Mathematics - 2017-01-08 (page 4 of 8)

arctic tern

06:00

[i|n] is already a function with a simple description, no need to express it in terms of other functions

TheGreatDuck

@arctictern It's not algebraically manipulable in order to truly state things regarding it.

we need to break it into smaller parts.

arctic tern

@TheGreatDuck your floor function expression is not manipulable

TheGreatDuck

it uses arithmetic operations though

arctic tern

and yes, [i|n] is totally manipulable

it is manipulated all the time in number theory

TheGreatDuck

really? give me an equality identity that relates it purely to known functions?

anyway regardless

arctic tern

06:01

@TheGreatDuck $$ \sum_d [d\mid n]f(d)=\sum_{d\mid n}f(d) $$

in fact, an identity I more or less just used while talking to euclid here in chat

it was literally used right under your nose

TheGreatDuck

no I mean an identity that represents something using [d|n] as something not using [d|n]

:p

arctic tern

why would you want to do that?

I mean, you can represent it by averaging dirichlet characters or complex exponentials

TheGreatDuck

because representing it as elementary function might make it trivial to prove things regarding it? Things we cannot seem to prove for the life of us.

arctic tern

but besides those two things there is no useful way to do it that comes to mind

euclid

@arctictern can you use telegram?

arctic tern

06:04

@TheGreatDuck no

@euclid what?

TheGreatDuck

fair enough

just seems like finding something better than either of our forms may be desirable

arctic tern

not going to happen. it's already perfect as-is, and no expression in terms of elementary functions (besides complex exponentials) will be useful.

TheGreatDuck

proven or an assumption you make?

arctic tern

experience and knowledge

the idea that the floor thing would be useful for theoretical manipulations is complete bonkers

Balarka Sen

just like how none of the known expressions for the prime counter function is useful

TheGreatDuck

06:06

i doubt there's any idea that's "completely bonkers"

Balarka Sen

lol

TheGreatDuck

that's not even relevant to mathematics.

euclid

@arctictern chat on telegram or skype. we can send pics of formula for better understand

arctic tern

@euclid I prefer to stay here on MSE.

euclid

ok

arctic tern

06:08

just wait for me to look at it tomorrow. you can always contact me later by responding to one of my comments posted at your question.

Balarka Sen

@TheGreatDuck that's just, like, your opinion, man

TheGreatDuck

@arctictern I was mostly drawing a connection with [d|n]. I'm saying using an elementary construction should be superior ultimately if we can find one that is manipulable.

arctic tern

you won't find one like the floor thing that is manipulable

euclid

@arctictern thanks.but i dont know when is your tomorrow because i dont know your time zone

TheGreatDuck

@BalarkaSen you claim that the life or death of a race is relevant to mathematics?

Balarka Sen

06:09

lolwat

arctic tern

@TheGreatDuck you said "any idea," if you wanted to quantify that to "any mathematical idea" you should have said so

TheGreatDuck

@arctictern "http://chat.stackexchange.com/rooms/36/***mathematics***"

Balarka Sen

i wasn't responding to your "relevant to mathematics" message, i was responding to the previous one

TheGreatDuck

I don't speak off-topic. ;p

arctic tern

@TheGreatDuck I was not responding to the room description, I was responding your claim about "any idea."

TheGreatDuck

06:11

i know

but we're in a room for mathematics

arctic tern

at any rate, chatroom is for both mathematical and general discussion alike

TheGreatDuck

what else would I be referring to?

arctic tern

as I wrote in the guidelines and room description

Balarka Sen

this is getting way too tedious

arctic tern

plus, obviously I am just making a point by being hyperbolic.

you are analyzing to death something you perfectly well understand

TheGreatDuck

06:12

a pretty pointless point seeing as how it goes off on a normal.

arctic tern

I just can't talk to you, can I?

sigh

TheGreatDuck

you can

geez

arctic tern

Nah. I find you very irritating. Making you one of about three people I've ever met on MSE over 5-6 years to do so.

