Mathematics - 2019-08-28 (page 1 of 2)

Ultradark

00:05

@LeakyNun Does the set of positive and negative primes have an algebraic structure $(G_{\Bbb P^{\pm}},*)$

my professor said it did but I can't figure out what it is

0

Q: Does the set of negative and positive primes have an algebraic structure?

Currently taking group theory. We just covered the group axioms. Here's my question: Does the set of negative and positive primes have an algebraic structure? I think I'd like to define the set as $$(G_{\Bbb P^{\pm}},*)$$ equipped with some binary partial function $(*).$ I think the identi...

yuvraj singh

03:30

@MartinSleziak can you prove triangle inequality for me

Secret

05:57

There exists a tamserf that does not reduce to an etiogical drift

As a result there exists no strawberry jam that proves the triangle inequality

corollary: we do not do someone else homework, and the dog will eat it if it does

Martin Sleziak

07:11

@yuvrajsingh That's rather vague question. Triangle inequality for absolute value? Triangle inequality for Euclidean distance?

In any case, whichever it is, I'm pretty sure many proofs can be found easily online (and probably even on the main site).

For absolute value: Proof of triangle inequality

This uses Cauchy-Schwarz inequality - so it works for any inner product space, but also for Euclidean distance: Proof for triangle inequality for vectors.

Maybe also: Proving the triangle inequality for the euclidean distance in the plane.

This one is again based on Cauchy-Schwarz: Euclidean distance proof.

Here is one more: Prove that Euclidean distance in $\mathbb{R}^n$ is a distance

yuvraj singh

@MartinSleziak basically how do relate triangle inequality with cauchy swchartz inequality

Martin Sleziak

In the context of inner product spaces? Or in the context of Euclidean distance?

yuvraj singh

Inner product of spaces

Martin Sleziak

For instructions how to render MathJax(TeX) in chat see this post on meta or go directly to robjohn's website

You want to show $\|x+y\|\le \|x\| + \|y\|$.

This is equivalent to $\|x+y\|^2\le (\|x\| + \|y\|)^2$ or $$\|x+y\|^2\le\|x\|^2 + 2\|x\|\cdot\|y\| + \|y\|^2.\tag{*}$$

yuvraj singh

@MartinSleziak yes

Martin Sleziak

07:19

Now if you work with an inner product $\langle \cdot,\cdot\rangle$, then you have $\|z\|^2=\langle z,z \rangle$.

So now you simply use this in both sides of $(*)$ and manipulate a bit.

So $(*)$ is equivalent to:

$$\langle x+y,x+y \rangle \le \langle x,x \rangle + 2\|x\|\|y\| + \langle y,y \rangle$$

$$\langle x,x \rangle + 2 \langle x,y \rangle + \langle y,y \rangle \le \langle x,x \rangle + 2\|x\|\|y\| + \langle y,y \rangle$$

$$2\langle x,y \rangle \le 2\|x\|\|y\|$$

$$\langle x,y \rangle \le \|x\|\|y\|$$

The last one is exactly Cauchy-Schwarz.

So it only remains to check whether each step can be reversed. (We want to get from the last inequality to $(*)$.)

I'll have to go afk - I have some work to do.

Anyway, I think what I wrote here is basically what you can see in the posts I linked.

Have a nice day!

ÍgjøgnumMeg

Morning

ÍgjøgnumMeg

09:07

A category having "enough injectives" means that you can always embed into an injective object right?

Leaky Nun

10:00

@ÍgjøgnumMeg ja

Secret

Suppose we have ZF except the following:

1. Nuke the axiom of infinity

ÍgjøgnumMeg

@Leaky cool danke :)

Secret

2. Nuke the law of excluded middle

3. Nuke nonconstructivity

4. Require all definable functions to be predicative

(I.e they can only be built from the axioms and empty sets)

The challenge here is:

Is $\omega$ predicative in this system?

Secret

10:45

1

A: Why is replacement true in the intuitive hierarchy of sets?

Suppose that as each stage $S$ is completed, we take each $y$ in $x$ which is formed at $S$ and complete the stage $S_y$. When we reach the stage at which $x$ is formed, we will have formed each $y$ in $x$ and hence completed each stage $S_y$ in $\mathbf S$. This is actually a circular justi...

It does seemed that if we nuke the existence of "countable" in principle 2, the whole thing is dead, and we cannot even reach $\omega$

Zermelo set theory

Zermelo set theory (sometimes denoted by Z-), as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences from its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering. == The axioms of Zermelo set theory == The axioms of Zermelo set theory are stated for objects, some of which (but not necessarily all) are called sets, and the remaining objects are urelements and do not contain any elements...

