Mathematics - 2023-01-13

shintuku

01:18

anyone know what sort of differential equation is given by $\dot x = -\beta xy^\alpha$; $\dot y = \beta xy^\alpha - \frac{1}{\sigma}y$ ?

copper.hat

05:39

@XanderHenderson i take 1.3g of tylenol arthritis most days. used to be ibuprofen, but that apparently mucks with your stomach and causes acid reflux, for which yet another drug was prescribed. my liver has probably survived much worse than tylenol...

not a fan of drugs of any sort other than exercise induced endorphins

David Raveh

06:32

How do I see latex in chat?

never mind, figured it out

I learned the basics of modular arithmetic, and how we define equivalence classes of integers by $a\equiv b \ (\text{mod} \ n)$. I see questions involving this all the time on MSE. What are the applications of this; why is it important to study?

Raffallo

07:19

Is there any way to solve AX=0 and get a result different than all 0?

Ted Shifrin

07:31

You’ve said nothing about the $K$’s and $L$’s, so of course.

Raffallo

https://docplayer.net/54105018-Color-adjustment-of-orthophotos.html
page 50

There is way to solve it by making at least one image as reference image, but in my case it wouldn't be always good, so I'm curious if there's any other way to solve it

Raffallo

07:49

There is also something called "Regularization",, but I'm not sure how should I use it, since I'm using a little bit different definitions. I'm not operating in colors, but in coordinates and tranformations

Raffallo

09:08

can be Landweber Algorithm used here? If yes, how could I do that?

The Tralfamadorian

11:27

Hello everyone, I will soon write my Bachelor thesis and am looking for some suggestions. I am interested in things like tensor algebra, exterior algebra, clifford algebra... I know that some of these concepts are used in differential geometry / differential forms. Does anyone have an idea for a topic that combines these two things?

user4539917

13:28

in English Language & Usage: Multi-Layered Discourse Room, 1 min ago, by CowperKettle

The Physics Principle That Inspired Modern AI Art

one potato two potato

15:13

If $f:S^1\times S^1\to S^1\times S^1$ is one of $f(x,y) = (\pm x,\pm y)$ or $f(x,y) = (y,x)$ then $f_*:H_2(S^1\times S^1)\to H_2(S^1\times S^1)$ is an identity map?

one potato two potato

15:26

Ignore the above chat

AMDG

15:58

@DavidRaveh If you dig around a bit, you will find relationships to the primes in there.

Gais, gais, I found a way to automatically make anything interesting: just say it's related to the primes.

jk but there are in fact some really interesting things regarding primes and modular arithmetic

inspectorG4dget

Hi everyone, it's my first time in this room, so please let me know if I make a misstep. I have a question for you fine folks and I'm hoping someone can point me in the right direction

I have a matrix of pairwise distances between points in an N-dimensional Euclidean space. There is a way to compute a coordinate matrix from this distance matrix. But I don't see a name for this algorithm and I'd rather not have to implement it from scratch. Anyone know what it's called? better yet, does anyone have a link to a library that implements this?

AMDG

I like how, in this room, everyone reveals their incapacity to answer by saying nothing.

inspectorG4dget

16:15

new development: apparently what I'm trying to do is Part1 of "Multidimensional Scaling"

robjohn

@AMDG that incapacity might simply be absence. On the other hand, would you prefer everyone who is either not interested in a question or unable to answer a question to make a statement to that fact?

David Raveh

@AMDG I am looking for applications outside of pure number theory; are there applications to more practical areas of math like Analysis, PDE, Linear Algebra, Topology, etc.?

AMDG

@robjohn Yes, if present, because silence is not communication per se. If it can be called communication at all, it is totally implicit and impossible to interpret even in context unless you're a mind reader. I wonder how many people, seeking help, have over-stayed their welcome because of this (not just here, but anywhere).

On that note, a bot that mentions "everyone in the room has been silent for 5 mins" wouldn't be a terrible idea either.

Ok maybe 5 min is a bit too short but you get the point

@DavidRaveh :joy: you make it sound like its only applications are within number theory by saying that. It's universally applicable. However, I can't help you beyond, "There's more than just number theory" I'm afraid.