TheGreatDuck

"you are analyzing to death something you perfectly well understand" Why do you assume I know that [d|n] cannot be expressed in other terms? Not everyone has studied number theory that rigorously.

arctic tern

I was not talking about [d|n], I was talking about you responding to the jews comment

TheGreatDuck

06:14

I wasn't really. I was just telling you not to take my post that literally.

as we're in a math chat and I'm obviously referring to math

I actually found the post quite annoying, but whatever. Let's just drop it.

Secret

This paper really needs an introduction, otherwise why would R modules with finite number of subalgebra interesting or useful

arctic tern

@Secret people like classifying things, subalgebras are things, it'd be interesting to some people to know when there are only finitely many

user400188

So anyway: I've been wondering for a while if there is a field of study in mathematics that talks about information loss. For instance; when a weaker theory becomes a stronger one we have restricted ourselves to a more specific case. If there a name for the topic of studying what is lost when we do things like this?

Secret

I see

arctic tern

@user400188 ask a logician / model theorist / set theorist

TheGreatDuck

06:20

@user400188 You mean a study of historical archaicnesses that we lose due to advancing further? Granted, not all things are just "tossed away" but it sounds like what you mean?

user400188

the logic room has been empty every time I've entered it

TheGreatDuck

^^^

user400188

That sounds like it actualy; ill look it up

TheGreatDuck

note: I'm just trying to describe it.

arctic tern

@user400188 are you using the word "theory" in a technical sense?

TheGreatDuck

06:21

better to look at how math was studied in the past

for instance

to the greek's numbers and distances (as in an actual space in real life) were one and the same

to them the number 1 was literally a line

(in geometry)

euclid

@arctictern please tell me why $\mathop \sum \limits_{m = 1}^d {e^{2i\pi mf(n)/d}}$ when we have $\chi (n) = {e^{2i\pi af(n)/(p - 1)}}$

TheGreatDuck

whereas now the "definition" of 1 is the number of elements in the set containing the empty set

arctic tern

@euclid Suppose $\psi$ is a generator and $\psi(n)=e^{2\pi i f(n)/d}$. Then $\psi^m(n)=e^{2\pi i mf(n)/d}$.

user400188

I'm not talking about history here; sorry

arctic tern

The set of $\chi$ satisfying $\chi^d=1$ is precisely $\{\psi^0,\psi^1,\cdots,\psi^{d-1}\}$.

TheGreatDuck

06:24

@user400188 so you wish to see what happens when we add new axioms to a system that restrict it's various rules?

specifically: what rules are lost?

Secret

Formal systems?

user400188

Now its getting hard to define what i'm looking for becuase I dont want to refer to the wrong thing

What I'm really looking for is information loss in a proof

TheGreatDuck

"information loss in a proof"

lolwat?

user400188

like lets say the proof starts with something then gets restricted and loses a lot in order to prove something

TheGreatDuck

when we prove something we gain knowledge.

user400188

06:27

I meant in the steps working it out

TheGreatDuck

can you give a concrete trivial example?

user400188

Most of those that ive given to others have left them confused: so I think it might be a bad idea...

Null

@TheGreatDuck lets build a great wall between USA and Mexico :)

TheGreatDuck

@Null excuse me?

what are you talking about?

user400188

But if I were to give one: it would be something like function composition When one function gets its domain changed and often restricted.

Null

06:29

@TheGreatDuck you said there are no ideas that are bonkers^^

TheGreatDuck

@user400188 I doubt you'd confuse me.

euclid

@arctictern why$ \psi(n)$ is $\psi (n) = {e^{2\pi if(n)/d}}$ as agenerator?

TheGreatDuck

@Null this is a math chat. Don't go off topic and be weird.

though to be fair trump did have a great idea

Null

@TheGreatDuck Let |x|<0

TheGreatDuck

(given proper funding it would be an awesome solution)

arctic tern

06:30

@TheGreatDuck topics are not restricted to math

TheGreatDuck

@arctictern but it's absurd to assume a statement isn't most likely referring to math.

user400188

I can never tell when these Trump Jokes are serious or sarcastic.

