/sci/ - Science & Math

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>>11152037

Which book is "easier" Herstein's Topics in Algebra or Lang's Linear Algebra.

Do I need to know Linear Algebra to go through an Analysis book, specifically Rosenlicht's

>>11166717
Hersteins is better. You don't need linear algebra to learn analysis. But an understanding of spaces might help.

>>11166717
>Which book is "easier"
>Do I need to know
Why are undergrads like this?

>tfw no absolute infinity in ZFC

>>11166717
>>11151480

>>11166749
I plan on going through all of them but I have to determine in what order.

Casey, an infinite set is a set whose elements can only be counted with an unbounded iterator.

How do I prove [latex]u_{0}=2;u_{n+1}= u_{n}-\frac{1}{u_{n}}[\latex] is not bounded?

>>11166757
>[math]u_{0}=2;u_{n+1}= u_{n}-\frac{1}{u_{n}}[\math]
I'm retarded

>>11166759
>[math]u_{0}=2;u_{n+1}= u_{n}-\frac{1}{u_{n}}[/math]
This is embarrassing

Let [math]I[/math] be the incentre of acute triangle [math]ABC[/math] with [math]AB\neq AC[/math]. The incircle [math]\omega[/math] of [math]ABC[/math] is tangent to sides [math]BC[/math], [math]CA[/math], and [math]AB[/math] at [math]D[/math], [math]E[/math], and [math]F[/math], respectively. The line through [math]D[/math] perpendicular to [math]EF[/math] meets [math]\omega[/math] again at [math]R[/math]. Line [math]AR[/math] meets [math]\omega[/math] again at [math]P[/math]. The circumcircles of triangles [math]PCE[/math] and [math]PBF[/math] meet again at [math]Q[/math].

Prove that the lines [math]DI[/math] and [math]PQ[/math] meet on the line through [math]A[/math] perpendicular to [math]AI[/math].

Prove that the fraction [math]\displaystyle \frac{21n+4}{14n+3}[/math] is irreducible for every natural number [math]n[/math].

In a plane there are [math]100[/math] points, no three of which are collinear. Consider all possible triangles having these points as vertices. Prove that no more than [math]70\%[/math] of these triangles are acute-angled.

Prove that a sequence [eqn]\displaystyle 0\longrightarrow A \stackrel{\alpha}{\longrightarrow}B\stackrel{\beta}{\longrightarrow}C\longrightarrow 0[/eqn] of [math]R[/math]-modules is exact if the induced sequence [eqn]0\longrightarrow \text{Hom}_{R}(C,N)\stackrel{\beta^{*}}{\longrightarrow}\text{Hom}_{R}(B,N)\stackrel{\alpha^{*}}{\longrightarrow}\text{Hom}_{R}(A,N)\longrightarrow 0 [/eqn] is exact for all [math]R[/math]-modules [math]N[/math].

>>11166749
curse of the low iq

what is the [math]0=1[/math] supposed to represent?

>>11166836
alright so apparently it's a chart of cardinals, but still looking for what is represents

>>11166772
Jesus Christ that looks really fucking hard. I have no idea how to even start that monstrosity.
>>11166782
O MY BASIC OPPOSITE CATEGORY RESULTS, LEND ME YOUR STRENGTH

>>11166644
hey how about I make a set that contains all natural numbers

>>11166839
another chart. if anyone could provide some information that would be great.

Sup boyos, I could use some help on a fucking problem that is currently eluding me. Let R be an integral domain with identity, and let [math]\mathbb{F}[/math] be the field of fractions of R. R can be considered as a subring of [math]\mathbb{F}[/math] via the canonical map [math]r \mapsto (r,1)[/math]. Let S be a subring of [math]\mathbb{F}[/math] containing R. I am trying to prove that the field of fractions of S is isomorphic to [math]\mathbb{F}[/math]. I've been working on this problem for a few fucking hours and I've no idea how to crack it. Any ideas?

>>11166892
Universal property.

