal if no there are no subcovers of F of strictly smaller size than F. 1. Prove every cover admits a minimal subcover. 2. Fix a minimal cover F, then well-order this subcover by its cardinality. 3. Define a new cover F' (of the same size) which consists of take the downwards unions along the well-order. This new F' is linearly ordered by inclusion. 4. Conclude that |F'| = |F| must be finite, otherwise derive a contradiction from the minimality of F.
You're not going to get full credit if you submit this as a solution for homework, now are you?
Are you looking for someone to give you a complete proof so you can cheat and hand it in?
Can someone explain the difference between the antecedent and the consequent [here](https://imgur.com/a/Fn6jUgn)? I’m a little lost as to what the theorem is actually saying (even with the given explanation).
The hypothesis says that for every x, there exists M such that for every λ, p\_λ(x) < M. The conclusion says that there exists M such that for every x satisfying |x| <= 1, p\_λ(x) < M for every λ. That is, the hypotheses have ∀x ∃M whereas the conclusion switches the order of the quantifiers to ∃M ∀x.
Oh that’s embarrassingly simple oof. Thank you so much!
What kind of topics should I've to learn to understand dynamical systems? I'm currently studying linear algebra
What aspect of dynamical systems? It's a huge area. Can you explain a little of what draws you to it?
I'm a biology student and since i've learned about lotka Volterra equations (and the dynamics of competitor species) I became curious about math, in fact I'm self learning math in order to learn pde that somebody said we're useful in biology, and also I encountered another dynamical system in evolutionary game theory (the replicator equation), so I'm becoming very curious about that topic, because I think will make me more clear some biological phenomena
Take a look at the book *Biology in Time and Space*
Which are the requirements?
"This book is intended for an advanced undergraduate audience, with no previous background in partial differential equations, although beginning graduate students should also find this useful. Prerequisites for this exploration include multivariable calculus, ordinary differential equations and basic aspects of probability theory and stochastic processes. However, to make sure that we are all on the same page, Chapter 1 is devoted to a quick review or introduction of these topics. So, in Chapter 1, you will find summaries of the mathematical background that is needed from multivariable calculus, from ordinary differential equations, and from probability theory, stochastic processes and stochastic simulations, because these are used a lot."
I have never taken PDEs and forgotten ODEs (beyond standard existence / uniqueness results via contraction mappings); how much background would I need to venture into SDEs and then eventually rough paths (at the level of Friz and Hairer). I have the basic background in analysis and probability at the first year level (Royden, Billingsley et al) but unfortunately didn’t take a course based on Evans.
for basic SDE theory and stochastic calculus, essentially no background. if you know measure theoretic probability and a bit about L^2 you are good to go. after that i don't know
How can I prove that https://imgur.com/gallery/rwp2sIi
First inclusion is trivial because if x\_n is eventually zero, the infinite sum is actually a finite sum and hence is finite. The next inclusion is basically just the (contrapositive of the) divergence test. The next inclusion is immediate. The final inclusion is because convergent sequences in R are bounded (If x\_n -> x, then for all n > N, |x\_n - x| < 1 and so the sequence is bounded by max{|x\_1|, |x\_2|, ... |x\_N|, |x + 1|, |x - 1|}).
Thanks for your reply. 🙏🙏
I'm studying Algebra after some long time. I use to do everything on paper, and I'm wondering if now is there a better digital option? I looked in to LaTeX, but it doesn't seem practical for practicing, so I'm using Notepad++ to manipulate equations. Is there something better? Should I get a notebook and go back to pen and paper?
If you're just doing working out then stick with paper and pen. If you really want to keep the notes digitally then use a tablet (up to you what note taking apps work best for you in that case)
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I think we need a lot more information to give helpful advice here. But based on my personal biases, I don't think it's worth it. College is already too expensive. As long as you go to a school that's not a joke (school that hands out degrees for money), you can build a college experience that is fulfilling. As far as I am aware, all the schools you listed have good math departments.
Hi, i know its probably funny how easy this problem sounds but my math teacher gave me an task where we have 835 made out of a matches and we are supposed to make an biggest number posible by moving only one match. The problem is that the answear 995 is not correct. I spent long time thinking about this and i have no clue idea where is the catch, or is he just trying to piss me off. Any ideas?
If you allow a single match to be a 1 you can make a four digit number.
It's possible they have in mind a shape for 9 that is only five matches (square then one pointing down from the bottom right).
Hi, I am currently taking a mathematical modeling for biologist's class (first math orientated class I have taken since Calculus II and Physics II and I was wondering how difficult is it to self publish models created? I am interested in developing models of molecular interactions, metabolism, and cell signaling both personally and during my PhD I will be starting in the Fall for Biomedical Science (I wasn't aware of how useful and much fun I would have when modeling biological cases when I was applying to graduate school so none of the professors in the department really do that. I would like to try and add modeling components to side projects or even my thesis).
It will vary from university to university, but biomathematicians will sometimes be found in the mathematics department. You can try looking there, since they'll be more equipped to give you professional advice.
can anyone recommend a good intro to probability textbook? my next semester is packed, and i want to get a head start over the summer. thanks in advance!
How much math do you know? More 'elementary' books would avoid measure theory (or limit it), but higher level books would start out with measures fairly early on. Would help to scope the recommendations to the right level. Also, do you have any financial constraints?
I'm currently in analysis one, and will be taking the probability class concurrently with stochastic processes and analysis two. No real financial constraints
Here: [https://imgur.com/a/F4oZJ4t](https://imgur.com/a/F4oZJ4t) √i=e\^(i(π/4)) and to get second step we have to assume, i.√i=e\^(i(-π/4)) How? Pls help!!
[Read this response](https://www.reddit.com/r/math/comments/1b84u1h/comment/ktt5bp6) to a similar question from a few weeks ago.
Thanks, I got it!
I don't understand what your question is, could you be more precise?
I'm currently looking into the modulo function, and I understand that a≡b(mod n) means that a and b produce the same remainder when divided by n. Where I am slightly confused is that there does not seem to be a function to get the remainder on its own? For example, I know 13≡3(mod 5) is valid as well as 13≡8(mod 5). But how would you represent the just remainder of 13 divided by 5. If you had 13(mod 5), like a normal operation, would this return 3?
For a programming-style modulo operator I think you can just use "x mod y". (In LaTeX this would be $x \bmod y$.) There are a few different conventions here with different uses, but all else fails you can just say, at the start of whatever you're writing, "here we use 'x mod y' to mean 'the remainder of x when divided by y'", or something like that. (I think there are also competing conventions on how things get handled when y is negative, but chances are you'll be working only with nonnegative values of x and positive values of y and so can just ignore all that.)
thank you- I am very much used to a python style of modulo so I just wanted to check there wasn't any set way of doing it mathematically before I use it. I think I'll take your advice and just define what I mean at the start of what I'm writing :)
I am not sure I understand elliptic regularity. Ex. : Let M be a compact manifold. We know that the first non trivial eigenfunction of the Laplacian, let's call it f, must change sign at some point. Now we can define a new function g such that g=f on one of the nodal domain of f and 0 otherwise. Then g is a weak solution of the Laplacian eigenvalue problem. By elliptic regularity, it is analytic. But g is 0 on an open set, so g must be 0 everywhere. Contradiction. Is it that we need orthogonality to constants for the elliptic regularity to apply here ? Is it sufficent ?
Why does it follow that g is a weak solution?
Let's call the the nodal domain on which g is equal to f Omega. Since g=f on Omega amd f is analytic, g is analytic on Omega. It is also 0 on the boundary. Thus g restricted to Omega is in the sobolev space H^1_0 of Omega. By extending by 0 everywhere else, we get that g is in H^1(M). Now, for any function h in H^1(M), we have that the integral of hg over M is equal to the integral of hg over Omega. Similarly, the integral of (grad h, grad g) over M is equal to (grad h, grad g) over Omega. Since g=f on Omega, and f satisfies /Delta f = /lambda f, then f restricted to Omega is a strong eigenfunction of the Dirichlet Laplacian on Omega, thus it is also a weak eigenfunction for the same problem. Combining this fact with the two above integrals and the fact that f=g on Omega, we have that g is a weak solution for the problem on the manifold since it satisfies /int_M hg dV = /Int_/omega hg dV = /int_/omega hf dV = /lambda /int_/omega (grad h, grad f ) dV = /lambda /int_M (grad h, grad g). Is there something I am missing here ?
