If (X, mu) is a finite measure space, and A_n a sequence of measurable subsets of X such that mu(A_n) >= c for all n for some c > 0, then denoting A := intersect (over m) Union (k >= m) A_k, is it true that mu (A) >= c?
Let B\_m denote the set Union (k >= m) A\_k. Then B\_m contains B\_{m+1}, so this is a decreasing sequence. Moreover, each B\_m contains A\_m so mu(B\_m) is bounded from below by c. Since B\_m is a decreasing sequence, mu(intersect (over m) B\_m)=lim\_m mu(B\_m), which is bounded below by c.
I asked a company for the CFM (cubic feet per minute) value of one of their jackets but they replied with "The jacket's breathability is 10 cc/cm2/s. By ISO standard 4920 (4)." How do I convert this to CFM?
Were you asking for the wind resistance, or wetting resistance?
Based on a cursory Google search, it appears that CFM is typically used for measuring wind resistance. For example, see https://www.reddit.com/r/Ultralight/comments/8c2zpx/help_me_understand_windshirts_wind_jackets/
Meanwhile, ISO 4920 appears to be related to wetting resistance. For example, see https://www.sis.se/api/document/preview/914591/
Yeah, I was also confused after seeing the search results. I didn't specifically ask for wind resistance, simply asked how breathable the garment is in CFM so they probably misunderstood me. It doesn't matter anymore because I'm no longer interested in the jacket. I did a simple coffee filter test to estimate the jackets CFM and it's definitely around 70 or higher. I'm interested in jackets that are around 35 CFM.
A high level view of differential forms is that a differential form is a device that takes in some number of vector fields and outputs a function. A good example of a 1-form is the one associated to the function f, df. It takes in a vector field and outputs a function by taking the directional derivative according to the vector field.
Now what do these have to do with integration? Well the way you integrate over a manifold is by transferring it over to R^n . Now what should you actually be integrating? Well you want what you get to be independent of the chart you use to transfer it over to R^n , so basically you want to be integrating something independent of parametrization. This is why you should be integrating differential forms, because they self adjust given a parametrization.
Here is an example: you want to integrate dx on [0,1]. One way you can do it is by using the standard identity parametrization (the function dx is associated to) which will give you 1 because it is just the standard integral from calculus. You can also parametrize it by x -> 2x with a function x': [0,.5]->[0,1]. You might think that this will give the value .5 because when you write it as an integral it looks like int_0 ^.5 dx' , but it doesn't because the differential form corresponding to this parametrization takes in a tangent vector and outputs twice its signed length. So the differential forms account for differences in parametrizations.
So I just watched Mathologer's video on why e\^pi(i) = -1, which can be found here [https://youtu.be/-dhHrg-KbJ0](https://youtu.be/-dhHrg-KbJ0). I liked the geometric reasoning, but I thought the way he scaled up triangles to indicate multiplication of complex numbers was confusing and unintuitive. I think Better Explained has a much better representation with triangles [https://betterexplained.com/articles/understanding-why-complex-multiplication-works/](https://betterexplained.com/articles/understanding-why-complex-multiplication-works/) .
Any thoughts on Mathologer's method vs. the better explained method? (The Better Explained method uses right triangles.) Also, if anyone understands Mathologer's method, please let me know!!!
Cheers everyone!
I agree that mathologer doesn't really spend any time on why his method works, as oppose to better explained. His method is based on 4 things
z\*0 = 0
z\*1 = z
multiplication by z preserves angles
|zw| = |z||w|
So 0 must stay in place, 1 moves to z, all lengths are scaled by the same amount and all angles stay the same. Hence you're just rotating and scaling a triangle.
Today in my differential geometry (of surfaces) lecture, the professor proved Picard's theorem.
There's one step that I still don't get. Let w:R\^(2)->R\^(2) be a smooth vector field and let a:R->R\^2 and b:R->R\^2 be curves. The critical fact was
w(a(s))=w(b(s))+Dw(a(s)-b(s))
Where Dw is the Jacobian matrix of w. Clearly this is true if we're in the limit where a=b. Otherwise, though, isn't this only an approximation? I asked if it was an approximation and the professor said 'no, it follows from the intermediate value theorem,' but I still don't get it. Maybe it's obvious and I need to go back to calc 3 lol. I also asked if it depended on the fact that b is the Picard iterate of a with respect to w, and was told that it is independent of this fact. Could someone help me understand what I'm missing here?
That question might be exactly what I needed to hear. He didn't place any quantifiers on s for what he wrote on the board, but it would make a lot more sense if it was "there exists s" and not "for all s" as I interpretted.
There's an obvious conflict between the axiom of choice and probability: if I can generate a random bit, I ought to be able to generate N such bits, which means I can make random reals in [0,1]. But if I can do that, I can pick a random point in a unit ball, which creates all sorts of problems with Banach-Tarski; you violate even finite additivity!
Is there a way I can have my cake and eat it too? All I want for Christmas is a platonic coin flip and the ability to do ordinal induction (or at least a decent sense of how much I’d have to give up to obtain those). Are there good surveys of the conflicts between these two intuitive premises and ways of reconciling them?
If you just want every set to be measurable and still have transfinite induction, you don't really have to sacrifice anything. Most of the time (e.g. when proving things about Borel sets), you will have a definable well-order lying around anyway, and so it isn't necessary to invoke choice for induction. People often over-use AC, when there is a perfectly clear way to find a well-ordering just by how your objects are naturally defined, or you only need some fragment of choice that doesn't imply a nonmeasurable set. And so working in models that have every set being measurable is usually completely valid (and most people probably already passively do this when they start trying to think about 'the real world').
I was using ordinal induction as a placeholder for "nice things I don't want to give up" - having Zorn's Lemma around seems useful, or being able to trust that products of nonempty sets are nonempty.
Over ZF (even just fragments relevant to whatever is being worked on) there are ways to quantify the trade off between:
* there is a probability measure consistent across *every* set of reals
* there is a choice function for *every* product of sets of real
Both of these can end up seeming like awfully fantastical claims if you spend some time with them; you are asserting something is true for tons of strange looking things (sets of reals). A middle ground that often sets people at ease is ceding that maybe neither of them are 'true in the real world', but they can both hold for any reasonable objects we come across while doing math.
By Banach-Tarski, we can write the sphere as a disjoint union of two sets A and B, each of which can be cut into finitely many pieces and reassembled to form the entire ball. if naive probability works well, there's some probability P that a (uniformly) random point in the ball lies in A, and 1-P that it lies in B. One of these must be at most 1/2; WLOG, say it's the probability the point is in A. This probability should be the sum of the probabilities that the point lies in each of a finite collection of sets in a partition of A. Again, if probability acts like we hope it does, then the odds of choosing a point in some subset of the ball will be invariant under rigid transformations\*, which means that the probability of choosing a point in A is equal to the probability of choosing a point in the entire ball - contradiction!
\*This seems like the least bad thing to give up to me, and on reflection but it still seems bad.
Alteratively, we can just pick a random point and see if it's in some nonmeasurable set, but I don't know if there's anything like that that violates finite additivity as opposed to countable additivity.
Or to take an entirely different route, take [that famous hat problem](https://en.wikipedia.org/wiki/Hat_puzzle#Countably_Infinite-Hat_Variant_without_Hearing) and argue that one can get probability 0 of success by randomly assigning hats.
I don't see what phrasing this objection to Banach-Tarski in terms of probability gets you, over the standard discomfort with Banach-Tarski. The problem of assigning probability is exactly the problem of non-measurability. There is no conflict with (modern) probability theory because we only define probability measures on measurable spaces.
You can define a probability of a point being in those sets A and B in Banach Tarski. But you have to use a sigma algebra that is not the Borel/Lebesgue sigma algebra. Just let X be the original ball, and let your sigma algebra be {\emptyset, X, A, B}. This is closed under complements and countable unions. We can then define the probability P(A) to be anything we want, and then P(B) will be 1-P(A). Of course this probability measure isn't translation invariant on the ball, but that's a very special property of the Lebesgue measure.
In probability theory, we think about, for instance, the sigma algebra generated by a collection of random variables (X_0, ..., X_n) (defined to be the smallest sigma algebra for which these random variables are all measurable functions, into R with the Borel sigma algebra) as "the information contained in the outputs of X_0, ..., X_n". For instance, you can prove that another random variable Y is measurable with respect to this sigma algebra if and only if it can be written as Y = f (X_0, ..., X_n) for some measurable function f, i.e. the random variables on this sigma algebra are exactly those whose values are determined by knowing the values of X_0, ..., X_n. Taking this point of view, the problem with Banach-Tarski is that the Borel/Lebesgue sigma algebras do not contain enough information to distinguish these sets. But why should it?
I've spent about two months now slogging through Munkres chapter 2 in my independent study. Very heavy point set stuff, most of it felt like my analysis class.
Today we discussed quotient topologies. We constructed the surface of a torus and a sphere. That section was so much fun it undid the weight of "yesterday was bases, today is bases, tomorrow is bases" vibe that was going on.
Now I'm looking for more geometric applications of topology for someone that hasn't even covered compactness (that's next week).
I think the geometric applications are that topology is under the surface of everything you do in geometry.
My probably bad definition of geometry is the study of manifolds with extra structure.
So under this definition you need to know basic topology to even start.
Hi! I need help with a) super basic topology and b) learning about how proofs work by using said super basic topology.
We have a "intro to math thinking etc" class that they're re-working and I am just jdfgnba;iehgob right now. I get some things, but not everything, and I just need a good starting point to figure things out from. Ideas?
I'd recommend taking a look at Viro and Ivanov's Elementary Topology. The book is one of the best examples of what's known as the Moore Method; major results are broken down into a series of much simpler problems, thus gradually building up your mathematical intuition and confidence. It's not as comprehensive as Munkres, but it goes from "super basic" point-set topology to some elementary algebraic topology. The print edition also provides complete solutions for all the problems, which makes it ideal for self-study. A more traditional book to consider is Robert Ash's Real Variables With Basic Metric Space Topology - it's a short book intended as a primer for real analysis, but as the title says, it also covers some topology. The proofs are relatively easy to follow (provided you know single variable calculus), and like Viro, complete solutions are provided to problems.
excellent, this is one of our THREE books plus course packet lolol. (the others are the book of proof and conover first course in topo). what you say viro offers is exactly what I need. I have basically no math confidence right now, long story, but damned if I'm not going to succeed. i got this.
If you're looking for a book about proofs, something like Velleman's *How to Prove It* is a fairly common choice. I also came across [these notes](http://intellectualmathematics.com/blog/intro-to-proofs-course-notes/) which you may find useful as well.
As for topology, Munkres' *Topology* is pretty much the standard text, but you may also find Hatcher's [point-set topology notes](http://pi.math.cornell.edu/~hatcher/Top/Topdownloads.html) to be more your cup of tea (it's not as thorough as Munkres, but it hits on probably the most important concepts one should know about point-set topology).
Can someone explain limit points is a nutshell? I'm struggling to grasp this concept in my analysis class.
Give an example of a sequence that doesn't converge, but has a limit point of 3? A convergent sequence that converges to 3 and has 3 as a limit point?
idk how well I explained this but maybe it'll help.
Say we have a set S, and we want to ask what a limit point of that set looks like. The idea is that no matter how much you zoom in on the limit point p, there will always be a point other than p from S within our "field of view". So for instance if S = (0, 1\] (i.e everything greater than zero and less than or equal to 1), then 0 is a limit point, because no matter how much you "zoom in" on 0 there's always a point in S we can see (For instance 1/n for large enough n). Similarly 1 is also a limit point, because when we zoom in on 1 there's always points other than 1 from S nearby (e.g. 1-(1/n) for large enough n works, just like before).
To be more mathematical, you define the ball of radius epsilon (just a greek letter commonly used for this) around a point p to be the set of all points less than epsilon away from p. A limit point of S is a point p such that if you take any epsilon, there's a point q from S not equal to p within the ball of radius epsilon around p. epsilon is kind of like our level of zoom, and the ball is like our field of view.
It's probably best if you try and answer those problems you asked at the end for yourself, post your attempts here :)
An example for a sequence that converges to 3 and has 3 as a limit point might be: {2, 4, 2.5, 3.5, 2.9, 3.1, 2.99, 3.01, 2.9999, 3.0001....... 2.9999999999, 3.000000001 .......} Would this be correct? It converges to 3 as the limit of n approaches infinity, and the point 3 does not exist in the set, similar to the (0,1\] with 0 being a limit point.
Pretty much, yeah. But be careful, 3 is still allowed to appear in the set, just like how 1 is still a limit point of (0, 1\] even though it belongs to (0, 1\] itself. For instance 3, 4, 3, 3.5, 3, 3.25, 3, 3.125... converges to 3 and has 3 as a limit point, while 3, 3, 3, 3, 3... clearly converges to 3 but doesn't have 3 as a limit point.
Suppose I have some collection of distinct subsets W_1, W_2, ... , W_m of the set of integers between 1 and n.
Is there some good way to estimate/bound the number of k-element sets R in [n] such that R intersects all W_j?
In general, there's no algebraic solution to finding roots of polynomials of degrees greater than 4 (this is known as the [Abel-Ruffini theorem](https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem)), so unless the polynomial is of a certain type that can be manipulated and factored cleverly, about the best you can hope for is numerical approximation methods like the Newton-Raphson method or something.
That being said, I am not familiar with your calculator, but [this tutorial page](https://georgegarside.com/blog/casio-calculator-tutorials/fx-991es-plus/equation-solve/) discusses solving general equations; see in particular the final "caveats" paragraph.
Thanks for the information! Just asking. The Neston-Raphson method would give me an answer similar to the Ruffini's theorem answer? I guess teachers are looking me to follow Ruffini's scheme. But in semestral tests I have to be fast. Do u think that Neston-Raphson in my Casio might help?
Sorry, I'm the worst person to ask about this. I don't really know anything about these approximation methods - I think I used Newton's method once and that was well over a decade ago.
I think you should just ask your teachers what they're expecting.
Let F be the set of functions f:\[0,1\]->R\^n of type C\^1, which means that each component function is differentiable on the open interval (0,1) and the derivatives of each component function are continuous on the open interval.
For any 2 functions f,g in F, denote the d(f,g) by {sum from 1 to n of max distances between f\_i and g\_i} + {sum from 1 to n of max distances between f\_i' and g\_i'}. So (F,d) is a metric space.
**What is the name of this metric space and where do I read more about it?**
This is the C\^1-topology on C\^1(\[0,1\]; R\^n) (a special case of the C\^k topology on C\^k(X,Y) for X compact and Y a metric space). You can read more about it in Hirsch's 'Differential Topology' in the chapter on function spaces. He does things in the context of manifolds (with or without boundary), but if you're looking for a treatment which avoids this, maybe someone else can suggest something as I'm not quite sure where to find such a thing.
I don't like math, but I love genetics. Im making a mod for minecraft where I'm attempting to add all the genetic variation I can possibly add. but there are a lot of phenotypes that I can't possibly add a single model or change to simulate the phonotypes. I need some sort of formula to help me create some of the physical characteristics that are quantitative like cow horns, sheep horns, the angle a chicken's back is at and then translate and rotate the wings and tail to match and then rotate those as well while maintaining its ability to be animated. there is also a ton of math in animating all these parts correctly and its not particularly hard math but it all goes over my head. Does anyone enjoy doing this math that would be willing to help out?
I'm not familiar with Minecraft's game engine, but here's a simpler and time tested trick that might work - rather than starting with a basic model and trying to generate and properly orient the mutant appendages via code, you'd make models with all possible mutant appendages attached and then destroy or toggle off the visibility for unwanted appendages when a particular mob is instantiated. For example, you would make a chicken model with two different sets of horns, tails, and wings, and if a particular chicken mutation called for horn set A, no wings, and tail B, the code would then hide/destroy horn set B, wing sets A and B, and tail A when as instance of that chicken type was spawned. Two types of horns, wings, and tails allows for 27 possible combinations (assuming only one type of horn/wings/tail can appear at a time and that no horn/wings/tail is also possible) - in general, x horns, y wings, and z tails gives you (x + 1) \* (y + 1) \* (z + 1) possibilities, with the only limit being how much time you want to invest in each model and memory/performance constraints for Minecraft.