Perturbative

@MikeMiller Thanks for your comments. Regarding the spectral sequences relevant in complex geometry measuring the difference in complex geometry and algebraic geometry, I would be highly interested in reading up about them if you could point me to a reference, even though I don't know any complex geometry or algebraic geometry at the moment, I could learn it along the way.

Secret

math.stanford.edu/~feferman/papers/predarith.pdf

lol ,never thought I will read Ferferman's paper

Secret

11:38

So it seems, one way to predicatively construct an infinite object, is to start with a finiteness axiom, defined in terms of what kinds of injective, bijective and surjective functions are allowed for the sets in a given universe

Thus consistent with the more philosophical exploration in the RudyRucker book, the essence of infinity is whether a process that is guarantee to continue will eventually complete the construction

Thus Absolute Infinite is precisely the largest inconsistently possible notion of a guarantee continuing process of which it can never go to completion, nor even get any closer to completion for each step

Thus, all notions of infinity are unified by the inability to be completed from below given a process that produce outputs that form part of it

Actually not so fast

in Logic, 26 secs ago, by Secret

Still, unless I am mistaken, it still uses contradiction in Theorem 8 in order to show that the pre N structure satisfy induction and hence an N structure. If excluded middle fails, is it still possible to recover the existence of a Dedekind infinite class that is a model of PA?

We have not ruled out contradiction

Apr 23 at 4:31, by Rithaniel

Here's a question for you: We know that no set of axioms will ever decide all statements, from Gödel's Incompleteness Theorems. However, do there exist statements that cannot be decided by any set of axioms except ones which contain one or more axioms dealing directly with that particular statement?

Thus the more precise formulation of this question is the following conjecture:

Nature of infinity conjecture: Given a formal system that is constructive, predicative (the notion that no stage of the process refers to objects that exists in the universal class which has not been proved to exist. See here for details), with no axioms of infinity or stronger/weaker variants, corollaries thereof, and no assumption of the existence of the class of natural numbers,

$\omega$ is impredicative.

s.harp

11:59

short question: In some lectures notes somebody defined an equivalence of categories to be a functor that has the properties: 1. It is full, 2. faithful, 3. and has representative image (its image contains an object of every isomorphism class in the target category)

this is different from the usual notion I know, where an equivalence is a functor $F$ which admits an "inverse" $G$ so that $F\circ G$ and $G\circ F$ both admit natural transformations to the identity

while it is clear to me how the definition via natural transformations implies the first, the other direction seems mysterious to me, is it really true like that?

in particular the "naturality" of $G\circ F \to \mathrm{id}$ (and the other direction) seems mysterious, but I also never really got a grip on natural transformations

Ultradark

does the primes have the groupoid structure

in terms of catetgory theory

Secret

personal.psu.edu/t20/talks/uconn1504/talk.pdf

I wonder what theory has proof theoretic ordinal $\omega, \omega +1$?

10

Q: Is set-induction relatively consistent?

One way to state the axiom of foundation is that the $\in$ relation on any transitive set is well-founded in the following sense: A relation $(X,\prec)$ is well-founded if for any subset $S\subseteq X$ which is inductive, in the sense that $y\in S$ for all $y\prec x$ implies $x\in S$, in fact $...

lo.logic constructive-mathematics set-theory

ok nvm, KP0 is not useful for our purpose

Ultradark

12:15

anyone?

this should be an easy question

ÍgjøgnumMeg

presumably not

Ultradark

why do you say that @ÍgjøgnumMeg?

ÍgjøgnumMeg

well assuming (algebraically) that your product is just multiplication, what are the inverses and what's the identity?

Ultradark

what product?

Secret

So, investigation so far suggests at least in classical logic, the nature of infinity conjecture is true (requires a notion of "not infinite", which arguably is good enough to be defined as an essence of finite and infinite). What remains to be found out is whether it holds for other logic systems predicatively and constructively, in particular, intuititionistic logic

ÍgjøgnumMeg

12:18

i.e. given $p, q$ prime, the product is $pq$ which is not prime (which is fine since closure is not an axiom) but you have no identity (1 isn't prime) and you have no inverses

Ultradark

addition

ÍgjøgnumMeg

okay but 0 is not prime

so you still don't have a groupoid

Secret

For anyone who are interested, the school of infinity I am in is called predicative finitism

Ultradark

I'm interested

Secret

> Basically predicativism, except you must constructively demonstrate that the set of naturals is an actual infinity (in set theoretic language, the class of natural numbers is a set)

Ultradark

12:23

@ÍgjøgnumMeg just take 0 out

ÍgjøgnumMeg

so what's your identity...?

