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Physics Meta

11:03

0

Q: No answers - is this question too hard?

I don't think that this question is too hard actually. There must be some way to solve this and I know there are plenty of people here with a solid background in physics and the mathematics thereof. What else further can I do to encourage someone to have a go?

discussion

Secret

Heisenberg group

In mathematics, the Heisenberg group, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ( 1 a c 0 1 b ...

important in quantum mechnics

Secret

11:26

Elementary group

In algebra, more specifically group theory, a p-elementary group is a direct product of a finite cyclic group of order relatively prime to p and a p-group. A finite group is an elementary group if it is p-elementary for some prime number p. An elementary group is nilpotent. Brauer's theorem on induced characters states that a character on a finite group is a linear combination with integer coefficients of characters induced from elementary subgroups. More generally, a finite group G is called a p-hyperelementary if it has the extension 1 ⟶ C ⟶ ...

They seemed to like primes so much. In here, in most p group examples and also the sylow theorems. Must sit down later and ready carefully on how composites causes trouble

ACuriousMind

Why would someone bother to write $+$ve and $-$ve with LaTeX instead of writing positive?! I don't know why this irritates me so much, but it does

user116211

I think it has been discussed earlier; but have to check the transcripts when...

Balarka Sen

@ACuriousMind Less characters to type.

ACuriousMind

@BalarkaSen But why not +ve and -ve

Why enclose the plus/minus sings in dollars?

Sanya

@ACuriousMind nicer font?

ACuriousMind

11:37

@Sanya I have trouble believing that someone who don't care that "+ve" is not a word would care about the font the + is in, but, well, maybe you're right.

Balarka Sen

Or maybe just instincts to write every symbol in latex.

Sanya

@ACuriousMind I can only speak for myself, but I tend to put numbers in equation environment to make them look nicer

user116211

Related but not same post by ACM:

user116211

Jun 4 at 13:08, by ACuriousMind

Why do people write "+ve charge" or "+ve direction"? They seem to mean "positive", but "+" does not mean "positi", so why do they append the "-ve" to "+" instead of just writing "+ direction", which would be a shorter abbreviation that carries the same meaning?

Secret

e.g. $C_4 \times C_9$ looks elementary? (C_9 is a cyclic p-group with order $9^1=9$, and gcd(4,9)=1)

Slereah

11:42

Wait

If I pick the sum of hyperreals $(1,0,0,0,...) + (0,1,0,0,0,...) + (0,0,1,0,0,...) + ...$

All terms are equivalent to 0

But shouldn't the sum be 1

Or even worse if I pick $(1,0,0,0,...) + (0,2,0,0,...) + (0,0,3,...)$

Then the sum would be infinite

ACuriousMind

@Slereah Well, did you show that taking the quotient to get the hyperreals commutes with infinite sums (or limits in general)?

Slereah

I'm still on chapter 3

Currently hyperreals aren't even defined as sequences

Just as some metamathematical extension of $\Bbb R$

ACuriousMind

Then I'd hold off on that question

Slereah

@yuggib halp

I'm guessing it doesn't work since that sum would be equivalent to the infinite sum of $(0,0,0,0,...)$

But I'm not sure how

Sanya

0

Q: In charging by induction a charged body mirrors its Energy to another body without transferring it, is that not a violation of Energy conservation?

This is more less like the ice pail experiment where I will recover the energy I used in charging the sphere when I contact the sphere to the container and extra electric energy when I Earth the container. This mirror effect can be amplified when you place a charged sphere in many other spheres o...

electrostatics experimental-physics energy-conservation

am I the only one who has no clue what this is about?

ACuriousMind

11:49

Nope, you're not alone :P

user116211

@Sanya What is mirrors?

Sanya

@MAFIA36790 I guess to duplicate onto something?

user116211

So reflection transformation?

Sanya

I guess ... not too sure about all of it

Xasel

Hey can somebody explain me what Nobel Prize for Physics 2016 is about in in laymen terms(terms I can understand)

user116211

11:56

Anyways, concentrating on my studies.

Xasel

"opological phase transitions and topological phases of matter""

Secret

Sylow theorems

In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Ludwig Sylow (1872) that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups. For a prime number p, a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group G is a maximal p-subgroup of G, i.e. a subgroup of G that is a p-group (so that the...

