Mathematics - 2018-06-24 (page 3 of 3)

The Great Duck

6:00 PM

@Secret exactly

Educ

@LeakyNun Yes continuous please I'm interested of what you saying

The Great Duck

basically what I found was that left and right derivatives were not equivalent to the differential fork definition for implied derivative

MatheinBoulomenos

@Liad so you know that $\tau(x)=x$ for all $\tau \in \operatorname{Gal}(L/K)$, or to put it differently, $x$ is contained in the fixed field under $\operatorname{Gal}(L,K)$

Leaky Nun

@Educ so if you have N, the pairs would be NxN, and then the equivalence relation would relate (a,b) to (c,d) iff a+d=b+c

The Great Duck

hence the paper essentially doesnt contradict itself per se, but flat out fails to accomplish it's goal

Leaky Nun

6:00 PM

then it's routine to check that this is an equivalence relation

Educ

yes

The Great Duck

the differential fork definition is the superior one

Leaky Nun

and then we call (NxN)/~ as Z

Educ

yes

The Great Duck

it makes more sense and has a more direct connection

to piecewise constants

Liad

6:01 PM

@MatheinBoulomenos got it, the fixed field of $Gal(L/k)$ is $K$ ..

Leaky Nun

@Educ graphically, when we relate (1,3) with (2,4) and with (3,5) and with (4,6) etc, we are relating points satisfying x-y = -2, i.e. y=x+2, where "-2" is from the outside perspective (from the inside we're still building Z)

Nicholas Roberts

@Liad by definition, an element in Gal(K/F) is an automorphism of K that when restricted to F (which is a subfield of K) acts as the identity

so for f in F, sigma(f) = f

Liad

that's the inclusion i already had O_o @NicholasRoberts

Leaky Nun

@Educ that's all I have to say

Educ

@LeakyNun that's why we define integer as (NxN)/~ as Z

Leaky Nun

6:03 PM

right

The Great Duck

@LeakyNun have you worked with differential algebrae before?

Educ

@LeakyNun I really apperciated your explanation

Leaky Nun

@TheGreatDuck no; @Educ thanks for your appreciation

Liad

@MatheinBoulomenos thanks!

Educ

@LeakyNun Thank you so much

@LeakyNun one things please before I give this definition (NxN)/~ as Z should I say "graphically, when we relate (1,3) with (2,4) and with (3,5) and with (4,6) etc, we are relating points satisfying x-y = -2, i.e. y=x+2, where "-2" is from the outside perspective (from the inside we're still building Z)"

The Great Duck

6:04 PM

:-/

MatheinBoulomenos

@Liad np

Leaky Nun

@Educ that's what I would call the motivation of such a definition

motivation means intuition

The Great Duck

is it against the rules to make a question asking for suggestions on differential algebra readings?

Educ

@LeakyNun yes I know that's what I need so I will say graphically ....

Leaky Nun

sure

but it's not a part of the formal definition

Educ

6:05 PM

@LeakyNun and use geogebra to draw your example

Leaky Nun

right

Educ

@LeakyNun what is teh formal definition do you haev any book or article that give intuition before the formal definition please

Leaky Nun

the formal definition is (NxN)/~

the graphical part is not the formal definition

I think a lot of books give intuition alongside formal definition

Educ

@LeakyNun yes it's just the motivation to definition, could you name one please (book)

Leaky Nun

name a book? I don't know, just find any book

Educ

6:07 PM

@LeakyNun real analysis or in algebra textbook ?

The Great Duck

@Educ what level of real analysis?

introductory

i know a good one for that

Leaky Nun

math.hawaii.edu/~pavel/aluffi_321/NZ.pdf @Educ

The Great Duck

same one I used when I took it last spring

Educ

@TheGreatDuck give me the name please whatever the level

@LeakyNun I will check right away

@TheGreatDuck name it please

The Great Duck

im getting there

slow down

geez

amazon.com/…

@Educ

Educ

6:12 PM

@LeakyNun Unfourtunelly there is no draw

@TheGreatDuck Thank you I will check it now

The Great Duck

ok

Leaky Nun

@Educ just search more then. I searched construction of the integers

Educ

@LeakyNun I already did but with no luck now I will check the book that @TheGreatDuck gave me

The Great Duck

ok

i find it hilarious that the noise it makes when someone @'s you is the sound of a bow string.