>>11166914
kys

>>11166770
-2*(21n+4)+3*(14n+3)=-1

>>11166859
hey how about you can't lol

found an explicit well order of the reals, but i think i'll keep it to myself

>>11166914
That's what I thought my dude. I used the map [math]\phi: R \rightarrow \text{frac}(S)[/math] defined by [math]\phi(r) \coloneqq (r,1)[/math], and then this induces (by the universal property) a map [math]\psi: \mathbb{F} \rightarrow \text{frac}(S)[/math] defined by [math]\psi(a/b) = \phi(a)\phi(b)^{-1}[/math], but this latter map is not an isomorphism... In particular, it isn't surjective. That's why this problem has been bodying me my dude

>>11166947
Ahem:
No anon, use [math]i : S \hookrightarrow \mathbb{F}[/math].

>>11166859
All right then, write it out for me

>>11166892
You want to build a map from frac(S) to [math] \mathbb{F} [/math].
Elements of S are fractions [math] \frac{a}{b} [/math], so elements of frac(S) are double fractions [math] \frac{ \frac{a}{b} }{\frac{c}{d}} [/math].
Elements of [math] \mathbb{F} [/math] are fractions.
What is an obvious way to view double fractions as single fractions? Why is this ring isomorphism?

>>11166959
le inclusion map? Fuck... I didn't think about that. By the universal property the induced map is [math]\psi(a/b) \coloneqq i(a) i(b)^{-1}[/math] for [math]a,b \in S[/math]. This map is injective, as the kernel is trivial. To show surjectivity, R is contained in S, so any element in [math] \mathbb{F} [/math] is the product [math](a,1)(1,b) [/math] for [math]a,b \in R \subseteq S[/math], so surjectivity can be shown as well. It's also a ring homomorphism, so gg. This looks great, but I feel like I'm "leaving some elements" out by using the inclusion map... Thanks a heap though, anonymous

what the fuck is a homology? why does /mg/ keep talking about it?

>>11167061
It's retarded bullshit that turns to be really useful in actually important problems.

>>11167055
>he does the smallest of first steps with the diagram and then concludes the proof with algebra
Please have mercy on my little hints, they don't deserve to be used like this. Do the full diagrammatic argument like a normal person.

>>11167073
wut is it

>>11167061
https://en.wikipedia.org/wiki/Homology_(mathematics)
A sequence of groups associated to some other object we want to study. Comes traditionally from topological spaces but can arise in other contexts. The definition is not intuitive so you're not really going to understand it without having actually gotten your hands dirty.
Homology/Cohomology are pretty powerful tools and relatively modern.

>>11167082
"sequence of groups associated", so isnt that kind of everything? wahts a topological space?

>>11167061
Singular homology:
Let [math]X [/math] be a topological space.
For each natural number [math] n [/math] consider the collection of all maps from the n-simplex to [math] X [/math] (we call these maps n-simplices in [math] X [/math] or simply n-simplices).
Let [math] C_n(X) [/math] be the free abelian group generated by these maps (n-simplices), so an element of [math] C_n(X) [/math] is a finite formal sum [math] \sum_i a_i \sigma_i [/math], for [math] a_i \in \Z [/math] and [math \sigma_i [/math] an n-simplex.
Define a map [math] d_n : C_n(X) \to C_{n-1}(X) [/math] by sending an n-simplex [math] \sigma [/math] to the alternating sum of its faces, [math] \sum_{i=0}^n \sigma|_[\hat{v}_i] [/math], where [math] \sigma|_[\hat{v}_i] [/math] denotes the face of [math] \sigma [/math] spanned by the vertices other than [math] v_i [/math].
Extend this map linearly so it's actually a group homomorphism.
Then a simple computation shows that [math] d_{n-1} \circ d_n = 0 [/math], so the image of [math] d_n [/math] is in the kernel of [math] d_{n-1} [/math].
Thus (since [math]C_n(X) [/math] is abelian), the quotient of the kernel by the image is a well defined group.
We call this group [math] H_n(X) [/math], the nth homology group of [math] X [/math].