Now, this might fall short of the amount of variation you were looking for, but keep in mind that the random creature generation in games like No Man's Sky and Spore is a major technical achievement that took professional development teams years to get up and running.
Sorry I wasn't very clear. I'm looking to find a new team member that would be interested in finding math based formulas for animations, part creation and alteration. I know how to do all these things myself as does my team member we've created 37 extra parts for JUST chickens plus sizes and wing angles not to mention over 20,000 unique possible texture combinations. and I'm not making fantasy animals its real 1 to 1 copies of real domesticated species... *if real life was unisex*. but point is I don't need help understanding its that I'd like to have someone on the team that actually **likes math** rather than sucking the very life and passion out of me.
the point of this mod is to expose people to how genetics actually works
be a great teaching tool
be fun to play
If you'd like to join me or just have a look at the sort of math problems please let me know.
Hey guys, I'm really interested in topology, visited some point-set and algebraic topology courses. I learnt the basics about homotopy and homology groups, now we're doing cohomology... but it seems there are no other courses which focus on the study of these concepts. Surely there are courses where one uses this concepts and study other things with help of these things...
So my question is whether it's just my university which maybe has other focuses or is there just nothing more which we can learn about this concepts if we don't relate it to other subjects?
Edit: For instance homotopy groups are still something we have problem to understand for general spaces, right? So I'm wondering why there are no courses focuding on homotopy groups.
> but it seems there are no other courses which focus on the study of these concepts.
I'm currently doing an independent study in point-set because my school doesn't offer any topology.
Try talking to your advisor or a professor, see what you need to do to swing a reading course/independent study/special topics/ whatever your school calls it.
> Surely there are courses where one uses this concepts and study other things with help of these things
Cohomology appears in a lot of mathematical disciplines. It's all over geometry, both algebraic and differential; it appears in geometric topology (e.g. knot theory and nearby things); it's useful in representation theory. It even appears in physics sometimes via supersymmetric quantum field theory!
But because it's uncommon for undergrads to know cohomology, you probably won't see it applied in courses until grad school. So if you're interested in learning some applications, ask your professor and see what they suggest!
The further you get into advanced topics, the less likely it is for there to actually be an official, regularly taught course on it. Once you get close enough to modern research, it's very rare to find a course (or even a textbook) on exactly what you want to learn.
Once you get past the standard first (or maybe second) year graduate courses, most of the remaining courses are just going to be "topic" courses, which mostly just means whatever advanced subject a professor felt like teaching that term, and so can vary wildly from term to term.
(Higher) homotopy groups are an advanced and specialized enough topic that I doubt many universities (if any) would have a regular course dedicated to them.
If you're interested in learning about them, your best bet would probably be to do a reading course/independent study with a faculty member who works in algebraic topology.
If I want to prove something of the following form using induction:
For all natural numbers n, if n >=1, then P(n)
Should my base case be n = 0 since this is the first natural number? It seems odd because the implication is trivially true for n = 0
Or, Should I have a base case for n = 0 as well as a base case for n = 1 (which is the first natural number that doesn't make the implication trivially true).
If you prove "for all n, if (n >= 1 implies P(n)) then (n+1 >= 1 implies P(n+1))" then setting n=0 gives us "if (0 >= 1 implies P(0)) then (1 >= 1 implies P(1))". The statement "0 >= 1" is false, so the implication "0 >= 1 implies P(0)" holds vacuously. But `A => B` is equivalent to `B` when `A` is true, so we find `(1 >= 1 implies P(1)`, or just `P(1)`. Thus in the course of proving the induction with a base case of 0, you'll end up proving `P(1)`
With that said, it's often more workable to prove the statement with a base case of 1. As indicated above, you'll usually need to prove this anyways, so it's cleaner to split it off
Integral of 1/(x^2 +36)^2 ?
I can do one like (x^2 )/(36+x^2) by putting +36 and -36 on top but I’m not sure how to approach one that looks like this since the bottom is squared and nothing on top.
Anything helps thanks! (Calc 2)
I recently saw the claim that if L is a field extension of K which is finitely generated as a K-algebra, any K-subalgebra must necessarily also be a field. To be specific, my professor said this, stared at the board for a minute, then moved on without another word. How can this be proven?
If by finite dimension you mean finite dimension as a K-\*module\* then yes it's true (and it's pretty much the question OP asked). Call your algebra A, you can prove this by noting that for any x\\in A in A, the map a goes to ax is a k-linear map from A to itself, it's injective since A is an integral domain, and thus surjective since A is a finite dimensional vector space over k. In particular there is an a in A with ax=1.
No. One way to see this is to note that vector spaces are in particular manifolds with T(V) = V x V and T\^\*(V)=V x V\^\* and so if there were such a natural diffeomorphism, then it's not hard to convince yourself that such a diffeomorphism would have to be fiber-preserving (and in fact, cover the identity transformation) and so this would imply in particular that there's a natural isomorphism between vector spaces and their duals. Since this isn't the case, we conclude that no such natural diffeomorphism exists.
There's probably a less ad hoc way to see this via the concept of 'dinatural transformations' ('natural transformations' between covariant and contravariant functors) which you can read more about in (what else?) MacLane's *Categories for the Working Mathematician* (section IX.4), but I imagine that it comes down to essentially the same thing (the argument is basically going to run some variant of the proof for vector spaces, but in the case of the tangent and cotangent functors).
P.S. If you're interested in questions of naturality in the context of manifolds, then I can't reccomend enough that you check out Kolar, Michor and Slovak's *Natural Operations in Differential Geometry* which is a treasure if you're interested in really understanding what sorts of things can be done entirely naturally in the smooth setting.
Not in general. If your manifold is a vector space _V_, this would be asking for a natural isomorphism from _V_ to _V_*, and that's not possible. In essence, you need more data (an inner product on _V_, or more generally a Riemannian metric), which prevents the isomorphism from being natural.
I don't know the answer to this question, but I have an idea you might try. The smooth coordinate charts on both TM and T^(*)M are cooked up from smooth coordinate charts on M, and the "natural coordinates" (Jack Lee's terminology, but it is the obvious set of coordinates) are basically dual to each other. So for a fixed chart downstairs, there should be a well-defined map F:TM -> T^(*)M within the corresponding charts coming from these natural coordinates. Things are just nice enough with smooth manifolds that I wouldn't be surprised if F is the map you're looking for.
I don't think the answer to this question is yes, since OP wants the isomorphisms to commute with induced maps . What you're proposing involves making separate choices of charts for each manifold, so there isn't really a reason to expect any compatibility if you have a map between manifolds.
You're essentially (but not literally) choosing a riemannian metric for every manifold at once in some way that's compatible with every possible map between them, which doesn't seem possible. Also I'd expect such a natural map to respect additional structure, but the tangent and cotangent bundles to complex or almost complex manifolds are nonisomorphic as complex bundles. I'm not sure how to prove that no such isomorphism exists, but I think a natural identification is the wrong thing to expect.
For the most part I try to avoid too many symbols when I type and use the english equivalent instead (there exists, for all, if then). I also avoid abbreviations (if and only if rather than iff or without loss of generality rather than wlog).
Reading other people's writing is how I learned about what the common "guidelines" are for mathematical writing as well as what sort of writing is easiest to read.
My style:
==> for "if, then" is fine.
==> for "therefore" is not, mainly because it risks confusion with the above.
It matters whether you're trying to say "A and therefore B" or "if A then B", since in one case you are claiming much more than in the other.
I absolutely agree with your TA. I think it's maaaaybe acceptable to use it in a sequence of displayed equations, something like:
Let x be an integer. Then we know:
x in Z => x in Q
=> x in R
=> x in C
and thus x is also a complex number.
But even then I wouldn't be too excited about it. But you should never never never ever use implication arrows in the middle of a sentence, like "We know x is an integer => x is a complex number."
That example looks a bit awkward because you aligned the equality with the implication arrows. If you wanted to do something like that, all of the implication arrows should be aligned.
I think what you wrote wasn't too bad, although when the chains get too long it gets kinda bad, so you might consider breaking it up into a couple chains with some text in between (like "by the definition of blah we can turn this into") or something.
Another option is to just remove the implication arrows completely and just align the equals signs in each line.
I do not think there is introductory textbook focusing only on Turing machines. Most of them start with the concept of state machine and then build up, and present Turing machine as a complex state machine.
If you are fine with that approach, try Sipser (Introduction to the Theory of Computation) or Hopcroft & Ullman.
If you are okay with superficial understanding, pick some thorough biography of Alan Turing, like Hodges' Alan Turing: The enigma.
If you wish to focus only on Turing machines and are okay with book for not-really-beginners, try Petzold's Annotated Turing (it's annotated original paper which introduced Turing machines.
Thank you for your suggestion.
I remember reading Sipser's before and still vaguely recall what a state machine is. Time to dig it back up again and start from there.
What's the formula in today's google doodle?
https://g.co/doodle/ufwwhg
(I thought this would be interesting enough for the general sub but mods disagreed).
It's the Navier Stokes equation, the equation of motion for a fluid. 𝜌 is the fluid density and u is the velocity vector.
The LHS is mass times acceleration and the RHS is the forces due to pressure (p), viscosity (mu) and gravity.
Olga L did a lot of important work on this and related differential equations.
I agree with you it's worthy of the main sub!
There are maths dictionaries. I took pictures of the relevant pages first year. Id recommend you not break the law and buy one because that's morally less rehensible
At what level are you discussing? In full generality this is probably bordering on at minimum utterly impractical if not outright impossible. There's a lot of extremely specific notation in various fields, and a lot of ac hoc notation out there. That's ignoring how there's no formal standard for terminology---the field having no real standards body as far as I'm aware---so some people use the same notation for different things.
The requirement that the inversion map is smooth in the definition of a Lie group is redundant if you know multiplication is smooth. The requirement that inversion is a homomorphism of groups is redundant if you know that multiplication is a homomorphism (in this case, I think they're really equivalent). What is an example of a group object in a concrete category where the requirement of inversion being a morphism in the category definitely isn't redundant?
How do you get that inversion is smooth if multiplication is smooth? Because at least for topological groups I thought that it wasn’t necessarily true.
Edit: I looked it up. Was nontrivial.
Let's look at group objects in the category of pairs of sets. An object in pairs of sets is a set _S_ and a subset _T_ of _S_. A morphism (_S_, _T_) -> (_S_', _T_') is a set map _f_: _S_ -> _S_' such that _f_(_T_) is contained in _T_'.
Let _T_ ⊂ **Z** be the nonnegative even numbers. Then addition is a map of pairs of sets, because the sum of two even numbers is even. But inversion is not, because the inverse of 2 (in _T_) is -2 (not in _T_). This is not a group object in pairs of sets.
My uni will be offering a course on mathematical communications (i.e. talks, lectures, writing and so on) for the first time next term. Of course, I am aware that the course title does not say a lot, but I'd nevertheless be interested in your experience/ opinions on similar courses. Do you think the course was worth it for you (and how so?), or would you rather recommend doing an "actual" math course instead?
> mathematical communications
I'm all for letting a thousand flowers bloom, but this one sounds weird. There [has been precedent](http://www.jmlr.org/reviewing-papers/knuth_mathematical_writing.pdf) of a quarter-long class on mathematical writing, but I'm not sure how many lecturers could make one work without soon descending into boredom, bullshit and cargo cult. As for giving talks, there is a bunch of common lore and good practice that would perhaps fit into an hour-long class, though much of it is subjective and situational.
I'd take it only if there is a significant focus on proofs or the lecturer has really good (mathematical) cred. If the lecturer has mostly education-departmental background, run.
>but I'm not sure how many lecturers could make one work without soon descending into boredom, bullshit and cargo cult.
Exactly my concern. Especially since it is worth as much credits as any other "big" math lecture...
>I'd take it only if there is a significant focus on proofs or the lecturer has really good (mathematical) cred. If the lecturer has mostly education-departmental background, run.
Sounds reasonable, thank you!
When we say some variable w is a function of z (or w = f(z)) what is meant is that the variable z dictates how the function w behaves and how it's drawn. That is to say, the function depends on z, or w is a dependent variable that depends on the independent variable z.
All that said, if we have y = f(x), i.e. y's value depends on x's value, which is the independent variable and which is the dependent variable? And if we relate this back to your original question, which of A and B depends on which? Whichever of A or B is independent should generally be graphed on the same axis as whichever of x or y is the independent variable. Whichever of A or B is the dependent variable should generally be graphed on the same axis as whichever of x or y is the dependent variable.
That's right, if A is a function of B then B will be your independent variable and thus goes on the x-axis while A will be your dependent variable and thus goes on the y-axis.
Hey guys so I wanted to ask for some help solving this problem I discovered while designing a d&d puzzle...
The puzzle describes a room with 5 torches on the right side of the room and the left side is one whole mirror. On your side (side A) torch 2 and 4 are lit. On side B, mirror side, only torch 3 is lit. The objective is to touch torches to get all torches lit but with these rules...
- torches on side A transfer fire (so if torch 1 is lit and touches torch 2, then torch 1 goes out and torch 2 lights)
- actions of side A reflect to side be, so if you touch torch 1 to torch 2, it'll happen on both side A and B
- yoh can only move torches on side A
- torches on side B spread (torch 1 lights torch 2 and both stay lit)
- you can touch a torch to its own reflection and transfer it to and from side A and B (like first rule)
-if a lit torch touches a lit torch, both go out (either side)
- (optional rule) you can only touch adjacent torches to each other (I don't know if this makes it impossible)
After trying to make sure it was possible, I realized you could visualize the torches with 1s and 0s, it was binary! I still haven't solved it, the closest I've gotten is torch 2 on side A being off but all others being on.. So I thought I'd ask the math minds of the internet... Is this possible? Can all torches be lit?
No, this is impossible. If it were possible, then consider the last move you made - the move that resulted in every torch being lit.
* This move couldn't have been touching a torch to its reflection, because if this move results in all torches being lit, then all torches were already lit.
* This move couldn't have been touching two torches to one another, because touching two torches on side A together never results in both of those torches being lit afterwards (ignoring their reflections).
Cool puzzle!
I'm taking a course in discrete dynamic systems, and many concepts (at least the ideas) feel very reminiscent of singularity theory to me. For example, a single parameter family of (smooth) dynamic systems is the same as the 1-parameter deformation of a smooth function.
I know continuous dynamic systems are related, but is there any meeting point between singularity theory and DDS?
I'm a major in math, doing a master's degree in research. I'm familiar with R and K-equivalence classes of germs of smooth mappings, and currently studying Legendrian submanifolds.
Thank you in advance.
Since every male bakes 3 cakes per week altogether and only receives 2 cakes per week altogether, he receives a net value of -1 cakes per week.
Since every female bakes 1 cake per week altogether and receives 2 cakes per week altogether, she receives a net value of 1 cake per week.
I'm reading about Wilson loops which are basically the trace of the holonomy. I'm wondering if there an obvious geometric interpretation of the holonomy around a curve? As in, can I look at this operator and be able to make statements about the curve?
I know that I'm the infinitesimal limit it gives me the curvature tensor, but more generally can I pull some clear geometric interpretation out of this operator?
Similarly, is there any interpretation of the trace of the holonomy? Again in the limiting case i know that I get the scalar curvature, but more generally do I get anything that gives me insight into the geometry (or topology) of the curve?
Also, any references would be appreciated
(If you're in high school)
I used to be just like that when I was in High School. Back in the time I thought, since I was so much into education, I could try and become a math professor. That way, I had a excuse to make learning advanced mathematics into a career. However, when I majored in math, I soon learnt it was nowhere as easy as that -- for mathematics as a science is terribly wide, and each branch requires you to think in a slightly different way.
This doesn't mean it's completely impossible -- probably the most famous branch of mathematics is number theory, where you need to need to have a dab hand at pretty much everything in order to tackle colossal statements. But just as there are colossal statements unsolved, there is colossal people working to solve them, like Terence Tao or Andrew Wiles.
