Mathematics - 2024-06-21

Novice

02:14

hello, is anybody familiar with proving that $\mathbb R$ is up to isomorphism the unique complete, ordered field?

Jakobian

03:03

@Novice yes

This should follow from each real number being determined by a Dedekind cut

Novice

my book takes $\mathbb R$ as axiomatic and one (long) exercise is about establishing an isomorphism between $\mathbb R$ and $\tilde{\mathbb R}$, another structure satisfying the axioms

I am near the end of the proof, I think, but struggling to establish that $f(x \cdot y) = f(x) \tilde{\cdot} f(y)$, where $\tilde{}$ refers to operations associated with $\tilde{\mathbb R}$

We define $f(x) := \sup(f(r) \colon r \in \mathbb Q \text{ and } r \leq x\}$, so I need to show that $\sup\{f(r) \colon r \in \mathbb Q \text{ and } r \leq xy\} = \sup\{f(p) \colon p \in \mathbb Q \text{ and } p \leq x\} \tilde{\cdot} \sup\{f(q) \colon q \in \mathbb Q \text{ and } q \leq y\} $

I am currently just trying to prove that RHS is an upper bound for the set on LHS. I think I have successfully addressed the cases $x, y > 0$ and $x, y < 0$, but currently struggling with $x < 0, y > 0$

I'm trying not to look for solutions on the web too much, but I saw some notes that discussed that for any $x \in \mathbb R$, we have $f(-x) = \tilde{-}f(x)$, so I am currently wondering if this could be useful.

(I imagine the idea is to try to reduce this case to the $x, y > 0$ case, somehow)

Jakobian

03:42

@Novice $f(r) = r$ for $r\in\mathbb{Q}$ if its a field homomorphism anyway

not sure what you're trying to do

did you prove that a complete ordered field has to be archimedean yet

i.e. there's no $x\in F$ such that $x > n$ for all $n\in\mathbb{N}$

Novice

that is given as a theorem for $\mathbb R$ in my text

let me just paste in the exercise (need a minute)

Jakobian

well, if we let $y = \sup \mathbb{N}$ then $y > y/2 \geq n$ for all $n\in\mathbb{N}$, contradiction

so it must be archimedean

Novice

I have to leave soon, but I'll try come back in a couple of days, if I haven't figured it out. I may also post a question about this

Jakobian

well the gist of it is that every element of a dedekind-complete ordered field can be recovered from natural numbers as supremum of some non-empty set $A\subseteq \mathbb{Q}$ bounded from above and with the property $x\in A, y < x\implies y\in A$

but to show this you need to know they are archimedean

@Novice I was happy to discuss the problem with you, but its your choice

Novice

04:07

I'm interested in discussing it but I have to leave my computer now. Thanks for the offer. I'm just super close to the end of this exercise and hoping I can wrap it up soon, and get on with the actual analysis

leslie townes

is getting on with the actual analysis even allowed around here? :)

Novice

hi, I've been drowning in my PhD program, but I decided that I want to learn some core mathematics For Real This Time, so I will try to devote some of my spare time to this book

Jakobian

04:25

The real in real analysis is real

user 85795

05:10

but is the analysis actual

Soumik Mukherjee

05:55

@Jakobian I hope this site is not as toxic as twitter.

*that site

Jakobian

does anyone see an easy argument for why there's no $f\in C^*(\mathbb{R})$ such that the ring $A$ generated by $f$ is dense in $C^*(\mathbb{R})$?

the easy part is important because I have methods to show this, but not easy ones

I think a method would be to choose some sequence $y_n$ and a continuous function $g$ with $g(n) = y_n$ such that you can't approximate $y_n$ by $p(f(n))$ for any polynomial $p$

psie

06:15

Consider $$p_{Y|X=x}(y)=\frac{p_{X,Y}(x,y)}{p_X(x)},\quad f_{Y|X=x}(y)=\frac{f_{X,Y}(x,y)}{f_X(x)},$$the two conditional distributions in the discrete and continuous setting. In some lecture notes I'm reading:

> In the discrete setting, the conditional distribution was defined as the distribution of our variable with respect to a different probability measure [the conditional probability]. In the continuous setting we have simply defined a new probability distribution without connection to the original random variable.