Consider, for example, that $2^x \bmod y$ can be used to compute the infinite binary expansion of any $a \in \Bbb{Q}$

The residues specifically

David Raveh

I guess that I am not really interested in these types of topics in math, it seems so pointless to me

I love abstract math for how it is used in physics

AMDG

16:31

Sounds like a you problem.

:P

@AMDG To further my argument, I would note that in most places where communication is essential to common operation, silence is usually cause for concern. ;)

David Raveh

indeed. I was forced to learn the material last semester, and I have no idea why

Thanks though

AMDG

Schools never teach in such a way that we can appreciate what is learned, nor truly grasp the material.

This is baffling considering we've been learning for thousands of years.

You'd think established institutions in 2022 could do something as simple as that.

David Raveh

Physics and Math is a very strong combo

all the time I learn something in math that has immediate consequences in my physics classes

AMDG

In reality, the core issue is that nothing is tailored to finding each individual's secular vocation and then making one's courses revolve around this. In practicing what one loves--only then can one truly find something appreciable about what he learns in things more theoretical, even if they have practical applications. Without a grounding in reality and existing understanding, all is meaningless.

I only came to love discrete mathematics and appreciate it because I got into algorithms research as a programmer.

David Raveh

never learned discrete math, what is it about?

AMDG

16:35

Well for one, congruence classes belong to it ;)

Discrete math is incredibly interesting to me. It cannot be solved by simply solving for a variable directly within an equation like you can with a real-valued domain.

You must instead rely on splitting atoms into subatomic particles; create a fusion reactor; and then and only then can you assemble an answer.

David Raveh

Analysis also doesn't have an equation to solve; how does it compare to that?

AMDG

Tell me about analysis. I don't know much about it.

David Raveh

One thing analysis deals with is convergence of sequences

AMDG

Convergence of sequences? Convergence in what way? You mean like a limiting sum?

David Raveh

like the sequence (1,1/2,1/3,1/4,...) converges to 0

so we define infinite sum as a sequence of partial sums

AMDG

16:40

I'm still not quite following. Do you mean the sum of all terms in the sequence, or that the values in the sequence approach a given value such as zero in this case?

David Raveh

the latter

AMDG

Understood :)

David Raveh

but how do we rigerously define $\sum_0^\infty x^n$?

AMDG

Well, hence why I called that a limiting sum ;)

An infinite sum is not equal per se to the value which it is equated to. The equivalence is in the limit only.

David Raveh

we define it by sequence $(1,1+x,1+x+x^2,...)$ and prove that this sequence converges

AMDG

16:44

If we wanted to say what an infinite sum actually equals, my intuition tells me that if $a$ is the limit of the sum, and $dx$ is the differential particle, then the true value of the infinite sum is $a - dx$.

David Raveh

I don't understand

AMDG

$dx$ is an infinitesimally small value.

David Raveh

Without analysis, the meaning of that statement is unclear

AMDG

You might think of it as a number with 1 at the end of an infinite sequence of preceding zeros succeeding a radix point.

I can't define it for you formally. That's beyond my capacity.

Xander Henderson

@AMDG Ack! No! Don't do that!

AMDG

16:47

@XanderHenderson Why the objection? XD

David Raveh

You will run into trouble if you define things like this, like $10 dx=dx$

Xander Henderson

If you want to work with infinitesimals (which is done in nonstandard analysis), you regard $\mathrm{d}x$ as a kind of "extended" real number, which has the property that $0 < \mathrm{d}x < a$ for any real number $a > 0$.

David Raveh

I've never seen that before

Xander Henderson

This is not the same as "an infinite number of zeros followed by a 1", as it is really impossible to make sense of "an infinite number of zeros" in a rigorous way.

@DavidRaveh It is roughly the way that Robinson defines infinitesimals in his text on nonstandard analysis.

David Raveh

in this extended system, do you allow for $\infty dx$ like a delta function?

Xander Henderson

16:50

Hyperreal number

In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form 1 + 1 + ⋯ + 1 {\displaystyle 1+1+\cdots +1} (for any finite number of terms).Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948.The hyperreal numbers satisfy...

AMDG

@XanderHenderson For some base $b$, and a positive integer $x$, $x = \sum_{n=0}^{\infty} b^n (a_n)$ where $0 \leq a_n \lt b$. Now all $x$ terminate at some highest power $b^q \leq x$, therefore, $0 = \sum_{n=q}^{\infty} b^n (a_n)$, therefore $a_{n\geq q} = 0$. Let "infinitely many zeros" be defined by this sum. Now $1$ is the smallest non-zero positive integer, and must also have infinitely many zeros preceding it.