>>11167091
>wahts a topological space?
https://en.wikipedia.org/wiki/Topological_space
It's like a really abstract generalisation of the notion of space. It's weak enough so that you have no notion of angle, distance, or lines, but you can still talk about things like continuity.
If you don't know what a topological space is then you've got a while to go before reaching homology anon.

>>11167150
>consider the collection of all maps
Here map is understood to mean "continuous function"

>>11167150
I really fucked this
>alternating sum of its faces
Should be [math] \sum_{i=0}^n (-1)^i \sigma|_{[\hat{v}_i]} [math]

>>11166772
trivial corollary of peano axioms
>>11166782
tranny

>>11167150
I was talking with a friend who knows algebra better than I do a few months ago, and he introduced me to the concept of a simplice. One of them was a triangle made of basis vectors. and i think the higher dimensional ones were too. I just remarked... so its a triangle? in confusion, but are you saying that shapes like triangle can be generalized to corresponding simplices in any given space?

What do topological spaces look like, both abstractly and symbolically. Are vector like objects still used for descriptions?

What if we used group theory on psychology? Gathered a list of interrelated illnesses and then arrange them to find operators and axes

https://www.youtube.com/watch?v=uJqbHaFqjmI
Could you have solved it, /sci/?

>>11167209
https://en.wikipedia.org/wiki/Simplex
A 0-simplex (point) is a single vertex.
A 1-simplex (line segment) is two vertices and the line segment between them
A 2-simplex (triangle) is three vertices, the lines segments between them, and the triangle bounded by the line segments
A 3-simplex (pyramid) is four vertices, the line segments between them, the faces between the line segments, and the volume between the faces.
In general an n-simplex is like an n-dimensional triangle.

A map from an n-simplex to a space is almost like viewing the n-simplex as embedded in your space (you can imagine a triangle on the surface of a sphere for example). The place where this breaks down however is that maps can send multiple points to the same point in your space, so the embedding can be a weird, twisted, self-intersecting one.

>What do topological spaces look like, both abstractly and symbolically.
Read the Wikipedia article I linked.

>>11166836
>>11166839
>>11166886
0=1 has the highest consistency strength because 0=1 implies Con(ZF+any large cardinal property). Kanamori is just being cheeky.

>>11167230
They would just keep redefining the illness to exclude the data that agrees with their objective.

Wait a minute..

>>11167230
What are the elements of your groups and what are the operations? What insight do you expect?

>>11167241
"map from n-simplex to a space" - how is the mapping chosen/what equation/set of equations defines a map. Like, we wanna map 2-simplex to spherespace, what equation defines the map.

>read wikipedia article
Woah. It looks like set theory organizing bundled of elements of a field, no space at all required. Then it just shows up in various geometries depending how the field is graphically laid out. Is that correct line of thinking?

Also what are topological spaces with algebraic structure about

>>11167272
Well im speaking loosely with the math terms cause the idea is vague so far, but I was basically thinking about how many illnesses seem to be diametric, like autism an schizo or depression and anxiety, and how they can be diametric on multiple seemingly different axes to different illnesses, and how the flipflop dysregulated nature of bipolar and glitchy error nature of OCD seem to be patterns that repeat in a lot of illness when distilled to their logical/defining essence

How do I define the set of all possible combinations of two sets, combinations as in n-tuplets of matching up elements. Would I list every single tuplet to define this object?

>>11167285
>how is the mapping chosen/what equation/set of equations defines a map
It depends on how you parameterise (label) the points of your n-simplex. The standard definition (pic related) allows you to describe points by their n+1 coordinates. Then you can describe a mapping as a continuous function of those coordinates. I'm not going to actually write one out for you because it's pretty tedious (try it yourself)
In practice with singular homology there are uncountably many of these maps, so we need to use other techniques to actually compute homology groups, rather than directly talking about explicit maps.

>organizing bundled of elements of a field
Be careful what you mean here, a "field" has a precise mathematical meaning, which has nothing to do with topological spaces (as well as a different meaning in physics).