What is meant by "without connection to the original random variable"? For me, the two formulas above seem identitcal in a sense. I don't see the conceptual difference between them.

Jakobian

@psie They mean that if we replace $P$ by $A\mapsto P(A|X = x)$ we get the discrete setting

psie

ok

Jakobian

yeah I don't know what they mean by "without connection" stuff

to be honest I don't see why would I care what they mean

doesn't seem one bit important

psie

06:32

@Jakobian well the continuous case is a bit special since we are conditioning on the event $X=x$ despite it having probability zero, right? That is how I interpret the subscript in $f_{Y|X=x}(y)$.

Jakobian

kinda

but hard to tell if they just meant this difference, they probably did

but I don't think they did a very good job at expressing themselves so I don't know

and its not that important

after all its not important how the author/s understand it

psie

12:45

I'm reading this answer and the author seems to use a.a. and a.s. at three and one occasion respectively somewhat purposefully. What is the difference?

According to Wikipedia, they seem to be the same.

Thorgott

13:03

I don't see why there should be a difference?

saying "$f=g$ a.s." should be to saying "$f(x)=g(x)$ for a.a. $x$" as saying "$f=g$" is to saying "$f(x)=g(x)$ for all $x$"

psie

ok, that clarified it, thanks. I conflated almost all and almost always. I think there is a small difference, namely the one you point out. Almost always I'd say is synonymous with almost surely, whereas almost all refers to the instances at which they equal. I understand now :)

I think the answerer meant almost all

Ryder Rude

13:18

@Jakobian do u find big bang theory more likely now

Jakobian

13:28

@RyderRude not really

Ryder Rude

@Jakobian oh

do u prefer the universes's age to be finite or infinite or dont care?

Jakobian

13:42

I care a little bit. But don't have a preference

Ryder Rude

oh

Obliv

15:20

I know dissecting wiki is not conducive to my learning but in the page on first order logic it says the difference between zero-th order (or propositional) logic and itself is the introduction of quantifiers over "non-logical" objects and relations. But in the propositional logic wiki page it says "It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them."

I'm guessing it's an error but maybe I'm just misinterpreting

I'm also not sure referring to free variables as "non-logical" is meaningful.

Thorgott

I don't see an error

Obliv

It says it introduces relations but prop. logic also has "relations" at least according to the wiki pages. I don't see how there's any difference between orders of logic besides what you're allowed to quantify

Also I thought relations were defined in set theory, not formal logic

Thorgott

15:37

ah, my guess would be the latter means relations in an informal sense and the former means relations in the formal sense as in set theory

Obliv

can't one of these versions of relations be used in propositional logic? And right above is a paragraph describing some possible meanings for relation

Thorgott

in any case, none of these sentences are precise mathematical statements and they're from Wikipedia, so, as you suggest, it's not worth worrying about

@Obliv I don't know what "used in" means here

Obliv

I thought maybe the nullary or unary relation could be the "version" of a relation that is similar to how we use relations in set theory or higher order logics. Though I don't even know how we use the set theoretic definition of relation in first order logic

Thorgott

yeah, so at this point I think you either have to get serious about this and pick up a book on logic or decide to not worry about it

Obliv

I took a look at one and it looked incredibly dense and dry. I wonder if there are any good modern books on the subject?

I don't want to go into the entire subject with excruciating detail though, I just want to learn enough so that I know what I'm talking about moving forward in math.

Jakobian

15:43

as is the subject

Ryder Rude

do u people think consciousness is material and why

Thorgott

I'm no logician, but I think the subject is rather dry by design

in any case, you don't need to know formal logic to know what you're talking about in math unless the field you are venturing into is precisely logic

or perhaps some other foundational areas

Ryder Rude

@Obliv i would say dont worry about formal logic yet if u r only learning it for other branches of math

Jakobian

or you're worrying about independence proofs

Thorgott

I consider that foundational, but yeah

Obliv

15:53

Well isn't an aspect of, if not the point of, mathematics to generalize true statements? So wouldn't it make sense to want to understand the underlying system/what it is we're actually doing?