Secret

Thus, the Nature of infinity conjecture is basically a reverse mathematics question on: Does predicative finitism exist

A positive answer will allow us to define a predicate that single out all infinite objects in the mathematical universe (including the absolute infinite)

and hence solving the question "what is infinity", "whether there is a possibility to expect physical infinity" once and for all

> Always remember, I am an idea extremist, if you want to convince me that X does not exist, it and I has to be done in such a way that regardless of what conditions in totality, X is impossible

Ultradark

there isn't an identity

Secret

and I will not hesitate to destroy the universe in achieving those goals

ÍgjøgnumMeg

So it's not a groupoid... i don't get the problem

lol

Ultradark

12:29

@ÍgjøgnumMeg yeah it's not a groupoid

Secret

In 2015 I have already shutting all naysayers of the existence of division by zero down into 100 repeats of the holocaust, by demonstrating a concrete example of a division by zero algebra (see relevant chatroom for details)

Ultradark

it has 1) invertibility and 2) commutivity

ÍgjøgnumMeg

right so what's the question?

Ultradark

@Secret look into translating infinite sets :)

Secret

that will not help solve the question

Ultradark

12:31

@ÍgjøgnumMeg Is there an 'algebraic structure' on the set of positive and negative primes

Secret

because what I am interested in is how likely we are expecting a physical infinity

and that requires a counterexample to Rithaniel's question, aka a counterexample to the nature of infinity conjecture

Ultradark

that sounds like pseudomath

Secret

If the notion of infinity is unavoidably circular in formulation, then we should not expect physical infinities exists

But if it is not, then there is a hope to reach it some time in the future

The answer to that conjecture will not give as any definite answer on whether physical infinities can exist, as whether that exist depends on what physics our universe ran on

But it will definitely help settle the debate between finitists and predicativists

by demonstrating that, there is a procedure that proves the existence of the naturals as a completed object, without any notion of nonconstructivity and predicativity such as excluded middle and infinity axioms

Five philosophical positions on infinity:

Ultrafinitism: Infinities, both potential and actual, do not exist and are not acceptable in mathematics.

Finitism: Potential infinities exist and are acceptable in mathematics. Actual infinities do not exist and we must limit or eliminate
their role in mathematics.

Predicative finitism: Potential infinities exist and are acceptable in mathematics. There exists at least one actual infinity that is constructive and predicative.

Predicativism: We may accept the natural numbers, but not the real numbers, as a completed infinite totality. Quantification over N is acceptable, but quantification over R, or over the set of all subsets of N, is unacceptable.

Infinitism: Actual infinities of all kinds are welcome in mathematics, so long as they are consistent and intuitively natural.

Secret

13:29

Update: 21820 just shot down almost everything I said 2 hours ago

in Logic, 33 mins ago, by user21820

@Secret I'm too lazy to do more than glance through, but I'm sure that part is cheating; it's just hiding induction inside (Card) because the other axioms axiomatize finite sets, and DedFin states rigidity of finite sets, and we get a class from WS-CA that cannot correspond to a finite set because it's not rigid, which relies critically on P-II to have non-pairs. So it's no better than ZFC's axiom of infinity that states existence of an inductive set.

Ferferman, you are a very cunning wit indeed

That means, the conjecture is still wide open

ugh

Alessandro Codenotti

@Ultradark It isn't a groupoid with 0 either

ÍgjøgnumMeg

Hey @Alessandro

Alessandro Codenotti

Hi @ÍgjøgnumMeg

ÍgjøgnumMeg

How's Italy?

