Mathematics - 2019-09-06

rapasite

00:05

@none math.ru.nl/~bosma/Students/CF.pdf have fun. I am happy if I was helpful to you even if I am not a mathematician ;)

none

Thank you, this seems very nice (and there's a chapter on decimal/continued fraction relationships).

Also, I think I figured out how to guarantee an interval around a number such that at least the first number of the continued fraction expansion coincides

(and for my purposes this will be enough for now)

manooooh

01:17

Hello guys!

I am trying to prove or disprove: $$ab\equiv x\pmod{n}\implies a\equiv x\pmod{n}\cdot b\equiv x\pmod{n}$$

Leaky Nun

First hint that body’s ‘biological age’ can be reversed - In a small trial, drugs seemed to rejuvenate the body’s ‘epigenetic clock’, which tracks a person’s biological age.

Akiva Weinberger

01:36

@manooooh I don't understand the statement

manooooh

@AkivaWeinberger I mean: knowing that $5^{22}\equiv1\pmod{23}$, can we replace it in: $(5^{22})^{71119}5^6\pmod{23}$?

To do that I think we must show these 2 results:

1) $ab\equiv x\pmod{n}\implies a\equiv x\pmod{n}\cdot b\equiv x\pmod{n}$

2) $a^b\equiv x\pmod{n}\implies(a\pmod{n})^b\equiv x\pmod{n}$

But I do not know if my notation is correct...

Akiva Weinberger

Your notation is not correct

Leaky Nun

@AkivaWeinberger hi

Akiva Weinberger

but $ab\equiv(a\bmod n)(b\bmod n)\pmod n$

in other words

Leaky Nun

@AkivaWeinberger reddit.com/r/musictheory/comments/4kob4l/…

Akiva Weinberger

01:40

if $a\equiv A\pmod n$ and $b\equiv B\pmod n$ then $ab\equiv AB\pmod n$

Leaky Nun

oh and did you watch my video

manooooh

@AkivaWeinberger shouldn't it be the other way i.e. if $ab\equiv AB\pmod n$ then...?

Akiva Weinberger

No. Take $a=2$, $~b=3$, $~A=1$, and $B=6$

Then $ab\equiv AB$ (because $6\equiv 6$) but $a\not\equiv A$ and $b\not\equiv B$

@LeakyNun Which one?

Leaky Nun

13 hours ago, by Leaky Nun

@AkivaWeinberger https://www.kapwing.com/videos/5d70ff1c2b9b4c001469a4ba

Akiva Weinberger

Oh yeah

Yeah

@manooooh If $5^{22}\equiv1\pmod{23}$ then $(5^{22})^{71119}5^6\equiv5^6\pmod{23}$

Leaky Nun

01:43

@AkivaWeinberger how do you propose we solve my problem

Akiva Weinberger

What problem

Tuning leading tones?

Also what is Kapwing

manooooh

oh, that is clear for me. Correct my implications: if $5^{22}\equiv1\pmod{23}$ then $(5^{22})^{71119}\equiv1^{71119}\pmod{23}$ and $1^{71119}=1$, so $(5^{22})^{71119}\equiv1\pmod{23}$, then $(5^{22})^{71119}\cdot5^6\equiv1\cdot5^6\pmod{23}$, so $(5^{22})^{71119}\cdot5^6\equiv5^6\pmod{23}$
Right?

Akiva Weinberger

Or is the "problem" that you end up somewhere else on the Tonnetz @LeakyNun

Leaky Nun

yeah

Akiva Weinberger

@manooooh Yes

manooooh

01:45

Thank you!

Leaky Nun

kapwing is a video-editing tool

also til it's called the Tonnetz

I was trying to transcribe the Prelude in C into just intonation

Akiva Weinberger

I thought it up one day (thinking about visualizing the rations of just intonation) and shortly thereafter found out that it was already a thing

Leaky Nun

I saw it and didn't know its name

Akiva Weinberger

I don't know enough about composing in just intonation to be honest

No experience

I can't even properly sing in just intonation (not consciously, anyway)

Microtones are hard

I am only capable of music'ing through piano and voice

Leaky Nun

just intonation is like some harmonic heaven that is hard to get to

yeah so why i sent you the reddit is that melodically the leading tone should be higher (than the 12TET), yet harmonically it should be lower (than the 12TET)

Akiva Weinberger

01:49

I wonder how closely a cappella(sp?) choirs stick to 12TET

I spelled it right

But I feel they'd be more likely to slide into purer ratios

but also a cappella choirs are more likely to slide down and end up a half-step lower than they started

(or at least all the ones I've been in are)

just 'cause singing high is more effort

Leaky Nun

really

I guess it wouldn't happen if enough people have perfect pitch

Akiva Weinberger

I have decent relative pitch

Once, after practice, my mom and I and two other people from choir practiced one of our songs and didn't drop at all

so we concluded that it was the rest of the choir that was dragging us down

(This was in the summer before I went off to college)

This is the only confirmed way to get random strangers to clap, by the way^

Sing harmonies in a public place

Leaky Nun

are you Ashkenazi or Sephardi?