Jakobian

I'm just thinking that for general topology for example, you somewhat care about foundational issues. I mean, axiom of choice is generally accepted, although topology is done without axiom of choice too. But even in ZFC. And I wouldn't say that this is venturing into foundational areas, but maybe I misunderstood you

Ryder Rude

the level of logic that is used in other branches of math is already understood intuitively @Obliv

u dont need to worry about it too much

formal logic is more useful for complex results like Godel's theorems

Thorgott

@Jakobian I mean, caring about foundational issues is much broader than caring about formal logic.

I've never had to care about formal logic in any of the math I've ever done

Ryder Rude

@Obliv e.g. have u yet faced problems in verifying the validity of proofs in other branches of math?

Obliv

Okay, I guess I'm just trying to be extra careful about the "intuitive" aspect of founding the base @RyderRude since we start with some pretty sketchy intuitions.

Ryder Rude

15:58

yeah... intuition is infamous for being wrong in math

Obliv

No, but the whole point is I wanted to understand what a "conservative" extension is (NBG from ZFC), finitely vs non finitely axiomatizable, primitive notions, logical comprehension

Thorgott

in practice, all the logic you need is understanding quantifiers, implications, law of the excluded middle, etc.

i.e. the stuff that you get taught in any 1st week of an introductory math course

Obliv

and I started to wonder about semantics, formal systems, formal language , and so on..

Thorgott

so it sounds like you do want to learn foundations

Ryder Rude

@Obliv it's great to be interested in this stuff, but note that, like @Thorgott said, it's not too useful in other branches of math

Jakobian

16:00

I guess my issue is that I don't fully understand what foundational areas refers to. If it means areas of foundations, or areas where foundations has big influence

Obliv

Yeah, I'm not just interested in mathematically true statements (they are stimulating to read like a book), but also studying meaning in a more wholesome way?

Jakobian

I agree with the latter but not the former (as general topology is an example)

Ryder Rude

@Obliv u probably need to read a book which covers this stuff

Jakobian

@RyderRude what do you mean by that

Ryder Rude

maybe an intro book, a non comprehensive one because u dont want too many details @Obliv

@Jakobian Bayesian probability comes to mind. people got it wrong for long

im sure there r many other things too... maybe the parallel postulate

Jakobian

16:05

I said: can you elaborate

you are inviting me to discussion about a point you never clarified

Obliv

I've tried reading a translated text of frege who contributed a lot to modern logic but it was very hard to understand I gave up pretty fast and started to browse wikipedia again.. :P

Ryder Rude

e.g. if a test is 99% accurate, intuition would conclude that u have 99% chance to have the disease @Jakobian

Obliv

@RyderRude Yeah I think I need something like schroeder for thermo, something that isn't very rigorous but isn't too informal either.

Jakobian

@RyderRude why is this "in math"

Ryder Rude

@Jakobian it is an intuitionistic reasoning about probability

@Obliv there is a really nice paper about Godel's theorem. u will like it too

Thorgott

16:07

@Jakobian give me an example of where formal logic is relevant in general topology?

Jakobian

@RyderRude but the setting isn't mathematics

Ryder Rude

@Jakobian it's more like applied math than pure math

Jakobian

@Thorgott to understand models and things like forcing

this can be used to give independence proofs about some topological things which can be true under CH but false otherwise

as a side note: I've seen a "logic" proof of Tychonoff's theorem recently using compactness theorem from logic

Didn't fully read it, but its cool you can apply it in such ways

Ryder Rude

@Obliv math.mcgill.ca/rags/JAC/124/GodelsProof.pdf it's an article for beginners

but it has math

Obliv

Hungerford says the primitive notions of NBG are class, membership, and equality. Also Russell's primitives are pretty interesting.