>no space at all required
Topos is literally the Greek word for space. Topology is an abstraction of the idea of space in the sense that it captures the notion of points being somehow related to each other. There's not really much point thinking about topological spaces until you've done basic analysis (metric spaces, etc). Before then the definition is incredibly opaque and doesn't seem motivated. Once you have done some analysis you see that often the basic properties of space which we actually rely upon are not distance or angles but open sets, and so it makes sense to forget about things like distance and angles and talk about spaces where you only have a notion of openness.

>Then it just shows up in various geometries depending how the field is graphically laid out.
???

>Also what are topological spaces with algebraic structure about
Think of for example the number line. The geometric properties of the number line behave well with addition/multiplication. Like if you have a number [math]x [/math] and add 1 to it, you get something close to what would happen if you picked a number close to [math]x [/math] and added 1 to that.

>>11167288
I'm not really seeing the connection to group theory. This sounds more like like data analysis which you might tackle with something like dimensionality reduction.

>>11167304
>n-tuplets of matching up elements
n-tuplets of pairs?
>Would I list every single tuplet to define this object?
Depends what you mean by define. If it's obvious that such an object exists then (to most people) it's enough to provide an unambiguous description. This wouldn't necessarily have to involve actually listing them out.

>>11167333
Well there could be operation groups, like some illnesses are associated with others under certain situation like time or cortisol threshholds. And I think data science is only needed to find the statistical essence of an illness through mris and stuff but i think psychology actually does a good job on distilling it, at least in my semi schizo headcanon

>>11167344
>n-tuplets of pairs
no, i guess combo is ambiguous. I guess what i meant here was selecting one element from each set and associating it with one other from every other set. so for n sets, you have bags of n elements, each one a unique(?) combo

>depends what mean define
define well enough that i can prove some other set has the same number of combos, or shares some of the same combos or whatever relationship

>>11167326
>Then you can describe a mapping as a continuous function of those coordinates.
But how is that mapping a simplex? Isnt that just mapping a space to another and then find out what happens to the basis vectors and thus the simplex(based on the bases) in the new space?

>field
im referring to field as a set of numbers which are associated to other numbers in the field under binary operation

>Topology is an abstraction of the idea of space in the sense that it captures the notion of points being somehow related to each other.
thats what i meant by bundled elements of a field, and no space at all required

>o>Then it just shows up in various geometries depending how the field is graphically laid out.
>???
Like, you have a set of n numbers, maybe some of them on different indices, referred to as a topology. Based on the way the number line is drawn and the indices are given geometric dimensionality, the topology appears as a space

>think of the number line
that confuses me on multiple layers

Do you as a mathematician think forwards or backwards? And by this I mean, do you try to see what properties an axiom lends itself to or implies, or do you check properties against an axiom? How do the required thinking styles differ, what are the benefits of either method?

Post examples of beautiful geometric proofs being algebraically disgusting and vice versa, if that ever actually occurs (im unsure)

>>11167460
>beautiful geometric proofs being algebraically disgusting
https://en.wikipedia.org/wiki/Nielsen%E2%80%93Schreier_theorem#Proof

>>11167487
they just smush H into P?

>>11167511
No

>>11167520
then what i dont get it

Barnett's infinity is the largest known infinity. A set is said to be "Barnett Infinite" if there exists a bijection between the set and the set of Barnett integrable functions [math] \mathcal{B} [/math]

[math] \mathcal{B}=\left \{f \in C(\mathbb{R})~:~\int \int \int f(x)~{\mathrm{d}^3 x}\leq \frac{-e^{i\pi}}{11.999...}\right\} [/math]

An interesting theorem is that a Barnett infinite set is infinitely more infinite than any countable set (Barnett, 2012).

>>11167528
The picture depicts a covering space
Read chapter 1 of Hatcher

>>11167529
0/10
Stale memes

>>11167531
i'm reading linear algebra done right by sheldon axler right now, is that a good follow up book?

>>11167546
Not really.
You need group theory and topology.