One day, I am going to plug everywhere that says p a composite number, and see how this whole thing will crumble in order to udnerstand why they can only use primes

Xasel

dead?

anyway will checkout chat later...

Balarka Sen

@Secret Take $A_4$, which is of order $12$. $6$ divides it, but it has no subgroup of such order.

Slereah

wait does Robinson ever actually define hyperreals as sequences

I'm not sure

MartianCactus

12:09

hey I have a problem

user116211

Everyone has problems.

Secret

@BalarkaSen Agreed with the counterexample

(However, it might be not very satisfying for me because I don't understand the underlying reason for all these counterexamples and how it arises (only knowing that there is a counterexample and how it is a counterexample). I will try to check myself on that later)

MartianCactus

whenever I learn something new in school, like magnetism, I want to know everything about it, and end up asking questions like why the magnetic fields even exist and stuff

and why they have the shape that they have

but all answers that I get is either of collage level physics or inclue calculus

user228700

@MAFIA36790 LOL XD

user116211

@MartianCactus Magnetic field lines have no real existence.

MartianCactus

12:11

so when I reach these levels will they teach me these things in advance and will it clear all my querries?

Xasel

@MartianCactus:the only solution is say bye to procrastination and work damn hard to get there

and at the end of the day

MartianCactus

@MAFIA36790 well then why do the iron nails align themselves in a specific pattern?

Xasel

become satified with answer that

physics never answer 'why'

MartianCactus

@Xasel yes but being a 10th grader..i don't think I can understand all that QM math

@Xasel ha nice joke

user228700

@MartianCactus I understand where you're coming from :-) I have the same "problem" but don't consider this to be a burden, because being curious is one of the best qualities for a human to possess.

MartianCactus

12:13

didnt newton ask why the apple fell? Not how

ACuriousMind

@MartianCactus Yes and no. Once you understand calculus (and a lot of other math) you will understand a lot more about physics, and how physical theories actually work. But your question of "Why does the magnetic field have the shape it has?" will simply be replaced by "Why do magnetic fields obey the equations they do?". And if you managed to find a theory that explains that, you'll ask "why" that theory is true...

user116211

@Xasel Why?

ACuriousMind

Physics is ultimately not meant to "answer why questions", but to provide models to predict what happens in reality.

4

MartianCactus

@Kaumudi but being curious and not able to find the solution after soo many questions is even harder

Xasel

@MAFIA36790:why is the question of eternity better left unanswered :P

user116211

12:14

@Xasel ;D

MartianCactus

@ACuriousMind so its an endless chain afterall

ACuriousMind

@MartianCactus For all I know, yes

MartianCactus

so noone knows why magnetic field lines exist?

Balarka Sen

No, because that's not a well-defined question.

ACuriousMind

What does that even mean?

Xasel

12:15

@MartianCactus:I can only refer you to books...the guys here can show you the way..but you have to walk it yourself...

ACuriousMind

Heh, @BalarkaSen, two posts, same thought :)

MartianCactus

why magnetic lines exist IS well defined

Balarka Sen

great minds think alike

MartianCactus

??

i mean the magnetic field lines just can't pop into existance

there has to be a reason

Xasel

well in phsyics nothing exist we just create mathemtical models to study the nature quantatively to some extent

Balarka Sen

12:17

That sounds equivalent to "why am I in the world? I must have a purpose"

user228700

@MartianCactus Yes, perhaps, but as you learn more, trust that many of the misconceptions that are the root of these ill-defined questions will become clearer and you will realize the meaning of what ACM has said, about the point of physics being to provide models to predict realty.

ACuriousMind

@MartianCactus Field lines are just fictions. You can't see a field line in reality. What you see is e.g. a compass pointing towards a wire. We invented the concept of "fields" and "field lines" to explain this and predict similar situations.

yuggib

@Slereah well, the sum $(0,1,0,\dotsc)+(0,1,0,\dotsc)$ is not a sum of hyperreals

Xasel

The "field lines" taught in many classes and used by physicists to visualize field strengths are purely to guide the eye; they don't have physical meaning.

MartianCactus

oh

Slereah

12:18

Why not

Sanya

Magnetic field lines do not exist. You see some effect in reality which you attribute to the field and its lines, but the field does not exist.