Jakobian

16:12

@RyderRude yes, sure, but I thought you were talking about intuition in mathematics as a whole

but you are talking about seemingly intuitive things in probability/statistics

Ryder Rude

@Jakobian this was the only infamous example i had in mind...

Jakobian

so it was just an example?

then you didn't explain yourself right

Ryder Rude

one more can be incompleteness i guess... but calling completeness intuitive would be a stretch

Jakobian

again, I was asking for a clarification and you didn't give me one

why invite me to a discussion so I get your point wrong instead of clarifying whats being discussed

Thorgott

@Jakobian I don't see how that is relevant

Ryder Rude

16:15

sorry... the original comment was not meant as a discussion prompt @Jakobian

Thorgott

I guess I don't consider questions about axiomatic dependence to be questions in the area of topology (or any other area besides foundations)

Ryder Rude

it was only a reply to Obliv @Jakobian

Jakobian

@RyderRude you could still clarify what you mean

Ryder Rude

i did. i meant things like Bayesian probability

Jakobian

do you mean that intuition is bad thing to go off of in mathematics

@RyderRude no you didn't

Ryder Rude

16:17

@Jakobian no

i meant that intuition has given us wrong results... thats all

Jakobian

I don't see how that makes anything clearer

Ryder Rude

but intuition is right too..it is an important tool

Jakobian

many undergraduates have bad intuition

so they come to wrong conclusions

Obliv

@Thorgott it's relevant to the extent that once it becomes relevant, it's typically not a good thing.

Jakobian

but what of it

Giving someone examples is not clarifying what you actually mean

Ryder Rude

16:19

@Jakobian so i was just stating the obvious thing that intuition cannot be infinitely relied on

Jakobian

infinitely relied on means always relied on?

Ryder Rude

it's not meant to be a too substantial statement

@Jakobian yes

Jakobian

this is not something about mathematics then, but everything

Ryder Rude

yeah...

Jakobian

@Jakobian but, what I think you should do is think about what I said here

Ryder Rude

16:22

it's just something i happened to say in the context of that conversation

Jakobian

because this is something I take huge issue with

Ryder Rude

okay

Jakobian

examples can direct someone to a position you're trying to take but they can never fully clarify it

Ryder Rude

i still think that examples r at least part of clarification...

@Jakobian yeah... maybe not fully

Jakobian

examples don't clarify anything, they only help to do so

I still have to assume what you mean

Ryder Rude

16:25

oh

Jakobian

this is the same thing that I see when physicists discuss mathematical problems

you haven't defined what you mean

giving me an example is a bad practice (for someone to properly understand you)

Ryder Rude

i will keep it in mind

Thorgott

@Obliv it just means you're switching areas

Obliv

I just meant that if formal logic becomes a concern to you while doing math, then maybe you have an underlying condition/brain injury or something.

As in, one must relearn how to think rationally or something.

leslie townes

empirically it wouldn't surprise me if people with "underlying conditions" and brain injuries were more likely than others to be interested in at least the trappings of formal logic and "learning how to think rationally"

Ryder Rude

16:34

@leslietownes why do u call these trappings

Obliv

I think even though natural language has its limitations, it can at least speak of things outside of formal language. Trying to logically deduce things outside of formal logic seems futile so we must come up with a base of self evident truths (which are as primitive as we can get them and distinct). Natural language can serve as a way of communicating in that realm of what we deem to be real and self-evident.

leslie townes

i don't understand that question. by the above, i meant to suggest that there is a difference between saying or feeling that one interested in 'formal logic' (or whatever name the precise choice of words is not the point), and having interest in the mathematical content that goes by that name

some people become very interested in symbols and eliminating human language and vague notions of undercovering how the brain 'really' works or eliminating error in their own mental processes as a result of some kind of onset of mental illness

i'm not trying to be funny here, i'm just saying, that's the point i was making

Ryder Rude

sorry i misinterpreted. i thought u were calling the field of formal logic as a trap which one should stay out of

leslie townes

when john nash lost his mind for a while, he did not actually "become interested in formal logic" in any real sense, although it may have looked kind of like that because he was mailing people postcards with stuff like "the number of god is 3485673496249563129856945634" on it

a lot of people babbling about 'formal logic' is closer to that than it is to anything that people who are in their senses care about

also a lot of latent mental health issues first become apparent around the same age that people are left by themselves in places like universities for the first time