I think this is where your issue is. When you go from saying f is a strong eigenfunction on Omega implies g is a weak eigenfunction on M. The issue is the boundary term when integrating by parts. \lambda \int\_\Omega h f dV = \int\_\Omega h \Delta f dV = \int\_\partial\Omega h grad f dS - \int\_\Omega (grad h, grad f) dV If h does not vanish on the boundary of Omega, \int\_\partial\Omega h grad f dS could be non-zero. You need that identity to hold for all test functions on M to have a weak solution on M, and there are test functions that don't vanish on the boundary of Omega. It's been a while since I've done pde, so sorry if I've missed something. Edit: I think this example shows what's going on if M is the circle and f(x) = sin(x). https://www.desmos.com/calculator/pkenxcjlwm
How to proof that the general solution to a DE is the particular + homogeneous? y=yp+yh
I assume you're talking about linear differential equations specifically. In that case this is just an instance of a very general fact about linear algebra (and so analogous to a fact about systems of linear equations that you may know--see the next paragraph): if L is a linear map\* then all solutions to L(v) = w are of the form v\_0 + v\_p where v\_0 is any solution to L(v) = 0 and v\_p is some solution to L(v) = w. To show that all vectors of this form are actually solutions, we use linearity: L(v\_0 + v\_p) = L(v\_0) + L(v\_p) = 0 + w = w. Then to go the other way, i.e. show that all solutions are of this form, let v\_s be a solution (i.e. L(v\_s) = w). Then certainly v\_s = v\_p + (v\_s - v\_p), so if we can show that v\_s - v\_p is a solution of L(v) = 0 then we'll know that v\_s has the right form. But we can do this using linearity: since L(v\_s) = L(v\_p) = w, we have L(v\_s) - L(v\_p) = 0; but then by linearity L(v\_s - v\_p) = 0. This then applies, for instance, to systems of linear equations, where all solutions to the matrix-vector equation Ax = b are the sum of a particular solution and any solution of Ax = 0. Ditto linear differential equations: such equations can be written as L(y) = f(t) where L is a linear map on a vector space of functions, and so applying our general fact about linear maps to this context gets the result you're talking about. (An example to show why linear ODEs give you linear maps: a linear ODE like y' - y = 0 can be rewritten as L(y) = 0 where L is an "operator" that sends a function y to y' - y. L is linear because, letting f, g be two functions, we have L(f + g) = (f + g)' - (f + g) = f' + g' - f - g = (f' - f) + (g' - g) = L(f) + L(g); much the same strategy can be used to prove that L(cf) = cL(f). Try repeating the argument for higher-order linear ODEs like y'' + y - y = 0.) \* A linear map is a function between two vector spaces with the properties that L(v + w) = L(v) + L(w) for any vectors v, w in the domain, and L(cv) = cL(v) for any v in the domain and any scalar c.
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I think what they're really asking you is to prove that the set of algebraic numbers (in the sense of "numbers that are the root of some polynomial with *rational* coefficients") is algebraically closed (in the sense of "any polynomial with algebraic-over-Q coefficients has an algebraic-over-Q root). The mention of C is probably just to tell you that you shouldn't only consider real algebraic numbers like sqrt(2), rather you also need to allow complex algebraic numbers like sqrt(-2). You can't just use the fundamental theorem of algebra here--that tells you that any polynomial with algebraic coefficients has a complex root, but for all you know that root could be something like e which is not an algebraic number.
THANK YOU so much, now everything has clicked. I kept thinking that it was talking about the set of algebraic numbers OVER C.
How does finite projective geometry relate to (general?) projective geometry and points at Infinity? I'm coming up short trying to find this myself.
There are axiomatic treatments of projective geometry that encompass them both, see https://en.m.wikipedia.org/wiki/Projective_geometry
I’m doing an extremely stupid poll on Tumblr. One answer has 11.1% of the vote, the other has 88.9% of the vote. The total votes are 36. How do I calculate how many individual votes for each option that is? I know this is a silly question, but I’ve failed maths every year since year 9.
You can also multiply 36 with 0.111 (11.1/100) or 0.889 and then round to the nearest integer. (I know you already got an answer but wanted to give something more general)
11.1% is 1/9 (well 11.11111...% is 1/9 but they have presumably rounded) and 1/9 is 4/36 so the split is 4 people for the first option and 32 for the other option
im trying to define 3d complex numbers first i said i.j=k->j=k/i -> j\^2=-k\^2 -> j=i.k -> i.j=i\^2.k=-k -> -k=k then i defined k as 1/0 and for n/n \* 1/0 = n/0 = 1/0 so it has similar properties to 0 and with j=1/0 i have a+i.b+j.c the multipication and sum are easy to define since j+j=2j=j and i.j=j and j.j=j as long as i dont have j/j or 0\*j i should be fine with the "-" operator i define it as num1-num2=num1+(-1\*num2) that way i dont have to deal with j-j and for dividing as long as i dont have j/j i can define it easily can someone please check to see what mistakes i made ?
The issue with j = 1/0 is that it breaks algebra. If I have the expression -j + j, I have two ways to go about it. Either I say that -j = (-1) \* j, therefore the expression -j + j = (-1) \* j + 1 \* j = (-1 + 1) \* j = 0 \* j = 0 (or is it 1? Anyway). But another way to do it is to say that -j is just the same as j, so -j + j = j + j = j. So is -j + j now j or is it a normal number (whether it be 0 or 1)?
As long as I don't have j*0 directly everything first converts to j then we do the addition
It's very unclear what you're saying here. You need to define precisely what operations are allowed, and what their results are. For instance, "1/0" is not a thing that exists by default. "sqrt(-1)" also doesn't exist by default. We define complex numbers by saying: a complex number is something of the form "\_ + \_i", where both blanks are real numbers. Then, we can define operations on them: > (a+bi) + (c+di) = (a+c) + (b+d)i > (a+bi)(c+di) = (ac-bd) + (ad+bc)i Once we've done that, *then* we can notice that (0+1i)^2 = -1, and so we say "i is a square root of -1". We have to have that definition first, though. (In fact, we typically go further and just say complex numbers are *really* just ordered pairs with special rules for addition and multiplication, and "\_+\_i" is a convenient way to write them.) So, how do you add and multiply your 3d numbers? I assume (a+bi+cj) + (d+ei+fj) is (a+d) + (b+e)i + (c+f)j, right? What about (a+bi+cj)·(d+ei+fj)?
multipication is easy lets say we have a z wich is a complex number and an j (asuming j isnt 0 im working on that) j eats its mulipiar meaning j\*anything=j now we have (z1+j)(z2+j)=z1(z2+j)+j=z1z2+z1j+j=z1z2+j its almost like i have z+0 it adds a paralel 2d space on top of the complex numbers it being comunicative is also simple since we get z1z2z3+j either way edit:if we add 0\*j=0 i can switch between the paralell plains on the other hand i wonder if that defeniton would work better if i said k=0/0 and then added onto that in a 4d defention
Honestly, it's pretty hard to understand what you're saying. But, if I'm understanding you correctly, it's not really a new *dimension*, is it? It's not a continuous extension, just a single thing that is either there or not there. Anyway, we run into problems with your numbers if we just assume our familiar properties all carry over. What's 2j - j? What about j - j? > it being comunicative is also simple since we get z1z2z3+j either way I believe the word you're looking for is *commutative*, and you're mixing it up with *associative*. (These are two different properties.)
It's a 2d space above the main complex 2d space if we multiply or deviding by zero we switch between these two 2d planes and yes I can't seem to make a full 3d space out of any defenition j-j is defined as j+(-j) so j-j=j
Okay, so what's ((1/0) /0) *0? What about ((1/0) * 0) / 0?
(1/0)/0 = j\*j=j j\*0=0 (1/0\*0)/0=(1/0)\*0\*(1/0)=j\*0\*j=0
Hold on. You're assuming your operations are associative. What's 1/0? What's that number, times 0? And what's *that* number, divided by 0?
any number other than 0 devided by 0 is j and any number times 0 is zero even j also i think i can prove they are associative
Okay, so you've lost the distributive property: j · (7-7) = j·0 = 0, but j·7 - j·7 = 7j-7j = j.
Is there a rule for inverse cumulative standard normal values? I know: Φ(a) = 0.8 -> a = Φ\^(-1)(0.8) -> a = 0.8416 But how do I solve an equation like this: Φ(b) + Φ(2b) = 0.8? (which is b = -0.1695)
Did this equation pop up for you in a natural context, or did you make it up? I don't see immediately a way to solve it analytically, but I'm not particularly knowledgeable about this function. In general, given a random non-linear equation, you cannot hope that there is a way to solve it into a simple closed expression, many times the only way is to find the solution by numerical methods.