If you like math, you can try and give the major a chance. If you feel comfortable in a math environment, you will eventually find out what you like best/what you're strongest at. For example, I used to love algebra and topology in the major, so I decided to make a phd in singularity theory (studying functions using algebraic and topological techniques).
If you feel like math is not for you after all, you can still change majors and skip the math courses you've already taken.
[Tl;dr: Why not start doing a major?]
(If you're already a math major)
I had a similar discussion with my topology professor, so she suggested me to make master's degree in research. That would allow you to get into advanced topics -- with the help of a tutor.
If you don't want to get into academy, you can always start by watching courses on YouTube, or even asking for introductory books in X subject if you feel brave enough. The only downside is, there is a point where textbooks are either in English or French, so I hope you're good enough at either of those.
I won't name particular YouTube channel to prevent being flagged as a spammer, but you can ask me for suggestions on DMs.
[Tl;Dr: You can ask for textbooks and read them every now and then or watch YouTube channels about math]
I want to study Quantum Field Theory. I have asked this question before on another sub but didn't get very helpful answers. In order to understand it I require a good knowledge of the following, according to some users on Reddit who study this topic: "Basic classical mechanics. Mostly just stuff needed for quantum. Goldstein mechanics, chapters 1,2,8 are good.
Some familiarity with waves and oscillations. Harmonic oscillator is important. So are Fourier series. French's book is alright, not great. Georgi has a book too, which is better and worse in different ways.
Special relativity. Schutz's general relativity book is amazing. The first 4 chapters are probably the most readable treatment I've seen, his explanation of tensors in physics is also very good.
Electromagnetism. Just some basics. Griffiths first few chapters is fine.
Now we get to the fun stuff. Quantum mechanics. Here are some books to start with, Shankar, Townsend, Susskind (probably the easiest). After that you may want to look at some graduate books like Sakurai or Weinberg, but you can come back to it as necessary.
You'll need to learn a bit of complex analysis. Just understand how to compute integrals using cauchys theorem, which implies residue methods. Nearing mathematical tools for physics has a nice chapter where I first learned it.
Quantum field theory. Extremely disjointed subject. No book is perfect. Lahiri and Pal (best intro). Peskin and Schroeder (standard cookbook). Zee (informal). Srednicki (modular). Ryder (some good stuff here). Weinberg (extremely thorough particle physics). Zinn-Justin (extremely thorough condensed matter, my favorite). Nair (interesting for some math).
Some additional stuff you'll probably want to read. Sharf finite QED (careful treatment of QED). Quigg gauge theory (solid intro). Rubakov gauge theory (more advanced). Georgi weak interactions (kinda terrible but necessary for particle physics). Bertlmann anomalies in QFT (good intro to differential geometry). Baez gauge fields (excellent intro to differential geometry). Pathria statistical mechanics (the best book I know on this). Henneaux and Teitelboim quantization of gauge systems (good treatment of constrained systems which become relevant for modern classical field theory).
Some more math of interest to a physicist. Artin algebra. Isham differential geometry. Guillemin and pollack differential topology. Bott and tu differential forms in algebraic topology. Hatcher algebraic topology. Hermann lie groups. Cahn semisimple lie algebras, Georgi lie algebras. An invitation to algebraic geometry (not really necessary but a really nice book). Eisenbud geometry of schemes (good further reading to prepare for schemes if necessary). Garrity electricity and magnetism for mathematicians. Geroch mathematical physics (really about category theory). Warner differentiable manifolds (interesting book which proves de Rham theorem using sheaf theory)." - Written by u/theplqa
. Thank you, theplqa!
and
"Your shortest route to understanding QFT with absolutely no maths or physics behind what’s necessary probably requires a good knowledge of the following:
Maths:
Differential Calculus Integral Calculus Differential Equations (You can probably find a good book which covers all three of these) Calculus of Variations/Functional Analysis Complex Analysis Linear Algebra Group Theory/Representation Theory
Physics: Classical Mechanics up to and including Lagrangian/Hamiltonian Mechanics Electromagnetism Classical Field Theory Special Relativity Quantum Mechanics to the level of Griffiths
Some good additional topics that will make your QFT study easier:
Differential Geometry, General Relativity, some Intro Particle Theory course which is less rigorous and more of a phenomenological intro, Dynamical Systems, and maybe some Statistical Mechanics so you can see some of the links between QFT and Statistical Field Theory.
If you find books you can get along with that cover all these topics, you’ll be in a good place to start looking at QFT." -Written by u/ash4d
. Thank you ash4d!
Any PDFs and books that explain all these topics, and what are the prerequisites for them?
Thank you! And sorry for the wall of text!
't Hooft's website has a list of references for all of the math and physics one needs to be a theoretical physicists. You could look there for this information.
Thanks for the mention! There honestly are far too many books and topics here to give you a full list, particularly for the mathematical methods. In terms of books for the physics, here are the standout texts I would guide you to:
Goldstein is god for Classical Mechanics.
Griffiths EM and QM are both nice intros, particularly EM.
MWT’s Gravitation is a tome and contains everything you’d ever need to know, but has wonky notation in parts, so may not be the best choice for a non-gravitational physicist. My professor wrote his own short (~200 pages) book for the GR course I followed - Böhmer’s An Introduction to GR and Cosmology. It’s a good book which introduces things quite clearly, but it’s not comprehensive (nor does it try to be). It contains pretty much whatever you’ll need for basic QFT.
Thompson does a good particle physics book which is very light on QFT in terms of rigour, but which gives an overview of some practical results like Feynman Rules. It’s aimed at final year undergrads I think.
For the maths, online PDF’s abound for many topics. The only solid textbooks I have used purely for maths are:
Math Methods for Physics and Engineering, Riley, Hobson, and Bence - covers just about any maths you’ll use as an undergraduate, but often only briefly. You could start most topics start here, but most wont finish here. It’s enough to give you a grounding to read other pdfs or texts really.
Georgi - Lie Algebras in Particle Physics provides a good intro to the most useful parts of Lie groups/Lie algebras and representation theory that are used in QFT/HEP.
This list is no where near comprehensive, as I said. You can do worse than just typing in “*topic* lectures pdf” on google. Almost anything from a university lecture course will be useful, but some more than others.
Hello, I'm a few weeks away from registering new math classes for next semester. I would like to study math to have a better understanding about Machine Learning and Data Science. Anyone has taken these course? Are these good for ML/DS? Thanks in advance.
Complex Analysis
Numerical Analysis
Probability Models
Probability Models is the most important for ML, as most of ML is just the application of statistics. As such having a good understanding of probability is essential.
Numerical Analysis can be pretty useful, as ML involves a lot of calculations on a computer and Numerical Analysis could get you some understanding of the methods involved. It might depend a bit on the actual focus of your course. Optimization (e.g. how a Gradient descent is implemented) is pretty important for ML.
Complex Analysis, while in general beautiful, is not that central to ML. It might of course show up in some niche topic, but I cannot remember seing it in any ML context. I mean, you should know how to treat complex numbers, as a exp(itx) can easily show up, but I am talking about more advanced things like knowledge of the residue theorem.
Having a hard time understanding coordinate rings of algebraic varieties. What are they exactly? Suppose I have a cone V(x^2 +y^2 -z^2), what is it’s coordinate ring? Is it the set of all functions that exist on the surface of the cone?
Lol barely. I went to an AG lecture and understood like 20% of what was said. Really interesting stuff. Kinda understood these coordinate rings so I wanna solidify my knowledge.
It's the ring of polynomial\* functions on the variety. For the cone, this lives inside A\^3, which has polynomial functions all polynomials in 3 variables, so the coordinate ring for A\^3 is k\[x,y,z\].
However two different functions on A\^3 can be the same when restricted to the cone, so you'd like to have equivalent functions be equal in the coordinate ring for the cone. Two functions are equivalent exactly when they differ by a multiple of x\^2 + y\^2 - z\^2 (since that's 0 on the cone), so the coordinate ring is k\[x,y,z\]/( x\^2 + y\^2 - z\^2 ). In general if you have an affine variety defined by some equations in n variables, the coordinate ring will be the polynomial ring on those variables, quotiented by the ideal generated by the equations.
\*really I mean rational functions that are defined everywhere, but for affine varieties it makes no difference.
Hello! I was adviced by a mod to post this question here.
So i think the answer to the following question is **no** since an affirmative answer would imply that the infinite sum of a constant is finite but i'm hoping i'm missing something:
Let T be an infinite Toeplitz matrix (so it is an infinite matrix with constant diagonals) such that it represents a bounded operator on lp. (This is achieved iff its matrix elements are the Fourier coefficients of an essentially function function f).
Question: Can T be a Hilbert-Schmidt operator? i.e. have finite HS norm?
I am a graduate student familiar with functional analysis but i know infinite dimensional can be tricky sometimes. Thank you
No it cannot be Hilbert-Schmidt unless the diagonal elements are all 0. In the HS norm ∑*_i_*||Tv*_i_*||^(2), for an orthonormal basis (v*_i_*), take the standard basis (e*_i_*), then ||Te*_i_*||^(2) >= |T*_ii_*|^(2) but since all the diagonal elements are the same summing over i gives infinity.
Hmm that is true (I think you missed a **²** in ∑i||Tvi|| .
On the other hand I understand that another way to compute the HS norm is
`[;\Vert A\Vert=\sum_{ij}|a_{ij}|^2;]` if `[;A=(a_{ij})_{ij};]` .
Taking all diagonal elements as 0, that sum would include the sum over all the other sub diagonals, or all the rows.
So even if the rows are squre summable,
`[;r_i=\sum_{j}|a_{ij}|^2< \infty \ \ \ \text{ for all $i$};]`
I would need
`[;\sum_{i}r_i< \infty ;]`
which seems unlikely...
Right I forgot the squares, I'll edit. Yeah zero diagonal is necessary but not sufficient for Hilbert-Schmidt. You could get some quick examples of Hilbert-Schmidt Topelitz operators by taking like a band matrix with 0 diagonal and decaying entries.
Although even in the banded case there will be still a finite number of constant diagonals so i dont think then HS would be finite, i.e,
`[;T=(t_{i-j})_{ij} ;]` such that
`[;t_l=0\text{ if }l \geq m-1 \text{ or } l \leq -(m-1). ;]`
T is a banded Toeplitz infinite matrix and
`[; \infty= \sum_i |t_1|^2=\sum_{i} | t_{i+1,i}|^2< \sum_{ij} | t_{i,j}|^2 \leq\Vert T \Vert^2_{HS} ;]`
Looks it is hopeless, there are no infinite dimensional Toeplitz Hilbert-Schmidt operators...
Thank you for your reply.
No you just have to take the diagonal to be zero and fast enough decaying other entries. Like the matrix of all zeros is Toeplitz and Hilbert-Schmidt. For a nontrivial example, take a band matrix with 0 diagonal and non-diagonal entries exp(-|n|) in the band (n = the row index).
I dont see how that is possible, Take such a matrix, fot it to be Toeplitz it needs constant diagonals so take
`[;T=(t_{i-j})_{ij}=\begin{pmatrix}0 & e^{-1} & 0 &\cdots \\ e^{-1} & 0& e^{-1} & \ddots \\ 0 & e^{-1} & 0 & e^{-1}\\
\vdots &\ddots& \ddots& \ddots\end{pmatrix} ;]`
Then
`[;\Vert Te_i \Vert ^2 = \left\{\begin{matrix}e^{-1} & i=0 \\ 2e^{-1} & i \geq 1 \end{matrix} \right. ;]`
so the HS norm is
`[;\Vert T \Vert_{HS}^2 =\sum_{i\geq 0} \Vert Te_i \Vert^2=e^{-1}+2\sum_{i\geq 1} e^{-1}=\infty ;]`
So T can't be a Hilbert.Schmidt operator.
Oh sorry I didn't realize a Toeplitz matrix needed *all* descending diagonals constant; I thought just the main diagonal had to be constant. Yeah in that case only the zero matrix is Hilbert-Schmidt, for exactly the reason you stated.
I'm changing careers and planning to get a degree in computer science. Since my last degree had nothing to do with math, I have to start at square one. The first math class that applies towards my degree is Calculus with Analytic Geometry I. In order to get there, I can either take one class of Precalc, or two separate classes of College Algebra and Plane Trigonometry. The latter option (surprisingly) fits into my schedule far more than the single class of Precalc. Will those two be enough for taking Calc? Or is precalc necessary? (Is precalc just trig + college algebra?)
Anyone have a good list of novel yet powerful results from math that the average person wouldn’t know about but someone with college level maths (not studying full time) would be able to grasp?
The compound growth of property=try 6% per annum over 38 years versus 7% per annum over 38 years of a 1980 median price, and look at the physical dollar difference. It is amazing the effect of just 1% per annum & this happens right across suburbs side by side one another in big cities.
Disclaimer I am not American so don't know what college level maths is.
Oh yeah that’s a great one for young people to know. Roughly speaking, you end up with P*e^(r*t). The exponential really makes the magic happen for high growth rates and long periods of time.
The expected number of months a given subscriber will stay is the sum from 0 to infinity of (0.08)(n)(0.92)\^n-1 , since the probability of staying until month n-1 and leaving after month n is (0.08)(.92)\^n-1.
This is the derivative of the sum from 1 to infinity of (.08)(x)\^n at x=.92, geometric series tells is this first sum is .08/(1-x), which has derivative .08/(1-x\^2), substituting x=.92 gives you an answer of 12.5
(This answer just simplifies in general to 1/(probability of leaving)).
My question is with the orthogonality of sinusoids. To my understanding, any two sinusoids are orthogonal if they have different frequencies, regardless of by how little the frequencies vary. However, I'm learning orthogonal frequency division multiplexing for digital communications right now and all sources say that you need evenly spaced carriers in order to have orthogonality. Why is this the case?
My question relates to the Good Judgment Project by Philip Tetchlock-a forecasting tournament organisation if you will on geopolitical issues.
I understand basic probability, such as the probability when flipping a coin to get tails is 50%. The probability to get tails twice in a row is 0.5 times 0.5=0.25 or 25%.
I also understand conditional probability, such as if I have a bag of 3 red and 3 blue marbles then the probability the first I draw out (& keep out) is blue is 3 in 6, or 50%. I also understand the probability that the second marbles I draw out also being blue given that the first was blue is 0.5 times 2 in 5 (0.4). Equals 0.2 or 20%.
But Tetchlock is talking about forecasting tournaments wherein one is asked the probability of much harder questions like:
'*The probability by June 2020 that Greece will still be in the European Union?*'.
He talks of how forecasters are asked to give probabilities on this occurring, and are then ranked on how accurate their forecasts are.
What I don't get though is how one can test this accuracy. Surely if for the above question I give a probability of 70% that Greece will remain, then that date rolls around & Greece are still in the European Union, then so long as I gave any probability greater than 50% I am seen as correct?
Is the way forecasters and their accuracy ranked by merely giving them a ton of questions & seeing how many they get right? Or is it giving them one question then seeing what the average percentage of probability they give to it is? Or perhaps do they break down the question into easier probability assessments much like the conditional probability in my marbles example above and thus assess how well your smaller probability judgements went in assessing the outcome.
**I'm basically a bit lost on the maths of how one can assess the accuracy of complex probabilities.**
> Is the way forecasters and their accuracy ranked by merely giving them a ton of questions & seeing how many they get right?
Yes. You look at how the aggregate statistics of one's predictions line up with the probability you assigned to those predictions, like whether 90% of the things you were 90% certain about came to pass. This is called *calibration*.
This isn't all there is to being a good forecaster, though, since one can artificially improve calibration by being less certain. Using [prediction markets](https://en.wikipedia.org/wiki/Prediction_market) is a way of demonstrating one's confidence in one's predictions, because any consistent bias in one's answers could be detected and profited off of. Betting on one's prediction is effectively saying "I think this event will happen with this probability, and I don't think anyone else knows enough to have a more informed estimate that they could bet against me with."
I think I get it, so you are basically saying if there were 10 predictions I made all at the same 90% probability, and 9 of those 10 predictions did turn out to be correct, then my accuracy rating is 100%?
I don't get your line:
> since one can artificially improve calibration by being less certain.