If you want to learn some group theory read Artin.

>>11167555
I think i'm more interested in topology because >tfw engii so what should I read for that

>300k starting

>>11166756
Who is Casey?

>>11167445
Mathematicians don’t work with axioms...
You have an idea of something you want to prove, then you try to see what conditions would be the best suited to proving such a result.
If that works, then you try to weaken the assumptions or strengthen the conclusion and make definitions to encapsulate the properties that you have used. This weakening of assumptions is what gave us the definitions and axioms that we have today. People did not sit around, write axioms and then try to draw conclusions from these axioms.

sorry bros I was wrong about riemann

>>11167714
>shitposts come from the internet

>>11167717
disregard that I suck cocks

Can /sci/ find the flaw in my proof? The theorem states: "Given [math]F,G[/math] closed subsets of a topological space [math]X[/math], and continuous mappings [math]f: F \to Y[/math], [math]g: G \to Y[/math] such that [math]f(x) = g(x)[/math] for all [math]x \in F \cap G[/math], show that the function [math]f \vee g: F \cup G \to Y[/math] is continuous".

(Probably bogus) proof sketch: Let [math]h = f \vee g[/math]. Given [math]C \subset Y[/math] closed, it's easy to show that [math]h^{-1} [C] = f^{-1} [C] \cup g^{-1} [C][/math]. Since [math]f,g[/math] are continuous and [math]C[/math] is closed, this must be a finite union of closed sets, therefore a closed set. I conclude that [math]h[/math] is continuous.

The problem is that my proof makes no use of the fact that [math]F,G[/math] are closed, so there must be something I'm missing here. It eludes me tho...

Hints appreciated!

>>11167946
f^-1 C is closed, but a priori only in F. you need it to be closed in the whole space

>>11167559
Learn analysis first.
Real analysis -> Metric spaces -> Topology

>>11167445
>do you try to see what properties an axiom
Didn't you say "you as a mathematician"?

I'm a retard. I battled depression for 10 years after HS before finally getting into uni. Someone spoonfeed me a nice beginner friendly book for analysis

>>11168024
Tao or Stein-Shakarchi

>>11168064
those were two easiest book downloads ever
thanks, now which one holds my hand the best and treats me like a kid or are both equivalent
note: this doesn't mean simple, it means guided

>>11168118
Probably Tao

>>11167205
>trivial
so, you don't know how to do it then.

Feeling really, really down. Someone cheer me up pls.

>>11168184
https://www.youtube.com/watch?v=6xHzZRnilLA

>>11168184
Do you have any publications? You can't hope to get into a top program with just a 'good resume'.

>>11168184
You have back-up schools, right? Surely you didn't go all in on fucking Stanford.

Yes, energy for 200 years till we have material degradacy issues if fucking sufficient as perpetum mobile.

>>11167285
dude the guy you're listening to has autism. he has no idea what he's trying to explain.
a topological space is simply a set where we can talk about things being "attached to" or "near" other things. an example which everyone learns first is called a metric space, which is just a set where you can measure the distance between any two points. there are topological spaces which are not metric spaces, but they tend to be pretty weird.
a topological space doesn't really care about shape, just about the nearness. for that reason people call topology "rubber sheet geometry" because you can twist and bend a topological space, and as long as you don't tear it or glue it together it doesn't really change.
so just think about it like some weird wavy surfacey/curvy thing.
now, what's homology good for? turns out it's really hard to tell how many holes a topological space has. why is it that we cant bend/twist a donut into a sphere? cause the donut has a hole. but how do we describe the hole mathematically?
one way is homotopy, which is probably a lot easier to understand than homology.
but the other way is homology. homology lets you slap a simplex into your space and ask what simplices it can be written as. well, if you slap a triangle onto a donut which goes around the hole, then try to break it up into other triangles, one of these triangles still has to go around the hole (there's a proof needed there). but on a sphere, if you slap a triangle on, you can just pull all the vertices together and make it super small til the triangle is a single point. this is what homology is distinguishing (via a "quotient from a boundary map").
i'll be honest with you, it takes a while to learn all the math to make the formal description make sense. but an intuition is important as a first step.