Balarka Sen

@ACuriousMind He's probably asking why the field lines look like they way they do, whatever that means.

yuggib

because to sum hyperreals you have to sum equivalence classes

Slereah

But then which representation do you pick

Xasel

Check this oyt:van.physics.illinois.edu/qa/listing.php?id=27163

yuggib

12:19

@Slereah the one you wish, but you have to be coherent

Xasel

5

Q: Do electric and magnetic lines of force physically exist?

As per my imagination any thing can't impose force on the other by not giving even a touch(i,e action at a distance). So I thought there must be some physical existence of lines of force. Although virtual particles hypothesis has been proposed. Up to my knowledge virtual particles are just said t...

quantum-mechanics electromagnetism forces electricity electrostatics

MartianCactus

also I don't really get the concpt of volt..

Xasel

colt?

Slereah

Same rep for same number?

MartianCactus

edited

ACuriousMind

12:19

The concept of colt would be easy: Point at someone you want dead, pull trigger ;)

yuggib

@Slereah the sum of two equivalence classes $[a]$ and $[b]$ is the equivalence class $[a+b]$

Slereah

Also Robinson indeed has an extension of Hilbert spaces to hyperreals

MartianCactus

XDXD

Slereah

Which is nice

Xasel

hmm.... do you understand the concept of energy and mathematical constuct to study it's evolutionn with time like potentiaal energy

Slereah

12:20

Much better than the ultrametric ones

Balarka Sen

Guns and bombs are the easiest things in the world.

yuggib

@Slereah told you so

MartianCactus

@Xasel me?

Xasel

yeah

Slereah

Oh by the way

For all internal statements

Can I just transfer them dumbly

Or is there something to watch out for

Xasel

12:21

I suspect you haven't built base structure and planning to build burj-khalifa @MartianCactus

MartianCactus

@Xasel nah

yuggib

@Slereah they hold only for internal objects

Xasel

I suspect you haven't built base structure and planning to build burj-khalifa @MartianCactus

Slereah

Well yes

But can I

Or do I have to watch out for like

user228700

@MartianCactus: When you learn physics at school, it is truly very difficult to properly understand everything, because the whole picture is not introduced to us. Eg: It was very difficult for me to understand why a perpendicular force does no work on the body but only later in my stidies did I realize why this is.

Slereah

12:21

The order of the statement

user228700

It can be difficult to say "OK, I'll learn it properly later", but if u don't have the resources to teach yourself the more advanced stuff, it will have to do. That's mostly how I made it through high school...

Xasel

first learnt he basics bro..no need to hurry

Slereah

Or something

MartianCactus

sooo why did the school teach us that? Before teaching the imp. stuff bout the base

yuggib

i.e. if you say "all subsets of reals blah blah" this transfers to "all internal subsets of the hyperreals blah blah"

Xasel

12:22

because most of them teach you learn definition and pass the exams

Slereah

That much I got, yeah

MartianCactus

so after high school..all those questions start to clear away?

Slereah

quite unfortunate since most interesting subsets are external

By the way are all finite subsets of R internal?

If I just define them by their members

yuggib

yes

Slereah

I suspected

user228700

12:23

@MartianCactus Yes. These questions arise because of misconceptions that in turn arise because of the whole picture not being introduced 'cause it's too advanced.

MartianCactus

so first they teach u how things work..then they teach you why things work

Balarka Sen

@ACuriousMind Though, I am curious whether there is a mathematically rigorous context in which you can show why the magnetic field has index 2 at it's only zero (aka where the magnet is).

That might sound a bit naive, but I admit I know nothing of physics.

user228700

@MartianCactus What do u mean by "how things work" vs. "why things work"? Can u provide an example of this that u have encountered in ur studies?

MartianCactus

um...taking the magnetic thing...how the magnetic fields work and why they exist are 2 different questions

Secret

It's the same idea in mathematics, we need to learn how things work, before we can start to understand (and prove) why
For physics, there is no proof, but there are experiments to test the model

For classical electromagnetism, magnetic fields are the result of special relativity

ACuriousMind

12:29

@BalarkaSen Well, from what do you want to show that?

MartianCactus

so special relativity and electromagnetism are linked/?

Balarka Sen

@ACuriousMind That's exactly what I am asking, if there is a workable axiomatic framework in which you can show that.

Xasel

Bro, you seriously need to learn all the things from gorund-up again if you want more understanding...