Educ

hahaha

@TheGreatDuck bow string bow string hahah

The Great Duck

6:15 PM

...

mutes tab

Rudi_Birnbaum

Hi all!

In a vector cross product $\vec{a}x\vec{b}=\vec{c}$, when I know $\vec{c}$ and $\vec{b}$, how do I get the possible $\vec{a}$'s?

The Great Duck

(removed)

Educ

@TheGreatDuck there is no set of Z there is : N, Q and R

The Great Duck

um what

Educ

@TheGreatDuck in the book you juste gave me wait I will show you

The Great Duck

6:22 PM

oh

well yeah

it's real analysis

they're just getting to the point

which is constructing R

Jason Kim

Oh,,

Secret

I will read and think about the rest tomorrow. I just finished writing the discussion and methods of some paper to be checked by my supervisor

The Great Duck

@Secret what was the discussion about?

if I may ask?

Educ

@TheGreatDuck Yes That's why I need algebre Textbook do you know any good one which explaine the construction of Z

Secret

It's part of my chemistry PhD. We are pumping out our first collaborative paper very soon

The Great Duck

6:25 PM

lol

@Educ all you're wanting is the construction of Z?

that's it?

Educ

Yes

The Great Duck

not real analysis stuff?

wikipedia

Integer

An integer (from the Latin integer meaning "whole") is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 1⁄2, and √2 are not. The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, …), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, …). This is often denoted by a boldface Z ("Z") or blackboard bold Z {\displaystyle \mathbb {Z} } (Unicode U+2124 ℤ) standing for the German...

go to "construction"

that's all you need to know

it's just ordered pairs of natural numbers

trivial

Educ

@TheGreatDuck I already checked not helpful if you know any good book. books make things easy

The Great Duck

a book on the construction of the integers?

Educ

I know @LeakyNun already gave me idea about that but I'm looking for good book that explained step by step with figures and some draw if possible

The Great Duck

6:27 PM

a book purely on that subject?

Educ

@TheGreatDuck yes please

The Great Duck

that doesnt exist

there isn't enough material for more than 5 or 6 pages

if you're looking for number theory or something beyond just construction that would make sense

but the construction of the integers is just a simple straightforward definition

wikipedia actually gives pretty much the whole enchilada on it

Educ

@TheGreatDuck Okay thank you I will use "graphically, when we relate (1,3) with (2,4) and with (3,5) and with (4,6) etc, we are relating points satisfying x-y = -2, i.e. y=x+2, where "-2" is from the outside perspective (from the inside we're still building Z) "as motivation before I give definition of Z

The Great Duck

are you familiar with equivalency classes?

what that means?

Educ

@TheGreatDuck yes I already familiar with that

The Great Duck

6:30 PM

ok

cause the integers are just defined as the equivalency class of ordered pairs of natural numbers whose differences are equal

that's all it is

Educ

my problem is write motivation with figure using geogebra for definition of z

The Great Duck

wait

is this a school thing?

Educ

@TheGreatDuck Yes but before I give that definition I would like to write motivation for it with some figure

The Great Duck

post the actual prompt please

and a figure as motivation won't make sense methinks

there are two ways of defining integers technically

one is an ordered pair like i just said

the other is an ordered pair of either 0 or 1 and then the magnitude number

0 or 1 basically representing plus or minus

but doing that makes thing act weird

(0,0) = (0,1)

Educ

@TheGreatDuck I'm going to stick with the first way to define Z and I would like to write presentation about that

The Great Duck

6:34 PM

ok

here's a good way to explain it

we are defining the integers as the set of differences of natural numbers

and we represent differences as an entity by using ordered pairs

Educ

(a,b) yes

The Great Duck

yes but there is no geometric motivation here

it's just that

Educ

what about geometric motivation should I try it with geogebra to attract the attention of students