Alessandro Codenotti

Hot mostly

ÍgjøgnumMeg

13:42

ew

Secret

Also, there's actually sort of something similar to what I called predicative finitism above

Pre-intuitionism

In the mathematical philosophy, the pre-intuitionists were a small but influential group who informally shared similar philosophies on the nature of mathematics. The term itself was used by L. E. J. Brouwer, who in his 1951 lectures at Cambridge described the differences between intuitionism and its predecessors: Of a totally different orientation [from the "Old Formalist School" of Dedekind, Cantor, Peano, Zermelo, and Couturat, etc.] was the Pre-Intuitionist School, mainly led by Poincaré, Borel and Lebesgue. These thinkers seem to have maintained a modified observational standpoint for the...

but they have not ditch the law of excluded middle

ÍgjøgnumMeg

@Alessandro still haven't found a place to live hahaha

Alessandro Codenotti

woops

I also found a place in the middle of September for Bonn last year

(which turned out to be awful though, so I moved at the end of the semester)

Ultradark

this is ridiculous. I asked an honest question and people just hate on it for no reason. (no one here)

ÍgjøgnumMeg

Fair, I'm hoping that I can get a place in halls

I applied early enough

Ultradark

13:54

I said algebraic structure :((((((((((((((((((((((((((((((((((((((((((((

Alessandro Codenotti

commutative semigroupoid if you really want to give it a name

But note that addition is almost never defined

ÍgjøgnumMeg

They're a basis for the monoid $\Bbb Z^+$ with multiplication

kek

s.harp

@ÍgjøgnumMeg have you bothered the studentwerk for an update on your application yet?

ÍgjøgnumMeg

@s.harp not yet, I think I will by the end of this week

@s.harp so I just rang and she said the chances are very slim that I'll get a place in the Wohnheim

lol

ÍgjøgnumMeg

14:25

just gonna find a WG

Secret

14:49

Idea extremism: Given an idea X that is only mentioned in passing by. If I like X, I will raise the voice of X so much that the whole universe will have to solve it before it will go back to silent again, even if the universe will be destroyed in the process of computational overload

Predicative mathematics, is one such idea

Ultradark

From now on I will work alone and publish alone. I will publish $5$ groundbreaking papers in the next $5$ years all less than $10$ pages. I will start $3$ new fields. I will solve the prize problems in $1$ paper.

And I will place the new mecca of mathematics in Nancy, France.

s.harp

@ÍgjøgnumMeg damn

MatheinBoulomenos

@ÍgjøgnumMeg why though? Usually international students have priority afaik

Alessandro Codenotti

@MatheinBoulomenos They have priority in Bonn as well, yet they told me at the end of August that they didn't have a place for me so that's not really a guarantee

MatheinBoulomenos

I see

ÍgjøgnumMeg

15:09

@Mathein idk, because the application went in too late, but I applied in early july so idk when you'd have to apply

lol

MatheinBoulomenos

@ÍgjøgnumMeg idk either

ÍgjøgnumMeg

Oh well, I'm still on the list so if I get one I get one

but I'll be searching for a WG in the meantime

Alessandro Codenotti

In my experience finding a WG is very hard if you can't physically visit them

ÍgjøgnumMeg

Fair, what did you do when you moved? >P

MatheinBoulomenos

yes, I can confirm (not from experience but I know how things work), people want to see you in person before you get into a WG

ÍgjøgnumMeg

15:13

sad

MatheinBoulomenos

that also applies to regular flats here in Germany, you usually have to visit them personally beforehand

Secret

A short comment about large cardinals and actual infinities in general:

They control what functions in the theory are total, computable and rate of growth

These in turn control consistency proofs and other model theoretic machinery

Kinda like the speed of light in physics tbh

ÍgjøgnumMeg

@Mathein faaak then I'll have to go into some kind of temporary accommodation before I find a place then

Alessandro Codenotti

@ÍgjøgnumMeg I found a private Studentenwohnheim for the first semester, while I visited a few WGs before finding one for the second semester

With private I mean not one owned by the university

ÍgjøgnumMeg

Fair enough

I remember hearing that the student housing situation in Heidelberg is slightly worse than elsewhere

MatheinBoulomenos

15:19

let me phrase it that way: Heidelberg is such a nice city that a lot of people want to live there

ÍgjøgnumMeg

lol

Alessandro Codenotti

lmao

Secret

Guys I need a quick sanity check: What is the weakest set theory that can prove that $\omega$ is a set assuming we have the class of naturals to start with?

ÍgjøgnumMeg

lol furnished room with a bed that only a dwarf would be able to sleep on

Alessandro Codenotti

$\mathsf{ZF}^-$ is enough

Secret

15:22

ok

MatheinBoulomenos

@ÍgjøgnumMeg iirc you get additional money from your scholarship if your housing is expensive enough. Why aim for a WG, then?