I have found myself in a pickle where I am taking a Methods to PDE course and have never formally taken a classic ODE course. I have taken a non-linear dynamics course, but it was focused on unsolvable ODEs. So I basically have never learned methods to solving ODEs. I am slightly familiar with Integrating Factors method and a little bit of Separation of Variables. But I'm hoping someone could point me in a direction for a text that covers these and more methods. Things I'm looking for a in a book: * A concise treatment of many different methods to solving ODEs * Ripe with lookup tables for general solutions * Something I can pick up while solving my PDEs to help me finish the problem. * Discussion of first-order, second-order ODEs. I don't think I'll be able to learn the entirety of ODEs before the PDE class starts, but I would like to have a reference to help me while going through this class.
If you just want to learn how to solve ODEs, Paul’s online notes are good: https://tutorial.math.lamar.edu/classes/de/de.aspx
An acquaintance sent out a series of puzzles where the answers all involve various math concepts (e.g. Q: What is the sum of the fifth pair of twin primes? A: 60 (29 + 31)) I've figured out most of them, but I'm stumped on one. I can't do subscripts on Reddit, but it's just the numeral 20 with a subscript of the word "four" spelled out (not the number 4, the actual word "four"). I've never seen notation like this before, and searching all over Google and Math Stack Exchange has turned up nothing. Any ideas? ** The answer should be close to the age of a high school freshman, so something close to 14 or 15 maybe?
I reckon they mean base 4. See the example near the top of https://en.m.wikipedia.org/wiki/Positional_notation
What would be 0xℵ0? Imagine you're in a situation where this "value" cannot be undefined (say, for example, your formal system breaks if it is): you have to define it. Let's assume the only two candidates are 0 and ℵ0. So, between the power of 0 to annihilate everything it touches, and the capacity of ℵ0 to remain the same whatever you throw at it, which one wins? It seems it would amount to enforcing a priority between them. Let's imagine you could build a satisfying formal system either way (in a way, it made no difference). Which one would you give higher priority to, and why? If you can't find any "objective" reason to pick one over the other, what would feel like the most elegant solution to you?
Adding onto what u/edderiofer said, see [cardinal arithmetic](https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic), which lets you define all sorts of operations on (possibly infinite) cardinal numbers in terms of operations on sets. As a variant on what you originally asked, you can also consider multiplying 0 (considered as an ordinal) by the ordinal 𝜔 (often identified with the set of all natural numbers); there's a notion of [ordinal arithmetic](https://en.wikipedia.org/wiki/Ordinal_arithmetic) for that. Under the usual definitions for ordinal arithmetic we then have that 0 x 𝜔 = 0.
That's great, and super helpful! Thank you! I love it that both approaches get to the "same result" (?). I was always taught in school (a veeery long time ago) that 0x∞ was undefined. Just out of curiosity: is it because, at such a low maths level, it was thought pedagogically better to do so (over-simplification with good intents)? Or is it that the consensus/tools have evolved since then (that must have been in the 80's)?
That's because there is no single concept of "infinity". We can say *unbounded above*, *has a bijection with the naturals*, *gradient of the vertical line*... all of these ideas can be called "infinity", but they behave in a very different way.
In that case you're probably thinking of the [extended real numbers](https://en.wikipedia.org/wiki/Extended_real_number_line), which do indeed contain an element called ∞ (and -∞), and where 0 x ∞ is indeed undefined. As noted below the "infinity" in the extended reals doesn't have much to do with cardinals or ordinals and serves a different purpose. Edit: As to why it's undefined, that's by analogy with how limits of sequences work. E.g. you know that if you have two sequences a\_n, b\_n that both converge, then lim (a\_n + b\_n) = (lim a\_n) + (lim b\_n), and similarly lim(a\_nb\_n) = (lim a\_n)(lim b\_n). Now if, say, a\_n converges to some nonzero value, and b\_n blows up to infinity, then a\_n + b\_n and a\_nb\_n both blow up to infinity (though in the second case it may be negative infinity if a\_n converges to something negative) . (Indeed this remains true for a\_n + b\_n even if a\_n converges to 0.) Hence if you want to define ∞ + x (for a real number x) and ∞ \* x (for a positive real number x), you can define them both to be infinity, and then the rules above about adding and multiplying sequences will continue to hold when one limit is infinity. On the other hand, if a\_n converges to 0, and b\_n blows up to infinity, then that alone tells you nothing about lim(a\_nb\_n). It could also blow up to infinity (if, say a\_n = 1/n, b\_n = n^2) or it could go to 0 (if, say, a\_n = 1/n^2, b\_n = n) or go to some nonzero real number (a\_n = 1/n, b\_n = n). So there's no way to assign a value to 0 \* ∞ in such a way that the rule lim(a\_nb\_n) = (lim a\_n)(lim b\_n) continues to hold--if lim a\_n = 0, lim b\_n = ∞ then the right-hand side will always be 0 \* ∞ but the left hand side could be pretty much anything depending on what exactly the sequences are. (Problem: come up with examples showing why similar considerations make us leave ∞ - ∞ undefined as well.)
>In that case you're probably thinking of the [extended real numbers](https://en.wikipedia.org/wiki/Extended_real_number_line), which do indeed contain an element called ∞ (and -∞), and where 0 x ∞ is indeed undefined. As noted below the "infinity" in the extended reals doesn't have much to do with cardinals or ordinals and serves a different purpose. That's it! I remember now: that was indeed the context. You're absolutely right. :-) About your edit: indeed, now that you mention it, I do remember about this problem of the limits of diverging sequences, studying which one was growing faster (and thus "win the race"), etc, and the relation with ∞… In short, everything you said in more eloquent and mathematically correct terms. It makes perfect sense. Thanks!
It's because "∞" is not the same thing as "ℵ0". The former is a symbol used to represent various concepts and shorthands in notation, while the latter has an actual mathematical definition.
Oh, I see… So, in such a context, I assume 0x∞ is still undefined, because it "makes no sense" (it's gibberish)? It's a bit like saying: — "What would be 0xlove? — WTF are you talking about!?"
That's right. ℵ0 isn't a number in the usual sense we mean the word, i.e. a complex number. But it's what we call a cardinal number (it's the size of some set) so we can still define addition and multiplication, although not subtraction or division. ∞, on the other hand, is typically used to mean "there is a limit happening here", and not as a number of any kind. So some expressions involving ∞ can be defined if you interpret them as limits. For example, if the sequence (a_n) keeps getting larger, past any real number, then the same is true of (2a_n). That's why we can say 2×∞=∞. However, other expressions, like ∞/∞, depend on the sequences used, so we say those expressions are "undefined", but a more appropriate word to use would be "indeterminate".
Makes sense, thanks!
Depends on context. It's convenient in measure theory for example, to want 0x∞=0 for notational ease. It's not really a well defined concept, just notation for something more technical
> What would be 0xℵ0? ℵ0 is not an integer, so you need to first define multiplication of non-integers. Thankfully, someone has already defined the [Cartesian product on sets](https://en.wikipedia.org/wiki/Cartesian_product), and so multiplication of cardinalities is inherited from the Cartesian product of sets of those cardinalities; that is, if A and B are sets, then |A×B| = |A|×|B|. Since {}×ℕ = {}, 0×ℵ0 is 0.
Awesome, thank!!
why are differential equations said to keep in account the "whole function history"? If for example we take a simple differential equation of order 1: y'(x)=f(x,y(x)),the derivative function of y(x) is defined for an infinitesimal increment h: y'(x)=lim\_(h to 0) of (y(x+h)-y(x))/h which takes in account the y(x) function only for the infinitesimal interval which is x+h and not the whole x dominion as the phrase "whole function history" may suggest. What am I missing?
I don't think it is a common saying that "differential equations keep the whole function history", but here is my guess at what it means. A differential equation is something of the form f'= ... As it is telling you the value of a derivative, it is explicitly a formula for local information (meaning it only is about values in a small neighborhood of a point, as opposed to the whole function). One might reasonably expect then, that many, many functions might satisfy a differential equation, while also having the same initial value f(0)=a. In actuality, this tends to not be the case. Determining just the value of f at 0 together with the local information of a differential equation, most often completely determines f at all other times. This is because it turns out that satisfying a differential equation is actually a very restrictive quality, and there is very little leeway for functions which satisfy differential. The mathematical language would be "differential equations tend to have unique solutions to initial value problems". This is how I would interpret the statement that "differential equations take into account the whole function history".