The Great Duck

(a,b) + (c,d) is convenient then

there is no geometric motivation

Educ

@TheGreatDuck Okay I see

The Great Duck

6:37 PM

it's just convenient

simple

easy to work with

(a-b) + (c-d) = (a+c) - (b+d)

Educ

@TheGreatDuck Could not considered this upload.wikimedia.org/wikipedia/commons/thumb/8/8e/… as geometric motivation

The Great Duck

and so addition of integers is just addition of ordered pairs

that's not motivation

that's just a visual depiction of the equivalency class

motivation would be why they needed a definition

Educ

Ah I see now

The Great Duck

there's no geometric reason to define integers

they need to be defined so that they are defined

Educ

@TheGreatDuck I see now Thank you so much

The Great Duck

6:39 PM

geometry is irrelevant

@Educ I suppose one might be able to compare integers to the natural numbers by saying they are the set of 45 degree lines in N^2

but that's not really accurate

Educ

yes that's what I thought

The Great Duck

but no

they aren't defined as a set of lines

that would imply geometry defines math

geometry is what was used before sets to define math and arithmetic

we don't want that. That's crude and nonrigorous

Educ

@TheGreatDuck should write this " (NxN)/~ as Z should I say "graphically, when we relate (1,3) with (2,4) and with (3,5) and with (4,6) etc, we are relating points satisfying x-y = -2, i.e. y=x+2, where "-2" is from the outside perspective (from the inside we're still building Z)

before the definition ?

The Great Duck

no

listen

Educ

so I will give directly the definition

yes I'm all ears

The Great Duck

6:42 PM

wanna know who used geometry to define math?

Educ

Yes why not

The Great Duck

The Greeks

Educ

@TheGreatDuck yes and also in the middle age

The Great Duck

yeah and it's something that was considered archaic and horrible practice

it's just that nobody could come up with anything better

Educ

interesting

The Great Duck

6:44 PM

we use sets because it's better

Educ

yes

The Great Duck

we do not define math with geometry

that's archaic and it is even worse practice nowadays when we actually have a definition

Educ

I see so I will give them directly the definition of Z as N^2/ R

The Great Duck

precisely

math should define geometry

not the other way around

Educ

yes I agree with you

in the past they used to solve the quadratic equation with geometry perspective

The Great Duck

6:47 PM

well

solving something for geometric reasons is one thing

but one shouldnt define equations with geometry

Educ

I will give you the example later I don't have it now

The Great Duck

that's fine

it's not that important

Educ

Okay, can I ask you question : what is your area of expertise ?

The Great Duck

my area?

:-P

Educ

your filed: Algebra, analysis, combinatory,probability

The Great Duck

6:49 PM

im just a college graduate

Educ

Ah :)

@TheGreatDuck how many hourse do you spend in your study for maths

The Great Duck

0 now

I’ve been trying to find a decent differential algebra text for the past half hour or so just so I have something to keep myself busy at night

thats not going well

@Educ where are you in math?

Educ

7:11 PM

stochastic calculus

The Great Duck

sounds interesting

Educ

yes but I have to study more I forgot lot of concept

The Great Duck

describe it in one sentence

Educ

7:27 PM

hard to say

Simply Beautiful Art

The last -2 digits of 1/9 is 0.

The Great Duck

....

lolololol

@SimplyBeautifulArt

Simply Beautiful Art

:^)

Bob

7:44 PM

I was wondering what people think of the book "Probability, Random Variables and Random Processes". It is part of the Schaum series. I am finding the problems very hard.

Leaky Nun

8:32 PM

hi @Mr.Xcoder

Mr. Xcoder

hi

geocalc33

8:55 PM

Is (zeta(s))^(1/lnx) where x is in (0,1) and s is in (1, infinity) a group under multiplication

?

s is a real number

And so is x

Zeta(s) is the Riemann zeta function

Xander Henderson

@geocalc33 Well, does it satisfy the axioms of a group? If yes, it is a group. If not, no it isn't. How far have you gotten in verifying the group axioms?

geocalc33

I think it does but somebody said it wasn't

They said it wasn't a group because zeta(s)^(1/lnx) times zeta(t)^(1/lnx) isn't equal to zeta(s+t)^(1/lnx)

But when I multiplicationed those two functions the resulting function is in the group

So I'm slightly confused

Mike Miller

9:20 PM

What is an element of this group? A real number or a function on (0,1)?

geocalc33

Each element is a function

Mike Miller

Okay. Where is the identity element?

