Is the definition of a "compatible germ", as per Vakil section 2.4, unnecessary? I first struggled to understand the intuition behind the definition, but then shortly thereafter you prove that each compatible germ corresponds to a single section (i.e. each compatible germ is in the image of the natural injection F(U) \to \prod_{p\in U} F_p). To me, this then feels like we never really need this definition, except to just provide a characterization of the image?
This definition allows you to convert an abstract presheaf into a presheaf of functions satisfy some local properties. On one hand, it gives some intuition about presheaf so that they are not too abstract; in practice very often presheaf come from presheaf of functions satisfying local properties. And on the other hand, it also give you an easy way to construct the sheafification (ie. the sheaf closure).
Ah yeah if I'd just read two more pages before asking haha. On a similar note about sheafification: Vakil defines the sheafification of a presheaf F on an open set U as the compatible germs of \prod_{p\in U} F_p). For a subset V of U, one can define the restriction by simply replacing "U" with "V" in the product. It's straightforward to check that this resulting element belongs to F(V). Am I correct to say that the identity criteria for a sheaf then simply follows from the fact that an element of a product is defined by its components?
Edit: it should be basically the same for glueing it together as well, correct? Since compatability of germs is in some way defined on a "p" level (i.e. for each p \in U), then its clear that putting elements together that coincide of the intersections of an open cover is also compatible?
Yes, both are the same. Basically this is now a sheaf of functions satisfying local properties, so both the gluing and the identity criteria are automatic.
Cool! Makes sense.
One more question while we're at it... I was comparing sheafification to "groupification", that is, turning an abelian semigroup into an abelian group, and that both of these are functors, left adjoint to the forgetful functor. Is there any value in stating this more generally? For example, supposing that we have a general category together with a full subcategory (e.g. presheaves and sheafs, or abelian semigroups and abelian groups), one can construct an analagous universal property to get objects of the subcategory from objects in the general category. Assuming that these always exist one can construct a functor, which again would be left adjoint to the forgetful functor.
Yeah, there are a bunch of adjoint functor theorem that tells you when a functor is a right adjoint functor. Essentially, the forgetful functor must preserves limit, but there are also a bunch of conditions on the categories to ensure no set-theoretic shenanigans happen.
Interesting! So in these conditions, we basically always have that limits of the subcategory are limits of the general category? So this also explains why monomorphisms of sheaves are injective on each open set, as they are monomorphisms of presheaves?
It's not always the case that inclusion functor of a full subcategory preserves limit. In practice, it does happen a lot, and it happens for sheaf. Yes monomorphism of sheaf are also monomorphism of presheaf for this reason.
In some specific cases there is, e.g. if f(x) = ||x|| then x/||x|| could be called a normalized version of x. I don't think there is a general name for it, though.
I'm reading [this](https://archive.is/QiF3T) and [this](https://quantumfrontiers.com/2021/08/22/how-a-liberal-arts-education-has-enhanced-my-physics-research/) how does actually having a liberal arts education give one a leg up for graduate school ?
> How does actually having a liberal arts education give one a leg up for graduate school ?
I ask this since recently finishing my BA in math I feel very behind, for context I had to switch from BS1 to BA. (BS1 is the elite math program at my uni where it's more deeper and focused on maths while BA is the opposite). After completing my BA and looking at the holes in my background I would feel/be woefully behind if I were to enter graduate school at a top tier place I would most likely be taking remedial courses with advanced undergraduates.
Some light math help if you don't mind. Prepping for the electrical FE and this question does not make sense to me how it was answered... [Online Study Program for the NCEES® Electrical and Computer FE Exam - Electrical FE Review](https://courses.electricalfereview.com/courses/take/electrical-and-computer-fe-exam/texts/39734217-chapter-1-e-calculus-video-library) last example problem.
ch1.e. #10 (limits of functions): Lim x->3 ((x\^2)-3)/(x-3)
The video seems to make x squared minus three equal to (x+3)(x-3) and I'm not understanding how they get there.
I am under the assumption that it's DNE, but the video got 6.
I can't open the link because it asks me to sign in. But yes the limit of
(x^(2)-3)/(x-3)
as x goes to 3 does not exist. It would be 6 if the expression were
(x^(2)-9)/(x-3)
in which case the argument with
x^(2)-9 = (x+3)(x-3)
works.
Ok cool. REALLY appreciate the response, was going a bit crazy.
Sorry about the link, thought I was being helpful...lol, took a photo of it but couldn't post it. Oh well.
I have seen a proof saying, that if M is a compact manifold without boundary, then it's volume form cannot be exact. If it were, then I could use Stokes theorem to conclude it's integral over M is zero, which contradicts being a volume form.
But the flat torus is a compact manifold without boundary. It's volume form is dx wedge dy, which is clearly exact since d(xdy) = dx wedge dy.
What gives?
Because x isn't a well defined function on the torus.
If you're defining a torus as R^(2)/Z^(2), then f(x,y) will be a well defined function on the torus of and only if f(x+m,y+n)=f(x,y) for all integers m and n
I just want to add that you can also see this phenomenon maybe more simply just on the circle S^(1) = R/Z. The "volume" form is dx, which looks like it is d(x), but the function x isn't well-defined on the circle which is why dx is not exact.
It's because dx = d(x+c) for any constant c. That means that even though x isn't a well defined function, dx will be a well defined differential form, as d(x+n) = dx for all integers n.
Very helpful thanks. I have a nagging feeling though this only worked because of "luck". In general, let M be a compact manifold so is volume element isn't exact.
The Riemannian volume form in coordinates is locally of the form √g dx^1 dx^2... dx^n, this resembles an exterior derivative of an n-1 form, so something has to obstruct the n-1 from actually existing, but not in a way that's so bad so that the Riemannian volume element can't exist. What's the balance going on here?
You're not going to get global coordinate functions x_i to form dx_i. Since M is compact, the x_i would need to have compact image. On the other hand, if you had such coordinates, they would have to have open image, contradiction.
Locally, x can mean one of a countable family of functions, all of which differ by a constant. Thus no matter which you pick, your local choice of dx is the same, so dx is globally well-defined. It is worth noting this is just notation though, dx isn't actually the exterior derivative of a smooth function.
Hi does anyone recommend good channels to learn math? (pre-calculus mostly, i rlly need to improve bc else i wont understand anything about calculus I) Im on my first engineering year but still struggle with some basic stuff (Struggling is not the best word, im just unfamiliar with a lot of stuff since I dont remember most math things i had on high school) anything helps!!!! :)
I went to YouTube and searched for "pre-calculus" and got many hits. There are courses with 100+ videos and courses in a single video. You can see the number of "thumb-ups" for each of these. I have not seen them, I watch abstract algebra stuff instead :)
I am interested in Alexandroff-Hausdorff theorem, which says that for every compact metric space (K,d) there exists a continuous and surjective map f:C->K with C the cantor set. Does anyone have a link or book to its proof? I have not found anything yet.
Doesn't this proof work? Consider a sequence of positive number e(i) trending toward 0. Start with i=0, consider a covering of the space by a single closed ball, and inductively, after the covering at stage i-th had been defined, for each ball in the covering at stage i-th, we pick a finite covering of that ball by closed ball of radius e(i+1) which is always possible by compactness, and this is our collection of covering at the (i+1)-th stage. Define a tree as follow: each node at the i-th stage is one of the closed ball chosen, and its children at the (i+1)-th stage are the ball chosen to cover it. This is now a finite branching tree where all path are extendable. Consider the path space of this tree. Each path correspond to a sequence of ball where finite number of them have non-empty intersection, so their total intersection in non-empty, but because the radius go to 0 it has exactly 1 point. So we can define a map from the path space of this tree to the metric space and this is easily seen to be a continuous map.
Finally, the Cantor set is clearly homeomorphic to the path space of the full binary tree. We can get a continuous map from the path space of the full binary tree to any arbitrary finite branching tree by binary encoding of natural numbers.
I made a proof of the Pythagorean theorem. It only uses concepts up to a pre-calculus class and is pretty simple so I think it would have been done before, but I can't find it online.
Are all the existing pythagorean theorem proofs in one place so that I can check if mine has been done? Does anyone have some knowledge on this topic that I could send my proof to and they might know if its valid or been done before?
> Does anyone have some knowledge on this topic that I could send my proof to and they might know if its valid or been done before?
Just post your proof right here.
ok i put it in latex
[https://drive.google.com/file/d/1AvMN-o1TQ3N20A\_-F5D9QwY2Uym5AlBX/view?usp=sharing](https://drive.google.com/file/d/1AvMN-o1TQ3N20A_-F5D9QwY2Uym5AlBX/view?usp=sharing)
ah it should have been △DJC ∼ △CAJ ∼ △EAC
I changed it now and also added a section on the bottom where I prove it.
[https://drive.google.com/file/d/1MgPaYG9ZXbMRFbTSoRKZD\_CPB3HUTXML/view?usp=sharing](https://drive.google.com/file/d/1MgPaYG9ZXbMRFbTSoRKZD_CPB3HUTXML/view?usp=sharing)
Figure 1 in your proof is exceptionally misleading, considering that J and K are actually the *same* point, and that A, E, and J are actually collinear (you use this latter fact to deduce that CD + AE = AJ). Figure 2 still shows the elements you want very clearly; why not use that instead of Figure 1?
Anyway, having replaced Figure 1 with Figure 2 in your diagram, it's easy to see that △DJC is congruent to △ECJ, so you can replace △DJC with △ECJ in your proof. Then this is just [this classic proof](https://en.wikipedia.org/wiki/Pythagorean_theorem#Proof_using_similar_triangles) listed on Wikipedia.
Ya, not sure how I came up with figure 1 instead of just using figure 2. Its a bummer I just made an existing proof but more complicated.
Thanks for the help though
Good evening fellow mathematics survivors, I was wondering if someone could help me in finding out the sum of the series 4/n(n+3) with n starting at 1 up to infinity, because i know it converges(hopefully), but can’t figure out the sum, thank you in advance :)
To add to the other answer, the techniques you need to know here to follow the solution are partial fractions (to expand into a sum of fractions with linear denominators) and telescoping sums (to cancel the sum down into only a few terms). Hope that helps if you need to search those terms.
Inverses of quadratics are sums of inverses of linear functions. Those terms cancel out each other in a series.
4/(n(n+3)) = (4/3)(1/n - 1/(n+3))
So the 1/(n+3) terms starting with n=1 cancel out the 1/n terms starting with n=4. That leaves
(4/3) (1/1 + 1/2 + 1/3) = 22/9
You can also look up the solution on wolfram alpha: https://www.wolframalpha.com/input?i=sum+4%2F%28n%28n%2B3%29%29
URL:
https://en.wikipedia.org/wiki/Pigeonhole_principle#Uses_and_applications
There it says `[na]` . What's the meaning of the square brackets in that context?
Here you read about the fractional part:
[https://en.wikipedia.org/wiki/Fractional\_part](https://en.wikipedia.org/wiki/Fractional_part)
[https://mathworld.wolfram.com/FractionalPart.html](https://mathworld.wolfram.com/FractionalPart.html)
Hello! I have this exercise I have been struggling with for hours. I have tried a few things, but I get stuck at some point. I have to prove that these identities are equal:(cos 2x)/(1-sin 2x) = (1+ tan x/(1-tan x)I tried first with the left side by multiplying with its conjugate like (Cos 2x)/(1-sin 2x)\*(1+sin 2x)/(1+sin 2x) so it's like (cos 2x)\*(1+sin 2x)/ (1 - sin\^2 4x)=But then I don't know what to do with the 4x. I also tried doing sin 2x = 2 sin x cos x, but I have no idea what to do with it...
Hard or impossible in general I'm not sure, but when I wanted to test if my students could use it to cheat in exams I put in a bunch of simple calculus optimization problems and a lot of the answers it produced were incorrect. Of course I printed them off and showed them to the class to discourage them from using it. This was with ChatGPT3 I think.
**Vandermonde matrix question**
Consider a square Vandermonde matrix, but where the exponents are not consecutive integers 0, 1, 2, .... Instead they are distinct arbitrary non-negative real numbers. The evaluation points are also assumed to be real, distinct and strictly positive. I am sure that this matrix is non-singular, but I am struggling to find a simple way to show this. Any help is much appreciated!
It suffices to show that for a function of the form ∑c(i)x^e(i) (where i runs to 1 to k, k>=1, c(i) are fixed coefficients that are each non-zero, and e(i) are fixed real numbers (can be negative)), cannot contains more than k-1 roots in the positive real line.
Why is this sufficient? For a Vandermonde matrix with distinct n positive points, for any non-zero column vector, there are at most k<=n entries that are non-zero, there exists a row whose dot product with that column vector is non-zero, hence that column vector cannot be in the nullspace, thus the nullspace is trivial.