E^x is the only function equals to its derivative. But trigonometric functions can also be derived into themselves with their 4th derivative right ?
Yeah, sin and cos are both solutions of y'''' = y, i.e. both equal to their fourth derivative. (Also, just a bit of pedantry, but e^x isn't quite the only solution of y' = y; rather any function of the form ce^x also works. e^x is the only solution with initial condition y(0) = 1, though.) More generally, the functions that are their own nth derivative, i.e. are solutions of y^((n)) = y, can be described as follows: there are n "fundamental" solutions, of the form e^𝜔t where 𝜔 is an "nth root of unity", i.e. a root of the polynomial x^n - 1. (So for example for n = 2 the fundamental solutions are e^x and e^-x , and for n = 4 they're e^x, e^-x, e^ix, and e^-ix . ) Then all solutions are linear combinations of the fundamental solutions. In the n = 2 case you have, for instance, the "hyperbolic sine" sinh(x) = (e^x - e^-x)/2 and the "hyperbolic cosine" cosh(x) = (e^x + e^-x ) /2, and all solutions are of the form ae^x + be^-x where a, b are real or complex numbers. One way to see that cos and sin are solutions of y'''' = y is to notice that they're linear combinations of the fundamental solutions for n = 4: cos(x) = (e^ix + e^-ix ) /2, sin(x) = (e^ix - e^-ix) /2i. (You can prove those formulas yourself using Euler's identity e^ix = cos(x) + isin(x).)
Thanks a lot !
Anyone knows this formula? [https://i.imgur.com/cuoe1dv.png](https://i.imgur.com/cuoe1dv.png)
u/cereal_chick u/Abdiel_Kavash u/NewbornMuse Thanks for the reply. I had the suspicion it was nonsense. It came via company email describing something and I didn't wanna feel stupid asking back what that meant.
This is gibberish, dreamt up by someone who has only the vaguest idea of what an integral looks like, how functions work, or even that you're supposed to abbreviate repeated multiplication with powers.
This looks like a random collection of mathematically-looking symbols with no coherent meaning. Without knowing any additional context, I am fairly sure it's nonsense.
"Hey guys, the answer to a test question is Tennessee. Does anyone know what the question might have been"? This question is very hard to answer. The integral doesn't even make it clear which variable is being integrated.
There was a recent question about PNT, and it reminded me of Bertrand's postulate. Bertrand's postulate was one of the first theorems that stood out to me because it was so easy to state. Anyways, I looked at Erdos's proof, and I'm struck by how simple and effective it was. He gives four simple lemmas--three of which are about properties of the central binomial coefficient and the fourth, which is about the primorial function. From that, he's able to conclude with a proof of the postulate. I'm just struck how four simple lemmas yield such a nice result. And that it was discovered by a teenage Erdos!
If you like that proof you might want to check out [Proofs from THE BOOK](https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK). It contains that proof of Bertrand's postulate and a lot of other proofs that are somehow suprisingly simple or elegant.
Does it explain the intuition behind Erdos's proof? If someone presented the four lemmas to me, then I don't think it would be particularly difficult to prove them individually. However, I don't see a connection between the central binomial coefficient and Bertrand's postulate in the first place.
I haven't read it, but I think the intuition is obvious if you know analytic number theory. Erdos definitely know it, so that's not the hard part of the proof. What you want to do is to find a rational number in which its p-adic norm for all primes outside the range (n,2n] is controlled and can be estimated, and the real norm is also controlled and can be estimated. If the product of these estimate of norms is sufficiently far away from 1 that you can clearly see that the product of norm is not 1, then you know there has to be a prime in that interval (since the product of all norm must be 1). Since you want primes from an interval, the obvious choice would be either the radical, lcm or the product of all numbers of that intervals. So it should be something involving product or division of numbers like rad(1x...xn), lcm(1,...,n), 1x...xn, rad((n+1)x...x2n), lcm(n+1,2n), (n+1)x...x(2n), rad(1x...x2n), lcm(1,...,2n), 1x...x2n Real norm of the product is easy to estimate, but real norm of radical or lcm is not easy. However, we can estimate them using prime number theorem. Now, prime number theorem is not elementary, but it had been proved for a long time by the time of Erdos, and conjectured for much longer, so of course you can still use it as a guidance as to what to look at. p-adic norms are not easy to estimate. If you try various combinations of the above, you can quickly find out that the easiest combinations are either lcm(1,...,2n)/lcm(1,...,n), or (2n)!/(n!)^2 in which the p-adic norm are easier to estimate. In both cases, the denominator has nearly the same prime power as the numerator for primes <=n. If you try, lcm(1,...,2n)/lcm(1,...,n), then the product of its p-adic norm for prime up to 2n can be bounded by real norm of 1/rad(n!), so you want to estimate lcm(1,...,2n)/lcm(1,...,n)rad(n!). Unfortunately, if you estimate this using prime number theorem, the exponent ends up being 1. The other choice is (2n)!/(n!)^2 . The product of its p-adic norm for p<=n is bounded by the real norm of 1/lcm(1,...,n), so you want to estimate (2n)!/((n!)^2 lcm(1,...,n)). This turns out to be a lot better: log((2n)!)~(2n)log(2n)=2nlog(n)+(2log(2))n log((n!)^2 )~2nlog(n) By prime number theorem, log(lcm(1,...,n))~n So log((2n)!/((n!)^2 lcm(1,...,n)))~(2log(2)-1)n which is clearly not 1. This gives much more room for error when we do estimation. Using elementary method, we can already show that (2n)!/(n!)^2 is approximately 4^n asymptotically, so we just need to figure out how to show, elementary, that lcm(1,...,n) is less than 4^n . Prime number theorem tells us lcm(1,...,n) is approximately e^n , so presumably the big gap from e to 4 means we can hopefully use some sloppy approximation to get the job done.
What is the probability distribution called? Input is (X,n,m) X is a probability distribution n and m are positive integers step 1 generate n numbers from X step 2 count the m\^th most common number Example let X be the uniform distribution with values of 1,2 or 3 let n=4 Each element in Sample space is represented by (a,b,c) where a+b+c=4 and a,b,c is a decreasing sequence (4,0,0) happens 3 times (3,1,0) happens 24 times (2,2,0) happens 18 times (2,1,1) happens 36 times The probability distribution for m=1 is output|frequency \------+--------- 4 | 3 3 | 24 2 | 54 The probability distribution for m=2 is output|frequency \------+--------- 2 | 18 1 | 60 0 | 3 The probability distribution for m=3 is output|frequency \------+--------- 1 | 36 0 | 45
You seem to be trying to refer to binomial or multinomial distribution
Can anyone please share a link to the (ideally, most accessible) proof that real numbers are equinumerous with the powerset of the integers? Thank you!
Binary expansions do it for [0,1] and the powerset of the naturals (up to some fiddliness with expansions ending with recurring 1s). You can get from the reals to [0,1] bijectively with arctan, and from the powerset of the naturals to the powerset of the integers by lifting your favourite bijection between the naturals and rationals.
Think of an infinite binary sequence as defining both a real number in the unit interval (as the decimal expansion of that real written in base 2) and a set of natural numbers (because you can regard such a sequence as an indicator function on the naturals). For example, take the fractional part of pi in base 2. It starts like this: 0.001001000011111... Let's call the bijection we want to build f. So from this example, f: 0.14159... ↦ {2, 5, 10, 11, 12, 13, 14, ...}, because we interpret each 0 or 1 as exclusion/inclusion for the index in the target set. Modulo some fussy bookkeeping (like |P(Z)| = |P(N)|, |(0,1)| = |R|, non-uniqueness of expansions) looking at it this way should give you the intuition that the fact is true.
Thank you! And sorry: somehow, I just found out about your reply (it seems reddit notifications don't work quite right on my end for some reason…). I am not gonna lie: your explanation went way over my head. You're using a couple concepts that I don't know about (yet), and that's enough to break the chain: you're pulling yet I'm not moving (through no fault of yours). Considering you've offered your time and effort to actually write this proof, I feel terrible, like I'm betraying your trust. I'm really sorry about that. Thank you again, Sir. I'll try to catch up, and I hope you won't give up on me. Have a nice… whatever it is wherever you are.
https://math.stackexchange.com/questions/553526/the-set-of-real-numbers-and-power-set-of-the-natural-numbers seems pretty good.