You are just shooting arrows in the mist

MartianCactus

ha ground up? im in 10th class so i have barely started

Xasel

much better go buy a good pre-calc and calc books

ACuriousMind

12:30

@MartianCactus Well, I still don't know what exactly you mean by "why". Asking "why" can never have a satisfying answer: Either the reason stated has itself no reason (then why couldn't you stop asking why one step earlier?) or the chain of reasons continues to infinity, which also makes it useless.

2

@BalarkaSen Well, would knowing it is divergenceless suffice?

MartianCactus

by why i mean a hard proof of why they exist

idek for now i gtg

Xasel

and thoroughly do each question(for every wrong question properly explain what do did wrong and 'why it is wrong' and and what is the right solution and why)

ACuriousMind

@MartianCactus A proof starting from what givens?

Balarka Sen

@MartianCactus To write down a proof, you need axioms.

MartianCactus

but axioms are (accordint to my understanding ) guesses

user228700

12:33

I agree with Xasel. Like everybody has pointed out, you need to start from the basics. @MartianCactus: I promise you that these questions will cease to exist once u start studying all this much more thoroughly.

MartianCactus

things that we dont proof

@Kaumudi oh

ok im going to go now

Balarka Sen

@MartianCactus You seem to have a wrong idea of what a "proof" means.

MartianCactus

maybe i need to learn more before understandih this stuff

user228700

@MartianCactus Yes.

MartianCactus

k now bye!!

user228700

12:34

Bye :-)

Secret

SR and electromag are linked indeed, but you should be proficient on the stuff at your level before start tackling with the uni level stuff

MartianCactus

whats is uni level?

oh

Balarka Sen

@ACuriousMind I am not convinced divergence determines the index. Look at, like, a circular flow around the origin (eg $(y, -x)$ normalized appropriately). That's index $+1$.

Slereah

unicorn level

Xasel

@MartianCactus: mirtitles.org/2012/03/06/…

MartianCactus

12:37

ah

Slereah

Hm

MartianCactus

do you think i can understanf that stuff??

@Xasel i first need to learn functions

Xasel

I know just giving you a list which you have to go through

mirtitles.org/2013/09/13/…

^books for developing intuition about probabilities

for 3d gemoterey(basics):3d maths primer for gfx and game dev by Fletcher DUn

heather

@Xasel, the lists to go through are so you can understand what topic?

Xasel

?

MartianCactus

12:39

oh

heather

If you are trying to understand x, you need to know the stuff on this list, what is x?

What is MartianCactu learning

Xasel

he is trying to develop his mathematical base for atleast HS-level physics

mirtitles.org/2012/07/04/…

^Go through this book thouroughlt for trignomtry

heather

@Xasel, ah, okay.

Xasel

Induction in Geometry(dvevelop for intution for geometry):

mirtitles.org/2012/01/30/…

archive.org/details/MatveevMechanicsAndTheoryOfRelativit

@MartianCactus:mind you all these books are by russian authors

heather

@Xasel, your second link isn't working, I don't think.

Xasel

12:45

@MartianCactus you will be treated harshly..but if you go through..Nirvana on your way:mirtitles.org/2013/11/30/mechanics-and-theory-of-relativity-matveev/

second link:mirtitles.org/2013/11/30/mechanics-and-theory-of-relativity-matveev/

uhoh

@ManishEarth are you pingable? for 1 or 2?

MartianCactus

oh ok!!

heather

@MartianCactus, I might also recommend some of the books on Project Gutenberg's site, specifically this classic book about calculus, Calculus Made Easy.

MartianCactus

i barely have time to read these

school presurre

uhoh

@ManishEarth last one 3.

Slereah

12:50

I wonder if $^*\mathcal L^2$ is $\approx {} ^* \mathcal H$

Slereah

13:00

is "relatively compact" just fancy math talk for "it's finite in extent"

ACuriousMind

@Slereah no

"relatively compact" is another word for "precompact"

It means the closure of the set is compact

Slereah

is "precompact" just fancy math talk for "it's finite in extent"

I know

But wouldn't that be what that entails

ACuriousMind

I guess in metric spaces you might imagine those things to be "finite in extent", yes

Slereah

why would I not pick a metric space

you weirdo

With your point set topology

Balarka Sen

lol

Slereah

13:10

As a reference for more hyperreal hilbert space, Robinson mentions an unpublished dissertation

I am peeved

Sanya

13:27

Can I put figures from books on SE if I quote them correctly?

