Mathematics - 2021-08-04 (page 1 of 2)

12:00 AM

I got this working for a 2 point emitter, but speakers are typically circular or rectangular (and eventually 3D) but for the sake of simplicity I'm trying to model a theoretical ribbon tweeter

I figure I'm going to need some kind of integral

I think I could come up with the integral representing the actual wave that the the speaker produces, however I want to graph the amplitude of that wave with respect to frequency

copper.hat

12:51 AM

this is more physics than mathematics...

10 Replies

1:10 AM

Avra

Hello, is this a typo or I am wrong please:

$\int_{1}^{n} \log{x} \,dx \lt \sum_{1}^{n} \log{x} \lt \int_{1}^{n} \log{x+1} \,dx$

I am asking about the left side above

copper.hat

1:37 AM

$\log$ is increasing so you have $\sum_{k=1}^n \log k \le \int_1^n \log x dx \le \sum_{k=1}^{n+1} \log k$.

Avra

2:03 AM

@copper.hat. In the book I have it's written as I wrote above, so probably a typo?

AMDG

Is there a name for the partial sum of the natural numbers like there is for the products of the natural numbers (factorial)?

Avra

@AMDG. Yeah partial sums

AMDG

That's it?

Avra

@AMDG. It's called partial sums

AMDG

Huh. That's underwhelming.

Avra

2:05 AM

I don't know if there are new names in real analysis, number theory

AMDG

~~I propose summurial~~

Am I correct in categorizing quotients in general as a form of unbounded knapsack problem, and if so, doesn't that mean that all knapsack problems can be reduced to n-dimensional quotients?

Also, as I was out, I was thinking more about the applications of division. Well, if you can easily divide anything into equal parts, then that means you can use division to compute nth roots, and if you can compute nth roots, then you can compute the circular functions.

I was watching a Bloons TD Battles video on YouTube. I noticed tower defense games here, and most of the things I find to be fun challenges, are some form of NP-hard problems, but that this in particular is also a form of knapsack problem with vertices and edges.

And, if I had to guess, the kinds of problems are the easier side of NP-hard because as game mechanics, whether it be in something like RTS, RPG, or tower defense, or even an FPS, it can't be so difficult a problem that it becomes intractable in a real-time recreational scenario.

fia presheaf

2:33 AM

does the fact a cross cap can have a color as a fourth dimension all along the edge where it is cut and reconnected reflect the fact they are connected: a to a' and b to b' while in real-life i can not connect a to a' and just to b??

because i can not connect them, i was just thinking if its another color then it reflects another time where they are connected correctly ..

I also dont get why the inner radius of a hole is gone in the quotient space..

im going to have din din now if someone could help me i would be so grateful lol !

Alex

@AMDG triangular numbers

@AMDG What do you mean by quotients in general? Quotient groups, quotient spaces, quotients of types of spaces under types of actions?

AMDG

Why are they called triangular? Is it because circles of equal radius stack neatly into a triangle as you add n + 1 rows?

If that seems oddly specific, it's because I glanced at this on Wikipedia: upload.wikimedia.org/wikipedia/commons/1/1c/…

Alex

@AMDG It's because its the side length of the triangles, much like square numbers

AMDG

@Alex Well, I'm referring to quotients of the form $\frac{f(a_0, a_1, ... a_n)}{g(b_0, b_1, ... b_m)}$

Alex

This theme continues en.wikipedia.org/wiki/…).

@AMDG Where f and g are maps from what n+1 and m+1 dimensional spaces, to what space?

AMDG

2:49 AM

Well, since f and g result in numbers in the broadest sense that I can imagine, $f : a_n \mapsto \mathbb{C}^n$ and $g : b_m \mapsto \mathbb{C}^m$. However, for most practical use-cases, we're working in $\mathbb{R}$.

So that also naturally includes things like quaternions (I hope).

If not, then, s/C/H

Alex

Don't you mean $f:\Bbb C^n\to \Bbb C$ and $g:\Bbb C^m\to \Bbb C\backslash\{0\}$ at least?

Also it's kinda weird for $f,g$ not to be taken values from a common space, unless there's something specific you have in mind

AMDG

Hm, that seems to be a more sensible mapping.

Ok, so then what I mean is $f : \mathbb{H}^n \to \mathbb{H}$ and $g : \mathbb{H}^m \to \mathbb{H}$

copper.hat

@Avra I made a mistake in transcription above: $\sum_{k=1}^n \log k \le \int_1^{n+1} \log x dx $. Note the upper limit on the integral.

AMDG

@Alex I see. I'm not well-versed in spaces. Could you please enumerate some for me to pick from?

Alex

@AMDG I mean it would be more reasonable if $n=m$ for example

AMDG

2:58 AM

Well they all map to distinct but not necessarily unique values even if $n \neq m$, no?