We may then go on to say that, according to my description of $dx$, that $dx = \lim_{p\to +\infty} b^{-p} 1$, but idk /shrug

Xander Henderson

@AMDG That doesn't make any sense to me.

@AMDG That limit is 0.

AMDG

If you think about it, it makes sense in consideration of what the difference quotient is.

David Raveh

certainly if it is not $0$ then it isn't a real number

AMDG

"interstitial number" ecks dee

robjohn

17:00

@AMDG the icons in the attendance bar (non-mobile interface) is ordered by quiescence. Checking the front of the list says how long things have been quiet. After an hour of silence, the room puts up a notice that the last message was an hour ago.

David Raveh

@XanderHenderson I was asking before about applications of modular arithmetic to other fields in math like linear algebra, analysis, calculus, etc. You have any thoughts?

AMDG

@robjohn Unfortunately, it isn't always telling of whether or not someone is AFK or if someone knows. The room could've gone silent simply because the conversation has ended and there's nothing to talk about at the moment.

@XanderHenderson What exactly doesn't make sense? I'm curious. This was just kind of a silly attempt of mine to rigorously define "infinitely many zeros" contrary to your statement that it could not be rigorously defined. :)

I'm saying that every positive integer has a finite representation in some base, and that it can be represented as an infinite sum where, after the largest non-zero term, there are only zeros off to infinity.

Xander Henderson

@AMDG Sure, but you can't then tack an extra digit on to the left of all of those zeros. That doesn't make sense.

It runs contrary to the whole idea of what infinity is.

AMDG

Taking the number 1, it can be easy to see there are then infinitely many zeros to the left (as well as to the right, but meh). If we shift 1 to the right, the number of zeros between the 1 and the radix point increases, so if we could shift 1 by $b^{-\infty}$, then we would have what I'm thinking of.

Xander Henderson

But what does it mean to shift 1 to the right by $b^{-\infty}$? That's nonsense.

AMDG

17:07

Shifting a digit in base $b$ is the same as multiplying by a power of $b$.

If I multiply $1$ by $b^{-\infty}$, or rather, if one could, then we should expect infinitely many zeros between the radix point and the one.

Xander Henderson

@AMDG I have no problem shifting a digit a finite number of times. I have a real problem shifting a digit an infinite number of times.

David Raveh

lol "a 'real' problem"

Xander Henderson

@DavidRaveh Indeed. Or even a natural problem.

AMDG

So tru. That's why I tried to use a limit definition. Presumably, since the limit is evaluated to be zero, you can stick $b^{-n}$ there into any arbitrary sum where the index $n$ is and you'd get what I'm thinking I guess. If you like, you can think of it as an error term $\epsilon$ within every convergent sum.

David Raveh

my father has tried to disprove Cantor's diagonalization theorem by making a list beginning with infinite zeros and ending with a one.

AMDG

17:12

:thonking:

Xander Henderson

@AMDG An error term is fine. That is what I defined above: $\varepsilon$ is an object (a hyper-real number, if you like) which has the property that $0 < \varepsilon < a$ for any positive real number $a$.

AMDG

So we're thinking about the same thing this whole time. Epic.

I just am terrible at writing things formally because I have no practice.

Xander Henderson

@AMDG No, because I am not thinking of something which is a number with an infinite number of zeros followed by a 1.

I am thinking of an object which is larger than zero, but smaller than any positive real number.

David Raveh

Consider learning analysis, the entire point is to make these statements precise

Xander Henderson

It does not have a decimal representation.

David Raveh

17:17

@XanderHenderson I got Munkres book from the library, I will take a look at it tonight

AMDG

@XanderHenderson Ok, but... every number proper has a representation, and if we're saying that epsilon is "the smallest non-zero number of the reals" basically, then it makes sense from a numerical consideration that some number $\epsilon$, at least as you've defined it, would be like a number represented with infinite zeros between it and the radix point since there are infinitely many reals.

Xander Henderson

@AMDG Every real number has a decimal representation. This error term $\varepsilon$ is not a real number.

And there is a real problem thinking of this error term $\varepsilon$ as an infinite number of zeros followed by a 1. For example, what is $\varepsilon / 10$?