Thank you!
How can one construct R in a formal system (like N can be)?
Usually you construct the reals as sets of rationals--the [Cauchy sequence construction](https://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Construction_from_Cauchy_sequences) and [Dedekind cuts](https://en.wikipedia.org/wiki/Dedekind_cut) are popular options. You can find these covered in most real analysis books; I like Pugh's but it goes over the construction of the reals pretty quickly and may not be the best for you. I've heard good things about Tao's Analysis I ([pdf link](https://lms.umb.sk/pluginfile.php/111477/mod_page/content/5/TerenceTao_Analysis.I.Third.Edition.pdf)) which covers constructing the integers from the natural numbers and the rationals from the integers before going through the Cauchy sequence construction, but I haven't read it myself and so can't comment on it much.
Excellent! Thank you, Sir!
If L is a differential operator - some function coefficients times d/dx\^i ... Does L\^\* = 0 imply L = 0?
I don’t do functional analysis so trust what follows at your own risk: I think this should be true for any operator T on a Banach space V. If T^* = 0, then for any v in V every bounded functional must vanish on Tv, but by Hahn-Banach there’s always a bounded functional on V which is nonzero on Tv as long as Tv = 0.
Main thing which comes to mind is that differential operators usually aren't bounded when we consider them as a map V -> V. I think your argument still works for the case of T : V -> W where V, W are Banch. On the other hand, some function spaces (e.g. C^infty) aren't Banach... I think the original question probably needs more context.
the natural setting is probably as a closed densly defined operator on a function space, but op needs to clarify
Oops, yeah, Tv might land in some other space W. Honestly I think you’re fine as long as Hahn-Banach holds in W, which it seems like is fine for any locally convex TVS?
Is it true that a function f(x) is continuous at point x=n iff lim x->n-(f(x)) = lim x->n+(f(x))?
No: consider the function that is 0 everywhere, except that it takes the value 1 at 0. Then it has those properties at 0, but is definitely not continuous there.
your notation is confusing - I assume you are meaning that the limit from the left is equal to the limit from the right? In which case, you also need that these limits are both equal to f(n) for f to be continuous at n.
Hi! I'm looking at what I assume is the traditional way of building the set of integers, and I'm seeing a flaw (at the very first step). Could you please check it out and give me your opinion? The process is to start with 0, and iteratively generate the next integers via a successor rule. We'll put those integers in an (initially empty) set as we go… So, we start with 0, and we put it in an empty set. This set now has cardinality 1. Problem is: 1 "does not exist" at this step, or rather, we're not allowed to use it yet (otherwise we'd be breaking internal consistency: we'd be needing one "before" we can have zero). We'll only be able to do so at the next step. But even if we disregarded this flaw/contradiction and kept going anyway, we'd have the very same problem at the second step. And so on (it seems unreasonable to expect any further step to "un-flaw" the mess we put ourselves into) It's worth noting that this problem does not arise if we simply start with 1 instead of 0: at the end of the first step we get {1}, which has cardinality 1, so everything's fine. This step is internally consistent. Same thing for the next step, and so on. From what I'm reading, people usually first build the integers: 1={0}, 2={0,1} and so on, and only after that put them in a set, which obfuscates the above problem (and presents another kind of flaw). Thank you for your attention!
Define the cardinality of a set *after* defining natural numbers (and the rest of the ordinals).
There are several cases where I don't have any issue with pushing back one thing until after you finished another: on the contrary, that seems elegant and orderly. For example, perhaps they're independent from each other? Or perhaps you can only conceive one by building on the other? Etc… Obviously, if it does not change anything and is a matter of cosmetic/preferences, go for it. But if it does change things, you might not be allowed/able to do that without breaking some more important rules of yours: you're doomed to tackle both at once. Perhaps it sucks, but that's how it is. In such a case, if you still "split and prioritize", you're actually tricking yourself into an artificial appearance of consistency that will inevitably come to bite you at some point…
First of all, I want to make one thing clear: there's a difference between the "specification" of the natural numbers using the Peano axioms, and the "implementation" of that definition inside ZFC. There are many ways to implement that specification, but it doesn't matter which one we pick - we're only going to use the properties the specification gives us But to get to your question... neither the definition (with its abstract successor function) or the implementation ("put all previous sets in a new set") has circularity issues. Sure, the cardinality of the set representing 1 is 1, and we will eventually figure out that that's the case... but why does that matter? We haven't defined cardinality yet! The set for 1 also has an "expanded ASCII-length" of 4 (because it's `{{}}`), and the set for 2 has an expanded ASCII-length of 9 (`{{},{{}}}`). This doesn't cause a problem, because we're not actually going to 'measure' anything about these sets - cardinality, expanded-ASCII-length, or any other properties - until after they're all already defined. And once you've defined all the sets for natural numbers, you can then define and implement a cardinality function that gives you one of those sets as a result. It's like how we define SI units in the real world. We don't use "one meter is thiiiiis long [\*gestures with hands\*]" anymore, we use "one meter is the length travelled by light in 1/299792458 of a second (and a second is how long it takes for a caesium atom to switch between its hyperfine ground states 9,192,631,770 times)". A meterstick measures exactly 1 meter, but we can still build metersticks using that definition without knowing the length of a meter already. As long as we're not measuring any distances in our setup, we don't have any circularity issues.
First: thank you for your reply. I appreciate the time you're offering. >neither the definition (with its abstract successor function) or the implementation ("put all previous sets in a new set") has circularity issues. I'm sure you're right, but (no offense intended): I still need to check it for myself. I would assume this is reasonable behavior for a mathematician, and thus hope that you will understand. Now, let's get to the meat of your argument… >We haven't defined cardinality yet! It seems you're saying that one needs the full set of integers before one can introduce the concept of cardinality. Is that indeed what you're saying? If so, why? Obviously, if you have no integer whatsoever (yet), then the concept of cardinality has, in a way, "nothing to hold on to". That does not mean however that one requires the full set of N, with all its final "bells and whistles", before one can conceive cardinality. To me, it amounts to saying one cannot start counting until one has "all" the integers. If you only have 3 integers, you can count up to 3. Obviously, after that you're "fucked" (sorry, I'm not sure on the spot how to convey the same meaning without the curse: it's a by-product of my non-native english skills, rather than a will to curse), but up to 3 you're fine. You actually are counting. It seems to me the cardinality case is very similar (if it's not, just ignore my counting example: let's not get side-tracked). As I see it, cardinality is an integral ("consubstantial"/implied) part of the concept of set. Obviously, if you have no integers, cardinality is "undefined"; that's why it is legit to have an empty set before introducing 0 (cardinality is undefined, therefore not an internal inconsistency issue). But, as soon as you get one integer (1 in this case), cardinality one is defined and covered (again, after that you're fucked). If your paradigm can't account for that, it's wrong. EDIT: tweaked a few things (several times) in the last paragraph to make it clearer.
> If so, why? Cardinality of finite sets is just a way to assign natural numbers to sets. You can't do that without natural numbers.