@RyderRude not at all, just saying, sometimes, the person who seems most interested in formal logic on a college campus is (a) not a math major, (b) not maybe educated in math at all, (c) maybe in crisis

like, empirically, i think it's more common for that to happen than for people to get actually interested in mathematical logic

Obliv

Why not both? Lose your mind and reason correctly :D

Ryder Rude

16:44

oh. i havent come across enough people interested in formal logic to judge that statement...

Obliv

In any case, my notes are filled with hyperlinks to wiki and it's comforting to know that I'm not leaving many stones uncovered with the language I'm using to set up the math I'm working in. eye twitches I'm not crazy..

Ryder Rude

but it is common for physicists to relate Godel's theorem to unrelated stuff like theory of everything @leslietownes

leslie townes

if you say so

Thorgott

17:01

it is also common for them to not understand what that theorem says

(I don't either)

Jakobian

@Thorgott existence of P-points of βN, while is a question of topology, turns out to be equivalent to CH. There are many questions like that which come into an umbrella term "set-theoretic topology"

Another example is existence of pseudocompact non-compact T_6 space

Thorgott

and that needs forcing?

Jakobian

I'd have to double check to make sure

@Thorgott that under not CH there exists a P-point of βN requires forcing. The converse is an easy exercise

This was done by Shelah (the not CH implies P-point direction)

The existence of pseudocompact non-compact T_6 space, not sure if forcing exactly but proofs here involve such things as the diamond principle

Diamond principle

In mathematics, and particularly in axiomatic set theory, the diamond principle ◊ is a combinatorial principle introduced by Ronald Jensen in Jensen (1972) that holds in the constructible universe (L) and that implies the continuum hypothesis. Jensen extracted the diamond principle from his proof that the axiom of constructibility (V = L) implies the existence of a Suslin tree. == Definitions == The diamond principle ◊ says that there exists a ◊-sequence, a family of sets Aα ⊆ α for α < ω1 such that for any subset A of ω1 the set of α with A ∩ α = Aα is stationary in ω1. There are severa...

Existence of Suslin line is another example of problem in general topology solved by set theory

nickbros123

17:37

are any of you getting weird redirects when going to linearalgebras.com ? Seems like virus?

Jakobian

Why are you posting a potentially dangerous link

Mary E. Rudin: "Set theory and General Topology"

►

@Thorgott also see this lecture by Mary Rudin. One direction of there exist P-points in $\beta\mathbb{N}$ iff CH is by Rudin

(the arguably easier one)

Rudin is pretty relevant in general topology and set-theoretic topology

Joe

18:44

@Jakobian: I believe you can also prove using mathematical logic that given an uncountable cardinal $\Kappa$ and number $p$ (where $p$ is prime or zero), there is exactly one algebraically closed field of characteristic $p$ with cardinality $\Kappa$. So for example the complex numbers are the only algebraically closed field of characteristic zero with the same cardinality as the continuum. (Not that this is relevant to topology – I just thought it was an interesting application.)

I should have written \kappa instead...

Thorgott

@Jakobian I see

my impression is that whilst these results have statements about topological spaces, the proofs and methods used in addressing them have very little to do with actual topology

Shelah also has an algebraic topology paper about which I feel similarly

@Joe this does not require any non-trivial mathematical logic or set-theory beyond the axiom of choice

Joe

@Thorgott: Which proof do you have in mind?