It is also an infinite number of zeros followed by a 1, which implies that $\varepsilon / 10 = \varepsilon$.

That's bad...

AMDG

Every number can have infinitely many representations, so I really don't see a problem there in that consideration; I do see a problem if you want to work with it algebraically as if it were like any other number.

Xander Henderson

What do you mean by "every number can have infinitely many representations"?

And also, how does that address my point?

AMDG

So having every elementary binary operation involving $\epsilon$ be idempotent is fine by me.

Xander Henderson

17:23

Every real number can be represented by a decimal. Other kinds of numbers don't necessarily have (or need) such a representation.

@AMDG That doesn't work in analysis, where $\varepsilon^2$ is not the same as $\varepsilon$ (for example).

AMDG

@XanderHenderson We can create any arbitrary model and interpret it's value to be any value we desire.

Xander Henderson

@AMDG I don't understand. That doesn't make sense to me.

AMDG

Well you have to get more metaphysical to understand. There are infinitely many possible spoken languages that could exist; infinitely many possible symbols that we could write; all of which can be used to represent real being within reality, or some object of knowledge.

Xander Henderson

@AMDG I am trying to talk about mathematics, not meta physics.

AMDG

Likewise in mathematics, I could take a number which has objective value and treat it as a model for some other objective value.

Xander Henderson

17:25

@AMDG I don't know what that means.

AMDG

I could say that 0 is now 1, then that is implicitly a rule that for every number I have, I should subtract 1 from it to get its true value.

Xander Henderson

@AMDG I don't know what that has to do with anything I've said...

AMDG

@XanderHenderson The pythagoreans who I believe are to be credited with the birth of mathematics believed "all is number", and this is but one step away from Aristotlean philosphy and the concept of "essence", so what you're saying here makes no sense to me, but I digress... I'm saying anything in being can have an arbitrary association with another object of being, so likewise any number or mathematical object can have an arbitrary association with another mathematical object.

So mathematically, this arbitrary association is an effective equivalence, not necessarily an equivalence per se.

0 is not literally 1, I'm representing 1 as the number 0

or symbol 0 if that makes you happier

Xander Henderson

(1) I still don't understand, and (2) I don't see how this is relevant to anything I've said or asserted.

AMDG

Then I don't know what to tell you. You tell me what isn't clear about the capacity to make free associations between arbitrary things and I'll clarify; otherwise, I give up.

Xander Henderson

17:33

What is the relevance?

AMDG

0.0...01 is merely a model: a chosen, arbitrary association between this would-be, number-like representation of the value of $\epsilon$ as given by you, and the value of $\epsilon$ itself.

Yai0Phah

What do you want to talk about 0.0...01?

AMDG

And that it proceeds intuitively from my existing understanding of numerical representations of numbers as a direct analog of positive reals is why I chose it

Yai0Phah

Speak human languages.

AMDG

I guess what I'm trying to say is that my representation is chosen to be analogical rather than literal in the same way that one uses analogies to explain difficult or hard-to-grasp concepts.

Xander Henderson

17:39

@AMDG Okay, but it is a bad model, for reasons that I have articulated above.

Because it uses confusing plain English to mean something other than what that plain English seems to say.

I mean, you are welcome to say "When I say 'an infinite number of zeros followed by a one', what I actually mean is 'an object $\varepsilon$ which has the property that $0 < \varepsilon < a$ for any real number $a$'."

But saying "an infinite number of zeros followed by a one" implies certain things in terms of algebra and analysis which are contradicted by the more formal definition. So it is a bad model for infinitesimals.

AMDG

Ok, but no one takes an analogy and runs with it like it's 1:1, and a model can be arbitrarily refined to better model the reality. I mean, we're talking about the difference between having a literal equivalence between infinitely many terms in a sum, and the limit of that infinitely many terms.

Yai0Phah

Infinitesimals are usually functions in modern mathematics.

AMDG

Whatever we might like to define $\epsilon$ as algebraically, I'm saying $\epsilon$ is the value that makes the equivalence literal as opposed to only true in the limit, and that taking the limit is the same as adding $\epsilon$ onto the sum iff it is convergent and has infinitely many terms.

Yai0Phah

Or asymptotic.

AMDG

I guess, but I'm honestly lacking in terminology and an overall well-formed foundation here. Some things I've forgotten since school; other things, I've learned because I had a particular end in mind.

