Mathematics - 2021-05-03 (page 1 of 2)

yeah, it is positive semidefinite but fails to be definite.

you can form a quotient space on which it is definite, it isn't very interesting. it looks like the space of scalars with the usual product as the quadratic form. you lose the polynomial stuff.

more interesting elsewhere than here.

on which it induces something definite, i should say. it's a standard construction often seen when defining inner products on tensor products of vector spaces.

sinclair

Thank you leslie. I'm just learning Linear Algebra so I am not familiar with some of the concepts you've stated.

But thanks anyway.

2:10 AM

suffice it then to say that you correctly identified the problem. it has the other properties of inner products - i think all of them. but not the important one.

2:21 AM

I may have overcomplicated a proof that the product of two (upper) triangular matrices is an (upper) triangular matrix

Overcomplicated or not, it's not an interesting proof.

Yeah it's very dry

but I went columns by rows and argued from the sum of outer products

if something can be proved just by choosing notation for matrix entries and writing out the definition of matrix multiplication and sigmas and stuff, that's a fine way to do it. it might be helpful to later generalize the notion of 'upper triangular' in a way that doesn't refer to matrix entries but it is never going to be too exciting.

How without referring to matrix entries?

not sure how that would look

2:31 AM

things like operators that act in the appropriate way on a flag of subspaces.

Well, a flag is equivalent to coordinates, but sure.

Flags are almost geometry, so be careful!

kadison had a definition of a triangular operator algebra with respect to a maximal abelian subalgebra (something playing the role of diagonal entries). T was triangular if T intersect T* was A.

there T is an algebra, not an individual operator.

people were trying tons of generalizations of that sort of thing to prove or better understand existence of invariant subspaces for hilbert space operators.

Right.

and A is the MASA. i'm all over the place today.

that's cool

Karim Mansour

2:36 AM

Hi Ted

Hi Leslie

i don't understand it, but it's cool

at least proving the inverse of a UT matrix is UT is more involved than messing with indices

Well, not really.

Hi Karim.

oh no

its not that difficult

now I really might have overcomplicated it

2:45 AM

Did you use elementary matrices?

the characteristic equation

Cayley-Hamilton theorem

Rajorshi Koyal

432.72+513.79-786.51-8*x% of 320=0

I would like to have a faster and more effcient technique for this

I am making some calculation error somewhere!! I feel so!!

3:07 AM

ha, that's kind of clever.

a good theme is that if an operator is invertible and sits in a nice enough set, its inverse will too. not a theorem but a theme. the cayley hamilton idea kind of gets at that.

nice

you can also avoid full C-H. the positive integer powers of T are an infinite set in a finite dimensional matrix algebra so there is a linear dependence among them. if you multiply by an appropriate power of T^(-1) you get it as a polynomial in T (and I) even if you don't know know that the polynomial has anything to do with eigenvalues. i think that works.

@Corellian if $U$ is UT then so are $U^k$. Cayley Hamilton gives $U(U^{n-1}+a_{n-1}U^{n-2}+...+a_1 I) = - u_0 I$, so you can read off the inverse which is UT.

Rajorshi Koyal

Again I am unable to solve this a slight mentorship will be most welcome.

A little more prosaic is to show that if $U$ is UT and invertible then $\operatorname{sp} \{ e_1,...,e_k \} = \operatorname{sp} \{ U e_1,...,U e_k \}$.

Ireland's president (not equivalent to the US president politically) is a man of slight stature www.reddit.com/r/ireland/comments/n3b35y/michael_d_higgins_near_a_standard_sized_acorn_for/

3:22 AM

right.

ha, the 'avatar' of that subreddit is the hallucinogenic nightmare dream sequence of 'my lovely horse'

Cayley Hamilton is cute but gives no geometric insight.

when i was young our milk bottles were delivered by horse & cart

Hello

$$A\cup B=\{x\mid x\in A\lor x\in B\}\\A\cap B=\{x\mid x\in A\land x\in B\}$$
$$A \subset B\leftrightarrow \forall x(x\in A\rightarrow x\in B)$$
Have I correctly used the symbols?

works for me. sometimes people use $\iff$ and $\implies$ instead of the arrows, which sometime are given other meaning in symbolic treatments of math.

thank you for using \mid. a lot of people who like that style of notation forget to do that and it looks unpleasant to my eyes.