0celo7

@Slereah finite in extend = totally bounded

Which is equivalent to precompactness

Slereah

I knew it

heather

@Sanya, yeah, I think you're good. As long as you don't blatantly plagiarize =)

Slereah

can't pull one over on me @ACuriousMind

heather

@Sanya, oh, quick question:

0celo7

13:38

@Slereah in metric spaces

So I agree with ACM

Slereah

who cares for non-metric space

0celo7

A general topological space has no notion of size

Slereah

What's even a non-metric space

The long line?

The empty set?

heather

with Shankar's quantum mechanics, I'm thinking I need to learn calculus. As 0celo7 and others suggested. Do you have any good suggestions for calculus books/videos?

0celo7

@Slereah algebraic geometers

Slereah

13:39

I say tar and feather them

innisfree

What's meant by thermal equilibrium in Sakharov's conditions?

0celo7

@BalarkaSen I'm fairly sure you're correct

Else someone would have said it

Balarka Sen

I mean, I gave a c/e

Noah P

Hey, someone hear for a quick question?

heather

@NoahP, just go ahead and ask, no need to ask to ask

Noah P

13:45

Okay

Sanya

@heather that's a hard question ... all my calculus books were in German so I'm a bit lost. Let me get my Shankar and see what he does not cover first.

Noah P

Using coulombs law, can you work out the acceleration of an electron?

Eg. $F=\frac{kQ_1Q_2}{r^2}$

Sanya

the acceleration through what?

Noah P

And then use $F=ma$

An electric field

Sanya

an outer field, another charge, ...?

ah, then you do not need coulombs law

Noah P

13:47

Could I use it though?

heather

@Sanya, well honestly, I am in highschool geometry right now, and I have some knowledge of linear algebra, but I don't know how to do proofs, or anything. I was still able to slog through chapter 2, but chapter 3 just got me stuck because of the calculus.

Sanya

@NoahP I can't think of a meaningful way

ACuriousMind

@NoahP Coulomb's law describes the force between two charges. It has no relevance to a charge in an outer electric field.

Noah P

Right okay

So I have to use $eV=E$ do I?

Sanya

@heather but you were fine with chapter 1?

@NoahP can you explain your symbols?

Noah P

13:50

$e=$ charge on electron, $V=$ voltage of the field,$E=$energy of the electron post acceleration, which can be put in the form $E=\frac{1}{2}mv^2$

ACuriousMind

@NoahP How did you come up with that formula?

innisfree

How about Sakharov's laws? It's been bugging me

Noah P

@ACuriousMind My A-Level textbook did

heather

@Sanya, oh, excuse me, I gave the wrong chapter numbers: I slogged through chapter 1 and got stuck in chapter 2 because of the calculus. I was able to understand most of chapter 1.

innisfree

What does it mean by thermal equilibrium? And how to prove a departure is necessary?

0celo7

13:52

@BalarkaSen yeah

ACuriousMind

@innisfree What is "Sakharov's law"?

0celo7

In 3D I wonder if the index is like curl + divergence

Something nice like that.

ACuriousMind

@NoahP Well, but for what? You originally wanted to work out the acceleration of the electron. Where in that formula does the acceleration appear?

innisfree

Oh I meant his conditions for baryogenesis

heather

@ACuriousMind, google comes up with Andrei Sakharov, a Russian nuclear physicist, but it doesn't say anything about a law.

ACuriousMind

13:53

@heather I can google on my own :P

Noah P

@ACuriousMind It doesn't give anything over than the velocity, which is why I asked about the acceleration

heather

@ACuriousMind, sorry =)

ACuriousMind

@NoahP Yeah, okay. So what is the definition of "electric field"?

innisfree

In al proofs of his 3rd condition - departure from thermal eqm - it

ACuriousMind

Using $F=ma$ to get the acceleration was the right thought - you just need the right formula for $F$.

Noah P

13:54

A radial field, around a uniform charged particle

innisfree

In all proofs of his 3rd condition - departure from thermal eqm - it assumes that thermal equilibrium means Gibbs canonical ensemble. But that seems odd

ACuriousMind

@NoahP Hm, maybe I need to start differently. Your outer electric field, where does it come from?