TheGreatDuck

i am serious

user400188

I guessed that by the second comment

arctic tern

@euclid The set $\{\chi:\chi^d=\chi_0\}$ is a cyclic group, so there exists a generator $\psi$, where the set is now $\{\psi^0,\cdots,\psi^{d-1}\}$. Then $\psi^d$ is trivial, so $\psi^d(n)=1$, so $\psi(n)$ is a $d$th root of unity, so there exists a $0\le f(n)<d-1$ such that $\psi(n)=e^{2\pi i f(n)/d}$

TheGreatDuck

06:31

if the US had enough money and were incredibly prosperous and we could afford it a wall would be the best case solution

unfortunately we are far from that

user400188

Its just kinda odd that the jokes are the same as the reality

TheGreatDuck

(goverment-wise)

arctic tern

@TheGreatDuck then say that, don't tell people not to talk about certain topics.

Secret

Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They are variously defined, for...

But that's for topology only

TheGreatDuck

@arctictern I'm just telling him not to assume a post is going off topic. I never said it was illegal to go off topic. I only said it was absurd to assume it isn't on topic.

user400188

06:33

Is that for me?

TheGreatDuck

anyway why did you bring this back up?

Secret

You want a notion of restriction that apply to the level of formal systems, I think that's what you are trying to find

arctic tern

@TheGreatDuck You wrote the command "Don't go off topic and be weird." Don't command others not to talk about things. Go ahead and call other people's assumptions absurd.

TheGreatDuck

@user400188 could you try to confuse me now?

user400188

It does sound (judging by its cover) that its what im looking for

Hmm ok. OIne sec while I bring up a page for some notation

Secret

06:34

So for each step of some general object, for each step you progress, you restrict the general case in order to arrive at a narrow conclusion, which is the strong theorem notion that you are trying to seek

and you want to check how much cases will be thrown away at each step, which should be captured by the restriction

TheGreatDuck

@user400188 do you mean like if we had the 5 postulates of euclidean geometry and added a sixth saying distance between any points is finite always?

that would then result in toroidal geometry and a "loss of knowledge" because now less surfaces fulfill "euclidean geometry"

user400188

@TheGreatDuck Not really; but the topic should be able to describe that
@Secret That sounds exactly what I'm looking for

TheGreatDuck

@user400188 sounds like the study of the manipulation of axioms

I have not learnt the term

Secret

I don't know how one can generalise sheafs to objects that does not have a topology though, but from what you said you seemed to be on the right track

TheGreatDuck

I just know it exists because it would be weird if it didn't.

@Secret is that what the study of formal systems is? Axiom manipulating?

user400188

06:37

http://math.stackexchange.com/questions/2078380/operation-of-function-and-relation-with-their-domain-and-range/2079260#2079260

Originally I was looking for something that can qualitatively and quantitatively describe the information loss when performing the steps in the answer on this page.

TheGreatDuck

restriction of a domain as a result of composition?

user400188

But I'm also looking for soemthing more general. That was just a spesific case that made me think of the question

TheGreatDuck

i don't think the two are related

user400188

Two whats?

TheGreatDuck

euclidean -> toroidal geometry

and "restriction of a domain as a result of composition"

euclid

06:39

@arctictern thank you very much.

user400188

Yeah I'm not sure if they are

TheGreatDuck

you say what you desire to study would cover both

hence your subject must be incredibly broad

"information loss" sounds like a good term

user400188

Yes it is. It should be able to describe information loss in all mathamatics. I'm guessing it would have to be related to formal logic in some way

Idealy it would start with logic and work its way through the rest of math

TheGreatDuck

but unless one can make euclid's postulates into a function returning a set of potential surfaces... then your subject covers function mapping and axiom manipulating

those two things are totally different

i think your subject might cover ALL of mathematics

user400188

You don't really need to worry about the functions themselves I think. You would only need to consider the functions definitions and how they change as you continue your working

TheGreatDuck

06:42

probably best to ask it as a question using those two things as examples and maybe link to the things @Secret showed you

"You would only need to consider the functions definitions and how they change as you continue your working" f never changes. h(f) just has a different domain than f

in reality it's a comparison of domains and ranges

user400188

I'm still working on this question by asking it in chat and thinking for a few hours. If I ask it too soon in stack Ill just get a heap of downvotes for not describing it well

TheGreatDuck

nah

If you use the two examples here and describe it like you are it will make sense

even say that it's information loss in general

user400188

To them and not to me. I still need to read up on this and think on it for a while if I want some benifit out of the question

TheGreatDuck

chances are, they'll just say "no it's not formally studied cause it's too broad"

Secret

Sheafs seemed to cover your idea well provided that topological sets are involved (which should be the case for all continuous maps), but if you want something on the formal system level, I am not sure if there is actually a discipline for that (still looking...)