Since multiplication is defined pointwise, one of your functions should be the identity, 1 everywhere.

You should also be clear (I think I made a typo above): your elements are functions from what to what?

It's important to start as clear as possible. What are the elements? What is the result when they are multiplied? Then you can check axioms.

Bob

I recently posted a question related to probability: Here is the link: math.stackexchange.com/questions/2830675/…

geocalc33

9:38 PM

@MikeMiller the identity element is 1. My functions are from the real numbers to the real numbers. The elements are a family of functions

Mike Miller

Your elements are not defined on the real numbers, but rather a subset, and there is no value of x in (0,1) for which that spits out the identity element.

geocalc33

The result when they are multiplied is another function in the family

Mike Miller

You need to start by very precisely understanding what your elements are. Is it the function $f_x(s): (0,\infty) \to \Bbb R$ given by $\zeta(s)^{\ln 1/x}$? That is, the function parameter is $s$, and you allow them to vary by choosing the exponent, as determined by x?

geocalc33

The inverse elements/functions are not in x in (0,1)

Mike Miller

This is very frustrating. I am asking important clarifying questions (I legitimately do not understand what you're talking about), and you don't seem to be answering. B

geocalc33

9:44 PM

Yes the parameter is x

Mike Miller

If you can name 1) The set the elements live in, 2) How to specify a specific single element in that, 3) How to multiply your two elements, then I will say more.

The set the elements live in is not "functions $\Bbb R \to \Bbb R$", and the option I gave above is only one way to try to understand what you mean.

geocalc33

The inverse elements live in x in (1, infinity) y in (1, infinity) and x in (0,1) y in (0,1)

The non inverse elements live in x in (0,1) y in (0,1)

Zee

@LeakyNun how is a generating set not a generalization of a basis as Nicholas said above??

geocalc33

To specify an element, you would have for example: (zeta(2))^(1/lnx)

Where s is equal to 2

To multiply any two elements you would for example do: zeta(3)^(1/lnx) times zeta (4)^(1/lnx)

And the resulting function is in the unit square

So it looks like it's closed under multiplication

geocalc33

10:07 PM

@MikeMiller it's a group then?

Xander Henderson

10:21 PM

@geocalc33 If I understand you correctly, the set of functions that you are considering is the set $$ \left\{ f_x : (0,\infty) \to \mathbb{R} \ \middle|\ f_x(s) = \zeta(s)^{1/\log(x)}, s\in(0,\infty)\right\}.$$ Is that correct?

Mike Miller

No, but do not mistake my lack of response for agreeing with you: I got tired of engaging. I wrote out the details anyway.

Your elements, as far as I can tell (and it is quite difficult to tell what you are saying), are functions $(0,1) \to \Bbb R$, $f_s(x) = \zeta(s)^{\ln 1/x}$. A specific value of $s \in (1, \infty)$ defines an element. When you multiply $f_s(x) f_t(x) = \left(\zeta(s)\zeta(t)\right)^{\ln 1/x}$, you are trying to find out if there is some $r \in (0,1)$ such that $f_s f_t = f_r$.

The $\ln 1/x$ is totally extraneous here: the question is just whether or not $\zeta(s) \zeta(t)$ is of the form $\zeta(r)$ for some $r \in (1, \infty)$. Then, the question is just "what is the range of $\zeta: (1, \infty) \to \Bbb R$? Because $\zeta$ goes to $1$ as $s \to \infty$ and to $\infty$ as $s \to 1$, its range is $(1, \infty)$, and it is clear that the product of two numbers in $(1, \infty)$ is also in $(1, \infty)$.

So the product of two functions in your set is also in your set (because $(1,\infty)$ is closed under multiplication). However, you have no identity element ($\zeta$ does not take on the value $1$), nor does it have inverses (it does not take on any values in $(0, 1)$).