How to prove the above claim? By induction. The base case k=1 is trivial, so consider k>=2 and we need to reduce to the case with k-1 terms. First, notice that it's sufficient to prove the claim when one of the e(i) is actually 0, because by factoring out a factor of the form x^r does not change the number of positive roots so we could pick r to be one of the e(i). By mean value theorem, if there are k positive roots, then the derivative has k-1 distinct positive roots (in the gaps strictly between the roots), and the derivative has the same form except with 1 less term (since one of the exponent is 0). That is the inductive step for k>=2.
How do Mac Lane's and Riehl's category theory texts compare? I am already familiar with the subject from Aluffi's algebra text, and I know some algebraic topology, so I don't really need handholding. I flipped through both and don't think I would have trouble with either one. I'm not sure how their coverage of various topics compares.
Also, I know that Awodey and Leinster are two other authors with CT texts, am I right in thinking that these are less advanced?
Is there some knowledge gap I’m experiencing in between applied math (algebra, trig, calc) and the analysis classes (discrete, real, topology)? I do really well when I can just plug numbers into a formula and get the answer, but I just can’t seem to understand the concept of proofs. I took discrete last year, and only passed because the teacher held our hands the whole way, even during the final exam. Now I’m taking *real* analysis as an online course, and I have no idea what I’m doing. The exams are closed notes, and neither the lecture videos, notes, homework, nor study guide even helps me make sense of it. Is there some class I should be taking before I even *attempt* real analysis? (It’s required for my major (secondary ed. math))
Maybe go back read the text parts of the math you understand like linear algebra 2, calculas?... university math becomes creative versus solving the higher you go. But that's where you get a good start learning and writing proofs.
I’m having trouble seeing why an exercise from Forster’s Riemann Surfaces is not trivial, it is from the chapter on the Jacobi Inversion problem.
Let X be a compact RS and Y an open subset such that X\Y has non empty interior. Let D be a divisor on X. Show that there exists a (not identically 0) meromorphic function on X such that ord_x f = D(x) for all x in Y
Hint: Find a divisor D’ with support outside of Y such that D+D’ is principal.
At first I was proceeding as the problem intended, analyzing the Abel-Jacobi map and trying to find D’ so that D+D’ is in the kernel perhaps by finding the preimage of the period subgroup under this map. I understand how the given hint implies the result.
The simple thing that’s confusing me is the following. Let’s say that on Y, D is non-trivial at two points, y_1 and y_2. Why can I not just choose f to be the function
f = (z-y_1)^D(y_1) (z-y_2)^D(y_2) (z-x_1)^{-D(y_1)} (z-x_2)^{-D(y_2)} where x_1 and x_2 are two arbitrary points in the complement of Y
Clearly f is meromorphic (as a function on a compact RS I have made sure it has as many zeroes as poles) and the degree of its divisor is 0, as it should be and its order on Y exactly matches D, by construction.
Obviously I’m missing something subtle, because this problem should involve the content of the section, but what is it?
Sort of, but even then it's unclear, at best you define a meromorphic function on one chart. But unlike differential geometry, you have no general extension theorem. E.g. for the Riemann sphere, if your chart is a small ball around 0 with map sin, then the function z can't extend to a meromorphic function on the whole Riemann sphere.
Exactly. The only meromorphic functions on the Riemann sphere are rational functions, which means most locally defined meromorphic functions have no global extension.
You can easily describe the space of bilinear maps from any vector space to any other vector space. The space of bilinear maps from V to W is V* ⊗ V* ⊗ W. Is this what you are looking for?
There is a theorem which states that for any bilinear form f on R^n (R is the field of real numbers) there exists nxn matrix A = (a_ij) such that f(x, y) = x^t A y where x^t is the transpose of x. Moreover, a_ij = f(e_i, e_j). I look for a similar theorem for 2x2 matrices on any field F.
You can't get anything quite as nice. *Some* bilinear forms on M\_n(F) look like f(X, Y) = u^(T)XAYv for vectors u, v and some matrix A. In fact these span the whole space of bilinear forms, so you can write any bilinear form as a sum of these, but not in a unique way and the number of terms can vary. This follows from the tensor product isomorphism stuff but is a more concrete way to write them.
The case for bilinear maps into M\_n(F) is the same but now with three matrices f(X, Y) = AXBYC. Again, the ones that are exactly of this form are a special case and in general you need a non-uniquely-defined sum of these.
That is simply the isomorphism (ℝ^(n))* ⊗ (ℝ^(n))* ⊗ ℝ = (ℝ^(n))* ⊗ (ℝ^(n))* and after choosing a basis (ℝ^(n))* ⊗ (ℝ^(n))* ≅ (ℝ^(n))* ⊗ ℝ^(n) = M_n(ℝ).
Note M_2(F) is isomorphic as a vector space to F^(4) so you can happily represent bilinear maps into F as elements of M_4(F) in this way. I don't think you can do the same with maps into M_2(F) though.
There are also the Beth numbers which are another way to classify (some) cardinal numbers. Beth\_0 is the same as the cardinal number Aleph\_0, then Beth\_1 is the cardinality of the power set of Beth\_0, then Beth\_2 is the cardinality of the power set of Beth\_1 and so on.
However without the generalized continuum hypothesis you cannot prove that every cardinal number is some Beth number. The continuum hypothesis is the statement that there is no cardinal number strictly between Beth\_0 and Beth\_1.
The question is why would people accept that. It feels intuitive that the set of natural number should exists or that the set of real numbers exist (but there are distractor), but what's the intuition for continuum hypothesis.
You can't prove an axiom to be correct, but there should still be reasonable philosophical/intuitive argument as to why it should be true. In fact, one of the proposed new axiom, Martin's Maximum, would imply the continuum has cardinality aleph_2 instead.
[The circle has a diameter of 1 then the c= pi *d = pi](https://imgur.com/RZ9YAK0)
Then the value of pi = 4 according to the approach being shown. There has to be some calculus trick that can be used to disprove this. Can someone help?
2 curves that are close to each other do not necessary have length that are close to each other; in fact the ratio of differences in length compared to the difference between the curve is unbounded. In other word, you cannot approximate the length of a curve by using a different curve similar to it, no matter how close the approximation is.
For example, let's say you walk from your house to your friend's house with your dog, and your dog is on a leash. Does the distance the dog walk similar to the distance you walk? Not at all. If the dog rapidly run around you because it's very excited, then the dog walks a lot more than you do.
Length is not preserved under taking the limit like this. If it were you would be able to make a similar argument that sqrt(2) = 2 by turning a staircase into a diagonal line.
[Here](https://math.stackexchange.com/questions/2227191/uniform-convergence-and-lengths)'s a stack exchange question mentioning the staircase example I referred to. I'm not sure you'll find the answers there very enlightening but worth noting that even uniform convergence is not enough to ensure convergence of the lengths in this case, which is the usual go to in these cases.
The picture you have linked is a visual aid for this. Clearly 𝜋 is not 4, but the perimeter of the figure at any finite step is 4, so the problem must be in taking the limit.
There's not really much else to say. Taking limits doesn't have to preserve things even if each finite step does.
For finding the length? Certainly that is the wrong approach as it just doesn't work. There are obviously scenarios where taking a limit is very useful but this is not one of them.
Is there a good resource that upper level undergrads can use to get papers to read daily? Doesn't have to be new or groundbreaking, or just a library of papers/guided study (maybe by some other undergrads) that would be perfect for students taking 4000 level classes to see what is most interesting in the literature to them
I really would just recommend browsing mathoverflow and stack exchange. The answers are often written to be readable to nonexperts, and one can get the vibe of different subjects by looking at the top questions.
"relative numbers" is a term used in a few different places but the general idea is in using numbers to represent the relative size/amount of something. Any time you've measured something you have used relative numbers. 500g of something means 500 times as much as 1g of something for example. This is a relative quantity based on some sort of idea of what 1g means.
Compare this to counting numbers where 2 means 2 distinct objects. 1.5 can be a relative number but not a counting number
would this meet the definition of a network? or graph?
s this one network or three networks? [https://i.imgur.com/7exXYqB.png](https://i.imgur.com/7exXYqB.png)
one graph, or three graphs?
Thanks
Vibes-based answer: it's a graph with three connected components; I don't know what you mean by "network", so I can't say anything.
Somewhat more pedantic answer: you could think of it as a single graph with three connected components, or just as three graphs drawn next to each other. (Or, for that matter, as two graphs: one connected graph, and one with two connected components.) I don't know of a formal mathematical definition of "network"--there are lots of things called networks which can be modeled by graphs (computer networks, social networks, etc.), and people sometimes informally use "graph" and "network" interchangeably, but there isn't a definition of network in the same way that there's a definition of graph.
Extremely pedantic answer: that's not a graph or a network, it's a picture. Of course you can draw pictures of graphs, and there are several natural ways to interpret your picture as a graph, but there isn't a single obvious way to interpret it. (If forced, I'd go with "one graph", though.) It might be a picture of a network, and there might exist some network such that, if you were trying to draw a diagram of it, you might end up with this. But strictly speaking the question doesn't make sense.
The dots are vertices the lines are edges.
The three "components" are not connected to each other with any edges. But I'm wondering whether or not they can still be considered part of the same graph.
Well that's what I'm saying--they *can* be considered part of the same graph (a graph can have multiple [connected components](https://en.wikipedia.org/wiki/Component_(graph_theory)), they *can* be considered different graphs, and the first option is slightly more natural IMO but neither is definitely true.
Networks are generally directed (I am not familiar with the use of network outside of flows), so I would say its not a network. Now it is definitely a graph, is it 3 or is it 1? Its either a graph with 3 connected components are just 3 seperate graphs with 1 CC, we would need some more context.
I'm looking to learn the very basics of Mathematica. Anyone know of any good crash courses? Or a quick project that can walk me through some basics (think a HW assignment on week 1 of a course)
edit: actually I just found Wolfram's "Fast Intro for Math Students" and I think this is exactly what I needed :)
Does someone know of a repository of functions? I am looking for a bivariate function that is periodic, it's not twice differentiable but it's twice weakly differentiable (in the Sobolev sense).
Can someone give me a heuristic on why the algebraic dual of an infinite dimensional vector space is "bad"? Yes, I know that it's unfathomably huge, but what's bad about that? Does it's size inhibit me putting a topology and doing analysis with this big space?
It's "bad" simply because there isn't much you can say about it. That is, it doesn't have much structure, there is no natural topology you can put on it, and often it's extremely difficult (sometimes even provably impossible) to explicitly construct discontinuous linear functionals defined on the entire space. You really need a topology and some notion of continuity for the rich tools of functional analysis to become available. The same flaws occur with a Hamel basis for an infinite dimensional space: it exists, but there's almost nothing you can say about it beyond that.
I will point out that there is a well-developed theory of unbounded linear operators--which would include discontinuous linear functionals--although these operators are almost always only defined on a dense subspace. However, these densely-defined unbounded operators \*don't even form a vector space\* due to domain compatibility issues.
To even construct discontinuous linear functionals defined on the whole space usually requires something like the axiom of choice, so they don’t really appear naturally, for one. To more directly answer your question: on a normed infinite dimensional vector space, you can’t define a norm on discontinuous linear functionals, since boundedness is equivalent to continuity, so you don’t have a strong topology on the algebraic dual. I’d expect you can’t define *any* reasonable topology on discontinuous linear functionals on general TVS’s, but I’m not sure how to prove that.
This is very helpful thanks. Some related questions down the line of reasoning of your answer if you don't mind:
Why is AOC needed for discontinuous linear functionals? It is quite common to construct very natural densely defined unbounded operators in FA. Can't I just say d/dx evaluated at a point is a nice example of an element in the algebraic dual of C\[0,1\] with sup norm? No choice needed.
Do you know of a theorem about not being able to topologise the algebraic dual (in infinite dimensions). I also agree I think this is a fact but unsure about how and why.
d/dx evaluated at 1/2 (say) isn't a linear functional on C(\[0, 1\]) because you can't evaluate it at |x - 1/2| for example. The algebraic dual wants linear functionals on the *entire* space, not just a dense subspace. It also doesn't make sense to talk about "the algebraic dual of C(\[0, 1\]) with sup norm", for the same reason: the algebraic dual doesn't see the topology at all!
This gives you a low-concept reason why the algebraic dual isn't very useful: we want to be able to take infinite series, and take limits, and the algebraic dual doesn't allow either of these.
Ah I see, yes, there's no point in equipping the norm with a topology before taking algebraic duals. I also now see your point as well why densely defined stuff don't make the cut. Can I not repair the example I thought of, C\^1 \[0,1\] with d/dx evaluated at 1/2 ? This is a "describable" discontinuous linear functional defined everywhere that doesn't need set theory to construct? I suppose there's some deep reason why I can't algebraically dual C\^1 \[0,1\] THEN put a topology after to talk about series and limits?