Alex

@AMDG I don't know what you mean by that. I just mean that normally you would want to consider $f(x_0,\dots, x_n)/g(x_0,\dots,x_n):\Bbb C^n\to \Bbb C$ where $g:\Bbb C^n\to \Bbb C-\{0\}$

But sure, you can consider $\Bbb C^n\times \Bbb C^m \to \Bbb C$ if you want

AMDG

What I mean is that for any arbitrary number of inputs, there is only one output for either f or g.

Alex

For a fixed f,g, there are only n and m inputs, right (respectively)?

AMDG

Yes.

Alex

So why are you saying 'an arbitrary number of inputs'

AMDG

3:01 AM

Arbitrary number of inputs denoted by the variables n and m for f and g respectively is what I mean.

Wolgwang

What are the solutions of $4m^2+1-4m=4m^2+4$ ? $m=\frac{-3}{4},\infty$ ? O_o

Alex

@AMDG But once you fix an f and g those numbers are not arbitrary. Also n,m are not variables? They are natural numbers

AMDG

@Alex of course

Pain. I wish I had learned more terminology in school relevant to the applied mathematics that I now have to learn on my own.

@Alex A simple reason for why I'd want to generalize this (and learn how to properly describe that) I can explain in the simplest case for defining inverses of functions of n-dimensions to their respective counter-parts as a piecewise (n+1)-dimensional function where the extra variable maps to more unique values such as for defining inverses of complex functions by mapping every branch.

As a basic example, you could define inverses of the circular functions with values of cos, sin, or tan and the modulus of the angles to determine what the exact angle was if the modulus value is available. How that would be helpful? Not sure, though it would obviously be useful forensically to determine which of the infinitely many angles maps to a given vector that was used specifically.

Wolgwang

3:49 AM

@Wolgwang $m$ is the slope of a line...

AMDG

@Wolgwang I'll give ya a hint: that whole thing can be rearranged such that it can be solved with the quadratic formula.

Although iirc you probably don't need that to solve for the value of m. You can just do it algebraically.

In fact, you don't whatsoever because this is actually just a line in disguise, not a parabola.

Oof, my laziness in computation really doesn't help me lol

Wolgwang

Find all the common tangent to the circles $x^2+y^2-2x-6y+9=0$ and $x^2+y^2+6x-2y+1=0 .$

I am solving this.

AMDG

Ah.

Wolgwang

I have found the equation of direct tangents.
But while finding the equation of indirect tangent.
I have to solve $4m^2+1-4m=4m^2+4$

where m is the slope of the equation.
This should give two values of slope as there are two indirect tangents.
But...
Equation of one tangent is x=0 whose slope is $\infty$ (not defined)

AMDG

Subtract $4m^2$ from both sides to get $1 - 4m = 4$ which should be trivial to solve.

Wolgwang

3:56 AM

But it gives only one value of $m$ that is $\frac{-3}{4}$

How to find the fourth common tangent's equation? (Which is $x=0$)

AMDG

You haven't described the relationship of this equation to the circles you have listed. You also said you need to find a common tangent, not tangents.

Wolgwang

@AMDG I found the intersection of the common tangents ($(0,\frac52)$)

And then used the formulae $\text{Distance between a line and a point}=\dfrac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}$

And put the distance equal to the radius of one circle.

AMDG

I have never seen a formula that gives a distance between a line and a point, and that isn't exactly well-defined as-is. A line is a set of points. A point is a line in the degenerate case.

You must surely mean the distance between a point on a line and a point elsewhere on the plane.

And if so, then it's easier to just remember that it's $\sqrt{\Delta x^2 + \Delta y^2}$.

Wolgwang

Perpendicular distance aka shortest distance.

AMDG

$\text{shortest distance }\neq\text{ perpendicular distance}$, whatever perpendicular distance means.

I assume you mean to say the length of a perpendicular to a line that passes through a point.

Or rather the length of the perpendicular line segment that begins at some point on a line and ends at a given point.

Hm, I find that quite interesting how one gets these (regular) polygons from arrangements of uniform objects on a plane with rows or columns increasing or decreasing by one in number.

I bet there's a nice root-finding algorithm tied to that.

Wolgwang

4:16 AM

@AMDG Yes...

AMDG

Yeah, simpler is better. Just remember that you can define any right triangle on the xy-plane as two points defining the hypotenuse of length $c$ and the points make up two of the three vertices of the right triangle. Then the legs, respectively, are of lengths $\Delta x$ and $\Delta y$ such that $\sqrt{\Delta x^2 + \Delta y^2} = c$ by the pythagorean theorem.

And of course $\Delta x = |x_2 - x_1|$, $\Delta y = |y_2 - y_1|$.

For points $(x_1, y_1)$ and $(x_2, y_2)$.