Thorgott

fix a transcendence basis over the prime field, it follows that the algebraically closed field is the algebraic closure (which is uniquely determined up to iso by Steinitz) of a purely transcendental extension of the prime field

furthermore, if the cardinality of our field is uncountable, you can argue that cardinality equals the cardinality of that transcendence basis

existence of transcendence bases is where I use choice

actually, the second step uses some cardinal arithmetic, so perhaps I implicitly use choice there as well

Jakobian

19:00

@Thorgott sure, one direction of such proofs usually is set theoretical and has not much to do with the subject, it requires one to construct some model. But I think this is necessary evil

Joe

I see. The proof I was thinking of did actually use mathematical logic in a nontrivial way. Specifically, it shows that the theory of algebraically closed fields has quantifier elimination. Not the conventional route, I admit. But it essentially uses more logic and less algebra

By the way, every proof that every field has an algebraic closure needs to use some kind of choice principle, since it is consistent with ZF that there is a field without an algebraic closure.

Thorgott

yeah, it needs existence of maximal ideals

@Jakobian That's fair. It's just not something I'm typically inclined to call general topology when the methods used are not those of general topology. But this is not for me to judge, I guess.

Joe

I think there are proofs of the existence of the algebraic closure that don't use the theorem that every nontrivial ring has a maximal ideal. The statement about maximal ideals is equivalent in ZF to the axiom of choice, but apparently the statement that every field has an algebraic closure is much weaker than choice. (I don't know any of the details though.)

Jakobian

My stand on this is a bit utilitarian. Methods are irrelevant as long as the results are what we are interested in general topology. Instead of not being general topology, such methods make it a set theory flavoured general topology, or set-theoretic topology

Joe

Does anyone know of a good reference which covers diagram chasing? I was working out the details of the Snake Lemma today in Aluffi, but I found them to be much harder than he seemed to be suggesting...

(I mean the Snake Lemma in the context of complexes of R-modules, rather than in arbitrary abelian category – I haven't got to the more complicated stuff yet.)

Jakobian

19:15

diagram chasing is actually pretty intuitive

I don't think diagram chasing in arbitrary abelian categories is a thing

its theoretically possible since they are locally categories of $R$-modules I think (or something)

to explain diagram chasing I'd have to do it on some example

Joe

@Jakobian: From what I have read, there is a form of "diagram chasing" that works in arbitary abelian categories, e.g. math.stackexchange.com/questions/74871/….

But I'm just quoting what I've read, of course.

Jakobian

I see, you go with subobjects instead of elements

I was thinking of Mitchell embedding theorem

Jakobian

19:35

one wonders how this would look for the normal diagram chasing, instead of taking an element of a module you would take a submodule

Joe

19:49

@Jakobian: I have an example of something that potentially could use diagram chasing, if you care to take a look.

This is Aluffi's version of the Snake Lemma (I think he states it slightly differently to how it usually is).

The hypotheses of the theorem are that $0\to L_1\to M_1\to N_1\to 0$ is an exact sequence, and so is $0\to L_0\to M_0\to N_0\to 0$. Further, the maps $\lambda$, $\mu$, and $\nu$ are such that the diagram commutes.

The claim is that there is an exact sequence $0\to\ker\lambda\to\ker\mu\to\ker\nu\to\operatorname{coker}\lambda\to\operatorname{coker}\mu\to\operatorname{coker}\nu\to 0$.

I've worked out of all of the definitions of the maps, but I'm having trouble verifying that the sequence is exact at $\ker\nu$.

Thorgott

@Jakobian These ad hoc methods work sometimes, but not always. However, there is also a general method for diagram-chasing in arbitrary abelian categories by working with "generalized elements" (up to refinement). This goes back to Bergman, I think.

Joe

But I'm not sure if this is the kind of thing that you would use diagram chasing to prove.

Jakobian

@Joe so what you've shown already is that all the maps are well defined?

Joe

@Jakobian: Yes.

Thorgott

@Joe what step are you failing at? e.g. have you already shown that the composite is $0$?