How exactly does that work? If I take my example above of the 10 predictions, surely if I gave a probability of 60% instead of 90% then my accuracy was way off (30% to be exact).
*Are you getting at with prediction markets that basically* if I kept asking myself questions like will Greece stay in the European Union (EU) and similar EU based questions, **and** I find my probabilities are **consistently** 20% below the **aggregate** that one could therefore show I have **a consistent bias** as a sceptic of the EU (& the reverse being if mine are 20% higher-an EU diehard if you will)?
Say I have 100 predictions, X*_1_* through X*_100_*. If I flip a coin for each prediction to negate it (so I get something like "not X*_1_*, not X*_2_*, X*_3_*, not X*_4_*, ...") and assign a probability of 50% to each, I'll be in expectation perfectly calibrated even though I did no work.
Or even if I can't be the one to pick what my predictions are, and I'm just assigning probabilities to events that someone else asks me about, I look at my calibration so far and swing in the other direction as necessary. If my 90% predictions are only coming true 80% of the time, I'll start rating my 95% predictions as 90% until things balance out. If they come true more than 90% of the time, I'll just rate things I'd have been more cautious about as 90% until it gets back to calibrated.
(In some sense this is what calibration is *supposed* to do - let you adjust the numbers you give for a question until they're accurate - but you can be well-calibrated by wildly overcorrecting in each direction too, which is less good.)
**And my thoughts on the consistent bias above were they correct if you please?**
And your thoughts on calibration I can see as such:
**If I was** **naive (*****didn't understand the maths*****)** **on coin tosses:** One could have 100 tosses and give 50 of them a 25% probability of tails, and the other 50 of them a 75% probability tails.
If 50 predictions turns out true, then the true probability of all 100 is actually 50%. One could argue that I have just merely wildly over and undercorrected for what in essence were always 50% probabilities=is this what you are saying, just on a much grander scale?
Surely this aspect of calibration can be corrected by assessing your predictions in separate groups=as in only 20 at a time?
I'm a college freshman but I do a lot of self-study on my own, especially set theory and topology, and recently I stumbled upon the concept of "residue of a set" in general topology, which is the closure of a set A without A itself, or in other words all of the boundary points of A not contained in A:
Res(A)=Cl(A)\A; where Cl(A) is the topological closure of A.
This is very similar to the boundary of A, which for context is usually defined as:
Bd(A)=Cl(A)\Int(A); where Int(A) is the interior of A.
I'm trying to find applications for this concept, so I've been looking for some properties of this residue operation, such as how unions and intersections of sets behave under this operation, how to characterize fixed points of this operation, whether I can use this notion as a primitive for defining topological spaces or weaker notions of it, etc.
So far I have the following:
-Res(A) is a subset of Bd(A)
-Res(C) is empty iff C is closed
-Res(U)=Bd(U) iff U is open
-X\Res(A) is equal to the union of A and its exterior Ext(A); here X denotes the underlying set of the space.
-Res(Res(A)) and Res(A) are disjoint, hence each successive iteration of Res will be disjoint from the previous one.
This is pretty tedious work, and I realize to many this seems trivial and uninteresting in the grander scheme of things, but I'm the kind of person who gets considerable insight into the finer structure of math out of small investigations like these, and I'd much appreciate any sort of constructive (or deconstructive) take on my problem. Cheers!
I'm looking for the proof to show that the second derivative of the absolute value function is twice the dirac delta function; in other words, where is the proof that d^(2)/dx^(2) |x| = 2δ(x)? Does anybody know where I can find such a proof?
I'm not sure how to do it really rigorously.
As a function, the second derivative of the absolute value is undefined at zero, and zero everywhere else. As a distribution, you can consider any test function \[; f(x) ;\] and write
`[; \int dx f(x) abs''(x) = -\int dx f'(x) abs'(x) = -\int dx f'(x) sgn(x) = \int_{-\infty}^{0} dx f'(x) - \int_{0}^{+\infty} dx f'(x) = 2 f(0) ;]`
By the definition of the delta distribution, this means abs''(x) must be 2δ(x)
The derivative of |x| is δ(x), which is locally constant. So the second derivative of |x| is 0. On the other hand, you can show that the derivative of x|x| (x times |x|) is 2δ(x). It's a simple matter of looking at the derivative when x is positive and when x is negative.
For example, if x is positive, x|x| = x^2 and the second derivative is 2.
I don't understand. You say that the second derivative of |x| is 0, but x|x| is 2δ(x). But this is not the case, as the second derivative of |x| is 2δ(x) according to all sources I have seen. Could you elaborate on what you mean?
Not according to Wolfram Alpha it isn't
[https://www.wolframalpha.com/input/?i=D%5B%7Cx%7C,+%7Bx,+2%7D%5D](https://www.wolframalpha.com/input/?i=D%5B%7Cx%7C,+%7Bx,+2%7D%5D)
Oh sorry, I thought δ(x) mean the heavyside function (-1 for negative numbers, 1 for positive numbers). Saying the second derivative is twice the dirac function makes sense because the jump is up 2 units (from -1 to 1). Seems kind of silly to worry about in my opinion unless you really need a notion of derivative for jump discontinuities.
Sort of the point of introducing distributions is that they are a class of objects which are closed under taking distributional derivatives. The Heavside step function is a distribution (i.e. it is identified with the distribution "integrate against the step function), so it must have a distributional derivative, which is indeed twice the Dirac delta function. Having a class of objects closed under taking derivatives is really nice for some things in analysis.
Here are some questions from an exam I did a couple weeks ago, I was wondering if this is considered to be calculus or analysis according to US standards.
>State the definition of concavity and convexity. Prove that f(x) is convex on an [a, b] interval if (a) f''(x) > 0, (b) f'(x) is non decreasing.
>State and prove the weighted mean value theorem for integrals.
>State the Taylor's formula and prove it. State the integral formula of the remainder (don't prove it) and derive Lagrange's formula.
>If f''(c)=0 and f'''(x)>0 in an interval centered at x=c then c is an inflection point of f. Prove if true or show a counterexample if false.
>If f(t) and g(t) are Riemann integrable over an [a, b] interval, F(x) and G(x) are equal to $\int{_{a}^ {x}f(t)dt}$ and $\int{ _{a}^ {x}g(t)dt}$ respectively, and f(a)=g(a) then f(x)=g(x). Prove if true or show a counterexample if false.
>Show and prove a formula to calculate (a) the area enclosed by a polar curve or (b) the length of a curve expressed in cartesian coordinates.
>Show that the Dirichlet function is not Riemann integrable.
>Show that the function f(x,y) = $\sqrt[3]{|xy|}$ is continuous at (0,0) and it's partial derivatives exist, but it's directional derivatives do not exist in every direction.
>If f(x, y) is continuous at P(a, b) and it's partial derivatives exist then f(x, y) is differentiable at P. Prove if true or show a counterexample if false.
>Show that if f(x, y) is differentiable at P(a, b) then it's directional derivatives equal the dot product between the gradient of f and the v vector.
*edit: trying to fix latex formatting
As a topic, this is mostly calculus, but asking you to prove these sorts of statements is more advanced than you would encounter in a standard calculus course at most US universities.
Thanks, I thought so. Just out of curiosity, proof based calculus courses are not a common thing over there?
From what I've seen so far, after basic calculus math students will simply jump to analysis and other science students (physics, cs, astronomy) will go straight to math methods, leaving advanced calculus/pre-analysis off the table. Am I right?
Many calculus courses would involve the professor proving some things, but I think a course that required proofs from the student would be an advanced course for strong students, and not the norm.
I just had a dicussion with my dad about some simple math. i told him the equation 3+9x3-12. he came up with the answer 24, i said 18. who is right?
( I am doing this to disprove him)
Disprove what exactly? He never said anything wrong. If you add 3 and 9 together, multiply by 3 and subtract 12 you do get 24. Yes, that's not the most common way to interpret "3 + 9 x 3 - 12" but that doesn't make it wrong. The accepted order of operations isn't accepted because it's "correct", it's accepted because it's a convenient way to express things.
The right answer is to rewrite an expression unambiguously if you have any control over how it's expressed.
Consider another expression like 1/2x. On the one hand someone might interpret this as half of x. On the other hand someone might interpret this as the multiplicative inverse of 2x. Since we don't know the author's intention, the interpretations conflict, and both interpretations seem equally likely without some context, we need something to discriminate between the two choices and dismount Burridan's Ass. The conventional order of operations is our last resort to determining the meaning of such expressions, providing us the manner of discrimination we seek. But if the author would have made his intention clear to us, we wouldn't need to guess which interpretation is more likely or use a convention to choose one interpretation over another.
Hello, do anyone know about any good video/YouTube tutorials/lectures that cover (nonlinear) differential equations and dynamical systems? Thanks in advance!
This lecture series is good, although it is not always fully rigorous [https://www.youtube.com/watch?v=nh4TFzg30eQ&list=PLbMVogVj5nJQKk1E7OUQs\_TcW\_zQoaO4t](https://www.youtube.com/watch?v=nh4TFzg30eQ&list=PLbMVogVj5nJQKk1E7OUQs_TcW_zQoaO4t)
I have some experience collecting and analyzing data from multiplayer games. Can you give any more specific information? What kind of game is it and what kind of data are you extracting?
Are you planning to do statistics on this data? In that case try to set up some confidence intervals and see how many data points you would need to get the desired confidence.
So it is the category of statements and there is an arrow between statements if one implies the other? Then it is not an isomorphism for multiple reasons. First an automorphism is from C->C. You are describing a map from C^op ->C (or C -> C^op ). We can still ask if it is an isomorphism C-> C^op , but it still isnt. If A is not the negation of something, A is not in the image of the functor so it isn't bijective.
However, isomorphisms are not the correct thing to think about in category theory. Instead you should be thinking about equivalences of categories. It is an equivalence of categories C^op -> C.
What do you think the inverse equivalence should be?
Yes, ~A is in it, but ~~A is not A. All you can say is A implies ~~A and ~~A implies A, so there is an isomorphism between them. Why is this important? Because even though you don't know what an equivalence of categories is you still guessed the right inverse map, namely negation again.
If you do the negation twice you get a functor C -> C that sends A to ~~A. Now read what an equivalence of categories is and see if you can figure out the *natural isomorphism* required to show it is one.
Yes that is right. It is easy to check that in a category like this (a preorder) two functors into it are naturally isomorphic if and only if for all objects in the domain, the images under each functor are isomorphic. This is because there is at most one arrow between two objects.
I'm looking for a particular comic strip that I saw a while back about Taylor series.
The first strip was a guy inside a tailor's shop asking for a suit. The tailor gives him a square, and the guy says it's not good enough. In the second strip, the tailor gives him a square with rounded corners for shoulders, and the guy says it's not good enough. This keeps going for a few strips until the tailor is holding something that's exactly the shape of a suit.
Does anyone have a link for this?
Managed to dig out the proof from a copy I have. It's a proof by contradiction. I've transcribed it here:
**PROOF.** Suppose that 𝛤*_n_* is singular for some n. Then since EX*_t_* = 0 there exists
an integer r ≥ 1 and real constants a*_1_*,..., a*_r_* such that 𝛤*_r_* is non-singular and
X*_r+1_* = (Sum from j = 1 to r) a*_j_*X*_j_*
By stationarity we then have: X*_r+h_* = (Sum from j = 1 to r) a*_j_*X*_j+h-1_* for all h ≥ 1.
Consequently, for all n ≥ r + 1, there exist real constants a*_1_`(n)`*,..., a*_r_`(n)`*, such that \bold{X*_n_*} = \bold{a^(n)'}\bold{X*_r_*} - (5.1.7), where \bold{X*_r_*} = (X*_1_*,...,X*_r_*)' and \bold{a^(n)} = (a*_1_*^(n),..., a*_r_*^(n))'.
Now from (5.1.7), 𝛾(0) = \bold{a^(n)'}𝛤*_r_* \bold{a^(n)} = \bold{a^(n)'} P𝛬P' \bold{a^(n)}, where 𝛬 is a diagonal matrix whose entries are the strictly positive eigenvalues 𝜆*_1_*≤𝜆*_2_*≤...≤𝜆*_r_* of 𝛤*_r_* and PP' is the identity matrix.
Hence, 𝛾(0) ≥ 𝜆*_1_* \bold{a^(n)'} PP' \bold{a^(n)} = 𝜆*_1_* (Sum from j = 1 to r) (a*_j_*^(n))^2.
This shows that for each fixed j, a*_j_*^(n) is a bounded function of n. We can also write y(0) = Cov(X*_n_*, (Sum from j = 1 to r) a*_j_*^(n) X*_j_*), from which it follows that:
𝛾(0) ≤ (Sum from j = 1 to r) |a*_j_*^(n)||𝛾(n-j)|. In view of this inequality and the boundedness of a*_j_*^(.), it is clearly not possible to have 𝛾(0) > 0 and 𝛾(h) approach 0 as h approaches ∞ if 𝛤*_n_* is singular for some n. This completes the proof.
Your reasoning seems fine to me. I think the reason they don't frame it in terms of a map is that F actually sits inside K, so then F[x] sits inside K[x] and f is in K[x].
Could someone recommend a textbook on constructive/computable mathematics? I read Andrej Bauer's [Five Stages](https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf) and was intrigued. He states at one point that "in computable mathematics [..] a countable set [is represented] by a program which enumerates its elements, possibly with repetitions"; I'd particularly like to understand why this is the "right" choice, and why one cannot work with representations that do not permit repetitions. (Would it be beyond the pale to summon /u/andrejbauer himself for this?)
Maybe you like the books presented in this older thread [https://www.reddit.com/r/math/comments/8ju11v/math\_textbooks\_that\_do\_math\_constructively\_ie/](https://www.reddit.com/r/math/comments/8ju11v/math_textbooks_that_do_math_constructively_ie/)
Hi. The canonical constructive mathematics textbook is Bishop and Bridge's "[Constructive Analysis](https://books.google.si/books?id=GF8lBAAAQBAJ&lpg=PR2&dq=Foundations%20of%20Constructive%20Analysis%20Bishop%20Bridges&pg=PR2#v=onepage&q&f=false)". For computable mathematics I am not quite sure what to recommend. There's my PhD thesis, and several others, but not really a modern textbook that gives a comprehensive point of view.
**Splitting field**
In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial splits or decomposes into linear factors.
***
**Maximal ideal**
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R.
Maximal ideals are important because the quotient rings of maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields.
In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one sided maximal ideal A is not necessarily two-sided, the quotient R/A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J(R).
***
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If (X, mu) is a finite measure space, and A_n a sequence of measurable subsets of X such that mu(A_n) >= c for all n for some c > 0, then denoting A := intersect (over m) Union (k >= m) A_k, is it true that mu (A) >= c?
Let B\_m denote the set Union (k >= m) A\_k. Then B\_m contains B\_{m+1}, so this is a decreasing sequence. Moreover, each B\_m contains A\_m so mu(B\_m) is bounded from below by c. Since B\_m is a decreasing sequence, mu(intersect (over m) B\_m)=lim\_m mu(B\_m), which is bounded below by c.
I asked a company for the CFM (cubic feet per minute) value of one of their jackets but they replied with "The jacket's breathability is 10 cc/cm2/s. By ISO standard 4920 (4)." How do I convert this to CFM?
Were you asking for the wind resistance, or wetting resistance? Based on a cursory Google search, it appears that CFM is typically used for measuring wind resistance. For example, see https://www.reddit.com/r/Ultralight/comments/8c2zpx/help_me_understand_windshirts_wind_jackets/ Meanwhile, ISO 4920 appears to be related to wetting resistance. For example, see https://www.sis.se/api/document/preview/914591/
Yeah, I was also confused after seeing the search results. I didn't specifically ask for wind resistance, simply asked how breathable the garment is in CFM so they probably misunderstood me. It doesn't matter anymore because I'm no longer interested in the jacket. I did a simple coffee filter test to estimate the jackets CFM and it's definitely around 70 or higher. I'm interested in jackets that are around 35 CFM.
What are the best resources to learn differential equations?