I prefer $\implies$. $|$ vs $\mid$?

just for spacing?

3:35 AM

$B \setminus A \;=\; \{x\in B \mid x\notin A\}$

yeah sometimes people just throw | in there. i get it, it's on the keyboard, but yeah the spacing is wrong. it's typeset like an absolute value that sticks to one thing or another and not something that sits in the middle.

|| instead of \| is another one. although it is possible to take tex orthodoxy too far. i think \| is deprecated by some because it doesn't distinguish between left and right and that can interfere with some packages.

i do not carry my fondness for nice looking tex to the verge of intemperance.

Also is the statement "No element of $A$ is the element of $B$"
equivalent to $$\lnot \exists x(x\in A\rightarrow x\in B)$$?

that looks ok to me.

both as symbols and in terms of the meaning.

I have finally learnt to use these symbols :D

learnt. that's another one. like coloUr.

3:42 AM

like burnt

learned*

Thanks :-)

learnèd, as in quite the learnèd student

i like that one.

i like all of them, actually.

@Wolgwang I think you mean "No element of $A$ is an element of $B$"

@Wolgwang "learnt" and "learned" are both correct

i am considering a meta proposal to execute people who use $|$ instead of $\mid$ in order to align myself with the mse duma

learnded

the use of good english is went

3:49 AM

that is too extreme. how about a script that roams the site autocorrecting people and leaving snarky comments.

that's me

i still haven't recovered obviously

my delicate self

i find that sometimes people who are very officious in one context will be very pleasant in others. you never really know what to expect.

i am predictably unpredictable

mse is highly atomized despite the best efforts of people to make it more uniform and connected. which tends to lead to that sort of thing.

i have been observing a number of questions with no efforts whatsoever that seem to pass muster. i can't quite figure out the plan.

3:52 AM

i have been voting to close with some frequency these days.

i only do so when i feel the OP is a parasite, which i admit is a bit subjective

followed by hours of guilt and 10 hail mary's

i tend not to do vote to close if the question is interesting and not something we have all seen twenty times already. i have complained of this behavior but i am also guilty of it.

i have given up trying to be consistent and am trying to live with the guilt

the people who politely but firmly nudge interesting question askers to say more about what they have done are doing the lord's work.

petty & childish one moment, a bastion of reliable maturity the next

3:54 AM

it's original sin.

that was a concept that left the liturgical flank wide open to my assaults

limbo, ffs

haven't they done away with limbo?

were you raised in some book based belief system?

excuse my memory

i was not but my parents were raised catholic, and my wife and mother are at least nominally still catholic.

ahh, i sympathise

and identify

3:56 AM

my wife obviously isn't a great catholic because she married me and didn't know if limbo was still doctrine the last time i asked about it.

i think it is, but unfortunately my leading lights in that regard have passed away.

on his death bed, fr gilbert said "joseph, i cried & prayed all night for you; someone told me you are not joining the priesthood"

even at the end he had a sense of humoUr

what to do with the kid is still an open question.

tough one.

per google it is not doctrine anymore, and maybe never quite had official status. this is fairly new. the press release that clarified this point emphasizes that original sin is still out there.

depends on where you want her to go to school.

maybe?