Jakobian

19:57

and I assume you've shown that the squares are commutative

Joe

@Jakobian: Yes I've also done that. The only thing I'm having trouble with is verifying that the sequence is exact in a couple of places

@Thorgott: The composite of what, sorry?

Thorgott

the map into and out of $\ker\nu$

Jakobian

to prove a sequence is exact you need to first prove that its a chain complex

sometimes informally written $\partial^2 = 0$, where $\partial$ refers to the boundary maps

Joe

@Thorgott: Yes, I think I've done that already. If we write $\tilde{\beta_1}$ for the induced map $\ker\mu\to\ker\nu$, then I've already shown that $\operatorname{im}(\tilde{\beta_1})\subseteq\ker\delta$, which is equivalent to the composite being zero. The trouble I'm having is with the opposite inclusion.

Thorgott

ok, so start with an element $x$ in the kernel of $\delta$

spell out how you defined $\delta$ and what that tells you about $x$

diagram-chasing is a very "go with the flow" type of thing

Joe

20:12

So you "follow" the path that we took when defining the map $\delta$?

Jakobian

So we take $x\in \ker \nu$ with $\delta(x) = 0$. By definition, $\delta(x)$ is defined by first taking $y\in M_1$ such that $\beta_1(y) = x$ then $z\in L_0$ with $\alpha_0(z) = \mu(y)$ and finally $\delta(x) = z+\text{im }\lambda$

Thorgott

@Joe yup

Jakobian

that $\delta(x) = 0$ tells you that $z\in \text{im }\lambda$ so that there exists $q\in L_1$ with $z = \lambda(q)$

Now I think $w = \alpha_1(q)$ should be your desired element of $\ker \mu$

$\mu(w) = \mu\circ \alpha_1(q) = \alpha_0\circ \lambda(q) = \alpha_0(z) = \mu(y)$

okay instead lets take $t = y-w$. Then $t\in\ker \mu$

Then $\beta_1(t) = \beta_1(y)-\beta_1(w) = x-\beta_1(\alpha_1(q)) = x$

Joe

I will have a think about all of this. Thanks to you both

Jakobian

as you see here we have to invoke the complicated definition of $\delta$, and then even take a difference of two elements. So maybe it wasn't super easy

but the idea is this jumping around a diagram

Joe

20:25

I think the difficult thing I find in general is to know where to go "next" when doing the chase. That is, when I have an element of a given module, I don't know which map to/from that module to look at

psie

Let $X$ and $Y$ be jointly distributed. In my book, it is poorly stated, but they simply say that $$\mathbb{E}[g(X,Y)|X=x]=\mathbb E[g(x,Y)|X=x],$$where $g$ is some function. I wonder a) what kind of function (where does it map to? is it measurable?) and b) how one can prove this identity?

a) I believe it is a measurable function $\mathbb R^2\to\mathbb R$.

b) I guess one needs to consider discrete and continuous r.v.s. separately, but I don't know how really.

psie

20:48

Here's an idea for the discrete case. $$P(X=z,Y=y|X=x)=\frac{P(X=z,Y=y,X=x)}{P(X=x)}=\frac{p_{X,Y}(x,y)}{p_X(x)}=p_{Y|X=x}(y),$$if $z=x$ and $0$ otherwise. Hence $$\mathbb{E}[g(X,Y)|X=x]=\sum_y g(x,y)\cdot p_{Y|X=x}(y)=\mathbb E[g(x,Y)|X=x].$$

I feel like the last equality is a bit unmotivated.

Jakobian

@psie sure

by the way, $X, Y$ be jointly distributed, that doesn't really tell you anything

psie

I believe they mean either they are both continuous or discrete.