Omni man

6:08 AM

Most math students these days are on lsd

to excel on research

that's why they come up with so $#&* up stuff

hell yeah

Learning

7:20 AM

0

Q: Question on Markov-Chain GATE (ST)-$2021$

Question: Let $\{X_n:n \ge0 \}$ be a time- homogeneous discrete time Markov-chain with either finite or countable state space $S$. Then $1.$ there is at least one recurrent state $2.$ if there is an absorbing state, then there exists at least one stationary distribution $3.$ if all states are p...

can someone help me with this?

robjohn

@AMDG what Wolgwang says is correct. The distance between two geometric shapes is the infimum of the distances between any two points (which is well-defined).

@AMDG how can you say two things are not equal, when you don't seem to know what one of the things means?

If a shape has a smooth boundary, the distance from some point to that shape will be along a line segment that is perpendicular to that shape. That is where the term "perpendicular distance" originates.

Avra

12:39 PM

@copper.hat. Okay, so it should be also $\int_{1}^{n-1} \log{x}\lt \sum_{1}^{n-1} \log{x}$? But, the integral takes infinite intervals, so how the sum is larger than the integral anyway over same interval please? That the part that confuses me.

robjohn

12:50 PM

@Avra for any monotonically increasing $f(x)$, $\int_{a-1}^bf(x)\,\mathrm{d}x\le\sum\limits_{k=a}^bf(k)\le\int_a^{b+1}f(x)\,\mathrm{d}x$

therefore, $\int_1^{n-1}\log(x)\,\mathrm{d}x\le\underbrace{\sum\limits_{k=2}^{n-1}\log(k)=\sum\limits_{k=1}^{n-1}\log(k)}_{\log(1)=0}\le\int_1^n\log(x)\,\mathrm{d}x$

AMDG

1:07 PM

@robjohn My mistake.

Why do I even try...

robjohn

Trying is good. Being able to improve and move on is good, too.

AMDG

Yes, well, how exactly does someone fix the kind of problems that I have? I don't really have the time to go through all of the lower levels of mathematics up to the point that I need because I must work; yet my ignorance concerning these things as well as formal terminology is a hindrance to communicating my ideas to others save by handing someone my code and having them decipher it.

For all practical intents and purposes, the only formal, common interface or medium of expression that I have for both realizing and describing my ideas is code.

And, you know, it's one thing to make a mistake in private. It is another to pass ignorance to another in your attempts to teach them.

robjohn

It would be nice to go through all of "the lower levels of mathematics", but probably not necessary. At least knowing the areas of mathematics that apply to what you are trying to do would be useful.

AMDG

Heh. Well, let's see. Calculus would be incredibly useful, but as most things in this area and others in mathematics, my knowledge has many holes, so I might know one topic but not another in the same area since I've mostly only looked for and directly went to the things I need to know without necessarily looing at the knowledge surrounding that thing.

Complex analysis seems to be incredibly useful and necessary with regards to cheaply performing computations on vectors.

(Especially considering our hardware is practically made for vector and matrix operations).

And then whatever I need to describe my metacompiler because I can guarantee you, I didn't get my design from a book; I just made it up and know it works because of its principles and the things that have been made manifest to me in all things in general by the grace of God, but even such a simple system which I have learned, I cannot express to others so easily as a "system of origins" which also describes my metacompiler.

I mean I can describe it in simple way because of how many times I've learned how to describe parts of it through trial and error in communication, but that is unsatisfactory.

It's primary inputs and outputs are decision trees, but obviously, it goes into more detail, and I don't believe it is relevant to this conversation directly.

robjohn

1:24 PM

The more one knows, the more one can draw on in different contexts, so the chances of knowing something applicable is better the more you know. Knowing the right thing to solve the problem on which you're working is often a crap shoot. Einstein benefited greatly from the differential geometry that was being developed at the time.

AMDG

Yeah, I've noticed that.

The question for now, at least, is how am I supposed to effectively "catch up" to these areas of mathematics that I need? I've never really been a fan of khan academy's way of learning, and in part because it doesn't let me learn fast enough. The classroom was never a fun and engaging learning experience. So I don't know what to do and where to go. I'm obviously not going to waste thousands on university.

Peter John

How can I show non existence of the pair $(p,n)$ where $p$ is a prime number and $n\geq 2$ s.t. $p^n-n^p =2$?

Avra

@robjohn. Thank you, then there is a typo in the book as they did not mention it as your did

Peter John

I wonder if taking $\mod p$ helps

Avra

@robjohn. They just wrote it $\int_{1}^{n-1} \log{x}\lt \sum_{1}^{n-1} \log{x}$, which is wrong based on montonically increasing property you mentioned?