TheGreatDuck

06:44

@user400188 you made it clear in about two seconds

just say the following

"is there a subject in math that studies the difference between two domains/ranges of functions, and also the narrowing/expansion of provable resulting from adding a new axiom to a set of pre-existing axioms"

heck, I might ask that as it's something I tend to do

Secret

@DHMO This may be relevant to you, although not on the number theoric topic: en.wikipedia.org/wiki/Axiomatic_system

TheGreatDuck

though for me it's negating/changing/asserting a random statement that is not an axiom

and looking at the results.

user400188

I do that too Duck. Often I will also take a for all statement and make it a for some to see what happens. The for some is true because the for all is.

TheGreatDuck

the way I figure, manipulating any system of logic or math operator and turning it into another thingy and seeing as how it compares to the original

for instance

"if and only if a function is constant does it have a derivative of 0 for all real numbers"

Secret

This might be relevant: en.wikipedia.org/wiki/Categorical_logic

user400188

06:50

Your very good at finding resources Secret

TheGreatDuck

what if instead of constant it were... piecewise constant, periodic, etc.

A healthy knowledge of the piecewise constant one and how to use it to solve diff eq is a really convenient way to avoid the laplace transform (sometimes).

for the other times... well at least those are roughly a 1/3 of the time.

:p

user400188

That seems like adding information Duck. Also why would you avoid a laplas. They skip a lot of work

TheGreatDuck

take the diff eq:

y'' + 2*floor(x)y' + floor(x)^2y = 0

double root of -floor(x)

solution has form of

y = C(x)e^{-floor(x)x} + D(x)xe^{-floor(x)x}

C(x) and D(x) are piecewise constant functions such that the individual solutions are continuous

y = ce^{-floor(x)x + floor(x)^2/2 + floor(x)/2} + dxe^{-floor(x)x + floor(x)^2/2 + floor(x)/2}

^^^the final solution

I solved it with the method of the roots of the auxilliary equation in the alternate system and converted it back by finding a continuous solution

I cannot prove that works but it has never failed to work and so I choose very strongly to believe it works. I also know many things that back it up but I cannot necessarily express them.

feel free to solve that with the laplace transform @user400188 took me 30+ minutes

the other method is relatively trivial from my perspective

user400188

From the looks of it it definitely is trivial that way.

TheGreatDuck

and since 90% of the laplace transform needing equations involving piecewise constants use the heavyside function on the RHS...

it is usually incredibly easy by using the method of undetermined coeficients

user400188

06:58

Good point

TheGreatDuck

take y'' - y = h(x) where h is heavyside

the lone term according to the method is h(x)

the roots yeild complex numbers though

which means

Secret

As for me, I am looking for something even more general, more like a "category theory of axiomatic systems". Inspired by dynamical systems, I want a morphism on the system of axioms itself, so you end up with a very wild algebraic system where the axioms changes at each step of the computation in some determinstic fashion. However this is probably way too wild for my level thus I am going to worry about that later

TheGreatDuck

y = sin(x)C(x) + h(x) is one individual solution (for instance)

@user400188 I have yet to find a way to solve this functional equation for C when using sin or cos. e^x is trivial cause I can divide by e^x and preserve the LHS continuity.

user400188

@Secret That sounds like a mind fluck

TheGreatDuck

in that case

i default to the laplace transform

user400188

07:01

@Secret But also really cool

TheGreatDuck

to be fair, this was originally a way for me to avoid splitting the integral as I found it incredibly esoteric

Secret

I am used to mindfuck concepts: Time travel, higher spatial dimensions, division by zero etc. I love studying the unknown and get them explained

TheGreatDuck

I took differential equations last semester and realized the method applied to them and made more sense to use in that context.