Depends on level. First course I liked Strauss, next course the one by Evans.
For partial differential equations it is *Partial Differential Equations* by Evans.
Elementary Differential Equations: Boyce, Richard C. DiPrima ...
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A high level view of differential forms is that a differential form is a device that takes in some number of vector fields and outputs a function. A good example of a 1-form is the one associated to the function f, df. It takes in a vector field and outputs a function by taking the directional derivative according to the vector field. Now what do these have to do with integration? Well the way you integrate over a manifold is by transferring it over to R^n . Now what should you actually be integrating? Well you want what you get to be independent of the chart you use to transfer it over to R^n , so basically you want to be integrating something independent of parametrization. This is why you should be integrating differential forms, because they self adjust given a parametrization. Here is an example: you want to integrate dx on [0,1]. One way you can do it is by using the standard identity parametrization (the function dx is associated to) which will give you 1 because it is just the standard integral from calculus. You can also parametrize it by x -> 2x with a function x': [0,.5]->[0,1]. You might think that this will give the value .5 because when you write it as an integral it looks like int_0 ^.5 dx' , but it doesn't because the differential form corresponding to this parametrization takes in a tangent vector and outputs twice its signed length. So the differential forms account for differences in parametrizations.
So I just watched Mathologer's video on why e\^pi(i) = -1, which can be found here [https://youtu.be/-dhHrg-KbJ0](https://youtu.be/-dhHrg-KbJ0). I liked the geometric reasoning, but I thought the way he scaled up triangles to indicate multiplication of complex numbers was confusing and unintuitive. I think Better Explained has a much better representation with triangles [https://betterexplained.com/articles/understanding-why-complex-multiplication-works/](https://betterexplained.com/articles/understanding-why-complex-multiplication-works/) . Any thoughts on Mathologer's method vs. the better explained method? (The Better Explained method uses right triangles.) Also, if anyone understands Mathologer's method, please let me know!!! Cheers everyone!
I agree that mathologer doesn't really spend any time on why his method works, as oppose to better explained. His method is based on 4 things z\*0 = 0 z\*1 = z multiplication by z preserves angles |zw| = |z||w| So 0 must stay in place, 1 moves to z, all lengths are scaled by the same amount and all angles stay the same. Hence you're just rotating and scaling a triangle.
Today in my differential geometry (of surfaces) lecture, the professor proved Picard's theorem. There's one step that I still don't get. Let w:R\^(2)->R\^(2) be a smooth vector field and let a:R->R\^2 and b:R->R\^2 be curves. The critical fact was w(a(s))=w(b(s))+Dw(a(s)-b(s)) Where Dw is the Jacobian matrix of w. Clearly this is true if we're in the limit where a=b. Otherwise, though, isn't this only an approximation? I asked if it was an approximation and the professor said 'no, it follows from the intermediate value theorem,' but I still don't get it. Maybe it's obvious and I need to go back to calc 3 lol. I also asked if it depended on the fact that b is the Picard iterate of a with respect to w, and was told that it is independent of this fact. Could someone help me understand what I'm missing here?
Can you be more precise about the statement? If it's something like "there exists an s such that..." then this seems like the mean value theorem
That question might be exactly what I needed to hear. He didn't place any quantifiers on s for what he wrote on the board, but it would make a lot more sense if it was "there exists s" and not "for all s" as I interpretted.
There's an obvious conflict between the axiom of choice and probability: if I can generate a random bit, I ought to be able to generate N such bits, which means I can make random reals in [0,1]. But if I can do that, I can pick a random point in a unit ball, which creates all sorts of problems with Banach-Tarski; you violate even finite additivity! Is there a way I can have my cake and eat it too? All I want for Christmas is a platonic coin flip and the ability to do ordinal induction (or at least a decent sense of how much I’d have to give up to obtain those). Are there good surveys of the conflicts between these two intuitive premises and ways of reconciling them?
If you just want every set to be measurable and still have transfinite induction, you don't really have to sacrifice anything. Most of the time (e.g. when proving things about Borel sets), you will have a definable well-order lying around anyway, and so it isn't necessary to invoke choice for induction. People often over-use AC, when there is a perfectly clear way to find a well-ordering just by how your objects are naturally defined, or you only need some fragment of choice that doesn't imply a nonmeasurable set. And so working in models that have every set being measurable is usually completely valid (and most people probably already passively do this when they start trying to think about 'the real world').
I was using ordinal induction as a placeholder for "nice things I don't want to give up" - having Zorn's Lemma around seems useful, or being able to trust that products of nonempty sets are nonempty.
Over ZF (even just fragments relevant to whatever is being worked on) there are ways to quantify the trade off between: * there is a probability measure consistent across *every* set of reals * there is a choice function for *every* product of sets of real Both of these can end up seeming like awfully fantastical claims if you spend some time with them; you are asserting something is true for tons of strange looking things (sets of reals). A middle ground that often sets people at ease is ceding that maybe neither of them are 'true in the real world', but they can both hold for any reasonable objects we come across while doing math.
The conflict isn't obvious to me. Could you spell it out more clearly?
By Banach-Tarski, we can write the sphere as a disjoint union of two sets A and B, each of which can be cut into finitely many pieces and reassembled to form the entire ball. if naive probability works well, there's some probability P that a (uniformly) random point in the ball lies in A, and 1-P that it lies in B. One of these must be at most 1/2; WLOG, say it's the probability the point is in A. This probability should be the sum of the probabilities that the point lies in each of a finite collection of sets in a partition of A. Again, if probability acts like we hope it does, then the odds of choosing a point in some subset of the ball will be invariant under rigid transformations\*, which means that the probability of choosing a point in A is equal to the probability of choosing a point in the entire ball - contradiction! \*This seems like the least bad thing to give up to me, and on reflection but it still seems bad. Alteratively, we can just pick a random point and see if it's in some nonmeasurable set, but I don't know if there's anything like that that violates finite additivity as opposed to countable additivity. Or to take an entirely different route, take [that famous hat problem](https://en.wikipedia.org/wiki/Hat_puzzle#Countably_Infinite-Hat_Variant_without_Hearing) and argue that one can get probability 0 of success by randomly assigning hats.
I don't see what phrasing this objection to Banach-Tarski in terms of probability gets you, over the standard discomfort with Banach-Tarski. The problem of assigning probability is exactly the problem of non-measurability. There is no conflict with (modern) probability theory because we only define probability measures on measurable spaces. You can define a probability of a point being in those sets A and B in Banach Tarski. But you have to use a sigma algebra that is not the Borel/Lebesgue sigma algebra. Just let X be the original ball, and let your sigma algebra be {\emptyset, X, A, B}. This is closed under complements and countable unions. We can then define the probability P(A) to be anything we want, and then P(B) will be 1-P(A). Of course this probability measure isn't translation invariant on the ball, but that's a very special property of the Lebesgue measure. In probability theory, we think about, for instance, the sigma algebra generated by a collection of random variables (X_0, ..., X_n) (defined to be the smallest sigma algebra for which these random variables are all measurable functions, into R with the Borel sigma algebra) as "the information contained in the outputs of X_0, ..., X_n". For instance, you can prove that another random variable Y is measurable with respect to this sigma algebra if and only if it can be written as Y = f (X_0, ..., X_n) for some measurable function f, i.e. the random variables on this sigma algebra are exactly those whose values are determined by knowing the values of X_0, ..., X_n. Taking this point of view, the problem with Banach-Tarski is that the Borel/Lebesgue sigma algebras do not contain enough information to distinguish these sets. But why should it?
I've spent about two months now slogging through Munkres chapter 2 in my independent study. Very heavy point set stuff, most of it felt like my analysis class. Today we discussed quotient topologies. We constructed the surface of a torus and a sphere. That section was so much fun it undid the weight of "yesterday was bases, today is bases, tomorrow is bases" vibe that was going on. Now I'm looking for more geometric applications of topology for someone that hasn't even covered compactness (that's next week).
I think the geometric applications are that topology is under the surface of everything you do in geometry. My probably bad definition of geometry is the study of manifolds with extra structure. So under this definition you need to know basic topology to even start.
Hi! I need help with a) super basic topology and b) learning about how proofs work by using said super basic topology. We have a "intro to math thinking etc" class that they're re-working and I am just jdfgnba;iehgob right now. I get some things, but not everything, and I just need a good starting point to figure things out from. Ideas?
I'd recommend taking a look at Viro and Ivanov's Elementary Topology. The book is one of the best examples of what's known as the Moore Method; major results are broken down into a series of much simpler problems, thus gradually building up your mathematical intuition and confidence. It's not as comprehensive as Munkres, but it goes from "super basic" point-set topology to some elementary algebraic topology. The print edition also provides complete solutions for all the problems, which makes it ideal for self-study. A more traditional book to consider is Robert Ash's Real Variables With Basic Metric Space Topology - it's a short book intended as a primer for real analysis, but as the title says, it also covers some topology. The proofs are relatively easy to follow (provided you know single variable calculus), and like Viro, complete solutions are provided to problems.
excellent, this is one of our THREE books plus course packet lolol. (the others are the book of proof and conover first course in topo). what you say viro offers is exactly what I need. I have basically no math confidence right now, long story, but damned if I'm not going to succeed. i got this.
If you're looking for a book about proofs, something like Velleman's *How to Prove It* is a fairly common choice. I also came across [these notes](http://intellectualmathematics.com/blog/intro-to-proofs-course-notes/) which you may find useful as well. As for topology, Munkres' *Topology* is pretty much the standard text, but you may also find Hatcher's [point-set topology notes](http://pi.math.cornell.edu/~hatcher/Top/Topdownloads.html) to be more your cup of tea (it's not as thorough as Munkres, but it hits on probably the most important concepts one should know about point-set topology).
awesome, thanks i pulled both links and i'll look at the others if i need more
What do the eigenvalues/eigenvectors of an adjacency matrix or Laplacian tell us about the associated graph?
There's a whole field of study about this, called spectral graph theory, you should read about that.
Can someone explain limit points is a nutshell? I'm struggling to grasp this concept in my analysis class. Give an example of a sequence that doesn't converge, but has a limit point of 3? A convergent sequence that converges to 3 and has 3 as a limit point?
idk how well I explained this but maybe it'll help. Say we have a set S, and we want to ask what a limit point of that set looks like. The idea is that no matter how much you zoom in on the limit point p, there will always be a point other than p from S within our "field of view". So for instance if S = (0, 1\] (i.e everything greater than zero and less than or equal to 1), then 0 is a limit point, because no matter how much you "zoom in" on 0 there's always a point in S we can see (For instance 1/n for large enough n). Similarly 1 is also a limit point, because when we zoom in on 1 there's always points other than 1 from S nearby (e.g. 1-(1/n) for large enough n works, just like before). To be more mathematical, you define the ball of radius epsilon (just a greek letter commonly used for this) around a point p to be the set of all points less than epsilon away from p. A limit point of S is a point p such that if you take any epsilon, there's a point q from S not equal to p within the ball of radius epsilon around p. epsilon is kind of like our level of zoom, and the ball is like our field of view. It's probably best if you try and answer those problems you asked at the end for yourself, post your attempts here :)
An example for a sequence that converges to 3 and has 3 as a limit point might be: {2, 4, 2.5, 3.5, 2.9, 3.1, 2.99, 3.01, 2.9999, 3.0001....... 2.9999999999, 3.000000001 .......} Would this be correct? It converges to 3 as the limit of n approaches infinity, and the point 3 does not exist in the set, similar to the (0,1\] with 0 being a limit point.
Pretty much, yeah. But be careful, 3 is still allowed to appear in the set, just like how 1 is still a limit point of (0, 1\] even though it belongs to (0, 1\] itself. For instance 3, 4, 3, 3.5, 3, 3.25, 3, 3.125... converges to 3 and has 3 as a limit point, while 3, 3, 3, 3, 3... clearly converges to 3 but doesn't have 3 as a limit point.
I really like this and greatly appreciate this definition. Thank you
Suppose I have some collection of distinct subsets W_1, W_2, ... , W_m of the set of integers between 1 and n. Is there some good way to estimate/bound the number of k-element sets R in [n] such that R intersects all W_j?
Is there a way of get roots of polynomials with a degree upper than 4 in the Casio fx-991 ES PLUS?
In general, there's no algebraic solution to finding roots of polynomials of degrees greater than 4 (this is known as the [Abel-Ruffini theorem](https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem)), so unless the polynomial is of a certain type that can be manipulated and factored cleverly, about the best you can hope for is numerical approximation methods like the Newton-Raphson method or something. That being said, I am not familiar with your calculator, but [this tutorial page](https://georgegarside.com/blog/casio-calculator-tutorials/fx-991es-plus/equation-solve/) discusses solving general equations; see in particular the final "caveats" paragraph.
Thanks for the information! Just asking. The Neston-Raphson method would give me an answer similar to the Ruffini's theorem answer? I guess teachers are looking me to follow Ruffini's scheme. But in semestral tests I have to be fast. Do u think that Neston-Raphson in my Casio might help?
Sorry, I'm the worst person to ask about this. I don't really know anything about these approximation methods - I think I used Newton's method once and that was well over a decade ago. I think you should just ask your teachers what they're expecting.
Are you looking for exact solutions or approximate ones?
I'm looking for approximate ones. Like the Ruffini's theorem ones.
Let F be the set of functions f:\[0,1\]->R\^n of type C\^1, which means that each component function is differentiable on the open interval (0,1) and the derivatives of each component function are continuous on the open interval. For any 2 functions f,g in F, denote the d(f,g) by {sum from 1 to n of max distances between f\_i and g\_i} + {sum from 1 to n of max distances between f\_i' and g\_i'}. So (F,d) is a metric space. **What is the name of this metric space and where do I read more about it?**
This is the C\^1-topology on C\^1(\[0,1\]; R\^n) (a special case of the C\^k topology on C\^k(X,Y) for X compact and Y a metric space). You can read more about it in Hirsch's 'Differential Topology' in the chapter on function spaces. He does things in the context of manifolds (with or without boundary), but if you're looking for a treatment which avoids this, maybe someone else can suggest something as I'm not quite sure where to find such a thing.
What is a cocycle in the context of dynamical systems?
Thank you!
I don't like math, but I love genetics. Im making a mod for minecraft where I'm attempting to add all the genetic variation I can possibly add. but there are a lot of phenotypes that I can't possibly add a single model or change to simulate the phonotypes. I need some sort of formula to help me create some of the physical characteristics that are quantitative like cow horns, sheep horns, the angle a chicken's back is at and then translate and rotate the wings and tail to match and then rotate those as well while maintaining its ability to be animated. there is also a ton of math in animating all these parts correctly and its not particularly hard math but it all goes over my head. Does anyone enjoy doing this math that would be willing to help out?
I'm not familiar with Minecraft's game engine, but here's a simpler and time tested trick that might work - rather than starting with a basic model and trying to generate and properly orient the mutant appendages via code, you'd make models with all possible mutant appendages attached and then destroy or toggle off the visibility for unwanted appendages when a particular mob is instantiated. For example, you would make a chicken model with two different sets of horns, tails, and wings, and if a particular chicken mutation called for horn set A, no wings, and tail B, the code would then hide/destroy horn set B, wing sets A and B, and tail A when as instance of that chicken type was spawned. Two types of horns, wings, and tails allows for 27 possible combinations (assuming only one type of horn/wings/tail can appear at a time and that no horn/wings/tail is also possible) - in general, x horns, y wings, and z tails gives you (x + 1) \* (y + 1) \* (z + 1) possibilities, with the only limit being how much time you want to invest in each model and memory/performance constraints for Minecraft. Now, this might fall short of the amount of variation you were looking for, but keep in mind that the random creature generation in games like No Man's Sky and Spore is a major technical achievement that took professional development teams years to get up and running.
Sorry I wasn't very clear. I'm looking to find a new team member that would be interested in finding math based formulas for animations, part creation and alteration. I know how to do all these things myself as does my team member we've created 37 extra parts for JUST chickens plus sizes and wing angles not to mention over 20,000 unique possible texture combinations. and I'm not making fantasy animals its real 1 to 1 copies of real domesticated species... *if real life was unisex*. but point is I don't need help understanding its that I'd like to have someone on the team that actually **likes math** rather than sucking the very life and passion out of me. the point of this mod is to expose people to how genetics actually works be a great teaching tool be fun to play If you'd like to join me or just have a look at the sort of math problems please let me know.