@robjohn. If you can see, they did not change the boundaries of sum nor integral in the equation above

robjohn

1:33 PM

@Avra unless you notice that $\log(1)=0$

Avra

@robjohn. Oh!!

robjohn

it is a true statement, but possibly confusing

Avra

@robjohn. I got it, I am just very quick judger

@robjohn. Thank you!

robjohn

@Avra that was the reason for the underbrace in my equation

Avra

@robjohn. My mistake. You already explained it 100%

robjohn

1:38 PM

@PeterJohn what is $n^p\bmod p$?

Avra

One more question please, for $n + 2^{k+1} - 1$, is not the time complexity supposed to be $O(2^n)$ (exponential), why the time in this case of amortized analysis is just $O(n)$? given that $k=\log{n}$ please?

Peter John

@robjohn n

waaaait

robjohn

@PeterJohn and what is $p^n\bmod p$?

Peter John

@robjohn 0

-n = 2 mod p

robjohn

so you've reduced your search, at least

Peter John

1:41 PM

$p\mid n+2$

robjohn

If $n+2$ is greater than or equal to $2p$, it should be easy to show that the equation is false.

Avra

@robjohn. Another quick judge. I forgot that since $k= \log{n}$ we can subsitute that in the equation $n+ 2^{k+1} - 1 = n + 2n -1 = 3n \in O(n)$

robjohn

@Avra I am not sure I understand what you are asking.

Avra

@robjohn. This is about growth time.

@robjohn. Or growth complexity (running time comokexuty)

robjohn

I guess it all boils down to the relation between $k$ and $n$, which I don't know.

Avra

1:45 PM

@robjohn. I forgot that $k = \log{n}$

It's given in the question

robjohn

Then that handles it?

Avra

when we substitute $k$, we can get $O(n)$ growth rate

Peter John

@robjohn Does $n\geq 2(p-1)$ show something?

Avra

@PeterJohn. I guess $p<n$, otherwise we will get negative numbers

@PeterJohn. So, we should constraint ourselves with possible pairs where $p<n$

@PeterJohn. But we have a case though $2^3 - 3^2 = -1$

So I don't think it's valid to assume that

@PeterJohn. How about prime factorization theorem? Can you use it to represent $n$ and then go from there?

Peter John

I think that will make the problem more complicated

Using the fact that $p\mid 2+n$ will be helpful

Avra

1:57 PM

@PeterJohn. Why $2+n$ and not just $p | n$, since $p$ is a prime number?

@PeterJohn. I tried to facrize it using fact that $n = dp+r$, where $r$ is remainder and it gives the same result as above, $p^p (p^d - d^p)$

it just complicates result

Peter John

what is $p^p(p^d-d^p)$?

Avra

I just substituted $n = dp +r$, where $d$ is the divisor and $r$ is the remainder or $n \mod{p}$

Peter John

2:19 PM

@robjohn Is your word some kind of obvious thing?

Avra

2:36 PM

@PeterJohn. How about l'HOSPITAL RULE? Why not try to change terms and then apply the rule to see if you can get something from it?

Peter John

2:48 PM

Catalan's conjecture

Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu. The integers 23 and 32 are two powers of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the only case of two consecutive powers. That is to say, that == History == The history of the problem dates back at least to Gersonides, who proved a special case of the conjecture in 1343 where (x, y) was restricted to be (2, 3) or (3, 2). The first significant...

Avra

3:05 PM

@PeterJohn. Interesting.

Easy proof: vixra.org/pdf/1906.0374v1.pdf

robjohn

@PeterJohn We want to show that $p^n-n^p=2$ is impossible, so if $n=kp-2$, consider $p^{kp-2}-(kp-2)^p$

Dividing both sides by $(kp)^p$, we have $\frac{p^{(k-1)p}}{k^pp^2}-\left(1-\frac2{kp}\right)^p$.

If $k=1$, this is $\frac1{p^2}-\left(1-\frac2{p}\right)^p$ which is less than $0$

Avra

@robjohn. Wow! Amazing, but you you choose $n=kp-2$ please?

Peter John

@robjohn What if $k\neq 1$?

robjohn

3:20 PM

If $k\ge2$, then $\left(\frac{p^{k-1}}{kp^{2/p}}\right)^p-\left(1-\frac2{kp}\right)^p$ is greater than $0$ and multiplied by $(kp)^p$ is much bigger than $2$

@Avra remember that $n\equiv-2\pmod{p}$

Peter John

@robjohn I don't get it. For $k\geq 2$ you just divide and multiply $(kp)^p$

SAJW

is abc a weaker password than ab1?

robjohn

@PeterJohn that gives you the same thing. we want to show it is greater than $2$

Avra

@SAJW. I guess, because c has 26 possibilities while 1 has only 9

Peter John

@robjohn Yes but you first divide both sides by $(kp)^p$ and multiply by $(kp)^p$ and suddenly bigger than $2$

Avra

3:27 PM

So the probalbility of gettinc abc is much lower than ab1

leslie townes

let's not miss the forest for the trees, those are both astonishingly bad passwords.