Akiva Weinberger

Ashkenazi

Leaky Nun

so you'd be having Shabbos tomorrow?

Akiva Weinberger

01:56

(Fun fact: I once new a Sephardi with the last name Ashkenazi)

@LeakyNun What, as opposed to Shabbat?

Leaky Nun

cool

yeah

Akiva Weinberger

I alternate

Shabbat is Modern Israeli Hebrew

I'm not very consistent

Also, until 8th grade, the school I was in was 90% Sephardi

It was in a neighborhood with a large Syrian Jewish population

so, outside of home and shul (synagogue) I would have heard "Shabbat" a lot more often

Leaky Nun

hmm I can't find a reading of genesis in liturgical ashkenazi pronunciation on youtube

I've heard it before in person

Akiva Weinberger

Bereishis boro elokim es hashomayim v'es ho'oretz

Leaky Nun

they pronouncs sav with a "s" right

Akiva Weinberger

01:59

Something like that

@LeakyNun Yeah

Leaky Nun

and the stress is on a different position

can you indicate stress?

Akiva Weinberger

I think the only difference in the first pasuk is "beREIshis" instead of "bereSHIT"

so

Beréishis boró elokím es hashomáyim v'es ho'óretz

Leaky Nun

ok

Akiva Weinberger

Also it might be eloykim instead of elokim?

Not sure

Probably varies based on which European country it's from

And "o" is [ɔ]

(except for in "elokim")

Fun fact

In Modern Hebrew

Personal names tend to get Ashkenazi stress

So for example

My sister's name is Tova (TO-va)

and the feminine form of the word "good" is tova (to-VA)

Similarly, SHA-lom (the name) and sha-LOM (peace, hello/goodbye)

TIK-va (name) vs tik-VA (hope)

etc

(also yes my sister's name literally means "good")

Leaky Nun

I see

Akiva Weinberger

02:09

There's a saying "Gam zu letova" which means "This too is for the best" but can also mean "This too is for Tova"

She likes to bring this up

Leaky Nun

cool

in Hebrew alphabet?

Akiva Weinberger

גם זו לטובה

Or just her name is טובה

Leaky Nun

I see

Akiva Weinberger

Your name is "ken", yeah?

Written כן it means "yes" and written קן it means "nest"

Leaky Nun

my name is Kenny

Akiva Weinberger

02:13

"Kenny" would be "my nest"

Leaky Nun

lol

Akiva Weinberger

(though with the accent on the last syllable)

kení

Leaky Nun

so like this:

Levi Screaming "Kenny" For 1 Hour

►

Akiva Weinberger

けに or けんに I wonder

Queni

Leaky Nun

google translate says ケニー

Akiva Weinberger

02:16

Ah

Kenii

Never understood why English -y becomes Japanese -ii in loanwords

It's not a long vowel in English

Then again, my dialect of American English doesn't have vowel length to begin with, so the whole thing is mysterious

Leaky Nun

I have no idea either

Akiva Weinberger

What common loanword in English is part Hawaiian and part Greek @LeakyNun

Leaky Nun

hmm

is it like a food name or something

Akiva Weinberger

No

Leaky Nun

because I don't know food

Akiva Weinberger

02:28

Maybe "loanword" isn't the right word

but half is loaned from Hawaiian and the other half is from Greek

Leaky Nun

hmm

do I have one in my room

Akiva Weinberger

...Kinda?

Leaky Nun

hmm

interesting

ok give me the meaning of the greek part lol

Akiva Weinberger

Education

Leaky Nun

so pedagogy

or school

Akiva Weinberger

02:33

Pedagogy is close

Leaky Nun

ok I have no idea

tell me lol

Akiva Weinberger

Wikipedia

Leaky Nun

oh...

lol

Akiva Weinberger

Wiki means quick in Hawaiian

Leaky Nun

fair enough

Akiva Weinberger

02:37

Incidentally, [k] and [t] are allophones in Hawaiian

and are both spelled <k>

so I think wiki is pronounced [viti] in Hawaiian

Um

Never mind

Oh OK

Upon further research it seems they are in free variation

Same with [v] and [w]