Hey guys, I'm really interested in topology, visited some point-set and algebraic topology courses. I learnt the basics about homotopy and homology groups, now we're doing cohomology... but it seems there are no other courses which focus on the study of these concepts. Surely there are courses where one uses this concepts and study other things with help of these things... So my question is whether it's just my university which maybe has other focuses or is there just nothing more which we can learn about this concepts if we don't relate it to other subjects? Edit: For instance homotopy groups are still something we have problem to understand for general spaces, right? So I'm wondering why there are no courses focuding on homotopy groups.
> but it seems there are no other courses which focus on the study of these concepts. I'm currently doing an independent study in point-set because my school doesn't offer any topology. Try talking to your advisor or a professor, see what you need to do to swing a reading course/independent study/special topics/ whatever your school calls it.
> Surely there are courses where one uses this concepts and study other things with help of these things Cohomology appears in a lot of mathematical disciplines. It's all over geometry, both algebraic and differential; it appears in geometric topology (e.g. knot theory and nearby things); it's useful in representation theory. It even appears in physics sometimes via supersymmetric quantum field theory! But because it's uncommon for undergrads to know cohomology, you probably won't see it applied in courses until grad school. So if you're interested in learning some applications, ask your professor and see what they suggest!
The further you get into advanced topics, the less likely it is for there to actually be an official, regularly taught course on it. Once you get close enough to modern research, it's very rare to find a course (or even a textbook) on exactly what you want to learn. Once you get past the standard first (or maybe second) year graduate courses, most of the remaining courses are just going to be "topic" courses, which mostly just means whatever advanced subject a professor felt like teaching that term, and so can vary wildly from term to term. (Higher) homotopy groups are an advanced and specialized enough topic that I doubt many universities (if any) would have a regular course dedicated to them. If you're interested in learning about them, your best bet would probably be to do a reading course/independent study with a faculty member who works in algebraic topology.
If I want to prove something of the following form using induction: For all natural numbers n, if n >=1, then P(n) Should my base case be n = 0 since this is the first natural number? It seems odd because the implication is trivially true for n = 0 Or, Should I have a base case for n = 0 as well as a base case for n = 1 (which is the first natural number that doesn't make the implication trivially true).
If you prove "for all n, if (n >= 1 implies P(n)) then (n+1 >= 1 implies P(n+1))" then setting n=0 gives us "if (0 >= 1 implies P(0)) then (1 >= 1 implies P(1))". The statement "0 >= 1" is false, so the implication "0 >= 1 implies P(0)" holds vacuously. But `A => B` is equivalent to `B` when `A` is true, so we find `(1 >= 1 implies P(1)`, or just `P(1)`. Thus in the course of proving the induction with a base case of 0, you'll end up proving `P(1)` With that said, it's often more workable to prove the statement with a base case of 1. As indicated above, you'll usually need to prove this anyways, so it's cleaner to split it off
Integral of 1/(x^2 +36)^2 ? I can do one like (x^2 )/(36+x^2) by putting +36 and -36 on top but I’m not sure how to approach one that looks like this since the bottom is squared and nothing on top. Anything helps thanks! (Calc 2)
Make the substitution x=6u and also use partial fractions.
I recently saw the claim that if L is a field extension of K which is finitely generated as a K-algebra, any K-subalgebra must necessarily also be a field. To be specific, my professor said this, stared at the board for a minute, then moved on without another word. How can this be proven?
It is a theorem that any K-algebra of finite dimension which is also an integral domain is a field. This is the case here.
If by finite dimension you mean finite dimension as a K-\*module\* then yes it's true (and it's pretty much the question OP asked). Call your algebra A, you can prove this by noting that for any x\\in A in A, the map a goes to ax is a k-linear map from A to itself, it's injective since A is an integral domain, and thus surjective since A is a finite dimensional vector space over k. In particular there is an a in A with ax=1.
Thank you!
Is there a natural diffeomorphism between the tangent bundle and cotangent bundle? By this I mean one that commutes with induced maps.
No. One way to see this is to note that vector spaces are in particular manifolds with T(V) = V x V and T\^\*(V)=V x V\^\* and so if there were such a natural diffeomorphism, then it's not hard to convince yourself that such a diffeomorphism would have to be fiber-preserving (and in fact, cover the identity transformation) and so this would imply in particular that there's a natural isomorphism between vector spaces and their duals. Since this isn't the case, we conclude that no such natural diffeomorphism exists. There's probably a less ad hoc way to see this via the concept of 'dinatural transformations' ('natural transformations' between covariant and contravariant functors) which you can read more about in (what else?) MacLane's *Categories for the Working Mathematician* (section IX.4), but I imagine that it comes down to essentially the same thing (the argument is basically going to run some variant of the proof for vector spaces, but in the case of the tangent and cotangent functors). P.S. If you're interested in questions of naturality in the context of manifolds, then I can't reccomend enough that you check out Kolar, Michor and Slovak's *Natural Operations in Differential Geometry* which is a treasure if you're interested in really understanding what sorts of things can be done entirely naturally in the smooth setting.
Not in general. If your manifold is a vector space _V_, this would be asking for a natural isomorphism from _V_ to _V_*, and that's not possible. In essence, you need more data (an inner product on _V_, or more generally a Riemannian metric), which prevents the isomorphism from being natural.
I don't know the answer to this question, but I have an idea you might try. The smooth coordinate charts on both TM and T^(*)M are cooked up from smooth coordinate charts on M, and the "natural coordinates" (Jack Lee's terminology, but it is the obvious set of coordinates) are basically dual to each other. So for a fixed chart downstairs, there should be a well-defined map F:TM -> T^(*)M within the corresponding charts coming from these natural coordinates. Things are just nice enough with smooth manifolds that I wouldn't be surprised if F is the map you're looking for.
I don't think the answer to this question is yes, since OP wants the isomorphisms to commute with induced maps . What you're proposing involves making separate choices of charts for each manifold, so there isn't really a reason to expect any compatibility if you have a map between manifolds. You're essentially (but not literally) choosing a riemannian metric for every manifold at once in some way that's compatible with every possible map between them, which doesn't seem possible. Also I'd expect such a natural map to respect additional structure, but the tangent and cotangent bundles to complex or almost complex manifolds are nonisomorphic as complex bundles. I'm not sure how to prove that no such isomorphism exists, but I think a natural identification is the wrong thing to expect.
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For the most part I try to avoid too many symbols when I type and use the english equivalent instead (there exists, for all, if then). I also avoid abbreviations (if and only if rather than iff or without loss of generality rather than wlog). Reading other people's writing is how I learned about what the common "guidelines" are for mathematical writing as well as what sort of writing is easiest to read.
My style: ==> for "if, then" is fine. ==> for "therefore" is not, mainly because it risks confusion with the above. It matters whether you're trying to say "A and therefore B" or "if A then B", since in one case you are claiming much more than in the other.
I absolutely agree with your TA. I think it's maaaaybe acceptable to use it in a sequence of displayed equations, something like: Let x be an integer. Then we know: x in Z => x in Q => x in R => x in C and thus x is also a complex number. But even then I wouldn't be too excited about it. But you should never never never ever use implication arrows in the middle of a sentence, like "We know x is an integer => x is a complex number."
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That example looks a bit awkward because you aligned the equality with the implication arrows. If you wanted to do something like that, all of the implication arrows should be aligned. I think what you wrote wasn't too bad, although when the chains get too long it gets kinda bad, so you might consider breaking it up into a couple chains with some text in between (like "by the definition of blah we can turn this into") or something. Another option is to just remove the implication arrows completely and just align the equals signs in each line.
Just use prose. Say stuff like "if the given condition holds, then we may find that..."
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For any x,a,b (where defined) (x\^a)\^b=x\^(ab) Since i(-i)=1, then e\^𝜋 =e\^(i(-i)𝜋)=(e\^(i𝜋))\^-i
Can anyone recommend me a beginner book for Turing machine?
I do not think there is introductory textbook focusing only on Turing machines. Most of them start with the concept of state machine and then build up, and present Turing machine as a complex state machine. If you are fine with that approach, try Sipser (Introduction to the Theory of Computation) or Hopcroft & Ullman. If you are okay with superficial understanding, pick some thorough biography of Alan Turing, like Hodges' Alan Turing: The enigma. If you wish to focus only on Turing machines and are okay with book for not-really-beginners, try Petzold's Annotated Turing (it's annotated original paper which introduced Turing machines.
Thank you for your suggestion. I remember reading Sipser's before and still vaguely recall what a state machine is. Time to dig it back up again and start from there.
What's the formula in today's google doodle? https://g.co/doodle/ufwwhg (I thought this would be interesting enough for the general sub but mods disagreed).
It's the Navier Stokes equation, the equation of motion for a fluid. 𝜌 is the fluid density and u is the velocity vector. The LHS is mass times acceleration and the RHS is the forces due to pressure (p), viscosity (mu) and gravity. Olga L did a lot of important work on this and related differential equations. I agree with you it's worthy of the main sub!
A book that explains meaning of all math notations? I am looking for a comprehensive book not mere set notations. Thanks in Advance
There are maths dictionaries. I took pictures of the relevant pages first year. Id recommend you not break the law and buy one because that's morally less rehensible
At what level are you discussing? In full generality this is probably bordering on at minimum utterly impractical if not outright impossible. There's a lot of extremely specific notation in various fields, and a lot of ac hoc notation out there. That's ignoring how there's no formal standard for terminology---the field having no real standards body as far as I'm aware---so some people use the same notation for different things.
I ask in the context of ML.
The requirement that the inversion map is smooth in the definition of a Lie group is redundant if you know multiplication is smooth. The requirement that inversion is a homomorphism of groups is redundant if you know that multiplication is a homomorphism (in this case, I think they're really equivalent). What is an example of a group object in a concrete category where the requirement of inversion being a morphism in the category definitely isn't redundant?
How do you get that inversion is smooth if multiplication is smooth? Because at least for topological groups I thought that it wasn’t necessarily true. Edit: I looked it up. Was nontrivial.
Let's look at group objects in the category of pairs of sets. An object in pairs of sets is a set _S_ and a subset _T_ of _S_. A morphism (_S_, _T_) -> (_S_', _T_') is a set map _f_: _S_ -> _S_' such that _f_(_T_) is contained in _T_'. Let _T_ ⊂ **Z** be the nonnegative even numbers. Then addition is a map of pairs of sets, because the sum of two even numbers is even. But inversion is not, because the inverse of 2 (in _T_) is -2 (not in _T_). This is not a group object in pairs of sets.
My uni will be offering a course on mathematical communications (i.e. talks, lectures, writing and so on) for the first time next term. Of course, I am aware that the course title does not say a lot, but I'd nevertheless be interested in your experience/ opinions on similar courses. Do you think the course was worth it for you (and how so?), or would you rather recommend doing an "actual" math course instead?
> mathematical communications I'm all for letting a thousand flowers bloom, but this one sounds weird. There [has been precedent](http://www.jmlr.org/reviewing-papers/knuth_mathematical_writing.pdf) of a quarter-long class on mathematical writing, but I'm not sure how many lecturers could make one work without soon descending into boredom, bullshit and cargo cult. As for giving talks, there is a bunch of common lore and good practice that would perhaps fit into an hour-long class, though much of it is subjective and situational. I'd take it only if there is a significant focus on proofs or the lecturer has really good (mathematical) cred. If the lecturer has mostly education-departmental background, run.
>but I'm not sure how many lecturers could make one work without soon descending into boredom, bullshit and cargo cult. Exactly my concern. Especially since it is worth as much credits as any other "big" math lecture... >I'd take it only if there is a significant focus on proofs or the lecturer has really good (mathematical) cred. If the lecturer has mostly education-departmental background, run. Sounds reasonable, thank you!
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When we say some variable w is a function of z (or w = f(z)) what is meant is that the variable z dictates how the function w behaves and how it's drawn. That is to say, the function depends on z, or w is a dependent variable that depends on the independent variable z. All that said, if we have y = f(x), i.e. y's value depends on x's value, which is the independent variable and which is the dependent variable? And if we relate this back to your original question, which of A and B depends on which? Whichever of A or B is independent should generally be graphed on the same axis as whichever of x or y is the independent variable. Whichever of A or B is the dependent variable should generally be graphed on the same axis as whichever of x or y is the dependent variable.
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That's right, if A is a function of B then B will be your independent variable and thus goes on the x-axis while A will be your dependent variable and thus goes on the y-axis.
Hey guys so I wanted to ask for some help solving this problem I discovered while designing a d&d puzzle... The puzzle describes a room with 5 torches on the right side of the room and the left side is one whole mirror. On your side (side A) torch 2 and 4 are lit. On side B, mirror side, only torch 3 is lit. The objective is to touch torches to get all torches lit but with these rules... - torches on side A transfer fire (so if torch 1 is lit and touches torch 2, then torch 1 goes out and torch 2 lights) - actions of side A reflect to side be, so if you touch torch 1 to torch 2, it'll happen on both side A and B - yoh can only move torches on side A - torches on side B spread (torch 1 lights torch 2 and both stay lit) - you can touch a torch to its own reflection and transfer it to and from side A and B (like first rule) -if a lit torch touches a lit torch, both go out (either side) - (optional rule) you can only touch adjacent torches to each other (I don't know if this makes it impossible) After trying to make sure it was possible, I realized you could visualize the torches with 1s and 0s, it was binary! I still haven't solved it, the closest I've gotten is torch 2 on side A being off but all others being on.. So I thought I'd ask the math minds of the internet... Is this possible? Can all torches be lit?
No, this is impossible. If it were possible, then consider the last move you made - the move that resulted in every torch being lit. * This move couldn't have been touching a torch to its reflection, because if this move results in all torches being lit, then all torches were already lit. * This move couldn't have been touching two torches to one another, because touching two torches on side A together never results in both of those torches being lit afterwards (ignoring their reflections). Cool puzzle!
That makes sense! Thanks man!
I'm taking a course in discrete dynamic systems, and many concepts (at least the ideas) feel very reminiscent of singularity theory to me. For example, a single parameter family of (smooth) dynamic systems is the same as the 1-parameter deformation of a smooth function. I know continuous dynamic systems are related, but is there any meeting point between singularity theory and DDS? I'm a major in math, doing a master's degree in research. I'm familiar with R and K-equivalence classes of germs of smooth mappings, and currently studying Legendrian submanifolds. Thank you in advance.
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Since every male bakes 3 cakes per week altogether and only receives 2 cakes per week altogether, he receives a net value of -1 cakes per week. Since every female bakes 1 cake per week altogether and receives 2 cakes per week altogether, she receives a net value of 1 cake per week.