SAJW

on the other hand 1 means the characters can be a-z or 1-9, that's what I think

leslie townes

longer is better. i don't know why so many sites go in for special characters.

Avra

@leslietownes. Exactly!

SAJW

on the other hand 1 means the digits can be a-z or 1-9, that's what I think

leslie townes

3:28 PM

i also hate how they tend not to tell you the rules until after you submit a proposal for a password.

SAJW

I think passphrases are superior to anything, given a big enough dictionary, but I could be wrong. at least they are easy to remember

Avra

@SAJW. 1 is a digit not charachter

robjohn

@PeterJohn not suddenly, look at what we have when $k\ge2$: $\frac{p^{k-1}}{kp^{2/p}}$ is big, but $\left(1-\frac2{kp}\right)^p$ is less than $1$

Peter John

How can I color the text? Anyone know the latex code?

robjohn

What did you want to $\color{#C00}{\text{color}}$?

Avra

3:37 PM

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}

\usepackage{xcolor}

\begin{document}

This example shows different examples on how to use the \texttt{xcolor} package
to change the colour of elements in \LaTeX.

\begin{itemize}
\color{blue}
\item First item
\item Second item
\end{itemize}

\noindent
{\color{red} \rule{\linewidth}{0.5mm} }

\end{document}

make sure to include {xcolor} package

What is the type of document you have?

Check this: overleaf.com/learn/latex/Using_colours_in_LaTeX

robjohn

@PeterJohn were you talking about MathJax for this site, or for documents, in which case, asking on the LaTeX site would be better.

Peter John

@robjohn MathJax for this site. I'm going to use it for asking question.

robjohn

@PeterJohn do you see how I added the color there?

Peter John

Oh I didn't see that. Thanks

robjohn

you can specify it as \color{#C00}{...} or \color{#CC0000}{...}

three nybbles or 3 bytes (giving more control)

you can also use pre-canned \color{red}{...}, though I forget all the color names

black, blue, brown, cyan, darkgray, gray, green, lightgray, lime, magenta, olive, orange, pink, purple, red, teal, violet, white, yellow

$\color{lime}{\text{lime}}\color{green}{\text{green}}$

leslie townes

3:51 PM

haha that's CGA all over again

and i'm here for it

soupless

Similar to quadratics and cubics where there is an "axis of symmetry" for a particular value of $x$, is there one for a general value of a general polynomial where the degree is greater than 3?

leslie townes

i don't think so. that seems peculiar to degree two.

now some algebraist will hop in and correct me.

soupless

For quadratics, the axis of symmetry is the vertex, and for cubics, the axis of symmetry is the inflection point?

(Polynomials of the form $P(x)$, by the way)

leslie townes

i've never heard of cubics having an axis of symmetry although if you want to define it that way it works for me.

Peter John

@robjohn Is it ok to just say $\frac{p^{k-1}}{kp^{2/p}}$ is big ? It seems like hand-waving claim

Jacob P.J

4:04 PM

Does anyone know why a fourier integral is integrated from 0 to infinity and not from -infinity to infinity??

leslie townes

it really depends on context. you do see integrals over the whole line, in fact, i think that's more common.

Jacob P.J

nope in the formula it's 0 to infinity

soupless

@leslietownes I mean, if $f_3(x) = a_3x^3 + a_2x^2 + a_1x + a_0$, then the inflection point must be $x = -\frac{a_2}{3a_3}$. Can we consider $x = -\frac{a_2}{3a_3}$ as the axis of symmetry of $f_3(x)$?

leslie townes

not e.g. here en.wikipedia.org/wiki/Fourier_transform but i can imagine applications where it would be.

0 to infinity is a common default for the laplace transform.

Jacob P.J

that's the fourier transform right? But for fourier integrals the limit is different

leslie townes

4:07 PM

soupless, you are welcome to define that although cubics tend not to actually be symmetric in the usual sense about that point.

Jacob P.J

@leslietownes I do understand that, because the integral is wrt time. But in fourier integral the integral the integration is wrt frequency

@leslietownes I'm sorry im kinda new to transforms , and hence i am having difficulty understanding simple stuff

leslie townes

for me fourier stuff is just a mystical adventure where you fall through it and come out on the other side with new results.

Jacob P.J

@leslietownes haha, in class the professor just wants us to mug up the formulas but I wanted to understand the story behind the formulas

leslie townes

i'm paraphrasing bruno de finetti, who worked with one of my professors. he said something like, nobody understands the fourier transform, it's this cave you wander into and then when you get to the other side maybe you've got something new.