4:00 AM

right now she's at a montessori where they worship rocks or trees or something.

the ceremonies are good and the social network excellent

i am sorry i did not do mine for that, but my wife is of evangelical extraction (dad was a sb minister)

i did not out of a stupid goal to be completely honest with my children

it may depend on whether we end up changing our geographic position. the high school my wife went to was mediocre.

mediocre with common goals is reasonable

it was not a hostile environment, i'll give it that. she was given a lot of encouragement. none of the stereotype of nuns whacking people with rulers.

they also maintained a level of discipline i would not have thought possible.

that is not necessarily too bad. i remember getting tutored in a convent and the door flew open and this tough but scared lad being chased by sister teresita who had a large club in her hand.

she had quite the rep, you did not want to dawdle when she was on the rampage

thankfully i was not educated by the nuns

4:04 AM

using clubs and threats of damnation are techniques that were not available in my high school.

strangely there were no eternal threats

mostly physical and some 'that;s the sort of vile pewrson you are' sort of thing

i may come around to it. academically mediocre is OK if it is the right environment. if my daughter gets stuck on anything she can just paste it into the chat and join the party.

besides she will stand out

good for self esteem

my wife has mixed feelings about it. she was her high school valedictorian but had not been prepared for much. first two years of berkeley were an adjustment that were bad for self esteem.

all fixed now.

the say a dublin man with an inferiority complex is someone who thinks everyone else is as good as him :-) (mostly told by non dublin folks)

4:07 AM

haha

some areas i am underconfident, some i am unshakeable

unrelated, but i like the line about an extroverted mathematician being one who looks at your shoes while talking to you.

laughing

passing that one on to my probabilist friend

i like the cayley hamilton trick for showing upper triangular

i learned a new trick

math.stackexchange.com/questions/20677/… is my favorite proof of the cayley hamilton theorem because of course it is.

you can do most of jordan canonical form with operator valued integrals too, although i probably wouldn't teach it that way.

very cute. t.b. (unfortunate userid) has some great answers/observations

4:20 AM

he used to post under his real name and then assumed an air of mystery.

that's a funny aspect of mse. when people change their names, it changes on all of their old stuff, but depending on how people were talking to them or referring to each other in comments, unless they were tagging, those don't updatte.

which can make for confusing reading.

some interesting history and interactions in the mse past, including my recent threatenor

maybe even tags don't update.

sometimes i recognize people i know who are using pseudonyms on the site because of mathematical or personal style.

i didn't follow that whole monica thing, but was wondering if i would follow suit

i don't think i know anyone on the site in person.

yeah that blew right by me.

well one person who i met in caffe med

4:24 AM

i think anybody who knew me in real life could recognize me from my posts. and even moreso from the nonsense i say in here.

i am sure i must know some people

the world of continuous & convex optimisation is not 'that' big

5:17 AM

I want to apply noether normalization lemma to $k[x, y] /(x y)$ but I dont know how to applly any suggestion?

5:40 AM

@IamKnull geometrically, your ring is the x-axis union the y-axis, and you want to project it to some affine line such that each fiber is finite

the x-axis won't work, because the preimage of 0 isn't finite; similarly the y-axis won't work

so you might want to project it to some diagonal line, which would correspond to taking $x+y$ as your transcendence basis

and you can verify that $1$ and $x-y$ generates your ring as a module over $k[x+y]$

hint: $(x-y)(x+y)^{n-1} = x^n - y^n$ and $(x+y)^n = x^n + y^n$

6:14 AM

@LeakyNun math.stackexchange.com/questions/3597751/… this is same question I think

Can you answer it, its a old unanswered question the same thing you answereed

6:25 AM

I need some help in noether normalization

Suppose that $k=\mathbf{R}$ and $A=\mathbf{R}[x, y] .$ Deduce that if $\mathfrak{m}$ is a maximal ideal of $A$, then $A / \mathfrak{m}$ is isomorphic to either $\mathbf{R}$ or $\mathbf{C}$. In the first case, note that $\mathfrak{m}=(x-a, y-b)$ for suitable $a, b \in \mathbf{R}$. Suppose now that $A / \mathfrak{m} \cong \mathbf{C} .$ The projection map $A \rightarrow A / \mathfrak{m}$ gives a surjective homomorphism:
$$
f: \mathbf{R}[x, y] \rightarrow \mathbf{C}
$$
where we are identifying $A / \mathfrak{m}$ with $\mathbf{C}$. Note that $\mathfrak{m}$ is the kernel of $f .$ Let $f(x)=\alpha…