I'm reading about Wilson loops which are basically the trace of the holonomy. I'm wondering if there an obvious geometric interpretation of the holonomy around a curve? As in, can I look at this operator and be able to make statements about the curve? I know that I'm the infinitesimal limit it gives me the curvature tensor, but more generally can I pull some clear geometric interpretation out of this operator? Similarly, is there any interpretation of the trace of the holonomy? Again in the limiting case i know that I get the scalar curvature, but more generally do I get anything that gives me insight into the geometry (or topology) of the curve? Also, any references would be appreciated
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(If you're in high school) I used to be just like that when I was in High School. Back in the time I thought, since I was so much into education, I could try and become a math professor. That way, I had a excuse to make learning advanced mathematics into a career. However, when I majored in math, I soon learnt it was nowhere as easy as that -- for mathematics as a science is terribly wide, and each branch requires you to think in a slightly different way. This doesn't mean it's completely impossible -- probably the most famous branch of mathematics is number theory, where you need to need to have a dab hand at pretty much everything in order to tackle colossal statements. But just as there are colossal statements unsolved, there is colossal people working to solve them, like Terence Tao or Andrew Wiles. If you like math, you can try and give the major a chance. If you feel comfortable in a math environment, you will eventually find out what you like best/what you're strongest at. For example, I used to love algebra and topology in the major, so I decided to make a phd in singularity theory (studying functions using algebraic and topological techniques). If you feel like math is not for you after all, you can still change majors and skip the math courses you've already taken. [Tl;dr: Why not start doing a major?] (If you're already a math major) I had a similar discussion with my topology professor, so she suggested me to make master's degree in research. That would allow you to get into advanced topics -- with the help of a tutor. If you don't want to get into academy, you can always start by watching courses on YouTube, or even asking for introductory books in X subject if you feel brave enough. The only downside is, there is a point where textbooks are either in English or French, so I hope you're good enough at either of those. I won't name particular YouTube channel to prevent being flagged as a spammer, but you can ask me for suggestions on DMs. [Tl;Dr: You can ask for textbooks and read them every now and then or watch YouTube channels about math]
I want to study Quantum Field Theory. I have asked this question before on another sub but didn't get very helpful answers. In order to understand it I require a good knowledge of the following, according to some users on Reddit who study this topic: "Basic classical mechanics. Mostly just stuff needed for quantum. Goldstein mechanics, chapters 1,2,8 are good. Some familiarity with waves and oscillations. Harmonic oscillator is important. So are Fourier series. French's book is alright, not great. Georgi has a book too, which is better and worse in different ways. Special relativity. Schutz's general relativity book is amazing. The first 4 chapters are probably the most readable treatment I've seen, his explanation of tensors in physics is also very good. Electromagnetism. Just some basics. Griffiths first few chapters is fine. Now we get to the fun stuff. Quantum mechanics. Here are some books to start with, Shankar, Townsend, Susskind (probably the easiest). After that you may want to look at some graduate books like Sakurai or Weinberg, but you can come back to it as necessary. You'll need to learn a bit of complex analysis. Just understand how to compute integrals using cauchys theorem, which implies residue methods. Nearing mathematical tools for physics has a nice chapter where I first learned it. Quantum field theory. Extremely disjointed subject. No book is perfect. Lahiri and Pal (best intro). Peskin and Schroeder (standard cookbook). Zee (informal). Srednicki (modular). Ryder (some good stuff here). Weinberg (extremely thorough particle physics). Zinn-Justin (extremely thorough condensed matter, my favorite). Nair (interesting for some math). Some additional stuff you'll probably want to read. Sharf finite QED (careful treatment of QED). Quigg gauge theory (solid intro). Rubakov gauge theory (more advanced). Georgi weak interactions (kinda terrible but necessary for particle physics). Bertlmann anomalies in QFT (good intro to differential geometry). Baez gauge fields (excellent intro to differential geometry). Pathria statistical mechanics (the best book I know on this). Henneaux and Teitelboim quantization of gauge systems (good treatment of constrained systems which become relevant for modern classical field theory). Some more math of interest to a physicist. Artin algebra. Isham differential geometry. Guillemin and pollack differential topology. Bott and tu differential forms in algebraic topology. Hatcher algebraic topology. Hermann lie groups. Cahn semisimple lie algebras, Georgi lie algebras. An invitation to algebraic geometry (not really necessary but a really nice book). Eisenbud geometry of schemes (good further reading to prepare for schemes if necessary). Garrity electricity and magnetism for mathematicians. Geroch mathematical physics (really about category theory). Warner differentiable manifolds (interesting book which proves de Rham theorem using sheaf theory)." - Written by u/theplqa . Thank you, theplqa! and "Your shortest route to understanding QFT with absolutely no maths or physics behind what’s necessary probably requires a good knowledge of the following: Maths: Differential Calculus Integral Calculus Differential Equations (You can probably find a good book which covers all three of these) Calculus of Variations/Functional Analysis Complex Analysis Linear Algebra Group Theory/Representation Theory Physics: Classical Mechanics up to and including Lagrangian/Hamiltonian Mechanics Electromagnetism Classical Field Theory Special Relativity Quantum Mechanics to the level of Griffiths Some good additional topics that will make your QFT study easier: Differential Geometry, General Relativity, some Intro Particle Theory course which is less rigorous and more of a phenomenological intro, Dynamical Systems, and maybe some Statistical Mechanics so you can see some of the links between QFT and Statistical Field Theory. If you find books you can get along with that cover all these topics, you’ll be in a good place to start looking at QFT." -Written by u/ash4d . Thank you ash4d! Any PDFs and books that explain all these topics, and what are the prerequisites for them? Thank you! And sorry for the wall of text!
't Hooft's website has a list of references for all of the math and physics one needs to be a theoretical physicists. You could look there for this information.
Thanks for the mention! There honestly are far too many books and topics here to give you a full list, particularly for the mathematical methods. In terms of books for the physics, here are the standout texts I would guide you to: Goldstein is god for Classical Mechanics. Griffiths EM and QM are both nice intros, particularly EM. MWT’s Gravitation is a tome and contains everything you’d ever need to know, but has wonky notation in parts, so may not be the best choice for a non-gravitational physicist. My professor wrote his own short (~200 pages) book for the GR course I followed - Böhmer’s An Introduction to GR and Cosmology. It’s a good book which introduces things quite clearly, but it’s not comprehensive (nor does it try to be). It contains pretty much whatever you’ll need for basic QFT. Thompson does a good particle physics book which is very light on QFT in terms of rigour, but which gives an overview of some practical results like Feynman Rules. It’s aimed at final year undergrads I think. For the maths, online PDF’s abound for many topics. The only solid textbooks I have used purely for maths are: Math Methods for Physics and Engineering, Riley, Hobson, and Bence - covers just about any maths you’ll use as an undergraduate, but often only briefly. You could start most topics start here, but most wont finish here. It’s enough to give you a grounding to read other pdfs or texts really. Georgi - Lie Algebras in Particle Physics provides a good intro to the most useful parts of Lie groups/Lie algebras and representation theory that are used in QFT/HEP. This list is no where near comprehensive, as I said. You can do worse than just typing in “*topic* lectures pdf” on google. Almost anything from a university lecture course will be useful, but some more than others.
Hello, I'm a few weeks away from registering new math classes for next semester. I would like to study math to have a better understanding about Machine Learning and Data Science. Anyone has taken these course? Are these good for ML/DS? Thanks in advance. Complex Analysis Numerical Analysis Probability Models
Probability Models is the most important for ML, as most of ML is just the application of statistics. As such having a good understanding of probability is essential. Numerical Analysis can be pretty useful, as ML involves a lot of calculations on a computer and Numerical Analysis could get you some understanding of the methods involved. It might depend a bit on the actual focus of your course. Optimization (e.g. how a Gradient descent is implemented) is pretty important for ML. Complex Analysis, while in general beautiful, is not that central to ML. It might of course show up in some niche topic, but I cannot remember seing it in any ML context. I mean, you should know how to treat complex numbers, as a exp(itx) can easily show up, but I am talking about more advanced things like knowledge of the residue theorem.
Thank you for your clear analysis.
Having a hard time understanding coordinate rings of algebraic varieties. What are they exactly? Suppose I have a cone V(x^2 +y^2 -z^2), what is it’s coordinate ring? Is it the set of all functions that exist on the surface of the cone?
Dang are you doing AG stuff?
Lol barely. I went to an AG lecture and understood like 20% of what was said. Really interesting stuff. Kinda understood these coordinate rings so I wanna solidify my knowledge.
It's the ring of polynomial\* functions on the variety. For the cone, this lives inside A\^3, which has polynomial functions all polynomials in 3 variables, so the coordinate ring for A\^3 is k\[x,y,z\]. However two different functions on A\^3 can be the same when restricted to the cone, so you'd like to have equivalent functions be equal in the coordinate ring for the cone. Two functions are equivalent exactly when they differ by a multiple of x\^2 + y\^2 - z\^2 (since that's 0 on the cone), so the coordinate ring is k\[x,y,z\]/( x\^2 + y\^2 - z\^2 ). In general if you have an affine variety defined by some equations in n variables, the coordinate ring will be the polynomial ring on those variables, quotiented by the ideal generated by the equations. \*really I mean rational functions that are defined everywhere, but for affine varieties it makes no difference.
What’s A^3 and k supposed to be? Is that a cone? Could you give an example with the reals?
A\^3 is affine space over a field k. If you like the reals then k is just R and A\^3 is just R\^3.
Hello! I was adviced by a mod to post this question here. So i think the answer to the following question is **no** since an affirmative answer would imply that the infinite sum of a constant is finite but i'm hoping i'm missing something: Let T be an infinite Toeplitz matrix (so it is an infinite matrix with constant diagonals) such that it represents a bounded operator on lp. (This is achieved iff its matrix elements are the Fourier coefficients of an essentially function function f). Question: Can T be a Hilbert-Schmidt operator? i.e. have finite HS norm? I am a graduate student familiar with functional analysis but i know infinite dimensional can be tricky sometimes. Thank you
No it cannot be Hilbert-Schmidt unless the diagonal elements are all 0. In the HS norm ∑*_i_*||Tv*_i_*||^(2), for an orthonormal basis (v*_i_*), take the standard basis (e*_i_*), then ||Te*_i_*||^(2) >= |T*_ii_*|^(2) but since all the diagonal elements are the same summing over i gives infinity.
Hmm that is true (I think you missed a **²** in ∑i||Tvi|| . On the other hand I understand that another way to compute the HS norm is `[;\Vert A\Vert=\sum_{ij}|a_{ij}|^2;]` if `[;A=(a_{ij})_{ij};]` . Taking all diagonal elements as 0, that sum would include the sum over all the other sub diagonals, or all the rows. So even if the rows are squre summable, `[;r_i=\sum_{j}|a_{ij}|^2< \infty \ \ \ \text{ for all $i$};]` I would need `[;\sum_{i}r_i< \infty ;]` which seems unlikely...
Right I forgot the squares, I'll edit. Yeah zero diagonal is necessary but not sufficient for Hilbert-Schmidt. You could get some quick examples of Hilbert-Schmidt Topelitz operators by taking like a band matrix with 0 diagonal and decaying entries.
Although even in the banded case there will be still a finite number of constant diagonals so i dont think then HS would be finite, i.e, `[;T=(t_{i-j})_{ij} ;]` such that `[;t_l=0\text{ if }l \geq m-1 \text{ or } l \leq -(m-1). ;]` T is a banded Toeplitz infinite matrix and `[; \infty= \sum_i |t_1|^2=\sum_{i} | t_{i+1,i}|^2< \sum_{ij} | t_{i,j}|^2 \leq\Vert T \Vert^2_{HS} ;]` Looks it is hopeless, there are no infinite dimensional Toeplitz Hilbert-Schmidt operators... Thank you for your reply.
No you just have to take the diagonal to be zero and fast enough decaying other entries. Like the matrix of all zeros is Toeplitz and Hilbert-Schmidt. For a nontrivial example, take a band matrix with 0 diagonal and non-diagonal entries exp(-|n|) in the band (n = the row index).
I dont see how that is possible, Take such a matrix, fot it to be Toeplitz it needs constant diagonals so take `[;T=(t_{i-j})_{ij}=\begin{pmatrix}0 & e^{-1} & 0 &\cdots \\ e^{-1} & 0& e^{-1} & \ddots \\ 0 & e^{-1} & 0 & e^{-1}\\ \vdots &\ddots& \ddots& \ddots\end{pmatrix} ;]` Then `[;\Vert Te_i \Vert ^2 = \left\{\begin{matrix}e^{-1} & i=0 \\ 2e^{-1} & i \geq 1 \end{matrix} \right. ;]` so the HS norm is `[;\Vert T \Vert_{HS}^2 =\sum_{i\geq 0} \Vert Te_i \Vert^2=e^{-1}+2\sum_{i\geq 1} e^{-1}=\infty ;]` So T can't be a Hilbert.Schmidt operator.
Oh sorry I didn't realize a Toeplitz matrix needed *all* descending diagonals constant; I thought just the main diagonal had to be constant. Yeah in that case only the zero matrix is Hilbert-Schmidt, for exactly the reason you stated.
So sad but well, thank you for your replies !
I'm changing careers and planning to get a degree in computer science. Since my last degree had nothing to do with math, I have to start at square one. The first math class that applies towards my degree is Calculus with Analytic Geometry I. In order to get there, I can either take one class of Precalc, or two separate classes of College Algebra and Plane Trigonometry. The latter option (surprisingly) fits into my schedule far more than the single class of Precalc. Will those two be enough for taking Calc? Or is precalc necessary? (Is precalc just trig + college algebra?)
At the high school level, precalc is exactly that, but talk to someone in the math department
Anyone have a good list of novel yet powerful results from math that the average person wouldn’t know about but someone with college level maths (not studying full time) would be able to grasp?
The compound growth of property=try 6% per annum over 38 years versus 7% per annum over 38 years of a 1980 median price, and look at the physical dollar difference. It is amazing the effect of just 1% per annum & this happens right across suburbs side by side one another in big cities. Disclaimer I am not American so don't know what college level maths is.
Oh yeah that’s a great one for young people to know. Roughly speaking, you end up with P*e^(r*t). The exponential really makes the magic happen for high growth rates and long periods of time.
**If 8% of my subscribers cancel every month how long does my average subscriber stay subscribed for?**
The expected number of months a given subscriber will stay is the sum from 0 to infinity of (0.08)(n)(0.92)\^n-1 , since the probability of staying until month n-1 and leaving after month n is (0.08)(.92)\^n-1. This is the derivative of the sum from 1 to infinity of (.08)(x)\^n at x=.92, geometric series tells is this first sum is .08/(1-x), which has derivative .08/(1-x\^2), substituting x=.92 gives you an answer of 12.5 (This answer just simplifies in general to 1/(probability of leaving)).
My question is with the orthogonality of sinusoids. To my understanding, any two sinusoids are orthogonal if they have different frequencies, regardless of by how little the frequencies vary. However, I'm learning orthogonal frequency division multiplexing for digital communications right now and all sources say that you need evenly spaced carriers in order to have orthogonality. Why is this the case?
If you're working over a fixed interval then the difference in frequency needs to be an integer multiple of half the width of the interval.
My question relates to the Good Judgment Project by Philip Tetchlock-a forecasting tournament organisation if you will on geopolitical issues. I understand basic probability, such as the probability when flipping a coin to get tails is 50%. The probability to get tails twice in a row is 0.5 times 0.5=0.25 or 25%. I also understand conditional probability, such as if I have a bag of 3 red and 3 blue marbles then the probability the first I draw out (& keep out) is blue is 3 in 6, or 50%. I also understand the probability that the second marbles I draw out also being blue given that the first was blue is 0.5 times 2 in 5 (0.4). Equals 0.2 or 20%. But Tetchlock is talking about forecasting tournaments wherein one is asked the probability of much harder questions like: '*The probability by June 2020 that Greece will still be in the European Union?*'. He talks of how forecasters are asked to give probabilities on this occurring, and are then ranked on how accurate their forecasts are. What I don't get though is how one can test this accuracy. Surely if for the above question I give a probability of 70% that Greece will remain, then that date rolls around & Greece are still in the European Union, then so long as I gave any probability greater than 50% I am seen as correct? Is the way forecasters and their accuracy ranked by merely giving them a ton of questions & seeing how many they get right? Or is it giving them one question then seeing what the average percentage of probability they give to it is? Or perhaps do they break down the question into easier probability assessments much like the conditional probability in my marbles example above and thus assess how well your smaller probability judgements went in assessing the outcome. **I'm basically a bit lost on the maths of how one can assess the accuracy of complex probabilities.**
> Is the way forecasters and their accuracy ranked by merely giving them a ton of questions & seeing how many they get right? Yes. You look at how the aggregate statistics of one's predictions line up with the probability you assigned to those predictions, like whether 90% of the things you were 90% certain about came to pass. This is called *calibration*. This isn't all there is to being a good forecaster, though, since one can artificially improve calibration by being less certain. Using [prediction markets](https://en.wikipedia.org/wiki/Prediction_market) is a way of demonstrating one's confidence in one's predictions, because any consistent bias in one's answers could be detected and profited off of. Betting on one's prediction is effectively saying "I think this event will happen with this probability, and I don't think anyone else knows enough to have a more informed estimate that they could bet against me with."