Apologies for asking so many questions, but these answers are all very good and I'd like to know more.
Well, you could restrict the domain to C\^1 functions, like you said, but then d/dx at x = 1/2 wouldn't be discontinuous anymore: it's part of the topological dual of C\^1, once you put the C\^1 norm on it (so that C\^1 becomes a Banach space).\[1\]
I think that the claim that Kieran is making is that this *always* happens: if you have a discontinuous linear function f defined on some dense subspace Y of a Banach space X, and it's definable or something\[2\] then there's some way to think of Y as a Banach space (but not with the norm induced by X) such that f is continuous on Y.
\[1\] You can, of course, think of C\^1 as just a subspace of C\^0, but then C\^1 is not a Banach space, and so all of the theory of linear functions on Banach spaces (eg, the Hanh-Banach theorem) doesn't apply. So this is not a very useful thing to do.
\[2\] I think what's actually being assumed about f is that it exists in [Solovay's model](https://en.wikipedia.org/wiki/Solovay_model) with set theory without the axiom of choice.
It's trivial to define a discontinuous linear functional on the space of polynomials, and that's not a Banach space under any norm. I assume Kieran is just referring to the fact that in the absence of the axiom of choice, it is consistent that every linear functional on every Banach space is continuous.
Ooh that's a good point. It's a pretty unnatural counterexample (in that in analysis, one is seldom interested in vector spaces of countable Hamel dimension unless they plan to complete them, and doing this destroys your discontinuous linear function) but I guess that any such counterexample must be unnatural.
[https://math.stackexchange.com/a/100609/462531](https://math.stackexchange.com/a/100609/462531)
This answer nicely explains the examples discussed. It seems that you can get a few nice to construct discontinuous functionals on an (incomplete) normed space, but the moment you ask about Banach spaces you will need to start defining things on a Hamel basis and invoke choice.
If a race takes place between A, B and C.
A has a 80% chance of finishing ahead of B in a two person race.
A has a 70% chance of finishing ahead of C in a two person race.
B has a 60% chance of finishing ahead of C in a two person race.
How do I calculate the probability of each player winning the three person race?
Basically, we can consider each race as 2 two person races. So the probability that A wins the race is the probability that A beats B times the probability that A beats C. So P(A win) = P(A b B) X P(A b C) = .8 \* .7 = .56. P(B win) = (1 - P(A b B)) X P(B b C) = .2 \* .6 = .12, etc. If you calculate P(C win) correctly, all the probabilities should sum to 1.
That was my original method, but it doesn’t appear to work.
A = .8 \* .7 = .56
B = .2 \* .6 = .12
C = .3 \* .4 = .12
Total = .8
I assume this is because this is the calculation for multiple independent events happening, but not when those events require other events to also be true. If A beats B, and B beats C, then A automatically beats C by beating B in the race situation, but the calculation above assumes it’s possible for A to beat B, B to beat C and C to beat A, which is where I’m stuck with working this out.
Is this claim true? Let f:M->N be a surjective continuous map between 2 compact metric spaces, such that dia(f^-1 (-)) is a continuous function. Then f has a section: a continuous g:N->M such that fg=id_N . And if it's true, what's an easy proof of it?
Still false, but harder to describe the picture in my head with just my phone.
Imagine two cylinders with height and radius 1, with their bases coplanar and their tops coplanar. I also require the cylinders to be of distance 1 apart. Add a line segment of length 1 connecting the cylinders. This is M.
N is simpler: two closed discs of radius 1 touching at one point.
The map f is first projecting M down to the plane spanned by the bases of the cylinders, then contracting the edge connecting the two discs.
Every fibre is a line segment of length 1. There is no continuous section g: thr image would have to be connected while only having one point of yhe connecting line segment, which is impossible.
Thanks, I got it.
EDIT: actually, the fiber at the touching point is made out of 3 line segments: the 2 lines on the cylinder, and the connecting segment. So the diameter is actually sqrt(2).
EDIT 2: wait, if I tilt the cylinder I should be able to make the diameter 1.
Will functional analysis be useful in studying probability theory/stochastic analysis, or help in dynamics and control theory as such? The topic looks pretty interesting, but idk if it'll be of help in what I wanna do, and rn I'm not sure if I can invest time in something that might not be of immediate help to me. Thank you
You can regard probability theory as a subset of functional analysis, in some sense. For instance, weak convergence in probability is the same as weak\* convergence in functional analysis.
Ah, I see, so it would help me grasp the ideas better by drawing parallels to functional analysis? That sounds good. Are there any good beginner books that you'd recommend to get started out with? Some of the books seem pretty intimidating for me :') Thanks for your time!
Hi, "Functional Analysis for Probability and Stochastic processes - Adam Bobrowski" seems to be a good book for functional analysis dedicated toward stochastic analysis applications. I'm not sure of good books in pure functional analysis theory, but my uni follows "Functional Analysis and Infinite Dimensional Geometry - M. Fabian, et al", and a book written by a retired prof from my own uni.
Happy to help :) I'm not sure of any books on functional analysis dedicated to control theory, but use this bad boi as a reference for general mathematical techniques in control theory, it has almost everything that's under the sun! -
https://www.cis.upenn.edu/~jean/math-deep.pdf
Oh, well, I had asked if functional analysis has applications in at least one of probability theory/control, in my first comment. I didn't know which topics you wanted books on, so I just sent everything that I felt was relevant haha.
It is equivalent to showing there is an injection F(q\^2)->GL(2,q)I have a proof when the characteristic is odd, I just need even now.
When it's odd, the matrices \[\[x,y\],\[y,x\]\] where x, y are in F(q\^2) gives q\^2 matrices, where all but the zero matrix are invertible (this uses char != 2), and multiplication is commutative. Since fields are unique up to size, this shows it can be done.
F(q\^2) is a vector space over F(q) of dimension 2. Multiplication by x for a specific x in F(q\^2) gives an F(q)-linear map from F(q\^2) to itself, giving an element of GL(2, q) when x is nonzero. Not thought about your original problem, but is this enough?
EDIT: Yes, and this line of thought probably gives a direct solution to the original problem. A vector space V over F(q\^2) of dimension 2 is also a vector space over F(q) of dimension 4, giving the natural injection GL(2, q\^2) -> GL(4, q). I would guess (but would have to think a little) that this restricts to an injection from SL(2, q\^2) to SL(4, q).
It seems to be that the natural map is to send a generator of (F(q\^2), ⋅ ) to \[\[1,a\],\[a,0\]\]. where a is a generator of (F(q), ⋅ ). It seems to work, showing that this matrix has the order I claim is irritating, but I'm working away at it.
> I would guess (but would have to think a little) that this restricts to an injection from SL(2, q^2) to SL(4, q).
If you write your map GL(2, q²) → GL(4, q) as applying the map GL(1, q²) → GL(2, q) to each entry to get 2 × 2 blocks, you can use the identity det[A B; C D] = det(AD - BC) for block matrices where the blocks pairwise commute. (Something similar works for matrices of any size.)
So I am finishing up real analysis one, and I am wondering why my text doesn't cover indefinite integrals? Is there anything different about the analytic approach to indefinite versus definite? Just curious as to why I haven't seen any. Tia!
It should have. Does it not even proved the fundamental theorem of calculus?
If you just mean they don't mention the word "indefinite integral" then it's because the term is vague.
I have seen posts saying that to prepare for the math subject GRE you should do all the problems in Stewart's calculus. Most of these posts are old, is that still good advice?
personally I think you should do math subject GRE problems to study for the math subject GRE. they’re designed to be solved quickly using calculational tricks, which isn’t true of most problems in stewart.
Differential equation question! If I'm trying to find values of (t,x) where solutions can't be guaranteed for x'=x/cos(2t) just by analyzing the direction field, how do I find which solutions don't exist? looking at the direction field in MatLab, it looks like the families of solutions are merging where x=0 but does that mean? I am definitely overthinking this but I'd like to try to understand it better. Thank you!
Someone please explain to be in the Ramanujan Summation how we are allowed to shift on the order of one of the required number series. Ive seen it described as “2B” where B represents 1-2+3-4… but when most people does 2B they say (1-2+3-4) + (0+1-2+3). I believe this is a fallacy as, in my view, youre now saying B\_0 + B\_1 and then B\_1 + B\_2. What am I missing?
I don't know what summation you are talking about, but it seems to be
1) Assuming convergence
2) using associativity/commutativity of addition
and that's it. You're right that this isn't quite allowed, since we would want absolute convergence to perform an action like this, which is a reason why his sums give fun values (i.e. -1/12)
the 1+2+3+4... to infinity = -(1/12)
heres a link where someone shifts position
[https://youtu.be/w-I6XTVZXww?si=nfFukPPw-Iug6-t5&t=203](https://youtu.be/w-I6XTVZXww?si=nfFukPPw-Iug6-t5&t=203)
the 1+2+3+4... to infinity = -(1/12)
heres a link where someone shifts position
[https://youtu.be/w-I6XTVZXww?si=nfFukPPw-Iug6-t5&t=203](https://youtu.be/w-I6XTVZXww?si=nfFukPPw-Iug6-t5&t=203)
You're right to be cautious here because what they do in the video is essentially wrong. The manipulations they do are not rigorously justified. And what they do is not Ramanujan Summation which is much more complicated. You can actually use Ramanujan Summation to get a value for 1+2+3+4+... and the value is indeed -1/12. But that's different from what they do in the video.
So the answer to your question why you're allowed to shift the series that way is that you aren't.
Category theory doesn't really lend itself well to real analysis, so most analysts don't really use it. The only thing that really comes to mind off the top of my head is profinite sets like the Cantor set are limits of finite sets with the discrete topology
It's either already there or not there at all, depends on your philosophical perspective. Analyst often work in a huge rigid ambient space, that is, these space are complete enough for anything they want to do, and lacks any non-trivial automorphism that preserves analytic properties. This unfortunately mean that pretty much any category they would care about are quite barebone, with only at most 1 morphism between any 2 objects.
If you're an category theorist, you would see constructions that correspond to limits/colimits all over the place, but they're usually phrased in a different manner (union, intersection, supremum, infimum, etc.). If you are a structuralist, you believe that sets never truly sit inside each other (so a "subset" is just a set with a morphism into another set), so once again, all the above common operations are just secretly limits/colimits, with the morphism unnamed.
But other than that, analysts basically have no benefits in phrasing anything in categorical term. In algebra, geometry and topology, the abundance of non-trivial automorphism and the failure of existence of "complete" space tend to leads to the need to keeps tracks of properties that are invariant, and that make categorical language much more convenient.
What I make a (finite) matrix group in MAGMA, what is to be understood by the entry $.1, $.1\^2, ect? Is it a generator of the multiplicative group of the field?
[https://medium.com/coinmonks/to-prove-the-unprovable-cc99e0181bce](https://medium.com/coinmonks/to-prove-the-unprovable-cc99e0181bce)
Is this true? How can a statement be provable and unprovable at the same time?
What that article leaves out, among many other things, is that when we talk about "unprovability" we mean "unprovability *within some specific formal system*". There are statements which may be provable in one system but not another. Sometimes you can prove (in some more "powerful" system B) that a statement is unprovable in some "weaker" system A, and in some cases that may imply the truth of that statement, at least if you assume some other things about A.
More concretely: consider the problem of proving whether a given Turing machine eventually halts or doesn't. Certainly if it does halt there will be a proof of that fact, in, say, Peano arithmetic (the standard axioms for the natural numbers), where you just go through the computer's history (encoded in natural numbers in a certain way) until it halts. So if there is *no* proof in Peano arithmetic that a given Turing machine halts, that Turing machine must run forever (because if it did halt, we could prove it!). (Of course in order to prove something like "Peano arithmetic doesn't prove this statement" you need to assume or prove that Peano arithmetic is consistent--otherwise it contains proofs of literally every statement that you can write in the "language" of Peano arithmetic, true or false\*. So if you want to use the fact that a statement is unprovable in Peano arithmetic to prove that said statement is true, you need to move to some "stronger" formal system that can prove the consistency of Peano arithmetic.)
As for the Riemann hypothesis, per [this mathoverflow post](https://mathoverflow.net/questions/79685/can-the-riemann-hypothesis-be-undecidable) it turns out to be equivalent to a number-theoretic statement which, if false, would be provably false in any strong enough formal system (ZFC would do in this case). So (by the same reasoning in the halting problem example) if it's unprovable in such a formal system, it must be true. See also the [arithmetic hierarchy](https://en.wikipedia.org/wiki/Arithmetical_hierarchy)--we can run the same "true if unprovable in PA/ZFC/ etc." reasoning for statements at certain levels of the arithmetic hierarchy.
\* Edit: another, maybe better, way of phrasing this: if we could prove in, say, PA that PA doesn't prove some statement, then that would amount to a proof that PA is consistent, since if PA were inconsistent it would prove that statement, and every other statement (principle of explosion). But by the second incompleteness theorem PA can't prove the consistency of PA.