This question I am trying to attend but I guess I am too dumb for this one

Can any one help me here on how to go ahead

6:40 AM

Another question I can't attempt

Suppose that $\alpha$ and $\beta$ are both complex. Set $f_{x}=(x-\alpha)(x-\bar{\alpha})$ and define $f_{y}$ similarly. Note that $f_{x}, f_{y} \in \mathfrak{m}$ and thus $\left(f_{x}, f_{y}\right) \subset \mathfrak{m} .$ Let:
$$
B=\mathbf{R}[x, y] /\left(f_{x}, f_{y}\right)
$$
Show that the dimension of $B$ as a vector space over $\mathbf{R}$ is 4 . Write down an explicit basis of $B$. Deduce that $\mathfrak{m}$ strictly contains the ideal $\left(f_{x}, f_{y}\right)$. However, $\mathfrak{m}$ can be generated by adding one more polynomial $g$. Can you find such a polynomial (hint: use the …

Just in case any one like to discuss or give me some suggestion

Euler 2

7:26 AM

wow maxima is so nice

2

8:13 AM

@IamKnull Do you know what your name means in Swedish?

@Euler2 I've never used Maxima directly, but SageMath uses it for its symbolic algebra. And (allegedly) you can run Maxima scripts on the SageMathCell server: sagecell.sagemath.org

Euler 2

8:56 AM

@PM2Ring I have heard of sagemath. Never tried it, I think it would also be good

Q: Is there any known relationship of the complex root of two non intersecting curve eqations with the minimal distance between the two curve equation?

9:20 AM

@Euler2 If you're familiar with Python, it's very easy to get into SageMath, since it's built on top of Python. OTOH, SageMath is vast. I think it would take forever to learn all of its features. ;) So after doing the general tutorial (which doesn't take long), you just consult the docs to learn about particular features. The docs are fairly good, although they are a bit of a rabbit warren: it's not easy to organise such a huge body of information.

souraj ghosh

11:18 AM

hello all,
this is my first time messaging, I posted a question a while ago, I haven't got any particular answer, can I direct message anyone for this problem.
ps. let me know if I shouldn't talk about my questions like an advertisement.

2

Suppose the curve lines do intersect in every possible ways. Then all the roots will be purely real and the minimal distance between them will be zero. Now suppose the curve lines don't intersect at all. Then the solution of the two curve lines intersection point will be complex with non zero ima...

here is the question i posted.

12:24 PM

Is $ sum_{n=0}^0 (n+1) = 1? $ IE: if we sum over the same index, does it mean we do 1 step?

@MadSpaces I'm pretty sure you are not supposed to consider an empty sum.

@MadSpaces No, it means you do zero steps, and the result is the empty sum, which is zero. BTW, use \sum to get proper sigma notation, eg $$\sum_{n=1}^{10} n = 55$$

12:42 PM

Thanks.

What would be an example of a sequence that converges to zero, and a sequence that converges to infinity, however when both multiplied, the result leads to a sequence that does not converge against infinity or any other real number.

I am reading a book, and it was stated that such is possible without providing an example.

I guess the term would be (diverge to infinity) but whatevs :D

12:59 PM

maybe (sin n)/n and n.

sin(pi (n + 1/2)) if you want to make the lack of sequential convergence more obvious.

or n^2 if you want things to get more goofy.

1:20 PM

Oh i see. thanks

3:09 PM

@MadSpaces yes, $\sum_{n=0}^0a_n=a_0$, PM 2Ring is mistaken

Alright, thank you for the clarification.

3:48 PM

@MadSpaces Oops. Sorry about that. Thorgott is correct. I got confused... I blame lack of caffeine. :)

If a RO or mod could trash chat.stackexchange.com/transcript/message/57860346#57860346 I would be very grateful.