Not that one learns much in public school...

David Raveh

17:51

In what course is one likely to learn measure theory? Real analysis perhaps?

Xander Henderson

@Yai0Phah No? Infinitesimals, when they show up in modern mathematics (which is rarely) are number-like objects. They are not functions.

@DavidRaveh Yes.

Ted Shifrin

Usually graduate real analysis.

Xander Henderson

@TedShifrin Or at the end of undergrad real anal (UCR taught some elements of measure theory in the last quarter of undergrad real).

Ted Shifrin

Instead of multivariable analysis, I assume. Most US schools give both short shrift.

Xander Henderson

@TedShifrin Indeed.

David Raveh

17:54

Measure theory seems very important to functional analysis, which I am learning myself. I haven't learned measure theory yet; how should I go about learning it?

Yai0Phah

@XanderHenderson I mean, just like $x^{-1}$ as $x\to+\infty$.

Ted Shifrin

Integration theory more than measure theory, I would say.

Xander Henderson

@Yai0Phah That isn't an infinitesimal, though. It is a limit, which is a number.

David Raveh

@TedShifrin I thought Lebegue theory is all based on measure theory

Xander Henderson

$\lim_{x\to \infty} 1/x$ is $0$, by definition.

Yai0Phah

17:56

@XanderHenderson No, I mean the growth of that, so you can compare them via $\ll$, $\gg$ etc.

Xander Henderson

@Yai0Phah Okay, but $O(x)$ isn't an infinitesimal either (as $x\to 0$). It is a set of functions.

Yai0Phah

@XanderHenderson You can also see them as an equivalence class of functions, and in that sense (just like how we see objects in L^1), they could be seen as functions. I was not trying to make it completely precise.

Ted Shifrin

It usually is, but still the measure theory per se is not that important in functional analysis. There is also a development of the integral called the Daniell integral which avoids much of the measure theory.

David Raveh

What about Borel measure? does that show up in functional analysis?

Yai0Phah

The trick is that you only needs some formal property of integrals, and it is very infrequent to play with measure-theoretic constructions.

Xander Henderson

18:00

@Yai0Phah I don't think that they are equivalence classes, as they don't partition the function space.

For example, $O(x^2) \subseteq O(x)$ as $x \to 0$.

They are sets, but the sets don't partition the space, so they are not equivalence classes.

Yai0Phah

@XanderHenderson It is because you are using $O$, which is a partial order in essence, but one should use "equivalent" infinitesimals (i.e. $\Theta$).

Xander Henderson

@Yai0Phah Okay, but I still wouldn't call those "infinitesimals".

Yai0Phah

What do you want to call infinitesimals?

Ted Shifrin

@DavidRaveh i would worry more about a thorough understanding of linear algebra first.

Yai0Phah

Is there any important theorem in elementary functional analysis which involves Borel measures?

I don't remember any.

Xander Henderson

18:04

@Yai0Phah I would say that an infinitesimal is a hyper-real number which is larger than 0 and smaller than any real number.

David Raveh

You have a book to recommend on linear algebra?

Yai0Phah

Maybe Riesz--Markov representation theorem, but it is not always included in elementary functional analysis.

Xander Henderson

@DavidRaveh Hoffman and Kunze, maybe? What level are you looking for?

David Raveh

He said I should get a thorough understanding of linear algebra--not sure what that entails

Yai0Phah

@XanderHenderson I am not familiar with that, but it is basically an ultraproduct of real numbers, thus you can still view them as functions on $\mathbb N$, and modulo an equivalence relation given by an ultrafilter.

Xander Henderson

18:08

@DavidRaveh If you are looking to learn the recipes, there are any number of undergraduate texts out there with titles like "Linear Algebra". I've taught out of Anton, which is likely much the same as any other at that level.

If you are trying to understand the theory a bit more, then something like Hoffman and Kunze is probably a good text.

David Raveh

I took one graduate level linear algebra course, but I will have to look at the book to see if I am familiar already with the material

Xander Henderson

@DavidRaveh Oh. Then you should be largely set for funky anal.

Yai0Phah

What is graduate level linear algebra?

David Raveh

Linear algebra for grad students and advanced undergrads

very small class

Yai0Phah

No, I mean the content.