Jacob P.J

@leslietownes Although confusion, must admit the guy who came up with it must have big brains. In this age with all the technology and materials available I find it hard to understand the concept!

confusing*

leslie townes

4:11 PM

somewhere around here i have the complete works of fourier in translation. he was very, very sharp.

limited by his timing and circumstances, but very sharp.

soupless

@leslietownes Sorry, I didn't understand that. Can you explain it more, please?

user178758

::insert von Neumann quote here::

leslie townes

that exact quote. thank you.

user178758

(-:

leslie townes

soupless for a quadratic you can perform a reflection about the axis of symmetry and it maps the curve onto itself. i don't think it does this in the cubic case.

Jacob P.J

4:13 PM

@leslietownes I'm still not clear about the limits though :(

leslie townes

in fact, it can't do this. you've got one end of the curve running off to +infty and the other to -infty. can't swap those.

soupless

@leslietownes Oh, sorry. I forgot to redefine symmetry as functions that have a similar feature to odd or even functions.

leslie townes

someone's just sent around a poll about whether we suggest to the management an 'app-based therapy product' where we can talk to therapists on an app. how about making the job less s--tty.

Jacob P.J

8

Q: Difference between Fourier integral and Fourier transform

What is the difference between Fourier integral and Fourier transform? I know that for Fourier integral, the function must satisfy $\int_{-\infty}^\infty |f(t)| dt < \infty$, but what if I have a function that satisfies this condition: what does it mean to calculate Fourier transform and Fo...

fourier-analysis

leslie townes

one thing that's surprised me about the 21st century is how completely dystopian it is.

user178758

4:18 PM

Yeah, the third decade raised the bar on dystopian thinking.

robjohn

4:35 PM

@PeterJohn well, compute how big it is.

user178758

\o ____compute this distance___o/

Peter John

@robjohn I mean proof.

@robjohn btw the case $k = 1$, it's not less than 0 if p = 2

robjohn

yes. well you always have $3^1-1^3=2$

Peter John

@robjohn p = 3 also does not hold

robjohn

we know that $p=2$ is impossible because then $n$ must be even and then $p^n-n^p$ is a multiple of $4$

so $p\ge3$.

you need to compute how big these things are to see when the quantity is greater than $1$.

for large $p$, it is big, so you need to see when it is big

soupless

4:50 PM

(Leaving the room now, bye everyone!)

user178758

cya

Peter John

@robjohn what do you mean $p\ge 3$ I'm talking about the case $k = 1$.

robjohn

For $k=1$, we have $p^n-n^p=p^p\left(\frac1{p^2}-\left(1-\frac2p\right)^p\right)$ and the quantity in the parens is negative for $p\gt4$

so the only solution for $k=1$ is $3^1-1^3=2$

For $k\ge2$, there are none

Rover

5:09 PM

Peter John

@robjohn Now I can see for the case $k = 1$ but I don't have any idea to show the differnece in the case $k\geq 2$. Actually computation for check works I think but it's not a proof

shintuku

@Rover switch A for an actual 3x3 matrix and see what you get

Rover

@shintuku I get 3 equation of plane with unknown coefficient ,which can take values 0, 1.

shintuku

hm perhaps someone else might know

robjohn

5:24 PM

As I said, you need to check for where it is bigger, and handle the other cases individually. There will be only a small number of those cases.

Ted Shifrin

@Rover So what is your answer to the question?

What do you know about solutions of $Ax=b$?

Same goes for @shin.

Rover

@TedShifrin Hm.., they are point of intersection of 3 planes.

Ted Shifrin

But don’t you know some general theory… relating solutions of $Ax=b$ to solutions of $Ax=0$?

Peter John

@robjohn I think I proved it. Let me write it down

Rover

5:40 PM

@TedShifrin Nope

Peter John

Nono I didn't

Ted Shifrin

Can three planes intersect in $0$ points? $1$ point? $2$ points? What are the options?

leslie townes

5:59 PM

eww, geometry.

hope you are doing well, ted. my appetite is almost back to 100%.

Ted Shifrin

And your obnoxiousness is in full force again?

leslie townes

yes but i've outsourced that to my daughter.

Rover

@TedShifrin planes can intersect in 0 points 1 point and infinite points

Ted Shifrin

That should spare us, then.

leslie townes

this morning's innovation was insisting that her shoes, which she had just put on, were on the wrong feet, and swapping them. they were on the right feet.

Ted Shifrin

6:04 PM

Good, Rover. That’s exactly right. Now reread your question.

She can remove her feet and swap them.

Rover

Oh... it cannot have 2 solutions, so answer will be option 1, there will be no such matrix A

Ted Shifrin

Yup. Just need to think !

Rover

Yeah, thanks .