Noah P

I'm coming from a really basic stand point, we've only just covered electric fields at school

heather

@innisfree, I'm still not clear what you are talking about - to be clear, you are talking about Andrei Sakharov, right? And if so, are you talking about his work with baryon asymmetry?

0celo7

@ACuriousMind holy crap you got fugn #rekt by a kid

Do you need a burn kit

innisfree

13:57

Yes, he famously listed 3 necessary conditions for baryogenesis

ACuriousMind

@0celo7 What?

0celo7

@ACuriousMind destroyed!

heather

@innisfree, okay, one moment, asking the google...

ACuriousMind

@heather Lookee here

@0celo7 How? I think you didn't understand that conversation correctly

heather

@ACuriousMind, thanks!

Okay, found his paper as well.

Sanya

13:58

@heather yeah well, Lagrangian mechanics is .... a lot to handle. So basically you'll need to know integration, differentiation and differential equations a bit - which is not too bad really. I like math myself but I would, if you really want to advance a bit into QM soon, stick off the purely mathematical treatments and proofs for now and look at the calculation rules and convincing illustrations instead. Something in the style of this

physics.miami.edu/~nearing/mathmethods/… the first four chapter

ACuriousMind

@innisfree I think that what exactly is meant by "thermal equilibrium" in that context might make a good question for the main site

Sanya

unfortunately it takes differentiation and integration for granted

0celo7

@ACuriousMind you were beaten at your own game

innisfree

@acd cool ok

0celo7

Now you'll think twice about telling me to just "google it"

ACuriousMind

14:00

@0celo7 I have no idea what you are talking about. The point was that I asked "What is Sakharov's law" precisely because Google doesn't give direct hits when typing that in.

heather

@Sanya, my dad has the calculus textbook Calculus and Analytic Geometry by George Thomas, would that be good for learning about integration/differentiation?

@0celo7, yes, even my result didn't give anything about Sakharov's law...ACuriousMind found the helpful website

Sanya

@heather I don't know it :( but have a look - anything which tells you how to integrate and differentiate polynomials, trigonometric functions, exponentials and logarithms and motivates what is happening there should suffice

0celo7

@heather Maybe just do like the Germans

Calculus on Banach spaces right from the beginning

heather

@Sanya, okay, thanks for the links!

@0celo7, whaaat?

ACuriousMind

@heather He's trolling :P

0celo7

14:05

@ACuriousMind When a QM problem wants me to compute a random transmission/reflection coefficient, should I assume the incoming particle is just a plane wave?

or...something?

I'm very confused by the problem

ACuriousMind

@0celo7 yes

0celo7

@ACuriousMind Am I?

Sanya

@heather If I stumble accross sth useful, I'll let you know

for now I need to go

Balarka Sen

@0celo7 Dunno, I feel like there is some relation between index and curl.

0celo7

I'm sure one could craft a first course on calculus w/ Banach spaces

@heather 4 realz, Larson is really good

really, really good

heather

14:07

@Sanya, okay, thanks. Have a good day!

@0celo7, I just googled it again and found a free "AP" copy, so that's good, I'll look through that. Thanks!

0celo7

@ACuriousMind Stupid question: if $f:X\to Y$ is cont. at $x_0\in X$ and $X$ is a metric space, can I find a neighborhood of $x_0$ on which it is continuous?

I'm thinking yes

ACuriousMind

No

18

Q: Find a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous at precisely one point?

Is the function $f(x) = \lim_{k \to +\infty} \tan{kx}$ a right example? But I do not know whether this is a function at first. Is this a function? Could you give a correct real-valued function that is continuous at precisely one point? Thanks.

real-analysis general-topology

0celo7

oh right

that was on my topology homework :P

Xasel

Can somebody answer this:

1

Q: Approximating a polynomial to a piece-wise function

I was going through my introductory calculus book(for high-school student) by a Russian author(N.Psikunov) where I encountered a theorem without proof named: Weierstrass Approximation Theorem So how can we apply this theorem and apply it to piece wise functions?(any general approach?)(say a sim...

calculus real-analysis sequences-and-series complex-analysis polynomials

0celo7

well that's no good

back to the drawing board

Xasel

14:12

the question is dead since eternity

:(

0celo7

@Xasel learn real analysis.