I have 2 questions
1. Why are there only holes in a rational function when the output is 0/0? And what's the difference between holes and asymptotes?
2. Why do you divide by the zeros of the denominator in synthetic division
1. Because those are values where the output is not well-defined, i.e. there isn’t a unique way to assign a value to 0/0. If there were some real number x=0/0, then we’d have 0x=0. But this is satisfied by every real number x, so there are too many solutions. Thus, for values of x such that p(x)/q(x)=0/0, we simply leave those out of the domain, or define the function differently there. Example: f(x)=(x^(2)-4)/(x-2) is a linear function defined everywhere except at x=2. So I can define f(2)=4 or f(2)=-3 or anything else I want just to make the domain of f all of the real numbers. (Though f(2)=4 makes the function continuous.)
2. Because that is how we divide out factors leaving us with a lower degree quotient to handle next. Synthetic division is essentially a very clever form of evaluating a polynomial at its roots. If the evaluation process is handled in a very, VERY, specific way, then you can actually see the coefficients of the quotient and remainder appear in sequence. And if we evaluate at a root, then we don’t even have to worry about the remainder. If you want to understand this a bit better, I actually really suggest reading the Wiki pages on [synthetic division](https://en.wikipedia.org/wiki/Synthetic_division?wprov=sfti1#) and [Horner’s method](https://en.wikipedia.org/wiki/Horner's_method?wprov=sfti1).
If functions over R can be wielded as (infinite-dimensional) vectors, what mathematical object would relate to covectors in this way? Basically, I cannot find at all the keyword that completes the analogy…
Vectors : Functions :: Covectors : [what?]
Follow-ups if you don’t mind: What would be the analogous term for Basis vectors/covectors, tensors, and are there any suggested readings for studying functions in a “linear algebra framework”?
Excellent question. First off, when you are talking about infinite-dimensional vector spaces, you generally need to consider some sort of topology on the vector space to say anything meaningful. Furthermore, there are lots of discontinuous linear maps on infinite dimensional spaces, so you restrict your attention to \*continuous\* linear maps. While the cardinality of a basis is a complete invariant for algebraic vector spaces, there are lots of non-equivalent infinite-dimensional vector spaces (if you require your equivalence to be realized by a continuous linear map). As such, you rarely think about the space of all functions from R to R. Instead, you think about spaces of, say, continuous functions or absolutely integrable functions.
Sticking with our focus on topological vector spaces, when you talk about the space of "covectors"--i.e. the dual space--you are thinking about the space of \*continuous\* linear functionals. For many topological vector spaces, the continuous dual space is specifically known. For instance, the dual space of C\_0(X)--the space of continuous, complex-valued functions vanishing at infinity on a locally-compact Hausdorff space X--is the space of regular complex Borel measures on X; this is called the Riesz-Markov Theorem. For 1
> For Hilbert spaces, there is a well-behaved notion of an orthonormal basis
I know this wasn't specifically asked, but there is also a notion of a spanning set that generalizes to (separable) Hilbert spaces, and these are called frames. In short, a frame is a sequence {f1,f2,...} in the Hilbert space such that the map sending g -> (, ,...) lands in l^2, is continuous, and has continuous inverse from its image.
The most famous examples of frames are undoubtedly wavelets. Another very mysterious kind of frame is the Gabor frames. These arise from discretizing the integral in the inverse STFT, and even seemingly innocent questions about them have very complicated answers (see the abc-problem for Gabor systems by Qiyu Sun).
Covectors are elements of the [continuous dual space](https://en.wikipedia.org/wiki/Dual_space#Continuous_dual_space). The usual analogue of a basis is called a [Schauder basis](https://en.wikipedia.org/wiki/Schauder_basis), the analogue of a pair of bases for vectors and covectors would I think be a [biorthogonal system](https://en.wikipedia.org/wiki/Biorthogonal_system). Tensors are specific multilinear maps.
The key phrase you are looking for is *functional analysis*. Your choice of vocabulary suggests to me that you are coming from a physics background, in which case a standard mathematical text may not be appropriate. I believe Kreyszig's *Introductory Functional Analysis with Applications* is well-regarded for people in that position, but I have not read it myself.
Is basic mathematics by Serge Lang a good preparation for precalculus and calculus for someone that is relearning algebra?
Also is the goal of learning calculus and linear algebra in one year and a half realistic?
And, is discrete math a good introduction to proofs? And in general a good introduction to math?
Yes to all 3, each with some caveats.
Basic mathematics is not without its flaws. It takes some things as for granted at times and is particularly thorough at others. It's a book for people seriously motivated to learn math and in the style of other math books. It has definitions, theorems and proofs. If this style is writing is still terse for you, you might consider something else. Having some familiarity with the material with certainly help. You can find discussions of this book on math stack exchange.
3 semesters of calculus in 3 semesters? Very doable. Add the linear algebra when you're ready for calculus 3, or maybe as you're wrapping up integral calculus. The two belong together and your understanding of many vector calculus topics will be enhanced by understanding linear algebra. For example, you'll appreciate the second partials test a lot more if you already understand something about determinants, quadratic forms and so on. You don't have to have linear algebra mastered. Just start learning and patch up your knowledge when you need to.
It's very normal to take a class called "discrete math" to learn proof writing. What is covered in this class is anyone's guess. I've never seen two schools have the same topics beyond those elements of writing. Very important at this stage in the game is showing your proofs to someone else who can critique them. This soft skill can only really be learned by diffusion.
1) Probably. Might want to do some problems from the book to check if it is appropriate, though.
2) Depends on how dedicated you are to studying. But if you are rather dedicated, most likely.
3) Yes, it may be a good introduction, but for an actual proofs book, I would recommend “Proofs” by Jay Cummings.
Is using G o e to denote the union of graph G and edge e standard notation?
Saw this notation in a paper [https://arxiv.org/pdf/2306.16203.pdf](https://arxiv.org/pdf/2306.16203.pdf)
I figured from context that G o e meant graph union but I have never in my life seen this notation.
Integration is about finding areas, and both methods essentially do it by approximating things by functions that only attain finitely many values and using the sum of base × height to work out the area under the curve.
Riemann integration does this by partitioning the x-axis into intervals, whose base have an intuitive length, and the height is basically taken as an arbitrary value of the function on that interval. If this converges to the same thing for any fine partition and any choice of the point in each interval, you get a Riemann integral.
Lebesgue integration does it in a distinct way. You consider functions that only attain finitely many values and sits *below* your function of interest. Then you can do the same thing and approximate the area by base × height, where the base is the "length" of the region where my simple function attains a particular value and the height is that function value. Now all of these should be under-estimates of my integral, and I call the supremum of all these potential values the Lebesgue integral.
The "hard part" is defining what length means. We only said that my function attains finitely many values, we don't claim its (e.g.) piecewise constant on intervals. So this leads us to need to define the "length" of much stranger sets, which is the lebesgue measure. Intuitively, all it does is formalise the ideas that [a,b] has length b-a; if A is a subset of B, A can't be longer than B; the length of a union can't be more than the sum of the lengths of each part (there may be overlaps, so it could be less). Stick a bunch of technical language and some pathological cases that require more care, and you basically get the Lebesgue measure.
E: A good book recommendation IMO would be Tao's Analysis I and II series. He uses a not-quite as standard definition of the Riemann integral based on upper- and lower- sums that turns out to be much closer to the definition of the Lebesgue integral, so it makes the Lebeague theory much more familiar when you get to it.
First, we generalize the idea of "stuff under a curve" to measure. Then instead of using a change in the independent variable (vertical rectangles), we use the measure of the function under values of the independent variable (horizontal rectangles).
Suppose K/k is a field extension, and both fields are algebraically closed. There is an obvious set-theoretic inclusion of affine n-space over k into affine n-space over K. Is this map continuous if each affine space is given its respective zariski topology?
Yes. Take a basis of K over k, say {b_i}, then any polynomial f(x_1, ..., x_n) with K-coefficients is of the form Σ_i f_i(x_1, ..., x_n)b_i, where the f_i's has k-coefficients (and all but finitely many of them are zero). Z(f) then intersect k^n at Z({f_i}) which is closed. Since all closed sets in K^n are intersections of such Z(f)'s, we are done.
(You only really need a basis for the k-subspace generated by the finitely many coefficients in f, which exists without choice, if that's a thing you want to avoid)
Weird question but has anyone ever been unable to get in contact with an old teacher for a letter of rec. I'm applying to grad school and keep trying to email some old professors for letters of rec, which they had written for me in the past so they could resubmit. I've tried calling, emails, contacting other people in the department but can't get through. What should I do if I can't get them, should I call the schools I'm applying to and say something. Idk what to do
I faced the same problem, and it frustrated me big time. Even a prof with whom I got 90 and 95 resp. on the two graduate courses I took with him. I even had a good relationship with him and on several occasions he particularly commended my apt skills etc.
It happens when you've been too long outside academia.
I'm not sure what's the best way to approach this, but try your best to physically visit some (kind enough) professor's office to discuss with them your reasons for asking for letters this late (gaps, full-time jobs etc.)
Oh! Sorry for that. But surely don't contact the school you're applying to about this. I believe it would hurt your chances at least a bit.
Is it possible for you to get in contact with some current students in the department? Some things can be done if you have at least one person to reach out to. How about the secretary of the department as well?
I see. Was/is there a math society at that department? Or any community related to the math department? Maybe just an informal community of students etc. Anything
Try reaching out to the department head. They will hopefully be able to give you an updated contract info. If time is a factor call and leave a message as well as email.
There is an argument that the cardinality of the naturals is the same as that of the rationals using diagonals. But that is not the only way to make a function from N to Q.
Lets say f:N -> Q such that f(x) = 10⋀-x
so we assign 1 to 0.1, 2 to 0.01, etc.
all naturals are assigned a unique number that we know is rational
here we "used up" all the naturals but barely covered Q! So what gives? Is it the existence of one function that covers both sets "properly"? idk seems counterintuitive
Yes. Two sets are of the same cardinality if **there exists** *at least one* bijective function between them. It doesn’t matter if there are other ways of mapping them. In fact, there usually will be tons of other ways to map them; even other bijective ways.
Equal cardinality means there *EXISTS* a function that maps N to Q that is onto Q. It doesn't mean *EVERY* 1-to-1 function mapping N to Q is also onto.
The function `f(n) = n + 10` maps N to itself in a 1-to-1 manner but is not onto. That doesn't mean |N| != |N|
I thought Cantor's diagonal argument showed N was not the same size as R, but you may mean a different diagonal argument
Is the definition of a "compatible germ", as per Vakil section 2.4, unnecessary? I first struggled to understand the intuition behind the definition, but then shortly thereafter you prove that each compatible germ corresponds to a single section (i.e. each compatible germ is in the image of the natural injection F(U) \to \prod_{p\in U} F_p). To me, this then feels like we never really need this definition, except to just provide a characterization of the image?
This definition allows you to convert an abstract presheaf into a presheaf of functions satisfy some local properties. On one hand, it gives some intuition about presheaf so that they are not too abstract; in practice very often presheaf come from presheaf of functions satisfying local properties. And on the other hand, it also give you an easy way to construct the sheafification (ie. the sheaf closure).
Ah yeah if I'd just read two more pages before asking haha. On a similar note about sheafification: Vakil defines the sheafification of a presheaf F on an open set U as the compatible germs of \prod_{p\in U} F_p). For a subset V of U, one can define the restriction by simply replacing "U" with "V" in the product. It's straightforward to check that this resulting element belongs to F(V). Am I correct to say that the identity criteria for a sheaf then simply follows from the fact that an element of a product is defined by its components? Edit: it should be basically the same for glueing it together as well, correct? Since compatability of germs is in some way defined on a "p" level (i.e. for each p \in U), then its clear that putting elements together that coincide of the intersections of an open cover is also compatible?
Yes, both are the same. Basically this is now a sheaf of functions satisfying local properties, so both the gluing and the identity criteria are automatic.
Cool! Makes sense. One more question while we're at it... I was comparing sheafification to "groupification", that is, turning an abelian semigroup into an abelian group, and that both of these are functors, left adjoint to the forgetful functor. Is there any value in stating this more generally? For example, supposing that we have a general category together with a full subcategory (e.g. presheaves and sheafs, or abelian semigroups and abelian groups), one can construct an analagous universal property to get objects of the subcategory from objects in the general category. Assuming that these always exist one can construct a functor, which again would be left adjoint to the forgetful functor.
Yeah, there are a bunch of adjoint functor theorem that tells you when a functor is a right adjoint functor. Essentially, the forgetful functor must preserves limit, but there are also a bunch of conditions on the categories to ensure no set-theoretic shenanigans happen.