Leaky Nun

@AkivaWeinberger reddit.com/r/IsTodayFridayThe13th

Akiva Weinberger

isitthursday.org

YES

Leaky Nun

  var today = new Date();
  var wkday = today.getDay();
  function thursday() {
    if (wkday == 4) {
      document.write('<h1 class="yes">YES</h1>');
    }
    else {
      document.write('<h1 class="no">NO</h1>');
    }
  }

yay it's so simplistic

Akiva Weinberger

isitfriday.org

♫

twitter.com/shonk/status/1110706508277276672

Jif

shonkwiler.org/knots

Leaky Nun

02:52

cool

Akiva Weinberger

^I like this image but I don't know why

I know it has to do with the texture and the color choice

but I don't understand either of those on an intellectual level

Maybe I should learn graphic design at some point

Secret

Empty formal system:

Consider the following formal system

1. Empty language

2. Empty grammar

3. No axioms

Akiva Weinberger

nonverbal response

Secret

4. Empty inference rules

I wonder if the only theorem is empty

Akiva Weinberger

I think there are no theorems

The set of theorems is the empty set

Hm

Logical symbols aren't assumed to be implicitly included in the language

right?

Secret

02:58

that I need to double check, because some people do include them

Akiva Weinberger

So then we can't even assume the empty theorem is true

if they're not included

and this is just a set of rules for going from one sentence to another

If logical symbols are included then $\forall x,x=x$ should be true

and $\top$ if that's included

$\top\land\top$, $~\top\lor\bot$, etc

Tautologies, in other words

Secret

Ok, so for most definitions, symbols includes even the logic symbols, and the empty language has no strings at all, not even the empty string

Interestingly, the empty language pops up in many finite automata studies

1

Q: Is empty language a singly capacitated regular language?

A singly capacitated regular language is such that exists a deterministic finite automaton (DFA) which has a single accepting state. For example an empty language (whose alphabet is an empty set) is singly capacitated regular language and here's a DFA demonstrating this: I don't understan...

So I am guessing, because the empty language does not accept even the empty string, the formal system will be rejecting everything, not sure if that can be called a ⊥ though

I need to think about it...

Secret

03:19

1

Q: Can we deduce anything from the empty set of axioms?

If the set of logical axioms is empty and so is the set of non-logical axioms, then it seems we can't make any deduction.As the first formula in the deduction must belong to either sets or it it's deduced from previous formulas by means of a rule of inference (no such previous formulas even exis...

logic predicate-logic

Ok, so even without logic symbols, it can still derive a tautology

must be one of those weird things about vacuous truths...

Akiva Weinberger

How do you write a tautology without even logic symbols

> See Ch.2.3 The Logical Axioms : the logical axioms are :

the three equality axioms [see page 56]

two quantifiers axioms [page 57].

It seems they're assuming those logical axioms are present

I can't say $\forall x,x=x$ if I don't know what $\forall$ and $=$ mean

or, for that matter, $,$

or bounded variables

Hm

30 mins ago, by Akiva Weinberger

This being an orthogonal projection,

you could reflect it front-to-back without changing anything

like that bistable cube illusion

so there's two interpretations of that GIF

(It's the figure-eight knot which is chiral anyway though)

(or at least that's what the description said it is)

Secret

is that a trefoil?

Akiva Weinberger

I just said

The description said it was a figure-eight knot

Secret

o sorry, having conflicts with my brain that the image I see does not seemed to match up what I think of a figure 8 knot, while at the same time, it is not matching a trefoil

Probably a symptom of lack of sleep

5

Q: Why can't we generally replace inference rules with axioms?

Is there a big difference in having insufficient axioms and insufficient inference rules/proof procedure to have a complete theory? It seems like in many cases adding a new inference rule or a new axiom has the same effect. For example, consider a language with 2-place connectives $\rightarrow, ...

logic proof-theory second-order-logic

> Eliminating all RoI (even just those that are not Axiom Schema) would make it impossible to introduce new theorems to the theory, so you would never be able to prove anything except what is explicitly assumed.

1

Q: Can BNF consist of zero rules?

I am trying to use BNF to describe its own grammar to get used to it. What I can not find any information about is whether BNF must consist of zero or more rules or one or more rules. The difference would be: <grammar> ::= <grammar> <rule> | <rule> vs. <grammar> ::= <grammar> <rule> | "" Th...

context-free formal-grammars

Akiva Weinberger

03:36

Property (1) $A$

Property (2) $A\implies B$

So we can conclude $B$, right?

Well we need a rule of inference

Property (3) Given $A$ and $A\implies B$, we can get $B$.

So now we can conclude $B$, right?

But don't we need another rule of inference?

Property (4) Given $A$, $A\implies B$, and "given $A$ and $A\implies B$, we can get $B$" (property (3)), we can get $B$.

What the Tortoise Said to Achilles

"What the Tortoise Said to Achilles", written by Lewis Carroll in 1895 for the philosophical journal Mind, is a brief allegorical dialogue on the foundations of logic. The title alludes to one of Zeno's paradoxes of motion, in which Achilles could never overtake the tortoise in a race. In Carroll's dialogue, the tortoise challenges Achilles to use the force of logic to make him accept the conclusion of a simple deductive argument. Ultimately, Achilles fails, because the clever tortoise leads him into an infinite regression. == Summary of the dialogue == The discussion begins by considering the...