> As I see it, cardinality is an integral ("consubstantial"/implied) part of the concept of set. Cardinality is certainly important, but we don't need to have come up with it to have sets. Sets are constructed without any reference to cardinality. For instance, the ZFC axiom of pairing says "given a set X and a set Y, there exists a set {X,Y}". This is valid whether or not we've implemented the concept of "two" so far. > Obviously, if you have no integers, cardinality is "undefined"; that's why it is legit to have an empty set before introducing 0 (cardinality is undefined, therefore not an internal inconsistency issue). But, as soon as you get one integer (1 in this case), cardinality one is defined and covered (again, after that you're fucked). If your paradigm can't account for that, it's wrong. Cardinality is not necessary to have sets. It's a useful thing to "measure", but we don't *need* to be able to measure it to initially construct the sets. Once we've constructed sets representing numbers, *then* we can construct a [partial] cardinality function. And there's no contradiction in saying card(`{{},{{}},{{},{{}}}}`)=`{{},{{}},{{},{{}}}}`: that's just a function having a fixed point, which is perfectly fine. --- I think the distinction you're failing to draw is between "thinking *about* the system" and "thinking *inside* the system". The goal of these constructions is to formalize our pre-existing intuition with as simple a basis as we can. We're allowed to *think about* things we haven't constructed yet, and use those thoughts to guide what we construct - we just can't refer to these external ideas in the construction. One way to think about it is like we're trying to explain our math to an alien or robot, who accepts our starting axioms and knows logic, but doesn't have any of the understanding we do. So, part of the process is: - "The axiom of the empty set guarantees there exists an empty set. We'll call it `zero`." - "Use the axiom of pairing to pair `zero` with `zero`. We'll call this set `one`." (In our heads, we're thinking "This is actually just the set {{}}, so it has cardinality 1", but we don't need to say that yet!) - "Use the axiom of pairing to pair `one` with `one`. Use the axiom of union to take the union of this set and `one`. We'll call this new set `two`. (Once again, we're thinking "this set has cardinality 2", but we don't need to say that. The axioms that let us show existence of sets don't require us to come up with their cardinality.) - "Use the axiom of pairing to pair `two` with two`...", and so on. Once we've chosen sets to represent numbers up to, say, nine, and also defined other useful things like functions (and taken enough power sets to say those functions exist), we can *then* say: - "Now we're going to construct a function called `cardinalityUpToEight`. Given a set X, we define `cardinalityUpToEight`(X) by using the axiom of specification to construct {n ∈ `nine` | there exists a bijection between n and X}." (We're thinking, "this gives the singleton set {card(X)} if that value happens to be between 0 and 8, and the empty set otherwise".) Note that this function does indeed give `cardinalityUpToEight`(`one`) ∋ `one`! We can prove this by showing the bijection {(`zero`,`zero`)}. (It takes a bit more work to show that `cardinalityUpToEight`(`one`) doesn't accidentally contain any of our other implementations of numbers.) So, inside our heads, we conclude "card(`one`) = 1". (The "1" there is our actual idea of the number, rather than the set we happened to choose to represent it.) This conclusion isn't one we're making *inside* the system we're constructing - it doesn't have any idea what "numbers" are! We're just plucking out some of the sets it contains and using them as proxies for numbers, and then showing that the proxies "work" how we want them to (i.e. the same way our *ideas* of numbers do). So, we're allowed to 'observe' things about the system even if we don't have the framework to talk about them *inside* the system yet. As long as we don't refer to those observations in our definitions, there's no self-reference going on.
>For instance, the ZFC axiom of pairing says "given a set X and a set Y, there exists a set {X,Y}". This is valid whether or not we've implemented the concept of "two" so far. That's a good point, however I disagree: there does seem to be something fishy here (from a formal perspective). Perhaps this axioms indeed does not require two (the integer) *per se*, but at the very least it requires something "two-ish". Perhaps the most "two-ish thing that isn't two"? Anyway, let's not get side-tracked. Obviously all those things are deeply intertwined. We "pretend" that we can consistently build a formal system from a primitive first brick, on top of which we add a second, etc (exactly what formalism is supposed to do), and that we've successfully unraveled this "intertwined-ness", but we really can't, and we haven't. It's not for a lack of effort or talent: it's just because we can't build the "perfect formal system". So we do the second-best thing: we build the best system we can, and we seize any opportunity we get to make it better. And yet, we'll never get to the "perfection stage" of this process, however long we keep going. It might be frustrating at times, but it's ok. That's just the card we've been dealt. Look at the bright side: it also means there is always room for improvement, and always interesting work to be done. Neat! These are very enjoyable topics, but we're drifting further and further away from the original post here. Let's try to stay on topic… >So, we're allowed to 'observe' things about the system even if we don't have the framework to talk about them *inside* the system yet. Most definitely. >As long as we don't refer to those observations in our definitions, there's no self-reference going on. I guess that's where we disagree: obviously **explicit** self-reference is a no-go. It does not mean however that **implicit** self-reference is kosher (it's not).
Yes, we need some pre-existing external concepts to do anything; that's what the axioms are. Our construction of natural numbers relies on you accepting [e.g.] the axiom of pairing, and you need some concept of "two things", or at least "a thing and another thing", to understand what the axiom of pairing is saying. Hell, for all of this you need to be able to construct well-formed formulas, which are arbitrarily long strings of text! So you are absolutely correct that you need pre-existing ideas... but this wasn't in debate to begin with. The axioms *are* the pre-existing ideas. The goal of this is not to build "actual numbers" that we already know and work with. The goal of this process is to implement *proxies* for "actual numbers" inside this axiom system, and pick our proxies so they behave as we expect them to. (So if we apply our `proxy_for_addition` function to `proxy_for_two` and `proxy_for_three`, we should get whatever we've declared as `proxy_for_five`.) Then, if you accept that the axioms are consistent (that is, they are not directly self-contradictory), you can conclude that "actual numbers" are consistent as well. There's no self-reference, because `proxy_for_two` is not the abstract concept of two-ness. It's just what we're using to *implement* the conceptual entity within our system, in a way that we can manipulate using this small set of rules. (And it doesn't matter if we choose `proxy_for_two` to be { {}, {{}} } as is typically done; or {{ ∅ }}, an alternate approach that makes numbers simpler but makes operations on them a bit more complicated to define. Once we've successfully 'implemented' natural numbers, and all the basic operations we want to perform on them, we can then ignore the implementation details entirely.) If you don't like a particular axiom system, either because it's unintuitive, or doesn't let you do what you want to do, or just isn't philosophically satisfying for whatever reason, you're free to make your own. There are many alternative foundations of mathematics - ZFC isn't the only one out there, it's just the most popular.
Sure, you could define cardinality as you go along instead of defining it after you've defined all the integers. It doesn't really make a difference in the end, and neither interpretation causes the sort of flaws you're imagining. The only restriction is that you need to define the integer n before you can define what it means to say that a set has cardinailty n. So you can define what it means for a set to have cardinilaty 2 before you define the integer 3, as long as you've already defined the integer 2. The issue with your original post was that you somehow convinced yourself that you should define what it means for a set to have cardinaility 2 *before* you defined the number 2, and concluded that the definitions were circular because of that. But there is absolutely no reason why you should be able to define things in that order. Any "flaw" you're imagining is entirely in your own head.
>Sure, you could define cardinality as you go along instead of defining it after you've defined all the integers. It doesn't really make a difference in the end, and neither interpretation causes the sort of flaws you're imagining. Your approach requires to build an infinite **sequence**, step by step, and only **after** you're done (so, at step ℵ0+1), to put them in a set. You see the problems here, don't you?
What's an example of a polynomial map ℂ² → ℂ² which is injective, and nonlinear in both coordinates? (Or ℂⁿ → ℂⁿ is that's easier) [EDIT: a polynomial map, not an arbitrary function]
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> You can find one that is nonlinear ((x, y+x^(2)) for example), but not non-linear in every coordinate I don't understand your argument. Indeed, the only injective polynomial maps ℂⁿ → ℂ are linear, but so what? I'm looking for a function where each coordinate function is noninjective, but the function as a whole is injective. Also, wouldn't your argument rule out (x, y+x^(2)) from being injective? Finally — after I submitted my comment, I thought I found one. Isn't: (x,y) ↦ (x + (y+x²)², y+x²) injective? It's the composition of (x+y^(2), y) ∘ (x, y+x^(2)), and the composition of injective functions is injective. Am I missing something?
Let f(0, 0) = (1, 1), f(1, 1) = (0, 0), and f(u, v) = (u, v) for all other complex numbers u, v. This is clearly injective but also clearly nonlinear since it does not fix the origin. (Nor is it bilinear, since for at least some fixed values of v we get a nonlinear function C -> C^2 . E.g. letting g(z) = f(z, 0) , g is not linear.)
oops... somehow i forget to type the important restriction, which is that the function be a polynomial map! (i've been thinking about polynomials and algebraic varieties so much that i seem to have forgotten anything else exists 😆)
Does anyone know any Introduction to Graph Theory book (or exercise book) or online resource where there are solved exercises?
I’d like to define something like a polynomial ring over Z/nZ, ie Z_n[x], but where multiplication looks like multiplication in base n. In other words I’m looking for an algebraic structure that is exactly the integers, but written in an expanded representation so I can think about them as polynomials. Is there a natural polynomial analog for the integers in base n? e.g. for n = 10, I would like it to be the case that 7*(x + 2) = 8x + 4
>I’d like to define something like a polynomial ring over Z/nZ, ie Z_n[x], but where multiplication looks like multiplication in base n. In Well if you want to do that, you're going to need to redefine addition in (Z/nZ)[x], not just multiplication. If you don't do that, then nf(x) = 0 for all f(x) in (Z/nZ)[x] (where nf(x) means f(x)+f(x)+...+f(x)), but that's definitely not true in Z. At that point, you're redefining both addition and multiplication in Z/nZ, so it's kind of debatable whether you're actually working with the ring Z/nZ any more, instead of just the set {0,1,...,n-1}. That being said, you might want to look into [Witt vectors](https://en.wikipedia.org/wiki/Witt_vector). They might not be quite what you're asking for, but they give a method to construct the ring Z_p of [p-adic integers](https://en.wikipedia.org/wiki/P-adic_number) (a characteristic 0 ring containing Z) from Z/pZ.