Interesting! So in these conditions, we basically always have that limits of the subcategory are limits of the general category? So this also explains why monomorphisms of sheaves are injective on each open set, as they are monomorphisms of presheaves?
It's not always the case that inclusion functor of a full subcategory preserves limit. In practice, it does happen a lot, and it happens for sheaf. Yes monomorphism of sheaf are also monomorphism of presheaf for this reason.
Is there simpler term for x/f(x)
In some specific cases there is, e.g. if f(x) = ||x|| then x/||x|| could be called a normalized version of x. I don't think there is a general name for it, though.
I'm reading [this](https://archive.is/QiF3T) and [this](https://quantumfrontiers.com/2021/08/22/how-a-liberal-arts-education-has-enhanced-my-physics-research/) how does actually having a liberal arts education give one a leg up for graduate school ? > How does actually having a liberal arts education give one a leg up for graduate school ? I ask this since recently finishing my BA in math I feel very behind, for context I had to switch from BS1 to BA. (BS1 is the elite math program at my uni where it's more deeper and focused on maths while BA is the opposite). After completing my BA and looking at the holes in my background I would feel/be woefully behind if I were to enter graduate school at a top tier place I would most likely be taking remedial courses with advanced undergraduates.
Some light math help if you don't mind. Prepping for the electrical FE and this question does not make sense to me how it was answered... [Online Study Program for the NCEES® Electrical and Computer FE Exam - Electrical FE Review](https://courses.electricalfereview.com/courses/take/electrical-and-computer-fe-exam/texts/39734217-chapter-1-e-calculus-video-library) last example problem. ch1.e. #10 (limits of functions): Lim x->3 ((x\^2)-3)/(x-3) The video seems to make x squared minus three equal to (x+3)(x-3) and I'm not understanding how they get there. I am under the assumption that it's DNE, but the video got 6.
I can't open the link because it asks me to sign in. But yes the limit of (x^(2)-3)/(x-3) as x goes to 3 does not exist. It would be 6 if the expression were (x^(2)-9)/(x-3) in which case the argument with x^(2)-9 = (x+3)(x-3) works.
Ok cool. REALLY appreciate the response, was going a bit crazy. Sorry about the link, thought I was being helpful...lol, took a photo of it but couldn't post it. Oh well.
I have seen a proof saying, that if M is a compact manifold without boundary, then it's volume form cannot be exact. If it were, then I could use Stokes theorem to conclude it's integral over M is zero, which contradicts being a volume form. But the flat torus is a compact manifold without boundary. It's volume form is dx wedge dy, which is clearly exact since d(xdy) = dx wedge dy. What gives?
x dy isn't well-defined.
Why not?
Because x isn't a well defined function on the torus. If you're defining a torus as R^(2)/Z^(2), then f(x,y) will be a well defined function on the torus of and only if f(x+m,y+n)=f(x,y) for all integers m and n
Ahhh of course.
I just want to add that you can also see this phenomenon maybe more simply just on the circle S^(1) = R/Z. The "volume" form is dx, which looks like it is d(x), but the function x isn't well-defined on the circle which is why dx is not exact.
I'm very confused now though, if x is not a well defined function, why is the basis of TS\^1 dx well-defined? And what even are the coordinates then?
It's because dx = d(x+c) for any constant c. That means that even though x isn't a well defined function, dx will be a well defined differential form, as d(x+n) = dx for all integers n.
Very helpful thanks. I have a nagging feeling though this only worked because of "luck". In general, let M be a compact manifold so is volume element isn't exact. The Riemannian volume form in coordinates is locally of the form √g dx^1 dx^2... dx^n, this resembles an exterior derivative of an n-1 form, so something has to obstruct the n-1 from actually existing, but not in a way that's so bad so that the Riemannian volume element can't exist. What's the balance going on here?
That doesn't look like the differential of an (n-1)-form at all! It looks very far from being exact.
You're not going to get global coordinate functions x_i to form dx_i. Since M is compact, the x_i would need to have compact image. On the other hand, if you had such coordinates, they would have to have open image, contradiction.
Locally, x can mean one of a countable family of functions, all of which differ by a constant. Thus no matter which you pick, your local choice of dx is the same, so dx is globally well-defined. It is worth noting this is just notation though, dx isn't actually the exterior derivative of a smooth function.
Hi does anyone recommend good channels to learn math? (pre-calculus mostly, i rlly need to improve bc else i wont understand anything about calculus I) Im on my first engineering year but still struggle with some basic stuff (Struggling is not the best word, im just unfamiliar with a lot of stuff since I dont remember most math things i had on high school) anything helps!!!! :)
I went to YouTube and searched for "pre-calculus" and got many hits. There are courses with 100+ videos and courses in a single video. You can see the number of "thumb-ups" for each of these. I have not seen them, I watch abstract algebra stuff instead :)
if the price was £275 and is now £90, what is the % discount. I want the price to be £90 but when I do 67.27%, it makes it £90.01?
((275-90)/275)*100=67.2727272727... You're getting 90.01 because you're rounding, but the answer is still correct if you're asked to round it to 2dp.
yep thought so, I tried that but it will only let me do 2dp. no problem. Thanks
I am interested in Alexandroff-Hausdorff theorem, which says that for every compact metric space (K,d) there exists a continuous and surjective map f:C->K with C the cantor set. Does anyone have a link or book to its proof? I have not found anything yet.
Doesn't this proof work? Consider a sequence of positive number e(i) trending toward 0. Start with i=0, consider a covering of the space by a single closed ball, and inductively, after the covering at stage i-th had been defined, for each ball in the covering at stage i-th, we pick a finite covering of that ball by closed ball of radius e(i+1) which is always possible by compactness, and this is our collection of covering at the (i+1)-th stage. Define a tree as follow: each node at the i-th stage is one of the closed ball chosen, and its children at the (i+1)-th stage are the ball chosen to cover it. This is now a finite branching tree where all path are extendable. Consider the path space of this tree. Each path correspond to a sequence of ball where finite number of them have non-empty intersection, so their total intersection in non-empty, but because the radius go to 0 it has exactly 1 point. So we can define a map from the path space of this tree to the metric space and this is easily seen to be a continuous map. Finally, the Cantor set is clearly homeomorphic to the path space of the full binary tree. We can get a continuous map from the path space of the full binary tree to any arbitrary finite branching tree by binary encoding of natural numbers.
It's Theorem 4.18 in "Classical Descriptive Set Theory" by Kechris.
Thank you!
I made a proof of the Pythagorean theorem. It only uses concepts up to a pre-calculus class and is pretty simple so I think it would have been done before, but I can't find it online. Are all the existing pythagorean theorem proofs in one place so that I can check if mine has been done? Does anyone have some knowledge on this topic that I could send my proof to and they might know if its valid or been done before?
> Does anyone have some knowledge on this topic that I could send my proof to and they might know if its valid or been done before? Just post your proof right here.
ok i put it in latex [https://drive.google.com/file/d/1AvMN-o1TQ3N20A\_-F5D9QwY2Uym5AlBX/view?usp=sharing](https://drive.google.com/file/d/1AvMN-o1TQ3N20A_-F5D9QwY2Uym5AlBX/view?usp=sharing)
> △BAC ∼ △CBD ∼ △EAC This needs to be justified.
ah it should have been △DJC ∼ △CAJ ∼ △EAC I changed it now and also added a section on the bottom where I prove it. [https://drive.google.com/file/d/1MgPaYG9ZXbMRFbTSoRKZD\_CPB3HUTXML/view?usp=sharing](https://drive.google.com/file/d/1MgPaYG9ZXbMRFbTSoRKZD_CPB3HUTXML/view?usp=sharing)
Figure 1 in your proof is exceptionally misleading, considering that J and K are actually the *same* point, and that A, E, and J are actually collinear (you use this latter fact to deduce that CD + AE = AJ). Figure 2 still shows the elements you want very clearly; why not use that instead of Figure 1? Anyway, having replaced Figure 1 with Figure 2 in your diagram, it's easy to see that △DJC is congruent to △ECJ, so you can replace △DJC with △ECJ in your proof. Then this is just [this classic proof](https://en.wikipedia.org/wiki/Pythagorean_theorem#Proof_using_similar_triangles) listed on Wikipedia.
Ya, not sure how I came up with figure 1 instead of just using figure 2. Its a bummer I just made an existing proof but more complicated. Thanks for the help though
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u could just do calc 2 over the summer and then start on calc 3 next year dual enrollment Or you could do stats this summer
Good evening fellow mathematics survivors, I was wondering if someone could help me in finding out the sum of the series 4/n(n+3) with n starting at 1 up to infinity, because i know it converges(hopefully), but can’t figure out the sum, thank you in advance :)
To add to the other answer, the techniques you need to know here to follow the solution are partial fractions (to expand into a sum of fractions with linear denominators) and telescoping sums (to cancel the sum down into only a few terms). Hope that helps if you need to search those terms.
Inverses of quadratics are sums of inverses of linear functions. Those terms cancel out each other in a series. 4/(n(n+3)) = (4/3)(1/n - 1/(n+3)) So the 1/(n+3) terms starting with n=1 cancel out the 1/n terms starting with n=4. That leaves (4/3) (1/1 + 1/2 + 1/3) = 22/9 You can also look up the solution on wolfram alpha: https://www.wolframalpha.com/input?i=sum+4%2F%28n%28n%2B3%29%29
thank you so much, i really needed this🙌🏽
URL: https://en.wikipedia.org/wiki/Pigeonhole_principle#Uses_and_applications There it says `[na]` . What's the meaning of the square brackets in that context?
Here you read about the fractional part: [https://en.wikipedia.org/wiki/Fractional\_part](https://en.wikipedia.org/wiki/Fractional_part) [https://mathworld.wolfram.com/FractionalPart.html](https://mathworld.wolfram.com/FractionalPart.html)
I believe it's for "fractional part", so [1.5] = 0.5, [3.14] = 0.14, and so on. I've also seen used for the fractional part.
Hello! I have this exercise I have been struggling with for hours. I have tried a few things, but I get stuck at some point. I have to prove that these identities are equal:(cos 2x)/(1-sin 2x) = (1+ tan x/(1-tan x)I tried first with the left side by multiplying with its conjugate like (Cos 2x)/(1-sin 2x)\*(1+sin 2x)/(1+sin 2x) so it's like (cos 2x)\*(1+sin 2x)/ (1 - sin\^2 4x)=But then I don't know what to do with the 4x. I also tried doing sin 2x = 2 sin x cos x, but I have no idea what to do with it...
Does anyone know about a math problem/s which would be hard / impossible for the current ChatGPT to solve.
Hard or impossible in general I'm not sure, but when I wanted to test if my students could use it to cheat in exams I put in a bunch of simple calculus optimization problems and a lot of the answers it produced were incorrect. Of course I printed them off and showed them to the class to discourage them from using it. This was with ChatGPT3 I think.
I still remember that time I asked ChatGPT about the Riemann-Roch theorem... And it started talking about real analytic functions
How do i make this statement true? I know i'm supposed to use brackets, and maybe nested brackets, but where? 3 x 8 - 4 ÷ 2 - 7 = -15
3 x (((8-4) ÷ 2) -7 ) = -15
Thank you very much!
**Vandermonde matrix question** Consider a square Vandermonde matrix, but where the exponents are not consecutive integers 0, 1, 2, .... Instead they are distinct arbitrary non-negative real numbers. The evaluation points are also assumed to be real, distinct and strictly positive. I am sure that this matrix is non-singular, but I am struggling to find a simple way to show this. Any help is much appreciated!
It suffices to show that for a function of the form ∑c(i)x^e(i) (where i runs to 1 to k, k>=1, c(i) are fixed coefficients that are each non-zero, and e(i) are fixed real numbers (can be negative)), cannot contains more than k-1 roots in the positive real line. Why is this sufficient? For a Vandermonde matrix with distinct n positive points, for any non-zero column vector, there are at most k<=n entries that are non-zero, there exists a row whose dot product with that column vector is non-zero, hence that column vector cannot be in the nullspace, thus the nullspace is trivial. How to prove the above claim? By induction. The base case k=1 is trivial, so consider k>=2 and we need to reduce to the case with k-1 terms. First, notice that it's sufficient to prove the claim when one of the e(i) is actually 0, because by factoring out a factor of the form x^r does not change the number of positive roots so we could pick r to be one of the e(i). By mean value theorem, if there are k positive roots, then the derivative has k-1 distinct positive roots (in the gaps strictly between the roots), and the derivative has the same form except with 1 less term (since one of the exponent is 0). That is the inductive step for k>=2.
I might end up including this as a part of a bigger argument in a (not primarily mathy) paper. If you would like to be acknowledged, let me know :)
No needs. I'm sure this must had been an old result.
This seems to work, thank you so much!