Secret

Modus ponens is defined by how you combine property 1,2 to get 3. so 1,2 are the inputs and 3 is the output

I think 4 is implicit as you plug the inputs into 1 and 2

Akiva Weinberger

OK so:

4: given 1, 2, and modus ponens, we can get 3

5: given 1, 2, 4, and modus ponens, we can get 3

6: given 1, 2, 4, 5, and modus ponens, we can get 3

http://platonicrealms.com/encyclopedia/Carrolls-Paradox

@Secret If it turns out modus ponens was wrong, how would you know

You can't prove that modus ponens is right without using modus ponens

because you can't think without using modus ponens

3Blue1Brown has 2,070,834 subscribers

2^21 = 2,097,152

We're very close to another Q&A

Maybe a week away

Secret

Ah, I never thought about that before. That's a metasystem problem, because to interpret what modus ponens means without becoming circular, we need to interpret it outside the system we are using (where it is assumed to be valid)

Akiva Weinberger

I saw a story somewhere

Paraphrasing

"A philosophy professor challenged me to prove logically that I was awake and not dreaming"

"Eventually I came up with an airtight proof"

"and then I woke up"

Secret

The 3rd sentence sounds very meta system to me

Akiva Weinberger

03:47

I never finished Gödel Escher Bach

but somewhere in the first half of the book there's a sequence where the characters have a way to "push" themselves into a fictional world and "pop" themselves out of it

and if there's a book within a book you can push twice

and they mentioned that, on the level of reality, one of their friends "popped" and they don't know what happened to him

Weird book

I oughta pick it up again

Mary Star

10:27

Hello!!

We have that a function $f:\mathbb{R}\to \mathbb{R}$ is twice differentiable in $\mathbb{R}$ and it has infinitely many roots. To show that the equation $f''(x)=0$ has infinitely many roots, do we use the definition of the derivative?

Mary Star

10:39

Hey @LeakyNun !! Do you have an idea about the above?

Rithaniel

11:07

I think you can apply the mean value theorem to that, @MaryStar

aitía

12:21

Hello

Can someone help me with a true or false question?

Rithaniel

Perhaps. It really depends on what the question is.

aitía

"There is only one way to parametrically represent the solution set of a linear equation." T or F

Larry Eppes

hello

someone can help me about my question? math.stackexchange.com/q/3340939/49660

thanks, I got stuck for days

Semiclassical

13:36

Saw this question on the main site: Find the equation of a cone whose center is the origin and which contains the circle $(x-2)^2+(y-3)^2=5 \wedge z=3$.

there's an elementary answer there, via a nice parametrization of the cone

But it seems like there should be an answer which appeals directly to projective geometry. Something along the lines of: If we homogenize the circle via $(x,y,z)\to (x/t,y/t,z/t)$, we get $(x-2t)^2+(y-3t)^2=5t^2 \wedge z=3t$. We can then eliminate $t$ to get $$(x-2(z/3))^2+(y-3(z/3)^2)^2=5(z/3)^2=0\implies (3x-2z)^2+9(y-z)^2=5 z^2$$

Larry Eppes

$(2+\sqrt{5}\cos t, 3+\sqrt{5}\sin t, 3)s$ which $t$ and $s$ are parameters, and $t$ ranges from 0 to $2\pi$

Semiclassical

blah, last should have been $(x-2(z/3))^2+(y-3(z/3))^2=5(z/3)^2 \implies (3x-2z)^2+9(y-z)^2=5z^2$

Is that a legit way to derive this cone? It seems to work but I can't quite get the logic

Kumar Nilesh

the cone you are talking about is right circular?

Larry Eppes

yes

I just use the vector to describe the points on the cone.

fogive my english, if there is some confusing.

Semiclassical

that's essentially the same as the answer I mentioned, yeah: math.stackexchange.com/a/3346243/137524

I'm just curious if there's a projective way of looking at it

especially since "homogenize and eliminate $t$" is such a simple process

@KumarNilesh you could also take the initial problem as defining an (infinite) oblique circular cone: the base is the given circle and the vertex is the origin

but any infinite oblique circular cone is equivalent to some infinite right circular cone

Kumar Nilesh

14:14

can we attempt it considering cone as Quotient space of algebric topology

We know the circle generating required cylinder and the point on which the end collapses

Ultradark

15:28

Do hyperbolas always have two parts

we are learning about conic sections

For example $f(x)=1/x$ is a hyberbola

and it has a curve in the first QUAD. And the third quadrant

is $e^{1/x}$ a hyperbola

why do we work with hyperbolas with symmetry abou the y-axis, when we could just work with involutions

RScrlli