>>11168184
>applying to a shithole like stanfurd
you get what you deserve bitch

I don't think I want to move in with mommy this summer. I'm trying to get a programming internship.

What other kinds of jobs should I look for as a math major?

>>11168184
This seems like bullshit, although I don't know why you would make something like this up. Stanford hasn't even closed their application period yet; they're not going to be seriously looking at applications in November.
The only way anyone could conceivably be rejected this early is if they applied with a 2.4 GPA and got bounced off the computer or something stupid.

Anyone else graduate (undergrad), get a job, and subsequently just stop doing math? I told myself I wouldn't let it happen and that I'd go to grad school after working for a bit, but after working all day my motivation to sit and read a math text is nonexistent.

For which a does sin x = ax +b have exactly three solutions?

>>11167150
does the definition use simplices because of their topological properties?

>>11168561
Jokes on you, people with graduate degrees in math don't get jobs

>>11168588
69

>>11167529
>2012
holy shit i need a life

>>11167413
>Like, you have a set of n numbers, maybe some of them on different indices, referred to as a topology. Based on the way the number line is drawn and the indices are given geometric dimensionality, the topology appears as a space
That's the 'point', ha ha.

>>11168442
Thanks for trying to explain.
>homology lets you slap a simplex into your space and ask what simplices it can be written as. well, if you slap a triangle onto a donut which goes around the hole, then try to break it up into other triangles, one of these triangles still has to go around the hole (there's a proof needed there). but on a sphere, if you slap a triangle on, you can just pull all the vertices together and make it super small til the triangle is a single point. this is what homology is distinguishing (via a "quotient from a boundary map").
What's the significant difference? Is it that because the donut has a hole there has to be at least one triangle which crosses the boundary marked out by the inner circle of the hole, whereas on a sphere there is no such requirement? Which is why you can't just shrink the triangle on the donut down until it's a point - it has to cross the hole in order to have fulfilled some condition which has to be preserved?

>>11168588
sinx is curvy

ax+b is a line

they are equal where they intersect

b is irrelevant beyond some interval, since sinx is infinitely curvy

therefore 'a' can be interpreted as the angle the line makes with the x-axis and the origin, and instead move b as part of the 'phase' of sinx, so we have instead the problem sin(x+t)=ax

this angle cannot exceed the angle of the steepest tangent line of sinx for obvious reasons (1 solution always)

suppose t=0. then it is clear that any slight perturbation of the tangent line to a less steep curve will yield 3 solutions, 0 and two other points on opposite sides of the peak/trough, by symmetry of sin. it will jump to 5 solutions when this line is tangent at two other points of sinx

suppose t is such that sin(x+t)=cosx. then similarly the angle a has to be less steep than the first time it is tangent to cosx, but more steep than being tangent at the second peak of cosx

hopefully these examples are enough to make you think

>>11168720
No, not really. That's more like homotopy.
You have a surface. You take out your pen, and you draw a triangle in it (sides need not be straight lines). If you can use your pen to paint the "inside" of the triangle, there isn't a hole there. If you can't, then there is one. Of course, on a sphere or in some other surfaces, the choice of inside is arbitrary, but you can still choose one and fill it in.
You might think that in something like a torus there will be just too many triangles that can't be filled in, but we also consider two different triangles drawn in a surface to be "homologous" if we can consider them as a double boundary and fill in the area between them.
Of course, there's only so much of the intuition that can be explained without knowledge of the formalism. I personally recommend Fuchs-Fomenko.

>>11168720
homology groups do this:
>draw a boundary of a triangle on a surface. is it possible to fill the inside of the triangle ?

homotopy groups do this:
>draw a loop on a surface, think of the loop as a curve starting and ending at the same point. is it possible to shrink the loop into that point ?

these may seem like the same thing (and they actually are in many cases), but their higher dimensional analogues (triangles are replaced by simplices and loops are replaced by spheres) turn out to be very different. there's one more fundamental concept:

cohomology groups do this
>draw a vector field on the surface. can we find a function such that its gradient is this vector field ?

what the hell does this have to do with topology? well, the following thing is true: if there exists a vector field with zero rotation which is not a gradient of some function, then there must be a hole in the space.