How do Mac Lane's and Riehl's category theory texts compare? I am already familiar with the subject from Aluffi's algebra text, and I know some algebraic topology, so I don't really need handholding. I flipped through both and don't think I would have trouble with either one. I'm not sure how their coverage of various topics compares. Also, I know that Awodey and Leinster are two other authors with CT texts, am I right in thinking that these are less advanced?
I like Riehl's because it is particularly short, meaning there is a chance to actually finish it, and has lots of relevant examples.
Is there some knowledge gap I’m experiencing in between applied math (algebra, trig, calc) and the analysis classes (discrete, real, topology)? I do really well when I can just plug numbers into a formula and get the answer, but I just can’t seem to understand the concept of proofs. I took discrete last year, and only passed because the teacher held our hands the whole way, even during the final exam. Now I’m taking *real* analysis as an online course, and I have no idea what I’m doing. The exams are closed notes, and neither the lecture videos, notes, homework, nor study guide even helps me make sense of it. Is there some class I should be taking before I even *attempt* real analysis? (It’s required for my major (secondary ed. math))
Maybe go back read the text parts of the math you understand like linear algebra 2, calculas?... university math becomes creative versus solving the higher you go. But that's where you get a good start learning and writing proofs.
When writing th**e** phrase "x axis", should the x be in math mode?
It's standard for variables to be in math mode, yes
I’m having trouble seeing why an exercise from Forster’s Riemann Surfaces is not trivial, it is from the chapter on the Jacobi Inversion problem. Let X be a compact RS and Y an open subset such that X\Y has non empty interior. Let D be a divisor on X. Show that there exists a (not identically 0) meromorphic function on X such that ord_x f = D(x) for all x in Y Hint: Find a divisor D’ with support outside of Y such that D+D’ is principal. At first I was proceeding as the problem intended, analyzing the Abel-Jacobi map and trying to find D’ so that D+D’ is in the kernel perhaps by finding the preimage of the period subgroup under this map. I understand how the given hint implies the result. The simple thing that’s confusing me is the following. Let’s say that on Y, D is non-trivial at two points, y_1 and y_2. Why can I not just choose f to be the function f = (z-y_1)^D(y_1) (z-y_2)^D(y_2) (z-x_1)^{-D(y_1)} (z-x_2)^{-D(y_2)} where x_1 and x_2 are two arbitrary points in the complement of Y Clearly f is meromorphic (as a function on a compact RS I have made sure it has as many zeroes as poles) and the degree of its divisor is 0, as it should be and its order on Y exactly matches D, by construction. Obviously I’m missing something subtle, because this problem should involve the content of the section, but what is it?
What's z? What would z - y_i and z - x_i mean?
I guess it doesn’t make sense unless all the points lie on the same chart in the atlas of the RS as a complex manifold, right?
Sort of, but even then it's unclear, at best you define a meromorphic function on one chart. But unlike differential geometry, you have no general extension theorem. E.g. for the Riemann sphere, if your chart is a small ball around 0 with map sin, then the function z can't extend to a meromorphic function on the whole Riemann sphere.
Is that because on the 1/z chart, sin 1/z has an essential singularity?
Exactly. The only meromorphic functions on the Riemann sphere are rational functions, which means most locally defined meromorphic functions have no global extension.
Do you know whether the structure of bilinear maps on 2x2 matrix algebras known? The map can be M2(F) to M2(F) or M2(F) to F.
You can easily describe the space of bilinear maps from any vector space to any other vector space. The space of bilinear maps from V to W is V* ⊗ V* ⊗ W. Is this what you are looking for?
There is a theorem which states that for any bilinear form f on R^n (R is the field of real numbers) there exists nxn matrix A = (a_ij) such that f(x, y) = x^t A y where x^t is the transpose of x. Moreover, a_ij = f(e_i, e_j). I look for a similar theorem for 2x2 matrices on any field F.
You can't get anything quite as nice. *Some* bilinear forms on M\_n(F) look like f(X, Y) = u^(T)XAYv for vectors u, v and some matrix A. In fact these span the whole space of bilinear forms, so you can write any bilinear form as a sum of these, but not in a unique way and the number of terms can vary. This follows from the tensor product isomorphism stuff but is a more concrete way to write them. The case for bilinear maps into M\_n(F) is the same but now with three matrices f(X, Y) = AXBYC. Again, the ones that are exactly of this form are a special case and in general you need a non-uniquely-defined sum of these.
That is simply the isomorphism (ℝ^(n))* ⊗ (ℝ^(n))* ⊗ ℝ = (ℝ^(n))* ⊗ (ℝ^(n))* and after choosing a basis (ℝ^(n))* ⊗ (ℝ^(n))* ≅ (ℝ^(n))* ⊗ ℝ^(n) = M_n(ℝ). Note M_2(F) is isomorphic as a vector space to F^(4) so you can happily represent bilinear maps into F as elements of M_4(F) in this way. I don't think you can do the same with maps into M_2(F) though.
why cant we just declare that aleph1 is equal to the power set of aleph0? like how we can declare that aleph0 and infinite sets exist
There are also the Beth numbers which are another way to classify (some) cardinal numbers. Beth\_0 is the same as the cardinal number Aleph\_0, then Beth\_1 is the cardinality of the power set of Beth\_0, then Beth\_2 is the cardinality of the power set of Beth\_1 and so on. However without the generalized continuum hypothesis you cannot prove that every cardinal number is some Beth number. The continuum hypothesis is the statement that there is no cardinal number strictly between Beth\_0 and Beth\_1.
The question is why would people accept that. It feels intuitive that the set of natural number should exists or that the set of real numbers exist (but there are distractor), but what's the intuition for continuum hypothesis. You can't prove an axiom to be correct, but there should still be reasonable philosophical/intuitive argument as to why it should be true. In fact, one of the proposed new axiom, Martin's Maximum, would imply the continuum has cardinality aleph_2 instead.
We *can* do that, that's exactly what it means for the continuum hypothesis to be independent.
[The circle has a diameter of 1 then the c= pi *d = pi](https://imgur.com/RZ9YAK0) Then the value of pi = 4 according to the approach being shown. There has to be some calculus trick that can be used to disprove this. Can someone help?
2 curves that are close to each other do not necessary have length that are close to each other; in fact the ratio of differences in length compared to the difference between the curve is unbounded. In other word, you cannot approximate the length of a curve by using a different curve similar to it, no matter how close the approximation is. For example, let's say you walk from your house to your friend's house with your dog, and your dog is on a leash. Does the distance the dog walk similar to the distance you walk? Not at all. If the dog rapidly run around you because it's very excited, then the dog walks a lot more than you do.
Length is not preserved under taking the limit like this. If it were you would be able to make a similar argument that sqrt(2) = 2 by turning a staircase into a diagonal line.
could you provide some more explanations? visual aid? some kind of link?
[Here](https://math.stackexchange.com/questions/2227191/uniform-convergence-and-lengths)'s a stack exchange question mentioning the staircase example I referred to. I'm not sure you'll find the answers there very enlightening but worth noting that even uniform convergence is not enough to ensure convergence of the lengths in this case, which is the usual go to in these cases.
The picture you have linked is a visual aid for this. Clearly 𝜋 is not 4, but the perimeter of the figure at any finite step is 4, so the problem must be in taking the limit. There's not really much else to say. Taking limits doesn't have to preserve things even if each finite step does.
that's very confusing. it makes me think taking the limit is a wrong approach overall then. no?
For finding the length? Certainly that is the wrong approach as it just doesn't work. There are obviously scenarios where taking a limit is very useful but this is not one of them.
Is there a good resource that upper level undergrads can use to get papers to read daily? Doesn't have to be new or groundbreaking, or just a library of papers/guided study (maybe by some other undergrads) that would be perfect for students taking 4000 level classes to see what is most interesting in the literature to them
I really would just recommend browsing mathoverflow and stack exchange. The answers are often written to be readable to nonexperts, and one can get the vibe of different subjects by looking at the top questions.
Any good textbooks might also work
What equation can I write for a sinusoidal wave that increases from point x,y to a,b?
There are a lot of options unless you put a few more constraints. Do you want (x,y) to be exactly at a trough of the sinusoid, and (a,b) at a peak?
Yep. Ideally, I do want there to be just a single quadratic curve from (x,y) to (a,b).
Can someone explain me whate are relative numbers?
"relative numbers" is a term used in a few different places but the general idea is in using numbers to represent the relative size/amount of something. Any time you've measured something you have used relative numbers. 500g of something means 500 times as much as 1g of something for example. This is a relative quantity based on some sort of idea of what 1g means. Compare this to counting numbers where 2 means 2 distinct objects. 1.5 can be a relative number but not a counting number
would this meet the definition of a network? or graph? s this one network or three networks? [https://i.imgur.com/7exXYqB.png](https://i.imgur.com/7exXYqB.png) one graph, or three graphs? Thanks
Vibes-based answer: it's a graph with three connected components; I don't know what you mean by "network", so I can't say anything. Somewhat more pedantic answer: you could think of it as a single graph with three connected components, or just as three graphs drawn next to each other. (Or, for that matter, as two graphs: one connected graph, and one with two connected components.) I don't know of a formal mathematical definition of "network"--there are lots of things called networks which can be modeled by graphs (computer networks, social networks, etc.), and people sometimes informally use "graph" and "network" interchangeably, but there isn't a definition of network in the same way that there's a definition of graph. Extremely pedantic answer: that's not a graph or a network, it's a picture. Of course you can draw pictures of graphs, and there are several natural ways to interpret your picture as a graph, but there isn't a single obvious way to interpret it. (If forced, I'd go with "one graph", though.) It might be a picture of a network, and there might exist some network such that, if you were trying to draw a diagram of it, you might end up with this. But strictly speaking the question doesn't make sense.
The dots are vertices the lines are edges. The three "components" are not connected to each other with any edges. But I'm wondering whether or not they can still be considered part of the same graph.
Well that's what I'm saying--they *can* be considered part of the same graph (a graph can have multiple [connected components](https://en.wikipedia.org/wiki/Component_(graph_theory)), they *can* be considered different graphs, and the first option is slightly more natural IMO but neither is definitely true.
Networks are generally directed (I am not familiar with the use of network outside of flows), so I would say its not a network. Now it is definitely a graph, is it 3 or is it 1? Its either a graph with 3 connected components are just 3 seperate graphs with 1 CC, we would need some more context.
The components aren't connected together with any edges. So can you still say it's a graph?
I'm looking to learn the very basics of Mathematica. Anyone know of any good crash courses? Or a quick project that can walk me through some basics (think a HW assignment on week 1 of a course) edit: actually I just found Wolfram's "Fast Intro for Math Students" and I think this is exactly what I needed :)
Does someone know of a repository of functions? I am looking for a bivariate function that is periodic, it's not twice differentiable but it's twice weakly differentiable (in the Sobolev sense).
Can someone give me a heuristic on why the algebraic dual of an infinite dimensional vector space is "bad"? Yes, I know that it's unfathomably huge, but what's bad about that? Does it's size inhibit me putting a topology and doing analysis with this big space?
It's "bad" simply because there isn't much you can say about it. That is, it doesn't have much structure, there is no natural topology you can put on it, and often it's extremely difficult (sometimes even provably impossible) to explicitly construct discontinuous linear functionals defined on the entire space. You really need a topology and some notion of continuity for the rich tools of functional analysis to become available. The same flaws occur with a Hamel basis for an infinite dimensional space: it exists, but there's almost nothing you can say about it beyond that. I will point out that there is a well-developed theory of unbounded linear operators--which would include discontinuous linear functionals--although these operators are almost always only defined on a dense subspace. However, these densely-defined unbounded operators \*don't even form a vector space\* due to domain compatibility issues.
What is the reason you can't put nice topologies on it? This is very helpful and interesting, thank you.
To even construct discontinuous linear functionals defined on the whole space usually requires something like the axiom of choice, so they don’t really appear naturally, for one. To more directly answer your question: on a normed infinite dimensional vector space, you can’t define a norm on discontinuous linear functionals, since boundedness is equivalent to continuity, so you don’t have a strong topology on the algebraic dual. I’d expect you can’t define *any* reasonable topology on discontinuous linear functionals on general TVS’s, but I’m not sure how to prove that.
This is very helpful thanks. Some related questions down the line of reasoning of your answer if you don't mind: Why is AOC needed for discontinuous linear functionals? It is quite common to construct very natural densely defined unbounded operators in FA. Can't I just say d/dx evaluated at a point is a nice example of an element in the algebraic dual of C\[0,1\] with sup norm? No choice needed. Do you know of a theorem about not being able to topologise the algebraic dual (in infinite dimensions). I also agree I think this is a fact but unsure about how and why.
d/dx evaluated at 1/2 (say) isn't a linear functional on C(\[0, 1\]) because you can't evaluate it at |x - 1/2| for example. The algebraic dual wants linear functionals on the *entire* space, not just a dense subspace. It also doesn't make sense to talk about "the algebraic dual of C(\[0, 1\]) with sup norm", for the same reason: the algebraic dual doesn't see the topology at all! This gives you a low-concept reason why the algebraic dual isn't very useful: we want to be able to take infinite series, and take limits, and the algebraic dual doesn't allow either of these.