ÍgjøgnumMeg

@Mathein right, it's 250€ if my costs are more than 33% of the 850€ I already get

but housing is so expensive in Heidelberg!

lol

MatheinBoulomenos

ah I see

Heidelberg isn't the worst city in Germany, though. Munich or Frankfurt are worse, for example

ÍgjøgnumMeg

Yeah I bet

1 room apartments always look awful

tbh

brb! Time to walk home from work

Lucifer

Bvaria also

Secret

15:45

yesterday, by Semiclassical

cardinals are weird, so cardinal arithmetic is also weird

Here's something more weird: Induction itself is already strictly speaking, impredicative, thus we have an essential element in mathematics that is impredicative to start with

The field of strict predicativity, proposed by some guy in 1999, however is still too young to address these questions. I think the quest on the nature of infinity will pause in here for now

A couple of things can be summarised on the nature of infinity learnt so far:

1. There are basically 3 kinds of infinities: Potential, Actual, Absolute

Potential infinity is basically a shorthand for a process that goes on indefinitely, meaning for every step x, you can always find some step y later than x. Induction and halting theorem embody the essence of potential infinity

Actual infinity, is what happens when a notion of closure is imposed on potential infinity, so that the non ending process will be contained in an object, such that all its possible outputs are inside that object. All notions of transfinite cardinals, be it Russell, Dedekind, Alephs, Beths, Regular, etc. belong to this

Absolute infinite obeys the reflection principle, meaning that there is no single predicate that can single it out without referencing to some other kinds of infinity. But roughly speaking to give a feel for it, is that it is an unreachable endpoint which the never-ending class of cardinals and their generalisation tends to, but not only it can never be reached under any step, no single step is any closer to it. In set theory, the von neumman universe capture some notion of absolute infinite

2. The leap from finite numbers to $\omega$ can be argued to be as difficult as the leap from 0 to 1. This is because $\omega$ is an actual infinity, thus no finite process can reach it from below, though unlike the absolute infinite, you can get arbitrarily close to it with increasing sequences. With the exception of Quine new foundations, which cheat with the universal set, there is no way to establish its existence without circularity.

Likewise, going from 0 to 1 without exponentiation is impossible as no arithmetic on zero can allow you to reach 1 (set theory meanwhile allow you to reach it in one step via paring)

rapasite

hi can someone tell me why I don't see the latex fomula?

Secret

16:01

The next major leap is the church Kleene ordinal $\omega_1^{CK}$, which requires the law of excluded middle to ensure the countable ordinals form a linear order

After that, the next leap is $\omega_1$ which needs the powerset axiom, as it is regular thus sup cannot reach it. Continuing onwards, we eventually reach the first inaccessible cardinal, and then enter the large cardinal axiom territory

3. For every actual infinity that corresponds to a leap added into the theory, the theory becomes more impredicative as a cost. In exchange for impredicativity, more things become provable as the theory is strengthened

4. Actual infinities including the large cardinal axioms, thus control what functions in the theory are total, computable and their rate of growth. These in turn control consistency proofs and other model theoretic machinery

And therefore, the essence of all infinities are:
a. Involve a process that is guarantee to continue uninterrupted
b. There is some object that can hold the output of this process
c. Said object cannot be reached using the process alone. That is, they are "incompletable" using the process

It is c that is responsible, philosophically speaking, for the impredicativity that is introduced

ÍgjøgnumMeg

@Mathein there is a private Studentenwohnheim in Mannheim that has some places

maybe I'll go for that lol

ALannister

16:27

Can you accept one answer but award a bounty to another?

Semiclassical

sure.

ALannister

I might do that. two people who provided answers to a question I had a bounty for are starting to fight

I thought maybe it would appease them lol

Semiclassical

@ALannister given that LinAlg is getting up in arms about supplying code to an assignment, I sorta doubt it

ALannister

So awarding him the bounty wont get him to calm down?

rapasite

bounty? what you guys are talking about?

Semiclassical

16:30

i dunno. i try not to get too emotionally involved in this kind of thing tbh

ALannister

Well it's done

Semiclassical

if it's getting bad, get a mod involved and let them sort it out

ALannister

I gave LinAlg the bounty and accepted Cesareo's answer

Trying to be King Solomon here

Semiclassical

btw, I keep meaning to put up an answer to your other problem. (not for the bounty, but just because I find it interesting)

ALannister

Both of their answers are very instructive and helpful

rapasite

16:31

can I see the problems please ;)

ALannister

Ugh this is so stressful. Maybe I should just delete the question.