:p

Secret

I wonder if I can ever look directly at an lovecrafian creature without going insane...?

user400188

Hang on; if the axom changes at each step of the computation doesnt that make literaly everything correct and dare I say "pointless"?

TheGreatDuck

07:03

@Secret you cannot look directly at a lovecraftian creature as they do not give off visible imagery.

user400188

Secret you probaly would find that it was one of them tht put all those ideas in your head

@TheGreatDuck Anyway I gota go. I'll be sure to try your method sometime. See you later too @Secret

Secret

no, because the change is governed by a free variable you plugged in, thus in a sense it is a massive generalisation of f(x)

TheGreatDuck

@user400188 I would assume the axiom changes in a restricted way.

Secret

I am not sure how it will be useful, however

TheGreatDuck

dare I say... the axiom changing is governed by axioms?

(no context. just a thought)

user400188

07:04

It would have to be or laws of logic

TheGreatDuck

what if every time you construct a line the parallel postulate negates?

Secret

yeah, one can probably say that there are basically just another layer of axiosm governing an axioms, hence it is probably nothign something new. Does nto make it less mindfuck if you think about it

TheGreatDuck

in that sense one might say "the parallel postulate is true if and only if there are an even number of lines present"

an alternate changing axiom

user400188

Thanks for the help everyone bye

TheGreatDuck

but completely logical

Secret

07:05

bye

TheGreatDuck

(though absurd)

bye

@Secret shameless plug to a question vital to what me and @user400188 were discussing. They might find it intriguing. math.stackexchange.com/questions/2078995/…

Secret

One of the basic criteria for something called a derivative is that it is usually a measure of linear (1st order) change. Of course it is ok to have a derivative that does not obey some notion of linearity, but then it will definitely be something that can only be said to be analogous to a derivative. This does not mean it is not interesting, for I have not touched calculus for quite some time since year 3, thus cannto said much on how it behave other than a notion of change is measured

Free variables and bound variables

In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where substitution may take place. Some older books use the terms real variable and apparent variable for free variable and bound variable. The idea is related to a placeholder (a symbol that will later be replaced by some literal string), or a wildcard character that stands for an unspecified symbol. In computer programming, the term free variable refers to variables used in a function that are neither local...

There's a lot of parallels between computational science and mathematics, since they are both mathematics in a way

TheGreatDuck

@Secret Oh I know how the piecewise constant one behaves. I actually tricked myself into believing it was the integral for a while. I've been using it for integration methods for some time. The issue is that I wish to construct a set of derivative axioms that govern the properties of the derivative where all the statements are independent so that regardless of how I negate one statement I can conclude I have a consistent systems.

In fact, I upgraded my symbolic derivative calculator I needed for a computer graphics tube plot model with that feature because it prevented sharp corners from having undefined derivatives.

(cause the piecewise constant function contributing jumps is taken as zero instead of 0 with undefined points)

Secret

> The issue is that I wish to construct a set of derivative axioms that govern the properties of the derivative where all the statements are independent so that regardless of how I negate one statement I can conclude I have a consistent systems.

This is a much harder question

TheGreatDuck

i know

I have a strong feeling my axiom will require other axioms we might have never even considered otherwise

whether they are useful is another issue altogether

interesting... if I combine mine with the differential algebra axioms

the piecewise constant negation does not create a contradiction as far as I know.

so I guess in some instances I could use that

Secret

07:20

Characterising the minimum number of independent axioms you need to define a structure in general is a difficult question. For example, in the meadows paper I have read, the authors have spent 3 papers showing that their system only need 10 axioms of some given form. The wheel guys also did something similar via variety and equations to show that the axioms that define wheels must have a certain form in terms of polynomial of the variables.

The general discipline for questions like these is introduced in this wikipedia link https://en.wikipedia.org/wiki/Axiomatic_system which also relates …

TheGreatDuck

oh

so the issue is more than just reformulating the current axioms?

hmm

interesting

I might ask a new question. It might make my current question less... important

periodic function with certain restrictions/piecewise constant functions are the most important as their differential equations supposedly act like ours.