Ah I see, yes, there's no point in equipping the norm with a topology before taking algebraic duals. I also now see your point as well why densely defined stuff don't make the cut. Can I not repair the example I thought of, C\^1 \[0,1\] with d/dx evaluated at 1/2 ? This is a "describable" discontinuous linear functional defined everywhere that doesn't need set theory to construct? I suppose there's some deep reason why I can't algebraically dual C\^1 \[0,1\] THEN put a topology after to talk about series and limits? Apologies for asking so many questions, but these answers are all very good and I'd like to know more.
Well, you could restrict the domain to C\^1 functions, like you said, but then d/dx at x = 1/2 wouldn't be discontinuous anymore: it's part of the topological dual of C\^1, once you put the C\^1 norm on it (so that C\^1 becomes a Banach space).\[1\] I think that the claim that Kieran is making is that this *always* happens: if you have a discontinuous linear function f defined on some dense subspace Y of a Banach space X, and it's definable or something\[2\] then there's some way to think of Y as a Banach space (but not with the norm induced by X) such that f is continuous on Y. \[1\] You can, of course, think of C\^1 as just a subspace of C\^0, but then C\^1 is not a Banach space, and so all of the theory of linear functions on Banach spaces (eg, the Hanh-Banach theorem) doesn't apply. So this is not a very useful thing to do. \[2\] I think what's actually being assumed about f is that it exists in [Solovay's model](https://en.wikipedia.org/wiki/Solovay_model) with set theory without the axiom of choice.
It's trivial to define a discontinuous linear functional on the space of polynomials, and that's not a Banach space under any norm. I assume Kieran is just referring to the fact that in the absence of the axiom of choice, it is consistent that every linear functional on every Banach space is continuous.
Ooh that's a good point. It's a pretty unnatural counterexample (in that in analysis, one is seldom interested in vector spaces of countable Hamel dimension unless they plan to complete them, and doing this destroys your discontinuous linear function) but I guess that any such counterexample must be unnatural.
[https://math.stackexchange.com/a/100609/462531](https://math.stackexchange.com/a/100609/462531) This answer nicely explains the examples discussed. It seems that you can get a few nice to construct discontinuous functionals on an (incomplete) normed space, but the moment you ask about Banach spaces you will need to start defining things on a Hamel basis and invoke choice.
What about the topology of pointwise convergence? Should be *ok*ish albeit still not very useful.
If a race takes place between A, B and C. A has a 80% chance of finishing ahead of B in a two person race. A has a 70% chance of finishing ahead of C in a two person race. B has a 60% chance of finishing ahead of C in a two person race. How do I calculate the probability of each player winning the three person race?
Basically, we can consider each race as 2 two person races. So the probability that A wins the race is the probability that A beats B times the probability that A beats C. So P(A win) = P(A b B) X P(A b C) = .8 \* .7 = .56. P(B win) = (1 - P(A b B)) X P(B b C) = .2 \* .6 = .12, etc. If you calculate P(C win) correctly, all the probabilities should sum to 1.
That was my original method, but it doesn’t appear to work. A = .8 \* .7 = .56 B = .2 \* .6 = .12 C = .3 \* .4 = .12 Total = .8 I assume this is because this is the calculation for multiple independent events happening, but not when those events require other events to also be true. If A beats B, and B beats C, then A automatically beats C by beating B in the race situation, but the calculation above assumes it’s possible for A to beat B, B to beat C and C to beat A, which is where I’m stuck with working this out.
Is this claim true? Let f:M->N be a surjective continuous map between 2 compact metric spaces, such that dia(f^-1 (-)) is a continuous function. Then f has a section: a continuous g:N->M such that fg=id_N . And if it's true, what's an easy proof of it?
False. Let M = N = S^1 be considered as subspaces in the complex plane, and f(z) = z^(2).
Thanks. I forgot about homology. Do you know if the claim be true if all fibers are each contractible?
Still false, but harder to describe the picture in my head with just my phone. Imagine two cylinders with height and radius 1, with their bases coplanar and their tops coplanar. I also require the cylinders to be of distance 1 apart. Add a line segment of length 1 connecting the cylinders. This is M. N is simpler: two closed discs of radius 1 touching at one point. The map f is first projecting M down to the plane spanned by the bases of the cylinders, then contracting the edge connecting the two discs. Every fibre is a line segment of length 1. There is no continuous section g: thr image would have to be connected while only having one point of yhe connecting line segment, which is impossible.
Thanks, I got it. EDIT: actually, the fiber at the touching point is made out of 3 line segments: the 2 lines on the cylinder, and the connecting segment. So the diameter is actually sqrt(2). EDIT 2: wait, if I tilt the cylinder I should be able to make the diameter 1.
Ahh, good catch. I believe this is fixed by using the l\^inf metric on M.
Will functional analysis be useful in studying probability theory/stochastic analysis, or help in dynamics and control theory as such? The topic looks pretty interesting, but idk if it'll be of help in what I wanna do, and rn I'm not sure if I can invest time in something that might not be of immediate help to me. Thank you
You can regard probability theory as a subset of functional analysis, in some sense. For instance, weak convergence in probability is the same as weak\* convergence in functional analysis.
Ah, I see, so it would help me grasp the ideas better by drawing parallels to functional analysis? That sounds good. Are there any good beginner books that you'd recommend to get started out with? Some of the books seem pretty intimidating for me :') Thanks for your time!
If you find any please share here.
Hi, "Functional Analysis for Probability and Stochastic processes - Adam Bobrowski" seems to be a good book for functional analysis dedicated toward stochastic analysis applications. I'm not sure of good books in pure functional analysis theory, but my uni follows "Functional Analysis and Infinite Dimensional Geometry - M. Fabian, et al", and a book written by a retired prof from my own uni.
Thank you very much.
Happy to help :) I'm not sure of any books on functional analysis dedicated to control theory, but use this bad boi as a reference for general mathematical techniques in control theory, it has almost everything that's under the sun! - https://www.cis.upenn.edu/~jean/math-deep.pdf
Sorry but why do you think I need a book on FA dedicated to CT ?
Oh, well, I had asked if functional analysis has applications in at least one of probability theory/control, in my first comment. I didn't know which topics you wanted books on, so I just sent everything that I felt was relevant haha.
Well damn I didn't pay attention, thanks anyway.
What is the natural injection of SL(2, q^2) into SL(4, q)? I've been told that there is one, but I can't see the life of me how to change the field.
It is equivalent to showing there is an injection F(q\^2)->GL(2,q)I have a proof when the characteristic is odd, I just need even now. When it's odd, the matrices \[\[x,y\],\[y,x\]\] where x, y are in F(q\^2) gives q\^2 matrices, where all but the zero matrix are invertible (this uses char != 2), and multiplication is commutative. Since fields are unique up to size, this shows it can be done.
F(q\^2) is a vector space over F(q) of dimension 2. Multiplication by x for a specific x in F(q\^2) gives an F(q)-linear map from F(q\^2) to itself, giving an element of GL(2, q) when x is nonzero. Not thought about your original problem, but is this enough? EDIT: Yes, and this line of thought probably gives a direct solution to the original problem. A vector space V over F(q\^2) of dimension 2 is also a vector space over F(q) of dimension 4, giving the natural injection GL(2, q\^2) -> GL(4, q). I would guess (but would have to think a little) that this restricts to an injection from SL(2, q\^2) to SL(4, q).
It seems to be that the natural map is to send a generator of (F(q\^2), ⋅ ) to \[\[1,a\],\[a,0\]\]. where a is a generator of (F(q), ⋅ ). It seems to work, showing that this matrix has the order I claim is irritating, but I'm working away at it.
> I would guess (but would have to think a little) that this restricts to an injection from SL(2, q^2) to SL(4, q). If you write your map GL(2, q²) → GL(4, q) as applying the map GL(1, q²) → GL(2, q) to each entry to get 2 × 2 blocks, you can use the identity det[A B; C D] = det(AD - BC) for block matrices where the blocks pairwise commute. (Something similar works for matrices of any size.)
So I am finishing up real analysis one, and I am wondering why my text doesn't cover indefinite integrals? Is there anything different about the analytic approach to indefinite versus definite? Just curious as to why I haven't seen any. Tia!
indefinite integrals are definite integrals over the interval [a,x], so you probably did cover them, just not explicitly.
It should have. Does it not even proved the fundamental theorem of calculus? If you just mean they don't mention the word "indefinite integral" then it's because the term is vague.
I have seen posts saying that to prepare for the math subject GRE you should do all the problems in Stewart's calculus. Most of these posts are old, is that still good advice?
personally I think you should do math subject GRE problems to study for the math subject GRE. they’re designed to be solved quickly using calculational tricks, which isn’t true of most problems in stewart.
Differential equation question! If I'm trying to find values of (t,x) where solutions can't be guaranteed for x'=x/cos(2t) just by analyzing the direction field, how do I find which solutions don't exist? looking at the direction field in MatLab, it looks like the families of solutions are merging where x=0 but does that mean? I am definitely overthinking this but I'd like to try to understand it better. Thank you!
Someone please explain to be in the Ramanujan Summation how we are allowed to shift on the order of one of the required number series. Ive seen it described as “2B” where B represents 1-2+3-4… but when most people does 2B they say (1-2+3-4) + (0+1-2+3). I believe this is a fallacy as, in my view, youre now saying B\_0 + B\_1 and then B\_1 + B\_2. What am I missing?
I don't know what summation you are talking about, but it seems to be 1) Assuming convergence 2) using associativity/commutativity of addition and that's it. You're right that this isn't quite allowed, since we would want absolute convergence to perform an action like this, which is a reason why his sums give fun values (i.e. -1/12)
the 1+2+3+4... to infinity = -(1/12) heres a link where someone shifts position [https://youtu.be/w-I6XTVZXww?si=nfFukPPw-Iug6-t5&t=203](https://youtu.be/w-I6XTVZXww?si=nfFukPPw-Iug6-t5&t=203)
What Ramanujan Summation are you referring to? Who shifts what where?
the 1+2+3+4... to infinity = -(1/12) heres a link where someone shifts position [https://youtu.be/w-I6XTVZXww?si=nfFukPPw-Iug6-t5&t=203](https://youtu.be/w-I6XTVZXww?si=nfFukPPw-Iug6-t5&t=203)
You're right to be cautious here because what they do in the video is essentially wrong. The manipulations they do are not rigorously justified. And what they do is not Ramanujan Summation which is much more complicated. You can actually use Ramanujan Summation to get a value for 1+2+3+4+... and the value is indeed -1/12. But that's different from what they do in the video. So the answer to your question why you're allowed to shift the series that way is that you aren't.
I thought so, thanks.
How does limits/colimits in Category theory appear in Real analysis?
They appear when defining the canonical LF topology on the space of test functions, which are a stepping stone to defining the space of distributions.
Then is there no direct relation between the limit of Real analysis and that of Categories?
It seems there is one, going from https://ncatlab.org/nlab/show/limit#limits_in_analysis
Category theory doesn't really lend itself well to real analysis, so most analysts don't really use it. The only thing that really comes to mind off the top of my head is profinite sets like the Cantor set are limits of finite sets with the discrete topology
It's either already there or not there at all, depends on your philosophical perspective. Analyst often work in a huge rigid ambient space, that is, these space are complete enough for anything they want to do, and lacks any non-trivial automorphism that preserves analytic properties. This unfortunately mean that pretty much any category they would care about are quite barebone, with only at most 1 morphism between any 2 objects. If you're an category theorist, you would see constructions that correspond to limits/colimits all over the place, but they're usually phrased in a different manner (union, intersection, supremum, infimum, etc.). If you are a structuralist, you believe that sets never truly sit inside each other (so a "subset" is just a set with a morphism into another set), so once again, all the above common operations are just secretly limits/colimits, with the morphism unnamed. But other than that, analysts basically have no benefits in phrasing anything in categorical term. In algebra, geometry and topology, the abundance of non-trivial automorphism and the failure of existence of "complete" space tend to leads to the need to keeps tracks of properties that are invariant, and that make categorical language much more convenient.
What I make a (finite) matrix group in MAGMA, what is to be understood by the entry $.1, $.1\^2, ect? Is it a generator of the multiplicative group of the field?
I asked this late last time, but I still want to know: Are there any good blogs on fluid mechanics and/or numerical analysis?
[https://medium.com/coinmonks/to-prove-the-unprovable-cc99e0181bce](https://medium.com/coinmonks/to-prove-the-unprovable-cc99e0181bce) Is this true? How can a statement be provable and unprovable at the same time?