Ted Shifrin

You’re welcome.

robjohn

6:34 PM

@PeterJohn I have simplified the $k\ge2$ case so that we can show $n\gt p\ge3\implies p^n-n^p\gt7$ I have dropped all mention of $n=kp-2$. I now handle $n=p-2$ and $n\gt p$

Charlie

Are maps between complex and real vector spaces which are defined like $T(\alpha x_1+\beta x_2)=|\alpha|T(x_1)+|\beta|T(x_2)$ of any particular interest? Or maybe $F(av_1+bv_2)=e^{ia}F(v_1)+e^{ib}F(v_2)$? just curious, don't have to entertain me

Ted Shifrin

Every linear map has the first form. The second is not linear.

Mad Spaces

Hello ! A dirty physicist here which needs help ! my professor introduced the concept of Fourier serieses and Fourier integrals and then he connected it to Diracs delta distubution and derived the most ununderstandable thing there is, he did of course like a physicst, which means with neglect of all mathematical rigour, and since that how i understand, i am looking right now for a book about it

Can you please give me suggestion or atleast tell me in which subject i will find topics about this ?

leslie townes

folland has a good book in harmonic analysis on R^n.

Ted Shifrin

To make the delta function rigorous mathematics is very advanced.

leslie townes

6:38 PM

fell and doran have a book on more abstract groups, that might not be what you're going for.

Ted Shifrin

Better to stay physics-y.

Mad Spaces

Well , what is the best option for me to try atleast get a good understanding of all the maths he did in a bad way? i dont want to memorise equations

i dont understand his arguments at all, what should i read to help me?

Charlie

Might want to start with mathematical physics textbooks, physics tends to use fairly advanced maths concepts in a way that makes them accessible for physicists

Ted Shifrin

For a compromise, look at something like Strang’s Intro to Applied Mathematics.

Charlie

I mean you could always ask on the main physics or math site, there are plenty of very good mathematicians on the physics.se site

@TedShifrin Ok ty, I didn't expect them to be linear necessarily just wondered if it was a thing

Mad Spaces

6:41 PM

leslie townes i have found this book in my library modern techniques and their applications / Gerald B. Folland

Do you guys suggest this?

Ted Shifrin

Might require grad math background. You need to see the prereqs for the book.

leslie townes

it could be too mathy.

Ted Shifrin

Strang is a good suggestion for your interests and lots of good applied stuff in it. I taught a year-long undergrad course out of that.

Mad Spaces

i did some basic maths

Calculus and linear algebra

user178758

Ask your prof.

Ted Shifrin

6:43 PM

Graduate real analysis is way beyond calc. Look at Strang.

Peter John

@robjohn wow you proved it

@robjohn You didn't use the fact $n = kp -2$ for the case $n>p$

robjohn

@PeterJohn no, I mentioned that above. I had thought that the $n\gt p$ case was much simpler than the $n\lt p$ case, and it was.

15 mins ago, by robjohn

@PeterJohn I have simplified the $k\ge2$ case so that we can show $n\gt p\ge3\implies p^n-n^p\gt7$ I have dropped all mention of $n=kp-2$. I now handle $n=p-2$ and $n\gt p$

Peter John

@robjohn Thanks. Really appreciate.

lone student

7:14 PM

@robjohn Hello. Are you god ?

Yai0Phah

7:25 PM

@MadSpaces Stein and Shakarchi's Functional Analysis, Chapter 3, Generalized functions, is a good introduction and quite accessible to undergraduates (I think that a general sophomore major in math has no difficulty to digest).

Ted Shifrin

@Yai0Phah You’re saying that calculus and linear algebra — and no analysis whatever — is the prereq? You’re just wrong.

Yai0Phah

If I remember correctly, they did not mention topological spaces to deal with these objects.

Avra

Hello, is it allowed to take $n^c$ out where $c$ is constant from growth function $n^c \times O(\frac{\log{a}}{n^c})$, so that we have $ \frac{n^c }{n^c } \times O(\log{a}) = O(\log{a})$

Yai0Phah

Maybe I am wrong.

robjohn

@lonestudent not that I know of, and if I were, I would think that I would know.

shintuku

7:30 PM

i challenge you!

there, no one can say people may enter this room unchallenged

Ted Shifrin

Define “you.”

shintuku

the protagonist, of course

Thorgott

Folland's book is lovely, but it's on the advanced side

Ted Shifrin

Yeah, math majors and grad students tend to lose perspective when someone needs a less sophisticated resource.

@shin: Not me, then. I'm antagonist, through and through.

Yai0Phah

I have just glimpsed at the available pages on Google Books. Seemingly my impression is not quite incorrect - the treatment in Stein-Shakarchi is elementary.

Ted Shifrin

7:41 PM

Define elementary. Accessible to someone knowing just calculus and computational linear algebra?