Secret

To do this rigorously, you need to show that given a set of axioms (an axiomatic system) you introduced is consistent. Next you need to use some kind of deductive logic to figure out which axiom deduce which theorems to check their independence, then work your way through the signature of the algebra, the equations (that is the axioms) and many other things that I don't really understood, to establish the form the axiom should take for a given structure
and the minimum number of independent equations needed

TheGreatDuck

(my doubt is in a lack of experience with periodic things)

Secret

Your example works mihgt be because your differential equation has a periodic solution (or a superpositon of them). however not all differential equations have periodic solutions or their superpositions, thus you need to check carefully

TheGreatDuck

"To do this rigorously, you need to show that given a set of axioms (an axiomatic system) you introduced is consistent." I know that the differential algebra axioms are sufficient. They already exist.

@Secret I'm not claiming periodic with the differential algebra axioms is consistent. I'm saying I only care about those two as in the resulting system differential equations are solved in the same manner. The steps work out the same. Plus, anything else results in all functions having derivative 0 because of reasons so...kinda pointless.

actually things of the form f(x)^2 for arbitrary f also work

but technically it's a superset of absolute value which is a superset of periodic

:p

Secret

07:25

hmm ok

TheGreatDuck

hence if tacking them onto differential algebra's axioms results in consistency...

I literally no longer care about the other question for at least another 3-4 months

at that point I'll probably run out of things to look at in that region and find another system (or just move on to other recreational things like number theory)

euclid

07:44

@Ramanujan hi

Shashaank

@TedShifrin I have , I will say good learning of calculus , vector calculus , Fourier Series , Laplace Transform Differential Equations , a very little of PDE and extremely less of complex analysis. (trying trishtam Needham).In physics I've done Special Relativity and 4 vectors , trying to begin with Tensors . Presently 2nd year Undergraduate !

Ramanujan

@euclid hi!

TheGreatDuck

@Shashaank here's an interesting exercise for you. Solve y'' + 2*floor(x)y' + floor(x)^2y = 0

(I noticed differential equations and it's a neat problem to solve)

DHMO

@TheGreatDuck what the hell is that

TheGreatDuck

@DHMO a differential equation.

DHMO

07:50

I know

TheGreatDuck

okay, then I don't get what you mean.

euclid

@Ramanujan i have some questions in number theory can you help me?

TheGreatDuck

@Ramanujan I know there is an actual ramanujan irl that is famous. Are you that person or just another random person using famous names from history?

:p

Secret

A dead person cannot be reborn in exactly the same manner, or is it?

This Ramanujan however do like to ask iterated root questions as well as infinite series, though

TheGreatDuck

i wasn't really sure when he was around

i only hear about him in reference to more modern ideas.

euclid

07:55

@Ramanujan i have some questions in number theory can you help me?

TheGreatDuck

@Secret math.stackexchange.com/questions/2088483/… Just want to make sure it is the clearest possible.

Ramanujan

@TheGreatDuck I like most of mathematicians but I like Ramanujan the most,so iam just a 16 years old teenager who want to become like him

@euclid just ask,dont ask to ask

TheGreatDuck

ah

fair enough

Ramanujan

If not I then someone other will help

TheGreatDuck

@Ramanujan i think he was meaning more along the lines of "are you good at number theory"

wait...

16 years old and asking about infinite series

wow

ambitious. I like that. :)

Secret

07:58

rational periodicity might get around that superposition problem mentioned by DHMO

TheGreatDuck

the what?

euclid

@Ramanujan look at this please

0

Q: brun's method and primitive roots

let be $N(x) = \sum\limits_{d|p - 1} {\mu \left( d \right)} {F_x}(d)$ that $p$ is a prime number and ${F_x}(d) = \frac{{{x^2}}}{{4d}} + O(x{p^{\frac{1}{2}}})$ that $x+1=g(p)$ and $g(p)$ is the least primitive root modulo $p$. Applying Brun's method to $N(x)$ in conjunction with ${F_x}(d)$ in orde...

analytic-number-theory

Secret

recall how DHMO said periodic functions are not closed unde sums unless some condition is obeyed?

TheGreatDuck

oh yeah

i thought you meant just now

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$Mathematics$ Mathematics