@Xasel Stone-Weierstrass says that a function algebra that separates points is dense in the set of cont. functions wrt. the sup norm (on a compact metric space)

you can specialize this to $[0,1]$ with polynomials

Shing

14:30

@heather you don't know any calculus at all?

heather

@Shing, no, not really. I can't calculate anything, though I have some general idea of what things are, like the derivative is like rate of change so the derivative of position is velocity, and summations are used for series of large numbers to be added up or whatever.

Secret

Archimedean group

In abstract algebra, a branch of mathematics, an Archimedean group is a linearly ordered group for which the Archimedean property holds: every two positive group elements are bounded by integer multiples of each other. The set R of real numbers together with the operation of addition and the usual ordering relation between pairs of numbers is an Archimedean group. By a result of Otto Hölder, every Archimedean group is isomorphic to a subgroup of this group. The name "Archimedean" comes from Otto Stolz, who named the Archimedean property after its appearance in the works of Archimedes. == ...

Shing

@heather Not sure where you are from. maybe you should just start on high school calculus. the states of america has some quite good text books on high school calculus. There is really no need to rush in. after all, no one cares how fast you get there. they only care how good you are.

Secret

[Slereah] Find me a non archimedian group that is based on the hyperreals

David Z

@0celo7 that's not a very useful response, you know

heather

14:36

@Shing, I am from America, yes. But I don't exactly want to wait two or three years to learn calculus and the other stuff you can learn because you know calculus.

So I'm trying to teach myself now.

@Secret, aren't the hyperreals themselves a field that is non-archimedean?

Secret

It is, but I am interested in group theoric aspects of it

heather

@Secret, I don't suppose this is of any assistance?

Secret

I want a non archimedian group that is non archimedian not because there is a cyclic subgroup nor because the order relation is being defined as a vector addition to prevent the archimedian property to hold

Hmm... well I guess I might be able to extract a subgroup from that hyperreal field, let's see...

Shing

@heather high school calculus is easy. it took me only half year (probably just two months if you are at full speed). and that's probably enough for everything you will encounter in undergrad physics department. right after high school calculus, you can start just from Apostol or Spivak or Courant (or Hardy, but I have never read it). those are classic. Or study whatever physics you want (those usually need calculus)

0celo7

@DavidZ I disagree.

@Slereah is the resident non-standard analysis expert

Slereah

14:43

Am I

0celo7

correct

Secret

yup he is currently

David Z

@Shing True, though I wouldn't really suggest thinking of calculus as split into "high school calculus" and "college calculus". (Not that I think that's what you meant) I'd put it more like, learning enough of calculus to do well in undergrad physics probably isn't too hard.

Slereah

Hyperintegers with addition are non-archimedean

Secret

Meanwhile reading subgroup series to prepare myself to understand solvable groups, one of the most important type of groups in group theory

heather

14:45

@Shing, thank you - I found Apostol online for free. That'll be good to read through.

0celo7

dude, stop pirating books

Secret

(Not an expert) Is $(\mathbb{R}^*,+)$ a group? I can see 0 as the identity, but the infintesimals also act like zero for the real numbers in $\mathbb{R}^*$

heather

@0celo7, first, I'm just googling them and finding them online. Second, I'm a poor middle school student, and I can't pay for a bunch of hundred dollar textbooks.

0celo7

@heather I don't see how the second point is an excuse

you don't go around stealing cars either

Slereah

Infinitesimals in $^*\Bbb R$ have all the group properties with respect to addition

0celo7

14:47

and the first is exactly what pirating is...

Slereah

Not with respect to multiplication, tho, course

heather

@0celo7, sorry.

Secret

ok in that case $(\mathbb{R}^*,+)$ shoudl be a good example of a non archimedian group that is neither virtually cyclic nor defined via some vector addition like order relation (e.g. lexicogrphic order), thanks

David Z

@0celo7 well, not necessarily. Only if what you find is being distributed in violation of copyright law.

heather

@0celo7, besides, some of the books I read online were posted by the author online, like some of Gilbert Strang's textbooks, see here.

0celo7

14:49

ok

Physics Meta

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Q: editors of questions please do not add links which answer the question when editing it

It seems that here , while making the question more intelligible, the edit also answered it partially with a link, making the questioner not only negligent in previous research, but also stupid. As a result the person got two down votes. I had commented with a link , and got notified on his qu...

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