sure @rapasite

0

Q: Gauss-Newton Method not converging for my function

I need to solve the optimization problem $$ \min_{x\in \mathbb{R}^{3}}f(x) $$ where the function $f$ is defined as follows: $$ f(x_{1}, x_{2}, x_{3}) = \frac{1}{2}\left[\left(2x_{1}-x_{2}x_{3}-1 \right)^{2}+\left(1-x_{1}+x_{2}-e^{x_{1}-x_{3}}\right)^{2}+\left( -x_{1}-2x_{2}+3x_{3}\right)^{2} \ri...

optimization matlab nonlinear-optimization jacobian newton-raphson

and

Semiclassical

one neat bit: it seems to be the case that if one of the $r_i(\beta)$'s is a linear function of $\beta$, then the first iteration of Gauss-Newton will always project $\beta_0$ onto the plane $r_i(\beta)=0$

ALannister

0

Q: Matlab: Gradient and Hessian of a function with vector input of user specified size

I need to write a matlab m file that takes the following function of $x=(x_{1},x_{2},\cdots, x_{2n})$, and for $n=10$, $n=100$, $n=500$, $$f(x) = \frac{1}{2}\sum_{i=1}^{2n}i(x_{i})^{2}-\sum_{i=1}^{2n}x_{i}+\sum_{i=2}^{2n}\left[\frac{1}{4}\left(x_{i}+x_{i-1} \right)^{2} +\left(x_{i}-x_{i-1}\right)...

optimization vector-analysis matlab nonlinear-optimization hessian-matrix

Please don't anybody get mad at me for flooding. I'm not flooding or spamming. @rapasite asked me to post this links.

Semiclassical

so, for instance, Cesareo's seed is $\beta_0=(-0.18,-0.36,-1.54)$ and the first iterate is $\beta_1=(0.42,−0.25,−0.03)$

ALannister

I'd say I was just following orders, but that's never a good reason for doing something, and a lot of evil has been done by people just following orders.

Hopefully posting links in chat doesn't count as doing evil.

rapasite

16:35

I don't seem to be able to make latex work erf

ALannister

@rapasite what do you mean? You mean in here?

rapasite

$f(x) = \frac{1}{2}\sum_{i=1}^{2n}i(x_{i})^{2}-\sum_{i=1}^{2n}x_{i}+\sum_{i=2}^{2n}\left[\frac{1}{4}\left(x_{i}+x_{i-1} \right)^{2} +\left(x_{i}-x_{i-1}\right)

ALannister

You need to close your $'s

Semiclassical

and the linear function is $r_3(\beta)=-x_1-2x_2+3x_3$

ALannister

@Semiclassical just out of curiosity when you say $r_{i}$'s what are you referring to again? Or am I brain farting?

rapasite

16:37

aaa sorry I click the link everything is fine now lol

Semiclassical

ah, i'm using the notation in the Wiki page

ALannister

On my old laptop I had robjohn's chatjax all set up and everything was hunky dory. Now, every time somebody types something new, I have to render MathJax.

This is what I get for disappearing off the face of the earth for almost 2 years.

Semiclassical

so $f$ is a sum of squares and the $r_i(\beta)$'s are the functions being squared

ALannister

Yeah I remember that now from the wikipedia page

I've had a rough time for the past 2 years now and I'm finally ready to start doing math again.

I'm trying to anyway lol

Semiclassical

in which case, the neat observation is that $r_3(\beta_0)=-3.72$ but $r_3(\beta_1)=0$

and in fact $r_3(\beta_n)=0$ from then on

in practical terms, that means that every iterate after the first (i.e. $\beta_{n\geq 1}$ lies on the plane $-x_1-2x_2+3x_3=0$

ALannister

16:41

So that's the path of descent?

Semiclassical

which makes it substantial easier to visualize what's going on, since everything becomes 2D rather than 3D

well, it's the first step

ALannister

"path of descent" being a very imprecise way of putting it I'm sure.

I've seen graphical renderings of what happens in the method of steepest descent.

with the level curves and whatnot

It would be interesting to see that for Newton's Method or Gauss Newton as well

A little beyond the scope of what I need to do here but it would be interesting nonetheless.

Semiclassical

basically, the upshot is that, no matter what $\beta_0$ you choose, your $\beta_1$ will necessarily lie in the plane I just gave

and any future iterates will also lie in that plane

so the story from that point on is what is going on in that plane

as you've noticed, the iterates tend to move towards (1,1,1) along the line $x_1=x_2=x_3$

roughly speaking, the algorithm quickly deduces that the correct answer should be -somewhere- along that line

but it's much slower to figure out where on that line it actually is

But this is why I've been thinking I should write up an actual answer, lol

rapasite

interesting stuff :0

i did not use matlab in a long time

but you got your answer right?

now I am using the Julia language it is very good, but I guess your teacher want you to use matlab @ALannister

ALannister

Well, that would be a very informative actual answer, @Semiclassical.