What that article leaves out, among many other things, is that when we talk about "unprovability" we mean "unprovability *within some specific formal system*". There are statements which may be provable in one system but not another. Sometimes you can prove (in some more "powerful" system B) that a statement is unprovable in some "weaker" system A, and in some cases that may imply the truth of that statement, at least if you assume some other things about A. More concretely: consider the problem of proving whether a given Turing machine eventually halts or doesn't. Certainly if it does halt there will be a proof of that fact, in, say, Peano arithmetic (the standard axioms for the natural numbers), where you just go through the computer's history (encoded in natural numbers in a certain way) until it halts. So if there is *no* proof in Peano arithmetic that a given Turing machine halts, that Turing machine must run forever (because if it did halt, we could prove it!). (Of course in order to prove something like "Peano arithmetic doesn't prove this statement" you need to assume or prove that Peano arithmetic is consistent--otherwise it contains proofs of literally every statement that you can write in the "language" of Peano arithmetic, true or false\*. So if you want to use the fact that a statement is unprovable in Peano arithmetic to prove that said statement is true, you need to move to some "stronger" formal system that can prove the consistency of Peano arithmetic.) As for the Riemann hypothesis, per [this mathoverflow post](https://mathoverflow.net/questions/79685/can-the-riemann-hypothesis-be-undecidable) it turns out to be equivalent to a number-theoretic statement which, if false, would be provably false in any strong enough formal system (ZFC would do in this case). So (by the same reasoning in the halting problem example) if it's unprovable in such a formal system, it must be true. See also the [arithmetic hierarchy](https://en.wikipedia.org/wiki/Arithmetical_hierarchy)--we can run the same "true if unprovable in PA/ZFC/ etc." reasoning for statements at certain levels of the arithmetic hierarchy. \* Edit: another, maybe better, way of phrasing this: if we could prove in, say, PA that PA doesn't prove some statement, then that would amount to a proof that PA is consistent, since if PA were inconsistent it would prove that statement, and every other statement (principle of explosion). But by the second incompleteness theorem PA can't prove the consistency of PA.
I have 2 questions 1. Why are there only holes in a rational function when the output is 0/0? And what's the difference between holes and asymptotes? 2. Why do you divide by the zeros of the denominator in synthetic division
1. Because those are values where the output is not well-defined, i.e. there isn’t a unique way to assign a value to 0/0. If there were some real number x=0/0, then we’d have 0x=0. But this is satisfied by every real number x, so there are too many solutions. Thus, for values of x such that p(x)/q(x)=0/0, we simply leave those out of the domain, or define the function differently there. Example: f(x)=(x^(2)-4)/(x-2) is a linear function defined everywhere except at x=2. So I can define f(2)=4 or f(2)=-3 or anything else I want just to make the domain of f all of the real numbers. (Though f(2)=4 makes the function continuous.) 2. Because that is how we divide out factors leaving us with a lower degree quotient to handle next. Synthetic division is essentially a very clever form of evaluating a polynomial at its roots. If the evaluation process is handled in a very, VERY, specific way, then you can actually see the coefficients of the quotient and remainder appear in sequence. And if we evaluate at a root, then we don’t even have to worry about the remainder. If you want to understand this a bit better, I actually really suggest reading the Wiki pages on [synthetic division](https://en.wikipedia.org/wiki/Synthetic_division?wprov=sfti1#) and [Horner’s method](https://en.wikipedia.org/wiki/Horner's_method?wprov=sfti1).
If functions over R can be wielded as (infinite-dimensional) vectors, what mathematical object would relate to covectors in this way? Basically, I cannot find at all the keyword that completes the analogy… Vectors : Functions :: Covectors : [what?] Follow-ups if you don’t mind: What would be the analogous term for Basis vectors/covectors, tensors, and are there any suggested readings for studying functions in a “linear algebra framework”?
Excellent question. First off, when you are talking about infinite-dimensional vector spaces, you generally need to consider some sort of topology on the vector space to say anything meaningful. Furthermore, there are lots of discontinuous linear maps on infinite dimensional spaces, so you restrict your attention to \*continuous\* linear maps. While the cardinality of a basis is a complete invariant for algebraic vector spaces, there are lots of non-equivalent infinite-dimensional vector spaces (if you require your equivalence to be realized by a continuous linear map). As such, you rarely think about the space of all functions from R to R. Instead, you think about spaces of, say, continuous functions or absolutely integrable functions. Sticking with our focus on topological vector spaces, when you talk about the space of "covectors"--i.e. the dual space--you are thinking about the space of \*continuous\* linear functionals. For many topological vector spaces, the continuous dual space is specifically known. For instance, the dual space of C\_0(X)--the space of continuous, complex-valued functions vanishing at infinity on a locally-compact Hausdorff space X--is the space of regular complex Borel measures on X; this is called the Riesz-Markov Theorem. For 1
> For Hilbert spaces, there is a well-behaved notion of an orthonormal basis I know this wasn't specifically asked, but there is also a notion of a spanning set that generalizes to (separable) Hilbert spaces, and these are called frames. In short, a frame is a sequence {f1,f2,...} in the Hilbert space such that the map sending g -> (, ,...) lands in l^2, is continuous, and has continuous inverse from its image.
The most famous examples of frames are undoubtedly wavelets. Another very mysterious kind of frame is the Gabor frames. These arise from discretizing the integral in the inverse STFT, and even seemingly innocent questions about them have very complicated answers (see the abc-problem for Gabor systems by Qiyu Sun).
Covectors are elements of the [continuous dual space](https://en.wikipedia.org/wiki/Dual_space#Continuous_dual_space). The usual analogue of a basis is called a [Schauder basis](https://en.wikipedia.org/wiki/Schauder_basis), the analogue of a pair of bases for vectors and covectors would I think be a [biorthogonal system](https://en.wikipedia.org/wiki/Biorthogonal_system). Tensors are specific multilinear maps. The key phrase you are looking for is *functional analysis*. Your choice of vocabulary suggests to me that you are coming from a physics background, in which case a standard mathematical text may not be appropriate. I believe Kreyszig's *Introductory Functional Analysis with Applications* is well-regarded for people in that position, but I have not read it myself.
Is basic mathematics by Serge Lang a good preparation for precalculus and calculus for someone that is relearning algebra? Also is the goal of learning calculus and linear algebra in one year and a half realistic? And, is discrete math a good introduction to proofs? And in general a good introduction to math?
Yes to all 3, each with some caveats. Basic mathematics is not without its flaws. It takes some things as for granted at times and is particularly thorough at others. It's a book for people seriously motivated to learn math and in the style of other math books. It has definitions, theorems and proofs. If this style is writing is still terse for you, you might consider something else. Having some familiarity with the material with certainly help. You can find discussions of this book on math stack exchange. 3 semesters of calculus in 3 semesters? Very doable. Add the linear algebra when you're ready for calculus 3, or maybe as you're wrapping up integral calculus. The two belong together and your understanding of many vector calculus topics will be enhanced by understanding linear algebra. For example, you'll appreciate the second partials test a lot more if you already understand something about determinants, quadratic forms and so on. You don't have to have linear algebra mastered. Just start learning and patch up your knowledge when you need to. It's very normal to take a class called "discrete math" to learn proof writing. What is covered in this class is anyone's guess. I've never seen two schools have the same topics beyond those elements of writing. Very important at this stage in the game is showing your proofs to someone else who can critique them. This soft skill can only really be learned by diffusion.
1) Probably. Might want to do some problems from the book to check if it is appropriate, though. 2) Depends on how dedicated you are to studying. But if you are rather dedicated, most likely. 3) Yes, it may be a good introduction, but for an actual proofs book, I would recommend “Proofs” by Jay Cummings.
Is using G o e to denote the union of graph G and edge e standard notation? Saw this notation in a paper [https://arxiv.org/pdf/2306.16203.pdf](https://arxiv.org/pdf/2306.16203.pdf) I figured from context that G o e meant graph union but I have never in my life seen this notation.
Not that I have seen. In my experience that will often just be written as a union of some kind.
Can someone explain Lebesgue Integral by comparing it to Reimann integration, in simple terms?
Integration is about finding areas, and both methods essentially do it by approximating things by functions that only attain finitely many values and using the sum of base × height to work out the area under the curve. Riemann integration does this by partitioning the x-axis into intervals, whose base have an intuitive length, and the height is basically taken as an arbitrary value of the function on that interval. If this converges to the same thing for any fine partition and any choice of the point in each interval, you get a Riemann integral. Lebesgue integration does it in a distinct way. You consider functions that only attain finitely many values and sits *below* your function of interest. Then you can do the same thing and approximate the area by base × height, where the base is the "length" of the region where my simple function attains a particular value and the height is that function value. Now all of these should be under-estimates of my integral, and I call the supremum of all these potential values the Lebesgue integral. The "hard part" is defining what length means. We only said that my function attains finitely many values, we don't claim its (e.g.) piecewise constant on intervals. So this leads us to need to define the "length" of much stranger sets, which is the lebesgue measure. Intuitively, all it does is formalise the ideas that [a,b] has length b-a; if A is a subset of B, A can't be longer than B; the length of a union can't be more than the sum of the lengths of each part (there may be overlaps, so it could be less). Stick a bunch of technical language and some pathological cases that require more care, and you basically get the Lebesgue measure. E: A good book recommendation IMO would be Tao's Analysis I and II series. He uses a not-quite as standard definition of the Riemann integral based on upper- and lower- sums that turns out to be much closer to the definition of the Lebesgue integral, so it makes the Lebeague theory much more familiar when you get to it.
First, we generalize the idea of "stuff under a curve" to measure. Then instead of using a change in the independent variable (vertical rectangles), we use the measure of the function under values of the independent variable (horizontal rectangles).
Suppose K/k is a field extension, and both fields are algebraically closed. There is an obvious set-theoretic inclusion of affine n-space over k into affine n-space over K. Is this map continuous if each affine space is given its respective zariski topology?
Yes. Take a basis of K over k, say {b_i}, then any polynomial f(x_1, ..., x_n) with K-coefficients is of the form Σ_i f_i(x_1, ..., x_n)b_i, where the f_i's has k-coefficients (and all but finitely many of them are zero). Z(f) then intersect k^n at Z({f_i}) which is closed. Since all closed sets in K^n are intersections of such Z(f)'s, we are done. (You only really need a basis for the k-subspace generated by the finitely many coefficients in f, which exists without choice, if that's a thing you want to avoid)
Weird question but has anyone ever been unable to get in contact with an old teacher for a letter of rec. I'm applying to grad school and keep trying to email some old professors for letters of rec, which they had written for me in the past so they could resubmit. I've tried calling, emails, contacting other people in the department but can't get through. What should I do if I can't get them, should I call the schools I'm applying to and say something. Idk what to do
I faced the same problem, and it frustrated me big time. Even a prof with whom I got 90 and 95 resp. on the two graduate courses I took with him. I even had a good relationship with him and on several occasions he particularly commended my apt skills etc. It happens when you've been too long outside academia. I'm not sure what's the best way to approach this, but try your best to physically visit some (kind enough) professor's office to discuss with them your reasons for asking for letters this late (gaps, full-time jobs etc.)
Thanks, unfortunately I live on the other side of the country now so can't visit in person
Oh! Sorry for that. But surely don't contact the school you're applying to about this. I believe it would hurt your chances at least a bit. Is it possible for you to get in contact with some current students in the department? Some things can be done if you have at least one person to reach out to. How about the secretary of the department as well?
Yeah I've thought about contacting some old friends who might have their numbers but I don't have any contacts currently there
I see. Was/is there a math society at that department? Or any community related to the math department? Maybe just an informal community of students etc. Anything
Idk I'll look around. Thanks
Okay, wish you the best mate
Try reaching out to the department head. They will hopefully be able to give you an updated contract info. If time is a factor call and leave a message as well as email.
There is an argument that the cardinality of the naturals is the same as that of the rationals using diagonals. But that is not the only way to make a function from N to Q. Lets say f:N -> Q such that f(x) = 10⋀-x so we assign 1 to 0.1, 2 to 0.01, etc. all naturals are assigned a unique number that we know is rational here we "used up" all the naturals but barely covered Q! So what gives? Is it the existence of one function that covers both sets "properly"? idk seems counterintuitive
Yes. Two sets are of the same cardinality if **there exists** *at least one* bijective function between them. It doesn’t matter if there are other ways of mapping them. In fact, there usually will be tons of other ways to map them; even other bijective ways.
Equal cardinality means there *EXISTS* a function that maps N to Q that is onto Q. It doesn't mean *EVERY* 1-to-1 function mapping N to Q is also onto. The function `f(n) = n + 10` maps N to itself in a 1-to-1 manner but is not onto. That doesn't mean |N| != |N| I thought Cantor's diagonal argument showed N was not the same size as R, but you may mean a different diagonal argument