Sha Vuklia

@LeakyNun Are u busy? ;x

@ShaVuklia depends on your question

Sha Vuklia

Well, I think it's a basic algebra question? Say you have two polynomials $f,g\in K[x]$ (here $K$ is a field), such that $f$ does not divide $g$. How can I show then that there exists an $x_0\in\overline K$ such that $f$ vanishes at higher order than $g$ at that point? @Leaky

you can factorize $f$ and $g$ into monic linear polynomials in $\overline{K}$

$f$ divides $g$ iff for every $x_0 \in \overline{K}$ the factor $(x - x_0)$ has higher or equal order in $g$ than in $f$

Sha Vuklia

4:03 PM

oh oops

ok

thx!

np

Alexandru Ionut

hi everyone! any numerical analysts here?

porridgemathematics

5:46 PM

could someone sanity check something very simple for me

if we take $S^2$ and remove $S^1$, where $S^1$ is the equatorial circle, is the resulting space homeomorphic to two open discs?

Quin

yes

Sure, an easy way to see is map to the plane and then the open disk.

yeah, what remains are two of the standards charts

6:24 PM

@copper.hat Not sure what you mean here. Are you doing stereographic projection from a point of the equator?

we're slicing basketballs in half and stomping on them.

very vidid picture, I like it

at least that's what i'm doing. it's for my new math ed youtube channel.

@TedShifrin that was what i had in mind.

stomping, so violent

the video ends with me throwing both pieces in a hoop. still getting that down because i want swishes.

6:30 PM

So the equator circle maps to a line in the plane, and the two disks map to half-planes.

@leslietownes if you want a slice of contemporary irish humoUr (i had to look up the word vagazzle, and some is a bit local events/knowledge) listed to the 'race' commentary at 04:00-05:00 on play.acast.com/s/the-mario-rosenstock-podcast/…

i'm afraid to click. is it race like horses?

:)

yep. the problem is that i have set expectations now, which is a killer

its something you could play in front of your daughter.

she will just ignore it.

it is a take on the sort of race commentary you would hear on the radio at the weekend.

really more an illustrative example rather than a belly laugh

i love all the turns of phrase people use when calling a race

some implicit swipes at politicans, i love that sort of humor/commentary

6:39 PM

that is a funny match for my fourth cup of ginger tea

rosenstock is very good

an acquired taste perhaps

black market vagazzle

covid commentary

i hate recommending stuff because it never lives up to the implicit expectations

i have a feeling i am missing out on a lot of good podcast/streamed comedy. it's something of a golden age if you know where to find the good stuff.

yeah, i was afraid of that, its a bit local. i have to pay attention and i keep up with local news

sry.

it really was more of an example, we were talking about irish comedy/commentary a few says ago

Care to comment?

i would have missed the (presumably) typo completely.

i can add nothing but to echo your comments.

6:46 PM

another victim of leibniz notation used in DEs

There's a first time for everything. I remember an exercise in Thomas when I was first learning calculus that amounted to showing $d^2f$ is not a tensor.

the great thing about being a student is that you never make mistakes

Blame all nonsense on poor Leibniz.

i have never found an entirely satisfactory notation for anything other than the entire Frechet derivative or the derivative with respect to a scalar.

a bit extreme.

i have a general preference for coordinate free stuff

but they are essential

It's hard to define most functions in a coordinate-free manner.

6:51 PM

for F(x,y,z) I like F_1 for the x partial, F_11, F_112, etc. for higher order and mixed derivatives.

the coordinate is there but i don't need to spend a whole variable on it in my notation, let alone all the d's. it gets an index.

Unless $f$ is a vector-valued function with components.

i like the ${ \partial \over \partial x}$ bit, but it is cumbersome and you are tied to the $x$ part which is confusing for students, but ${ \partial \over \partial 1}$ is more confusing :-)

Quin

I've found that no matter what notation I seem to adopt for these things, situations come up where being inconsistent in notation is the best option. I guess it just means that one (i.e. I) cannot try to always rely on notation and actually need to explain what im doing with words

The former is only confusing when people (including plenty here) butcher the chain rule.