I am familiar with Stein/Stakarchi's books, although not that particular one. They all are advanced undergraduate level, assuming real analysis.

shintuku

no real undergraduate math books if someone hasn't done real analysis

Yai0Phah

No, but somebody who complained about non-rigor of physicist's approach is not those who just know calculus and computational linear algebra.

Ted Shifrin

Well, you're wrong. This particular person has that background.

Yai0Phah

"Real analysis" - I don't know whether it is somehow implicit, but seemingly the knowledge of Lebesgue integral is not needed.

schn

@robjohn , why is it so hard to determine the uniform convergence of the estimate?

Ted Shifrin

7:42 PM

I didn't say Lebesgue integral, did I?

Real analysis is baby Rudin or the equivalent.

shintuku

maybe even a more introductory real analysis book would be a good place to start if someone just has calc and linear algebra

Ted Shifrin

The asker, in this case, is trying to make sense of his physics lectures in real time. A good applied math book is a better suggestion than the notion of learning basic analysis to understand the delta function as a distribution.

Anyhow, I'm tired of this.

Yai0Phah

Well, the volume "Real analysis" of Stein-Shakarchi is about Lebesgue integral.

Ted Shifrin

I'm not talking about the third book in their four-book series.

lone student

@robjohn Hope you checked the link. The answers given can be criticized technically, but philosophy should not be made about the personality.

shintuku

7:46 PM

is it good btw? i need to do probability distribution integrals soon and i'm looking for a measure theory reference

Ted Shifrin

It is written as an advanced undergraduate text for Princeton undergraduates. That makes it a graduate-level text.

Avra

@shintuku. Measure theory is used a lot in probability theory, especially statistical inference

Yai0Phah

Then I don't know any book which is at Baby Rudin's level, talking about distributions in an elementary fashion.

Ted Shifrin

I'm talking about below baby Rudin. No notion of metric spaces or limit proofs.

No careful definition of the Riemann integral.

Sorta like telling someone who's had basic linear algebra and multivariable calculus to read Volumes 1 and 2 of Spivak's Differential Geometry to learn "basic" differential geometry. Doesn't work at all.

Yai0Phah

Well, as for differential geometry

Seemingly Needham's new book is somehow basic and intuitive.

Ted Shifrin

7:58 PM

I haven't seen it. I'm happy with my own "elementary" text.

Thorgott

first two volumes of Spivak aren't too bad :P

Yai0Phah

I was a TA of a math course for engineers (sophomores). It was very difficult to let them understand the uniform convergence.

robjohn

@schn that depends on the circumstances of the question, which is why I encourage the examination of particular instances where little-o is used.

@lonestudent I did, but I don't want to get into an argument where people's minds seem to be made up already. It is frustrating and usually a waste of time.

The question involved was short, but it was a question posed by someone who was confused about a point, not a regurgitated PSQ.

Since it had been brought to my attention, I thought that I would provide a more explanatory answer.

Yai0Phah

8:17 PM

@shintuku I think that you could directly go with textbooks on probability. Most of them cover the necessary measure theory they need.

I have no background in probability, so my opinions are not that reliable. For example, for probability, I don't think that you need much about Radon measures.

Balarka Sen

@MadSpaces What course is this

Hall's quantum mechanics book is good, nevertheless. It's meant for math-oriented physicists, and genuinely contains good math.

Possibly too advanced for you, on second thought.

shintuku

Yai0 swell, thanks for the tip

Yai0Phah

Personally, I self-learnt the probability from Kai-Lai Chung's A Course in Probability Theory.

Slate

@shintuku If you really need a measure theory reference specifically, I've found Axler's Measure, Integration, and Real Analysis to be clear to read. (But otherwise I'd defer to Yai0Phah.)

Yai0Phah

I heard that Durrett's book is fashionable but I did not read that book seriously.

shintuku

8:34 PM

noted, thank you!

Yai0Phah

You could also look at mathoverflow.net/a/52764 and the following comments if you are really interested in measure theory.

pritchard

Hi, I am wondering: If we are given $B$ and need to show that $C\Leftrightarrow D$; is that equivalent to assuming $C$ and showing $B\Leftrightarrow D$? Thank you!

robjohn

8:50 PM

@lonestudent Peter John came to chat and asked if this answer was valid. It was, but to explain why I could either post here, or answer the question. I figured that answering the question would be useful to more people. I guess that the question may disappear entirely, now that I've brought it attention.

lone student

@robjohn Technically speaking, I'm not equipped to speak. My criticism is to philosophize about personality. "God" parable and etc. There's a lot of ranting in that room. They deleted many of my/others' non PSQ questions. Even the newly elected moderator found it unfair. When I make a criticism, they say, "We will flag you". And suddenly my best questions are downvoted. Too many comments are full of arrogance. You did not read, unfortunately..

robjohn

@lonestudent I did read. I see that I have too many "oops".

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$Mathematics$ Mathematics