Everyone agrees that topology is the best field in mathematics?

>>11168843
Point-set topology is a dead field, you gotta be more specific

>>11168843
definitely not. sure, everyone loves learning about fundamental group and "le rubber sheet klein bottles", but believe you me that this is just the honeymoon period.

>>11168843
that's not how you spell set theory

>>11168861
Honestly this, but I still like it.

>>11168872
>set theory
Those are not maths.

>>11168952
how so

>>11168561
What job did you get anon?

Is the even naturals under addition isomorphic to the odd naturals under multiplication?

>>11169102
the evens are generated by a single element, the odds have infinitely many generators (I think)

>>11169102
>>11169110
>what are prime numbers

>>11169102
No, but the even naturals under addition are isomorphic to the naturals.

>>11169154
prove it

>>11169164
Send natural n to even 2n and send even 2n to natural n.
Trivially bijective.
Isomorphic because 2(a+b)=2a+2b and 2a+2b=2(a+b).

>the prof slips his directed derivative in my Jacobian
oh god oh I think gonna coom

>>11169032
Trivially so.

can someone explain the reflection principle in set theory? Apparently it is a construction that resembles every set. But is not every set?? why is this

people use the word 'trivial' far too often. i doubt they even know what it means in the context of proof.

>>11169237
It's short for "trivial and left to the reader as an exercise."

>>11168678
to be honest I'd be happy if I could just find a way to fit some kind of math into my life, even if it was just independent study.

>>11169040
Programming. I do a fair bit of linear algebra but that's about it math-wise on the job

>>11169237
it's short for "this seems true but I don't have a proof for it."

>>11169311
just pick up a damn math book. go on wikipedia and read about concepts way beyond your level, let your mind get blown, and get motivated. works for me

In traditional high school math notation, why does x-x=0, if x is in the set of all real numbers? Like, sure the two x's could be constrained to be equivalent for any given instantiation in an equation, but what if you're treating them like whole domains, the whole real number line subtracted from itself, like a y=x graph minus another y=x graph. Why do they match up, x per x. It seems like unrigorous terminology, how do I make sense of it?

is the fusion dance a mapping from 2 vectors onto another vector?

>>11169350
its output is a set that contains elements of both inputted sets, and new elements that can be considered as outputs of various operations on different elements of the inputted sets.

its more like the sum of two vector spaces but not requiring the full initial vector spaces and having more operations than just addition in the second stage

>>11169374
>having more operations than just addition in the second stage
plz explain to my room temp IQ brain

>>11169331
[math]x[/math] is an element of the reals, not the reals itself.

>>11169388
>[math]x[/math] is an element of the reals
So [math]x[/math] is an "imaginary particle", as they say?

>>11169385
Well, you're trying to meld stuff like personality traits. If you're Toriyami, you might be creative and instead of saying Vegeta has X vectors in the bad dimension and Goku has -X vectors in the bad dimension, aka X good vectors, producing 0 good or bad aka neutrality, you might create a different type of interaction for the conflicting emotion vectors. Maybe a new emotion or synergies playing on subvectors in Bad, different micro emotions under that umbrella will combine in Toriyami's mind and inspire emotion vectors that might not be produced by sums of each characters individual vectors.

>>11169392
no

>>11169388
So X is analyzed as specific instantiations? That makes sense, I think. But I see a lot of times people will treat entire functions as unique entities, like in my linealg book the author refers to polynomials as basis vectors, sums of z^n where z is in the subset of a field

it's just subtracting two functions of equal value you fucking brainlets jesus fucking christ

>>11169404
You're the brainlet. Nothing is "just" anything. Define subtracting two functions.