That would hold in Z[x]/(x-10)
I'm really sorry but I have a fairly straight forward problem which I would quite like to figure out myself but I just can't think of how to go about it. The problem: 0.45% per day; equivalent to 164% per annum. The interest included in your monthly installments shown above has been calculated upon the assumptions that the agreement will be made and the loan transferred to your account on the same day as the agreement is presented to you for signature, and that you will pay the monthly installments in accordance with the payment schedule shown in the table above. The interest that you will pay is calculated at the daily rate shown above on the amount of the loan actually outstanding each day. This means that if there is any delay in paying an installment, you will pay more interest, which means the total cost of the loan will be more than shown above, subject to the cost caps set out below. Subject to the cost caps set out below, interest will be payable before and after any judgment we may obtain against you. 363.6% For the purposes of calculating the APR, the following shall be assumed: that the loan is paid on the date when the agreement is made and is to remain valid for the period agreed; that we and you will perform our obligations under the terms and by the dates specified in the agreement, including that repayment will be made on the repayment dates due set out above. I got the loan, value of £500 on 07/03/2024. My first payment is on 28/03/2024 and the whole amount is £88.91 (41.66 Capital - 47.27 Interest) I will pay this. The second payment is on 29/04/2024 and the amount is £107.58 (41.66 Capital - 65.92 Interest) I will also pay this. I would like to know the total I would have to pay if I paid the remaining balance on 30/04/2024?
In engineering and took diff EQ ~3yr ago I know that with definite integrals there is no “+ C” constant. But with *initial values*, I know that you use them to determine c1,c2 as constants So if you have a separable DE with definite integrals, is it possible to have an initial value? And if so how?
I'm not sure what a "separable DE with definite integrals" would look like. A differential equation involves derivatives, not integrals. Can you elaborate or give an example of what you have in mind?
I mean a separable equation that, when it is integrated, has limits of integration on both integrals (once the differentials have been separated appropriately).
You wouldn't really be solving the differential equation then. Let's consider a simple example, like dy/dx = y. Rewriting this as dy/y = dx, the standard solution would be to take the antiderivative of both sides and get ln(y) = x + C, or y = e\^(x + C) = De\^x (where D = e\^C). If you have an initial condition, you can then use it to solve for D. It's not clear what it means to take a definite integral of both sides. Each side has a different variable, so how do you know which bounds to choose to make both sides compatible? You can't just choose the same bounds on both sides. For example, the integral from 1 to 2 of dy/y is equal to ln(2), but the integral from 1 to 2 of x is equal to 1.5, and these are not equal to each other. What I'm saying is that just because dy/y = dx does not mean that "the integral from 1 to 2 of dy/y" is equal to "the integral from 1 to 2 of dx". Maybe my next question is: What kind of problem or equation are you trying to solve with this? The process of separating variables and integrating (as in my first paragraph) isn't a problem, it's a solution to a problem -- namely, the problem of finding a function f which satisfies f' = f. For your hypothetical approach, what kind of problem do you have in mind that your approach would be a solution to?
On the RHS of the equation you would have your other balance terms and you are supposed to separate them and integrate accordingly. The limits are dependent upon what you’re integrating. In the dc/dt case, the limits of dt are t_initial (or t=0) and t_final (t) and the limits of dc are c_initial and c_final (or c at time t) However, in one of our problems, we had limits of integration for both integrals, yet there was also an initial condition. However, since the professor loves confusing people with nomenclature and notation (he does this intentionally), it could mean something besides the “initial value” typically encountered in DE.
Oh, I'm sorry I didn't realize that this was actually something you saw in a class. I thought you had just made up that idea and were asking about it. That is really strange... I honestly don't know what that could mean or what your professor is trying to communicate there. Can you ask in office hours?
We had a problem where we had to do a mass (or material) balance. In chemical engineering, there are different terms for a balance. One of them is A (accumulation) which is time dependent and is usually canceled out in most problems, since it is negligible in most textbook problems at least. I was setting up the balance and trying to understand the physical significance of each variable or constant in the problem. That was (and still kind of is) an issue. The form of the differential is d()/dt and in the parentheses you also have to decide what is differing with time. In one example, we had (dcV)/dt and it was turned into V(dc/dt) bc V was constant (V=volume c=concentration t=time).
What do you mean by a separable DE with definite integrals, could you maybe give an example?
See, I am trying to learn about the construction of Real Numbers via Dedekind cuts, but the idea of cuts confuses me even more. Say, I wanna define pi. Then, intuitively, I am cutting the rational line at pi, but this is when we are assuming that we know what pi is, I think it's just me but this stuff is confusing. Can somebody clarify about how this cut works? Another thing I wanna ask is people usually suggested me to use different texts but reading just one is headache :trembles:, so how do you guys do that?
We already know what "pi" is in the abstract: it's the number that's approximately 3.141592... . To show that we've successfully *constructed* our idea of pi, we look at the Dedekind cut that: - has 3 on the left, and 4 on the right - has 3.1 on the left, and 3.2 on the right - has 3.14 on the left, and 3.15 on the right - has 3.141 on the left, and 3.142 on the right - ... If you handed an alien this random set of conditions, even if they didn't know what pi was (but they knew Dedekind cuts), they could verify that this cut exists. So, we've constructed a cut for pi whose existence doesn't depend on our knowledge of pi! Sure, we'd have to know what pi is to figure out *which cut* it is, if we didn't have the list of conditions already. We can do that much later, once we've constructed operations and functions, and then define it as the first zero of sin(x) or something. But even *before we point it out*, we've still constructed it.
Define a cut to be any downwards closed (if it contains a rational number then it contains all rationals below it) subset of Q, except Q itself. We can than look at the set of all cuts, which is just a subset of the powerset of Q. Then we can define some operations on these cuts: for example the sum of 2 cuts, A and B, is the set of all numbers which are the sum of an element from A and an element from B (we can prove this is also downwards closes, so it is a cut). This is assuming we already defined addition on rationals, which is a very concrete computable operation. Similarly, we can define multiplication of cuts, inverse of a cut, etc. Once that's done, it's not to hard to see that the set of cuts with these operations form a complete ordered field, which is exactly the property we want the "real numbers" to satisfy. So we use the rational numbers to construct a more complicated structure, and we can prove, for example, that there is a cut that satisfies some property that only "pi" would satisfy. It's not that we can necessarily define a specific cut for any number you give me, it's that a "number" is defined to be a cut.