Jakobian

@psie as you know, the function $x\mapsto E[Z|X = x]$ is only really defined up to some set of measure zero

therefore I also have slight problems on how to interpret it in the continuous case

although the formula should intuitively be true in some sense

I'll try to consult some book

psie

21:05

ok

Jakobian

@psie Let $h$ be such that $h(x) = E[g(x, Y)]$, then $$\mathbb{E}[g(X, Y)|X] = h(X).$$ I believe this is what the formula is saying

psie

ok, I'll probably look at this again tomorrow, but thanks

Jakobian

well. This is something that I had a problem with in one class in university even, its a tricky equation to interpret.

so good luck with that I guess

EE18

21:29

I asked this a while back but I can't quite remember what the answer was

I am only working the exercises now

Is there an algorithm for finding the monic generator of a given polynomial ideal?

Im doing exercises and each time it's ad hoc/possible because it's "obvious" if that makes sense

leslie townes

EE18: how is the polynomial ideal "given"? (no specific form of presentation is implied by the words "polynomial ideal" alone, at least to my mind that just tells me what kind of ring the ideal is an ideal in)

Thorgott

if you are working in the polynomial ring over a field in a single variable, then there is a reasonable answer

leslie townes

and because we're implying that an ideal has a monic generator, polynomials over a field? or what

Thorgott

if you've seen the fact before that this ring is a PID, you will be familiar with the answer

EE18

I'm not quite sure I follow the quesiton Leslie. Are you asking what sort of set-theoretic description is given for the subset of $F[x]$? (In Hoffman and Kunze we only worry about fields)

Yes fields only, sorry

leslie townes

21:35

EE18: well, not "set-theoretic description" necessarily, but yes, a description. the algorithm needs an input

EE18

Here is an example I guess

(c) and (d) for instance are ideals

Finding their monic generator is easy/obvious

But I can imagine more complex ideals (maybe (e), I haven't started yet) where I can't just see the answer

The criterion I am using for finding monic generator right now is that if $M$ is the given ideal, then $M = dF[x]$ for $d$ the proposed monic generator

leslie townes

EE18: for much the same reasons that one might be able to "define" an integer without having a useful input to an algorithm, one might have something that counts as a "given" ideal in a ring without being able to feed it into any algorithm

EE18

I have a theorem which says this $d$ is unique. So I am "guessing and checking"

@leslietownes I think I see what you mean Leslie

Suppose I narrow the scope and say the ideal is that generated by some finite set of polynomials ($p_1F[x]+...+p_nF[x]$)

I know then that the monic generator thereof is the gcd of the polynomials $p_i$. Is there an algorithm in this case given the fairly explicit description of the ideal?

I imagine there is some analog of the euclidean algorithm. Will look this up

EE18

21:55

Maybe the answer will be in DF so I won't belabor the point. From HK I see that I can deal with the case $n = 2$ above via Euclidean algorithm

Is there an algorithm for $n > 2$?

leslie townes

22:06

i'm not sure if this is what you are asking, but e.g. gcd(f,g,h) = gcd(f,gcd(g,h)) allows you to compute gcds of tuples of polynomials in F[x] in terms of pairs of gcds in F[x]. that might not always be the best way to do it, but you can do it.

so if an ideal in F[x] is "given" by a finite list of generators, there isn't much more to know at the abstract level. the other problems are more about, what if an ideal is given as the kernel or range of a map with some structure (e.g. a ring homomorphism, or at least an R-linear map). those are also fairly general questions that i guess aren't answered by that, but then pose, how is the map "given."

EE18

That's exactly what I am asking :) I will have to think about that identity (it's obvious I think for integers but it's sending me for a spin for polynomials)

Thorgott

well, the general answer is that an ideal will be generated by any non-zero polynomial with minimal degree it contains

EE18

That's what I'm using now :) only just figured it out because that fact was sort of hidden in the proof

So basically my "algorithm" at the moment for the most general case is to find the minimum degree such that $f \in M$, and then find a monic polynomial in $M$ of that degree. The theorem I alluded to then guarantees uniqueness of said monic polynomial and also that it generates $M$

Thorgott

as leslie alluded to, an "algorithm" will only be so specific as the form in which you are given the ideal

but as far as "strategy" is concerned, this is as good as it gets

EE18

Fair play to you and leslie, I guess I am using "algorithm" loosely

like an engineer would, you might say

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