Yeah @rapasite. I must admit I have never even heard of the Julia language. If it's new it's because, like I said, I've been out of the saddle for a couple of years.

I had a death in the family that affected me extremely badly and I'm just now beginning to crawl out of a downward spiral of depression and self-destructive behavior.

rapasite

16:56

new and not so new the version 1.0 came not so long ago, but it is free and the community is amazing

ALannister

@Mr.Xcoder you look very fancy with your top hat.

I realized also that my avatar is deceased.

Perhaps I need a new one.

Semiclassical

here's a visualization:

rapasite

sorry to hear that , it's good that you came out

Mr. Xcoder

@ALannister Thanks, I guess?

Semiclassical

the red points are 40 initial seeds, and the blue points is the first step of Gauss-Newton

ALannister

16:58

@Semiclassical what do the red dots and blue dots mean?

er. Nevermind beat me to it

Semiclassical

the red points are chosen randomly in the cube [0,2]^3

ALannister

Thanks @rapasite

random seeds

Semiclassical

i'm a big fan of randomly chosen examples lol

ALannister

Are you a probability theorist at all?

Semiclassical

well, if you do physics and go down the statistical mechanics / quantum route, you pick up probability along the way

so while i'm not formally trained as such, it is a rather natural tool for me

so anyways. you start with a bunch of seeds (the red points), and after one iteration you end up with the first iterates (the blue points)

and what that plane shows is that all the blue points are on that plane

ALannister

17:01

That's cool. I like how everything fits together like that and people in one field use tools from a bunch of other different fields.

Semiclassical

yeah

if you do one more iteration with those same points, you get this

ALannister

Goes to show that a lot of the distinctions between different fields are kind of artifical anyway, and it's useful to learn as many disparate things as you can, becayse you never know when it might come in handy.

Semiclassical

rapasite

can you start with a point [red] and show lots of iterations [blue] ;)

Semiclassical

blue points are the same as before, but the green points are the iterates you get from using those

ALannister

17:02

So bascially this time, the blue points served as your seeds?

Semiclassical

right

ALannister

ah cool

Semiclassical

and now you can see that they're all concentrating onto the green line

ALannister

So you could keep going, this time usingthe green ones as your seeds if you so wished.

Yeah, I see that. That's the line you mentioned before

Semiclassical

right. let me do that

note that the point at the very center of the plane is (1,1,1)

so everything is approaching the middle along the (1,1,1) directions. but it's not exactly moving fast

there are basically three directions here: perpendicular to the plane, in the plane and perpendicular to the line, and along the line

the motion of iterates along the first direction is instant---everything goes to zero immediately. the motion of iterates along the second is slower but still fast: everything quickly moves to be along the line

it's the motion along the line that gives the slow convergence.

ALannister

17:06

I don't mean to sound dumb here, but it's approaching along those three directions separately? which is fastest?

oh I see.

Semiclassical

fastest is the perpendicular to the plane. (since everything in red immediately went to being in the plane, i.e. zero perpendicular distance)

ALannister

you know this is cool, and I think you should do it as an answer. I'd definitely upvote because it's awesome and informative, and I like visuals.

Semiclassical

yeah, that's why I'm planning :)

ALannister

You da man, @Semiclassical. You da man.

rapasite

@Semiclassical can you make the story of one point(red) and let say 30 iterations(blues) or it does not f yorcode tx

*fit your code

Semiclassical

17:16

that's what i'm working on at the moment

@rapasite there you go (color is red/green but w/e)

one thing to note there is that the manner in which the red point gets sent to the plane is a bit unclear to me. it's not orthogonal projection, as that picture attests

so the story from red-> first green is a bit unpredictable. from then on it's pretty simple tho

rapasite

17:42

what is the scale? that seem to converge pretty fast ;)

it is beautiful!