90% of the attempted proofs of Euler's thm on homogeneous functions I've seen are crap.

if tarantino did math movies he would butcher the chain rule.

i like compact notation, but if it gets in the way of clarity, particularly in a teaching situation, i do not get it

6:57 PM

i wouldn't teach my system, it's only for me. i know of only one book that uses it.

i have seen it a few places

No, it's semi+standard.

both the letters & the numbers (less frequently)

I'm partial towards Leibniz notation, cause I like working on manifolds

i hate discovering that 'my' things aren't mine.

6:58 PM

mostly in optimal control, lqr, etc

I use $f_u$ and $f_{uv}$ in my differential geo notes.

leibniz notation is a sharp knife, the best tool in experienced hands.

if i use fu i get reprimands

I don't.

as in for flocks sake

6:59 PM

you were just caling for your flock of sheep and typed into the wrong window.

sry, i'm kidding, its a national hazard

i need to put the chat window on a different computer, it so much more interesting than the windows that actually generate income

i have no life, clearly

hard to get out there these days.

i'll just stay among my windows.

Quin

this chat has become a sort of surrogate for sitting in my office chatting with other grad students about math that stop by lol

another week for the vaccine to take effect and i'll be back to super spreading

i remember offices.

7:06 PM

i'm thinking of going down to sunnyvale, just to visit the office. how sad

i need to collect some stuff and use their confidential bins to lose some tax records

sadly i no longer have my DEC VAX t-shirt

shredding was a big portion of my last trip in. had to hide all of the evidence.

plus i probably don't need my bank records from my student days

you definitely don't need them if you skipped paying your taxes that year.

unfortunately i have always paid them. sometimes minimised, but never avoided.

i was audited while a student, more than once. but it sounds much worse that it was (just supplying extra records).

something to do with rent and being a non resident. stupid stuff. my filing was fine. waste of everyone's time.

7:25 PM

a few modest proposals come up every few years to do away with some of the silliest stuff. i'm sure this session of congress is getting right on it.

'leadership'

@copper.hat That’s a serious schlep.

:-)

if it's anything like here the traffic has probably already gone back to 'normal.' it seemed like there were a few light months.

More than a few.

7:31 PM

i would avoid the main commute, mainly to meet some friends for lunch now that i & they are bill gated.

Kumar Shuvam

9:28 PM

@robjohn u there?

@copper.hat, @leslietownes u there?

9:53 PM

i'm not here

i was joking of course :-)

10:25 PM

One of my favorite lines from House at Pooh Corner: “Rabbit, are you home?” “He's not here.”

10:48 PM

i'm very randomly in and out right now. every few minutes i flip a coin.

In order to show that $c_0$ is a closed subspace of $\ell^\infty$ I guess I just have to show that it's closed under addition and multiplication, and that it actually is a subspace of $\ell^\infty$. Right?

you also need the other sense of 'closed,' in the norm topology of ell^infty.

Right

What would its boundary be?

it does not need to be closed under the pointwise multiplication algebra of ell^infty to be a 'closed subspace' (in context means linear subspace), although it happens to be one.

I'm doing functional analysis and topology at the same right now, so I must admit that I'm a bit confused.

Do we only care about being a subspace and about being closed here?

Nothing about being closed under addition and multiplication?

10:52 PM

there's too much unqualified closed. let's fix it.

to be a closed subpace of ell^infty, in the functional analysis sense, it needs to (1) be closed under the addition and scalar multiplication operations that ell^infty has as a vector space, and (2) closed in the norm topology of ell^infty.

Isn't (1) trivial?

that's probably still too many. closure in the sense (1) is closure under a binary operation, not the topological sense of closure

I mean if $x$ and $y$ converge to zero, then $x+y$ does as well, and so does $\alpha\cdot x$.

(1) is certainly simpler than (2), at least to me. yes.

i hesitate to wave the red flag of 'trivial' in front of a grader or instructor who has put this stuff in front of me, but that's exactly the point.

So, yes about what I just wrote?

10:57 PM

yes.

Maybe I need to a bit more careful in showing it

lone student

Hi everybody. How are you? I wish everyone is fine..

ok

but then it's only (2) that I find tricky