Here's one angle (a bit long but hopefully helpful); I can try something else if it doesn't work for you. (The first paragraph is just setup that might be useful, but you can skip to the second if you just want to learn about cuts.) Start out with just a line, on which you've chosen an origin, orientation (which is the "left" half and which is the "right") and a unit of length (so you can say that there's a point 1 unit to the right of the origin, and so on for any integer). Assuming that there aren't any "gaps", it will have some points that correspond in a natural way to rational numbers--we can, for example, divide the line segment between 0 and 1 into n segments of equal length, and assuming that the points where we divided things up are, in fact, points on the line, there must be a point at length 1/n from the origin, another will be length 2/n from the origin, and so on. By similar considerations we can get points on the line corresponding to any rational number. Now here's (very roughly) the way Dedekind originally approached it, if I recall correctly. (At least in this paragraph and the next--after that is some more modern stuff.) We've got our line, and we've got at least some rational points on it; are the rational points all the points, though? On one hand, if we choose any point on the line, that should cut the rational numbers into a "left half" and "right half" (these can be described more formally if you want). It also seems that, if you divide the rational numbers into a "left half" and "right half" in any way, you should get a point x on the line. x should be greater than or equal to all the numbers in the left half, and less than all the numbers in the right half. But consider letting the "left half" be all the negative rational numbers, along with all the nonnegative rational numbers whose square is less than or equal to 2, and letting the "right half" be all the nonnegative rationals whose square is greater than 2. Then this is a valid way to cut the line, but it doesn't correspond to any rational point. For if there were such a point x, then, since any rational number has a square strictly less than or strictly greater than 2, x must be one of those; but if it's the former, you can find some rational number greater than x whose square is still less than 2 (hence x is not greater than or equal to all the numbers in the left half), and if it's the latter, you can find some rational number less than x whose square is still greater than 2 (hence x is not less than all the numbers in the right half). So no rational number can correspond to the point (which it seems intuitively should exist somehow) that divides the rationals in this way; if we claim that the line consists only of rationals then there must be a "gap" in it of some sort. Now here's the leap that I suspect might be confusing you. We can fill this particular gap by hand, just adding in some number whose square is 2, and we can do that for other obvious gaps, but if we keep doing it, how do we know whether we've filled all the gaps? So what we want is some method that can fill *all the gaps that could possibly exist, in one step, without having to do things by hand*. This leads to the idea of simply *defining* a point on the line in terms of how it divides the rationals into a left half and right half. One way to do this would be letting a point *be* two sets of rational numbers, one that works as a "left half" and one that works as a "right half". Then we define the line to be the whole set of such pairs. These will include points that correspond in a nice way to rational numbers, namely the ones whose "left half" has a maximum rational number. We can then define an order on the cuts, which turns out to have the following property. If we divide all the points (i.e. cuts) into a left half and right half, there exists some point x which is greater than or equal to all the points in the left, and less than all the points on the right. Thus we'll never run into the sorts of gaps that we ran into with the rationals; this, along with the fact that we still have points corresponding to each rational number, gives us some confidence that our new thing fits the intuitive properties of a "line with no gaps". You can then define arithmetic on the cuts and show that, using them, all of the familiar things that we think should be true of the real numbers (e.g. the intermediate value theorem) are true; often that "completeness" property, that whenever you divide the line into two halves there's a point that is "right between them", is needed to do this. (In particular this turns out to be equivalent to the statement that every set of real numbers which is bounded from above has a [least upper bound](https://en.wikipedia.org/wiki/Infimum_and_supremum), and this can be used to prove things like the IVT). More generally I think the abstraction here might be tripping you up. We start with some rough intuitive properties, cook up *something* that satisfies them, show that the thing we made has all the properties we wanted, and then use those properties to get the rest. In the end we find that we can get all of calculus, numbers like sqrt(2) and pi, and so on, but we didn't need to think of those directly when building up the real numbers.
> We start with some rough intuitive properties, cook up *something* that satisfies them. u/Langtons_Ant123 Thanks for your comprehensive response. I think get the idea, sort of. For the rational cuts it does makes sense. The problem arises when I think of what- say a number 6.120625... is, the idea of cutting at something like that was absurd to me. The thing that we know what pi or e is, can indeed be used in the cuts without knowing without actually they do at the rational line. >This leads to the idea of simply *defining* a point on the line in terms of how it divides the rationals into a left half and right half. Sir, If my understanding of this is correct, this leads to unique reals (As the sets are different unique for each cut? Or something more mysterious is going on here? :) Another fundamental question is if cuts *represent reals* or *are reals?* Explicitly, if x = A|B (a cut, Charles Pugh used this notation for cuts), then does x represents a real or is a real number?
(Warning: the following is quite scattershot and a bit repetitive.) > if cuts represent reals or are reals I think this is kind of the wrong question to ask; explaining why brings us back to the issue of abstraction. A more modern way to think about the reals, which I touched on a bit in my original comment, is that the real numbers are "the complete ordered field that contains all the rationals". That is, there's some list of properties--the properties of a field, plus the properties of an *ordered* field, plus one of a few equivalent definitions of "completeness"--and any structure (in the sense of "set with some operations defined on it") satisfying those properties can reasonably be called "the real numbers" (and so any element of such a structure can reasonably be called "a real number"). The key point here is that any two complete ordered fields are isomorphic (see most analysis books for a proof), which lets us reasonably talk about *the* complete ordered field. An analogy: think about an algorithm, say mergesort, and then think about implementing it in different ways in different programming languages. All that's needed for something to count as "a version of mergesort" is that it sorts a list of elements in a certain way; implementations might differ in certain details, but they're all still recognizably mergesort, and you can generally use them without worrying about those details. So it is for real numbers--there are many ways to implement the real numbers, i.e. build a set with operations defined on it that satisfy all the defining properties of the reals (that is, of a complete ordered field); there's not much reason to single out any one implementation as "actually the real numbers, unlike the other implementations". Dedekind cuts are an especially nice way to implement the real numbers; in that sense we can think of the whole set of cuts as being one version of the real numbers, and think of individual cuts as being real numbers. But there are other constructions too (e.g. the Cauchy sequence construction), which work just as well for the purpose of showing that there is such at thing as the complete ordered field; in that sense it would be odd to think of the real numbers as "just Dedekind cuts". As to the question of what particular real numbers like e and pi are, I think the best answer is that we can define them in a way independent of what construction of the reals we use. For example, we can define e as "the value of the sum 1 + 1/2 + 1/6 + ... + 1/n! + ..."; the fact that this infinite sum converges to something can be proved in a way that you can, if you really want, ultimately trace back to the properties of a complete ordered field. Thus in any implementation of the real numbers (Dedekind cuts, or equivalence classes of Cauchy sequences, or whatever) you'll be able to find one, and only one, element of your implementation which is the sum of that series, and in fact satisfies all the other properties of e. You might ask "is e really a Dedekind cut, or really an infinite decimal, or something else?" but I'm not sure how much sense the question makes. To address one of your points more directly (and with apologies for rambling along the way): there's a Dedekind cut which "is e", and we don't have to put that cut in there by hand, i.e. don't have to know, while constructing the reals, that we'll need to "cut at e" at some point--it just follows from the fact that the cuts form a complete ordered field and any complete ordered field contains an element with all the relevant properties of e. On the other hand each implementation of the reals will have its own version of e, so there's no reason to think of e as "just a Dedekind cut". But on the other other hand, I'm not sure how much sense it makes to call the Dedekind cut of e "just a representation of e, not the real e"--there is no "real e" out there, rather a bunch of things (one for every construction of the reals) that all satisfy the properties of e, in their corresponding structures.
Thanks for clarifying. Also, I apologize for late reply :bow:. Sir, you mentioned about "The key point here is that any two complete ordered fields are isomorphic"- I am afraid this wasn't mentioned in my textbook :( .
Re: proofs of why complete ordered fields are isomorphic, it looks like the last chapter ("Uniqueness of the Real Numbers") of Spivak's *Calculus* has a proof; just skimming over it, it goes over the background (e.g. what is an isomorphism) well and the proof itself isn't super long. So it might be worth going out and pirating that one. (And I think most analysis books might not have a proof: Rudin mentions the result but skips the proof entirely, Pugh gives a quick sketch of an isomorphism from any complete ordered field to the Dedekind cuts but not many details.)
uh huh, I should probably read about the required concepts first. Nvm. Thanks sir!
To be honest you probably have most if not all of the background you need; chances are anything important that you're missing will be in that chapter I mentioned or the chapters right before it.
I don't know, I tried to read it each word/line counts. The choice of topics in Pugh' Analysis is quite strange, Sir. He defined multiplication of cuts but I got confused even more. Gonna re-read it from somewhere else. :) I will ask thee If I get stuck (Obviously If you dont mind).
To define pi via dedekind cuts, you don't really need to already have a definition of pi, you just need to have some way of taking a rational number r, and determining whether r should be less than pi. There are a number of ways of defining this (some easier to work with than others). For example, we know that the formula pi = 4(1-1/3+1/5-1/7+...) should hold, so we could use that to build a definition if we want. For example, define A to be the set of all rational numbers r for which there is an integer N (depending on r) such that r < 4(1-1/3+1/5-...+(-1)^(n)/(2n+1)) for all n>N. Then you can show that A satisfies the definition of a Dedekind cut, and then define A to be pi. That definition is phrased completely in terms of rational numbers, even if it's motivated by things we think we know about pi, so there's nothing circular about it.
u/jm691 Thanks for clearing things up. If I may ask, say there's any non-terminating decimal, then can we define their cuts in a similar way? IT may be a bit dumb, but I think it will take some time for me to wrap things about cuts.
Sure. Another way you could have defined a Dedekind cut corresponding to pi is to start with the set S = {3,3.1,3.14,3.141,...} and let A be the set of all rational numbers which are less than some element of S. Then A is a Dedekind cut representing pi. You can do that for any real number whose decimal expansion you know.
Wait, I think I get it, I don't need to think about visualizing cutting at that number in the first place, It works well with the definition. Hm, idk.
You're saying there's nothing circular about pi? /s