@XanderHenderson That is what I thought, but see the messages above yours: zeta(2)^{1/ln(x)} is an element of his set.

Xander Henderson

Oi...

Mike Miller

It's the other way around.

Xander Henderson

@geocalc33 First task: very carefully describe the elements of the set that you want to examine. Right now, I don't care about the operation, I just want to know what a typical element looks like. If it is a function, specify the domain and codomain.

geocalc33

A typical element lives in the first quadrant except x is not equal to 0 or 1 and y is not equal to 0 or 1

And the form of the elements is (zeta(s))^(1/lnx)

And the inverse elements are (zeta(s))^(-1/lnx)

Mike Miller

10:37 PM

The second bit you never mentioned earlier (as far as I could parse). It is important to list ALL ELEMENTS of a group. You also need to include the identity element, remember.

geocalc33

Oh yeah, the identity element is 1

Mike Miller

Once you do all this, yes, you are a group, and canonically isomorphic to $\Bbb R^+$, the positive reals under multiplication.

Sending a real $x$ to the function corresponding to the unique $\zeta(s) = x$ if $x>1$, to the identity if $x = 1$, and the unique $\zeta^{-1}(s) = x$ if $0 < x < 1$.

Xander Henderson

@geocalc33 This is nonsense. Elements of "the first quadrant" are ordered pairs, such as $(2, 4)$. If you say that "A typical element lives in the first quadrant," then you are saying that a typical element is an ordered pair of real numbers, i.e. an element of some subset of $\mathbb{R}^2$

So we are still on task one: describe the elements of the set, i.e. describe a typical element of the set in a rigorous manner.

geocalc33

All the elements are functions.

It made sense

Xander Henderson

Great. What functions? What are their domains and codomains? Again, rigorously describe the elements of the set...

geocalc33

10:42 PM

I believe I said that already

Xander Henderson

Well, I am still confused. If you don't want to clarify your definitions for me, then there is nothing more that I can say to be of help. Peace out.

Jason Kim

Find the unit digit of $0^9+1^8+2^7+3^6+4^5+5^4+6^3+7^2+8^1+9^0.$

*units

Xander Henderson

Found it. Now what?

Jason Kim

What is it?

Xander Henderson

Why do you want to know?

And what have you done in order to determine it?

Jason Kim

10:59 PM

I know the answer already using groups of four simplification

Xander Henderson

Okay... I don't know what that last phrase means, but bully! If you know the answer, why are you asking about it here?

Jason Kim

Actually I'm working on it right now...

It's 0+1+8+9+4+5+6+9+8+1=0

modulo 10

Xander Henderson

You might want to check your work; I don't get zero.

Jason Kim

Oh $1$

Jason Kim

11:30 PM

.

Fargle

11:54 PM

Hey chat!

Ted Shifrin

runs away to disappear from Fargle

Fargle

How are you, @Ted?

Ted Shifrin

doing fine, thanks, Fargle, and you?

Fargle

Excellent! Just came back from my first Pride.

Ted Shifrin

Oh excellent!

I will miss this year's in San Diego, but I've done a few dozen ...

Did you have a good time?

Fargle

11:56 PM

A great time.

Ted Shifrin

:)

Fargle

I'm a bit troubled by the commoditization of it all and so on, of course. Stonewall was a riot, not a World's Fair.

But it was a fun experience. Good people and lots of positivity, as you might expect.

Ted Shifrin

Well, yes, I don't disagree on that. But it's about continuing to knock down walls, especially with what's going on these days.

And feeling support.

MatheinBoulomenos

Hi @Ted

Ted Shifrin

hi @Mathein

MatheinBoulomenos

11:59 PM

it feels good to solve a modular forms exercise with algebraic number theory, although this is probably not what is expected

Ted Shifrin

I have no clue, @Mathein. I'm totally ignorant.

Fargle

@TedShifrin Amen to that. It's more vital than ever that disenfranchised voices are heard. That's part of why I ended up coming out--I felt a societal responsibility in this day and age.

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$Mathematics$ Mathematics