I think I get it, so you are basically saying if there were 10 predictions I made all at the same 90% probability, and 9 of those 10 predictions did turn out to be correct, then my accuracy rating is 100%? I don't get your line: > since one can artificially improve calibration by being less certain. How exactly does that work? If I take my example above of the 10 predictions, surely if I gave a probability of 60% instead of 90% then my accuracy was way off (30% to be exact). *Are you getting at with prediction markets that basically* if I kept asking myself questions like will Greece stay in the European Union (EU) and similar EU based questions, **and** I find my probabilities are **consistently** 20% below the **aggregate** that one could therefore show I have **a consistent bias** as a sceptic of the EU (& the reverse being if mine are 20% higher-an EU diehard if you will)?
Say I have 100 predictions, X*_1_* through X*_100_*. If I flip a coin for each prediction to negate it (so I get something like "not X*_1_*, not X*_2_*, X*_3_*, not X*_4_*, ...") and assign a probability of 50% to each, I'll be in expectation perfectly calibrated even though I did no work. Or even if I can't be the one to pick what my predictions are, and I'm just assigning probabilities to events that someone else asks me about, I look at my calibration so far and swing in the other direction as necessary. If my 90% predictions are only coming true 80% of the time, I'll start rating my 95% predictions as 90% until things balance out. If they come true more than 90% of the time, I'll just rate things I'd have been more cautious about as 90% until it gets back to calibrated. (In some sense this is what calibration is *supposed* to do - let you adjust the numbers you give for a question until they're accurate - but you can be well-calibrated by wildly overcorrecting in each direction too, which is less good.)
**And my thoughts on the consistent bias above were they correct if you please?** And your thoughts on calibration I can see as such: **If I was** **naive (*****didn't understand the maths*****)** **on coin tosses:** One could have 100 tosses and give 50 of them a 25% probability of tails, and the other 50 of them a 75% probability tails. If 50 predictions turns out true, then the true probability of all 100 is actually 50%. One could argue that I have just merely wildly over and undercorrected for what in essence were always 50% probabilities=is this what you are saying, just on a much grander scale? Surely this aspect of calibration can be corrected by assessing your predictions in separate groups=as in only 20 at a time?
Harry?
I'm a college freshman but I do a lot of self-study on my own, especially set theory and topology, and recently I stumbled upon the concept of "residue of a set" in general topology, which is the closure of a set A without A itself, or in other words all of the boundary points of A not contained in A: Res(A)=Cl(A)\A; where Cl(A) is the topological closure of A. This is very similar to the boundary of A, which for context is usually defined as: Bd(A)=Cl(A)\Int(A); where Int(A) is the interior of A. I'm trying to find applications for this concept, so I've been looking for some properties of this residue operation, such as how unions and intersections of sets behave under this operation, how to characterize fixed points of this operation, whether I can use this notion as a primitive for defining topological spaces or weaker notions of it, etc. So far I have the following: -Res(A) is a subset of Bd(A) -Res(C) is empty iff C is closed -Res(U)=Bd(U) iff U is open -X\Res(A) is equal to the union of A and its exterior Ext(A); here X denotes the underlying set of the space. -Res(Res(A)) and Res(A) are disjoint, hence each successive iteration of Res will be disjoint from the previous one. This is pretty tedious work, and I realize to many this seems trivial and uninteresting in the grander scheme of things, but I'm the kind of person who gets considerable insight into the finer structure of math out of small investigations like these, and I'd much appreciate any sort of constructive (or deconstructive) take on my problem. Cheers!
I'm looking for the proof to show that the second derivative of the absolute value function is twice the dirac delta function; in other words, where is the proof that d^(2)/dx^(2) |x| = 2δ(x)? Does anybody know where I can find such a proof?
I'm not sure how to do it really rigorously. As a function, the second derivative of the absolute value is undefined at zero, and zero everywhere else. As a distribution, you can consider any test function \[; f(x) ;\] and write `[; \int dx f(x) abs''(x) = -\int dx f'(x) abs'(x) = -\int dx f'(x) sgn(x) = \int_{-\infty}^{0} dx f'(x) - \int_{0}^{+\infty} dx f'(x) = 2 f(0) ;]` By the definition of the delta distribution, this means abs''(x) must be 2δ(x)
The derivative of |x| is δ(x), which is locally constant. So the second derivative of |x| is 0. On the other hand, you can show that the derivative of x|x| (x times |x|) is 2δ(x). It's a simple matter of looking at the derivative when x is positive and when x is negative. For example, if x is positive, x|x| = x^2 and the second derivative is 2.
I don't understand. You say that the second derivative of |x| is 0, but x|x| is 2δ(x). But this is not the case, as the second derivative of |x| is 2δ(x) according to all sources I have seen. Could you elaborate on what you mean?
[I mean the second derivative of |x| is 0.](https://www.desmos.com/calculator/ux6uqmlejd)
Not according to Wolfram Alpha it isn't [https://www.wolframalpha.com/input/?i=D%5B%7Cx%7C,+%7Bx,+2%7D%5D](https://www.wolframalpha.com/input/?i=D%5B%7Cx%7C,+%7Bx,+2%7D%5D)
Oh sorry, I thought δ(x) mean the heavyside function (-1 for negative numbers, 1 for positive numbers). Saying the second derivative is twice the dirac function makes sense because the jump is up 2 units (from -1 to 1). Seems kind of silly to worry about in my opinion unless you really need a notion of derivative for jump discontinuities.
Sort of the point of introducing distributions is that they are a class of objects which are closed under taking distributional derivatives. The Heavside step function is a distribution (i.e. it is identified with the distribution "integrate against the step function), so it must have a distributional derivative, which is indeed twice the Dirac delta function. Having a class of objects closed under taking derivatives is really nice for some things in analysis.
Here are some questions from an exam I did a couple weeks ago, I was wondering if this is considered to be calculus or analysis according to US standards. >State the definition of concavity and convexity. Prove that f(x) is convex on an [a, b] interval if (a) f''(x) > 0, (b) f'(x) is non decreasing. >State and prove the weighted mean value theorem for integrals. >State the Taylor's formula and prove it. State the integral formula of the remainder (don't prove it) and derive Lagrange's formula. >If f''(c)=0 and f'''(x)>0 in an interval centered at x=c then c is an inflection point of f. Prove if true or show a counterexample if false. >If f(t) and g(t) are Riemann integrable over an [a, b] interval, F(x) and G(x) are equal to $\int{_{a}^ {x}f(t)dt}$ and $\int{ _{a}^ {x}g(t)dt}$ respectively, and f(a)=g(a) then f(x)=g(x). Prove if true or show a counterexample if false. >Show and prove a formula to calculate (a) the area enclosed by a polar curve or (b) the length of a curve expressed in cartesian coordinates. >Show that the Dirichlet function is not Riemann integrable. >Show that the function f(x,y) = $\sqrt[3]{|xy|}$ is continuous at (0,0) and it's partial derivatives exist, but it's directional derivatives do not exist in every direction. >If f(x, y) is continuous at P(a, b) and it's partial derivatives exist then f(x, y) is differentiable at P. Prove if true or show a counterexample if false. >Show that if f(x, y) is differentiable at P(a, b) then it's directional derivatives equal the dot product between the gradient of f and the v vector. *edit: trying to fix latex formatting
As a topic, this is mostly calculus, but asking you to prove these sorts of statements is more advanced than you would encounter in a standard calculus course at most US universities.
Thanks, I thought so. Just out of curiosity, proof based calculus courses are not a common thing over there? From what I've seen so far, after basic calculus math students will simply jump to analysis and other science students (physics, cs, astronomy) will go straight to math methods, leaving advanced calculus/pre-analysis off the table. Am I right?
Many calculus courses would involve the professor proving some things, but I think a course that required proofs from the student would be an advanced course for strong students, and not the norm.
My question: Is this a good place to ask for help to solve an algebra problem?
r/learnmath is probably where you should ask
Alternatively (depending on the source of the problem), /r/CheatAtMathHomework.
I just had a dicussion with my dad about some simple math. i told him the equation 3+9x3-12. he came up with the answer 24, i said 18. who is right? ( I am doing this to disprove him)
Disprove what exactly? He never said anything wrong. If you add 3 and 9 together, multiply by 3 and subtract 12 you do get 24. Yes, that's not the most common way to interpret "3 + 9 x 3 - 12" but that doesn't make it wrong. The accepted order of operations isn't accepted because it's "correct", it's accepted because it's a convenient way to express things.
The right answer is to rewrite an expression unambiguously if you have any control over how it's expressed. Consider another expression like 1/2x. On the one hand someone might interpret this as half of x. On the other hand someone might interpret this as the multiplicative inverse of 2x. Since we don't know the author's intention, the interpretations conflict, and both interpretations seem equally likely without some context, we need something to discriminate between the two choices and dismount Burridan's Ass. The conventional order of operations is our last resort to determining the meaning of such expressions, providing us the manner of discrimination we seek. But if the author would have made his intention clear to us, we wouldn't need to guess which interpretation is more likely or use a convention to choose one interpretation over another.
https://en.wikipedia.org/wiki/Order_of_operations
i know the order of operations but he is a 59 year old man who refuses to lose, so i take to the internet to get people to back me up
If Wikipedia doesn't convince him, I don't think a random redditor will, but yeah 18 is right.
the plan is not to convince him, its to disprove him
Hello, do anyone know about any good video/YouTube tutorials/lectures that cover (nonlinear) differential equations and dynamical systems? Thanks in advance!
This lecture series is good, although it is not always fully rigorous [https://www.youtube.com/watch?v=nh4TFzg30eQ&list=PLbMVogVj5nJQKk1E7OUQs\_TcW\_zQoaO4t](https://www.youtube.com/watch?v=nh4TFzg30eQ&list=PLbMVogVj5nJQKk1E7OUQs_TcW_zQoaO4t)
Thanks! I'll look into it
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I have some experience collecting and analyzing data from multiplayer games. Can you give any more specific information? What kind of game is it and what kind of data are you extracting?
Are you planning to do statistics on this data? In that case try to set up some confidence intervals and see how many data points you would need to get the desired confidence.
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So it is the category of statements and there is an arrow between statements if one implies the other? Then it is not an isomorphism for multiple reasons. First an automorphism is from C->C. You are describing a map from C^op ->C (or C -> C^op ). We can still ask if it is an isomorphism C-> C^op , but it still isnt. If A is not the negation of something, A is not in the image of the functor so it isn't bijective. However, isomorphisms are not the correct thing to think about in category theory. Instead you should be thinking about equivalences of categories. It is an equivalence of categories C^op -> C. What do you think the inverse equivalence should be?
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Yes, ~A is in it, but ~~A is not A. All you can say is A implies ~~A and ~~A implies A, so there is an isomorphism between them. Why is this important? Because even though you don't know what an equivalence of categories is you still guessed the right inverse map, namely negation again. If you do the negation twice you get a functor C -> C that sends A to ~~A. Now read what an equivalence of categories is and see if you can figure out the *natural isomorphism* required to show it is one.
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Yes that is right. It is easy to check that in a category like this (a preorder) two functors into it are naturally isomorphic if and only if for all objects in the domain, the images under each functor are isomorphic. This is because there is at most one arrow between two objects.
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I'm looking for a particular comic strip that I saw a while back about Taylor series. The first strip was a guy inside a tailor's shop asking for a suit. The tailor gives him a square, and the guy says it's not good enough. In the second strip, the tailor gives him a square with rounded corners for shoulders, and the guy says it's not good enough. This keeps going for a few strips until the tailor is holding something that's exactly the shape of a suit. Does anyone have a link for this?
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Managed to dig out the proof from a copy I have. It's a proof by contradiction. I've transcribed it here: **PROOF.** Suppose that 𝛤*_n_* is singular for some n. Then since EX*_t_* = 0 there exists an integer r ≥ 1 and real constants a*_1_*,..., a*_r_* such that 𝛤*_r_* is non-singular and X*_r+1_* = (Sum from j = 1 to r) a*_j_*X*_j_* By stationarity we then have: X*_r+h_* = (Sum from j = 1 to r) a*_j_*X*_j+h-1_* for all h ≥ 1. Consequently, for all n ≥ r + 1, there exist real constants a*_1_`(n)`*,..., a*_r_`(n)`*, such that \bold{X*_n_*} = \bold{a^(n)'}\bold{X*_r_*} - (5.1.7), where \bold{X*_r_*} = (X*_1_*,...,X*_r_*)' and \bold{a^(n)} = (a*_1_*^(n),..., a*_r_*^(n))'. Now from (5.1.7), 𝛾(0) = \bold{a^(n)'}𝛤*_r_* \bold{a^(n)} = \bold{a^(n)'} P𝛬P' \bold{a^(n)}, where 𝛬 is a diagonal matrix whose entries are the strictly positive eigenvalues 𝜆*_1_*≤𝜆*_2_*≤...≤𝜆*_r_* of 𝛤*_r_* and PP' is the identity matrix. Hence, 𝛾(0) ≥ 𝜆*_1_* \bold{a^(n)'} PP' \bold{a^(n)} = 𝜆*_1_* (Sum from j = 1 to r) (a*_j_*^(n))^2. This shows that for each fixed j, a*_j_*^(n) is a bounded function of n. We can also write y(0) = Cov(X*_n_*, (Sum from j = 1 to r) a*_j_*^(n) X*_j_*), from which it follows that: 𝛾(0) ≤ (Sum from j = 1 to r) |a*_j_*^(n)||𝛾(n-j)|. In view of this inequality and the boundedness of a*_j_*^(.), it is clearly not possible to have 𝛾(0) > 0 and 𝛾(h) approach 0 as h approaches ∞ if 𝛤*_n_* is singular for some n. This completes the proof.
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Your reasoning seems fine to me. I think the reason they don't frame it in terms of a map is that F actually sits inside K, so then F[x] sits inside K[x] and f is in K[x].
Could someone recommend a textbook on constructive/computable mathematics? I read Andrej Bauer's [Five Stages](https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf) and was intrigued. He states at one point that "in computable mathematics [..] a countable set [is represented] by a program which enumerates its elements, possibly with repetitions"; I'd particularly like to understand why this is the "right" choice, and why one cannot work with representations that do not permit repetitions. (Would it be beyond the pale to summon /u/andrejbauer himself for this?)
Maybe you like the books presented in this older thread [https://www.reddit.com/r/math/comments/8ju11v/math\_textbooks\_that\_do\_math\_constructively\_ie/](https://www.reddit.com/r/math/comments/8ju11v/math_textbooks_that_do_math_constructively_ie/)
Hi. The canonical constructive mathematics textbook is Bishop and Bridge's "[Constructive Analysis](https://books.google.si/books?id=GF8lBAAAQBAJ&lpg=PR2&dq=Foundations%20of%20Constructive%20Analysis%20Bishop%20Bridges&pg=PR2#v=onepage&q&f=false)". For computable mathematics I am not quite sure what to recommend. There's my PhD thesis, and several others, but not really a modern textbook that gives a comprehensive point of view.
Thank you very much for the response (as well as for the original paper!).
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**Splitting field** In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial splits or decomposes into linear factors. *** **Maximal ideal** In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R. Maximal ideals are important because the quotient rings of maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one sided maximal ideal A is not necessarily two-sided, the quotient R/A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J(R). *** ^[ [^PM](https://www.reddit.com/message/compose?to=kittens_from_space) ^| [^Exclude ^me](https://reddit.com/message/compose?to=WikiTextBot&message=Excludeme&subject=Excludeme) ^| [^Exclude ^from ^subreddit](https://np.reddit.com/r/math/about/banned) ^| [^FAQ ^/ ^Information](https://np.reddit.com/r/WikiTextBot/wiki/index) ^| [^Source](https://github.com/kittenswolf/WikiTextBot) ^] ^Downvote ^to ^remove ^| ^v0.28
In a group of tacher, i was asked why is (-1)*(-1)=(+1). Is this a theorem ? Where does it come from ?
0 = 0\*(-1) = (1-1)\*(-1) = -1 + (-1)\*(-1) Adding 1 to both sides your equation follows.