David Raveh

18:20

Inner product spaces, spectral theory, dual space, etc.

A lot of it was proving theorems rigorously

Yai0Phah

Also tensor products?

Quotient spaces?

David Raveh

mentioned tensor products, but avoided it

not quotient spaces

Thorgott

that sounds like undergraduate material

Yai0Phah

That seems enough for elementary functional analysis, unless your functional analysis covers injective/projective tensor products.

David Raveh

I am an undergraduate (junior)

Yai0Phah

18:25

The classification of material is not important, since different schools have different syllabi.

David Raveh

We ended with exponentials of matrices and jordan forms

Vrouvrou

20:22

Hello

copper.hat

20:44

another unexplained random downvote

mick

@copper.hat where ?

i edited again.
getting close to decent lol

https://math.stackexchange.com/questions/4616600/classify-all-solutions-to-fxy-gfxfy-fx-fy

copper.hat

i get downvotes every now and then, always curious why, particularly on closed questions

mick

not from me

copper.hat

:-)

mick

my view on free will and math is this :

Is math discovered or invented ? Do we have free will ?

well just as our body picks to walk on a certain path but does not create the path itself , I believe the same is true for free will and math.

We discover the math landscapes we walk on but did not create them.
So we have free will but are tied in many ways.

Ted Shifrin

20:53

I am more upset by good answers that stay unaccepted, @copper. Rant on about that one!

mick

@TedShifrin you got a question removed or an answer unaccepted ?

maybe your answer was good but to advanced for the OP ?

Ted Shifrin

21:19

No. Not so. Just hundreds of unaccepted answers!

leslie townes

i accept them, ted. i just can't 'accept' them.

Ted Shifrin

You are under the influence of evil Munchkin.

lorenzo

21:49

@TedShifrin Hello, I have a quick question, if I may. If (as in Exercise 3.4.1 of Multivariable Mathematics) I have to find the tangent line of a given level curve at a prescribed point, for example $x^3+y^3=9$ at $\mathbf{a}=\begin{bmatrix}1\\ 2\end{bmatrix}$ would it be correct (even if not the fastest method) to first find the equation tangent plane to $f\begin{pmatrix}x\\ y\end{pmatrix}=x^3+y^3$

, in this case $z=3x+12y-18$, and then "slice" it with the plane $z=9$ and conclude that the equation of the tangent line is thus $x+4y=9$?

Ted Shifrin

You could do that, but you're just supposed to use the gradient! Super powerful technique.

lorenzo

22:09

So, if I have understood correctly, in this case it suffices to say that since $f:\mathbb{R}^2\to\mathbb{R},\ f\begin{pmatrix}x\\ y\end{pmatrix}=x^3+y^3,\ Df\begin{pmatrix}x\\ y\end{pmatrix}=\begin{bmatrix}3x^2 & 3y^2\end{bmatrix}$ the equation of the tangent line is
$$
Df(\mathbf{a})(\mathbf{x-a}) =0 \Leftrightarrow \begin{bmatrix} 3 & 12\end{bmatrix}\begin{bmatrix} x-1 \\ y-2\end{bmatrix}=0\Leftrightarrow 3(x-1)+12(y-2)=0\Leftrightarrow x+4y=9
$$, right?

Ted Shifrin

22:25

Yes.

lorenzo

Thanks

mick

0

Q: " Taxi cab primes "?

Taxi-cab numbers: sums of $2$ positive integer cubes in more than $1$ way. $1729, 4104, 13832, 20683, 32832,$ etc see : https://oeis.org/A001235 let $x$ be a cube-free taxi cab number. consider all the solutions $$x = a_i^3 + b_i^3$$ Now let $v(x)$ be the smallest common multiple of $a_1 + b_1,a_...

new question

Semiclassical

22:47

I don’t know what it is, but the comments here rub me the wrong way: math.stackexchange.com/questions/4617230/…

Ted Shifrin

The OP was confusing vectors and scalars. But, yes, Mark can be a pill.

copper.hat

23:05

@TedShifrin apart from the jumpsuit, i don't really care about the rep, but the negative without explanation gets to me :-)

the downvotes mostly seem to be on answers that were closed some time ago.

Ted Shifrin

I tell you — there are a few people who hate us.

copper.hat

:-) I am sure I have irritated a few.

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$Mathematics$ Mathematics