Semiclassical

the cube is [-1,3]^3

so yeah, it converges fast

it's slow relative to the other directions, though. it moves onto the plane in one step, and rapidly approaches the line x=y=z

so that's why it counts as slow. what's basically going on at that point is that I'm basically doing a 1D Newton's method for the problem $f(x)=x^4$.

but that's really really flat

so you get slow convergence relative to what you'd usually expect for newton's method (which is that the number of correct digits doubles each time)

this is more like getting a factor of 2 closer to the center after each step (ignoring red). still fast but not as fast

ok, gotta go for now

rapasite

Last night I had an idea for a Primality test any

anyone wanna try to follow my though with me?

see you @Semiclassical

it start like this: any row of the Pascal triangle is a power of 11

second: the second number of any row is prime if and only if it divide(the division result is a integer) all others numbers in this row

Tobias Kildetoft

17:57

How do you interpret the rows from 5 and on to be a power of 11?

rapasite

imagine every number in a row is inside a column and those column are the same as the one we use the write numbers

Tobias Kildetoft

but the 5th row is then 15101051, right?

rapasite

for example 12 is 1*10 + 2*1

Tobias Kildetoft

ahh, so you want the 10 there to be a "digit"?

rapasite

yeah

Tobias Kildetoft

18:04

Right, then it fits

rapasite

that mean for example if you make your pascal triangle in base 11 every row is a power of the base+1 so 12

Tobias Kildetoft

I had not noticed that before. That is a pretty cool little fact

rapasite

and the problem off making number become digit is resolve for the fith row

but for resolving the problem of row 13 you will need to use the base=1717 and every row become the power of 1718

so I think this method is getting out off hand pretty fast

I mean to find out that 13 is prime you need to compute $1718^13$ in base 1717 then try to divide every digits in the answer still written in base 1717 by 13

I mean $1718^(13)$

Tobias Kildetoft

18:20

you need {} to get it all in the exponent

rapasite

lol thx

so what do you think is the asymptotic curve of this method?

$1718^{13}$

Ted Shifrin

binomial expansion of $(10+1)^n$

rapasite

why and is it worst than exponential? thx

when you say 10+1 in witch base? does it matter?

funny fact when you take 1 as the base every rows is a power of 2 and since every "number column

in base 1 are the same the somme of every number in the nth row of the pascal triangle =$2^n$

I mean the sum not somme

user76284

19:11

Does the class of all ordinals, $\text{Ord}$, satisfy the von Neumann definition of an ordinal except for the fact that it’s not a set?

Perturbative

@MikeMiller I'm not sure if you remember our discussion a while back where I was questioning why the authors of the textbook I was reading defined the $E^{\infty}$ page of a homological spectral sequence as $E^{\infty}_{p, q} = \operatorname{colim}_{r \to \infty} E^r_{p, q}$. I think the answer (which I'm sure you alluded to previously) is this:

For first quadrant spectral sequences we don't need to introduce a colimit we can just define $E^{\infty}_{p, q}$ as $E^{r_0}_{p, q}$ where $E^{r_0}_{p, q}$ is the stablilizing term, i.e. $E^r_{p, q} = E^{r_0}_{p, q}$ for all $r \geq r_0$, and in first quadrant spectral sequences all terms eventually stabilize for $r$ large enough, so it make senses to do this.

But if we don't have a first quadrant (homological) spectral sequence then we don't have this luxury of our terms stabilizing for $r$ large enough and we have to resort to defining the $E^{\infty}$ page of a (homological) spectral sequence as $E^{\infty}_{p, q} = \operatorname{colim}_{r \to \infty} E^r_{p, q}$

Similarly for a convergent cohomological spectral sequence one has $E^{\infty}_{p, q} = \lim_{r \to \infty}E^r_{p, q}$

Sorry I forgot to mention that I'm assuming the homological spectral sequence I'm talking about is convergent

Mike Miller

19:29

I think I explained last time that you are making implicit assumptions. The only way to get a map $E^r_{p,q} \to E^{r+1}_{p,q}$ is to assume that $Z^{r+1} = Z^r$, so that this is a quotient map otherwise $E^{r+1}$ is a quotient of a submodule of $E^r$; that is, $d^r$ should be zero on $E^r_{p,q}$ for large enough $r$.

This will be true if the spectral sequence is contained in a half-plane, with the differentials eventually exiting that half-plane (i.e., homologically graded and the right half-plane or the bottom half-plane).

I objected to writing "colimit" without making those assumptions clear, though I probably did not state my objection very well.

Jizhan Huang

20:09

I totally don't understand why my questions are downvoted. :-(
https://math.stackexchange.com/q/3332972/695163
https://math.stackexchange.com/q/3331539/695163
I asked and edited them seriously.

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$Mathematics$ Mathematics