>>11169414
[math]f(x)-g(x)=(f-g)(x)[/math]

>>11169420
mind = memed

>>11169400
Why not? It seems quite imaginary to me.

>>11169420
This need not be well-defined at the singularity.

>>11169420
That doesn't really help. Say f = x and g = x,

f - g = x - x. Well what the fuck is x - x? If x is equivalent to a number its sensible, but what if you write x = (-inf,inf). What if you write x is some element of C. It gets confusing.

>>11169435
the constant function 0.

>>11169420
>420

>>11169438
How do you prove this if x isnt a number?

>>11169435
>If x is equivalent to a number
Why would it not be? Every concept is in some sense isomorphic to a number.

>>11169438
The is the Smough of constants

>>11169414
>Nothing is "just" anything.
What did he mean by this?

>>11169449
I think that's a "Wittgenstein" quote. Seems to be >>>/lit/-appropriate material.

>>11169445
If you allow x to be a set or any possible element of a set, then its not a number, certainly at least not in the former case. In the latter case, how do you notate that your equation will hold true for any instantiation of x

>>11169449
>>11169456
I never read Wittgenstein. What I'm saying is that saying "the definition of A is A" is pointless, you have to show how it fits into the system, interacts with operators and other elements.

>>11169460
look up what an image of a set is and fuck off with your dumb retarded posts.

>>11169331
"x-x = 0" literally means "for any real number x it is true that x-x = 0". clearly this is a true statement.

nobody gives a fuck what happens "what if you're treating them like whole domains...", because that's not what it means.

That doesn't really answer much. It talks about mapping a set to another set, evaluating every element. But some authors write "x is an element of R", which is different than saying "x is a set with domain equivalent to R." In the first case, the concept of image doesn't apply. In the second, we have f = x so the image of x is just... x. How does that help me figure out what x - x is? I guess if we're talking sets, its just the empty set, but that doesn't jive with the way it was graphically shown back in highschool to be the constant zero function.

read a book.

>>11169496
>for any real number x
How can x be equal to "any real number". I get how it can be a single number, or a set of numbers, or any other kind of abstract OBJECT. But, what is the object that you refer to as "any real number". Can you define it for me?

I'm sorry if I seem retarded or childish, I think I just have autism.

>>11169508
Which book? Every proof I've seen online or in various books that deals with functions or sets seems to take this thing for granted, and it confuses me every single time.

>I'm sorry if I seem retarded or childish, I think I just have autism.
first sensible thing you've posted so far.

>>11169518
>How can x be equal to "any real number"
in the exact same way like you can say "for any dog it is true that the dog has four legs"

>>11169522
Well autism is about nitpicking for rigor and precision, and so is math. And there's a lot of neuroplasticity in childish minds, good for making abstractions and generalizations.

>>11169528
>nitpicking for rigor and precision
>so is math

>>11169528
hate to break it to you, but subtracting functions isn't as deep as you're making it out to be.

>>11169527
Four legs is a concept I can define, so it makes sense. It is bijectable with the fingers on my hand without my thumb. "any real number" is not a concept I can define, so I don't understand it. I know what a real number is, I know what the set of real numbers is, its properties, etc etc. But "any" real number? I don't understand that word any. I understand the concept of "all real numbers", thats a set. But it doesn't work for the equation x-x=0, because its operating on a whole set with set arithmetic. So what does any mean?

>>11169528
>math is about nitpicking for rigor and precision

>Well autism is about nitpicking for rigor and precision, and so is math
oh highschool....

>>11169545
What are you talking about? Highschool is just memorizing things. A real math book is basically just endless proofs. Prove every step, based on first principles or based on things based on first principles. Airtight.

>>11169551
>Prove every step, based on first principles or based on things based on first principles. Airtight.

> A real math book is basically just endless proofs. Prove every step, based on first principles or based on things based on first principles. Airtight.

Okay okay you've had your merriments now can someone explain what the word "any" means

>>11169577
In the case you're freaking out about it means "no matter which"

To: 11169551
>A real math book is basically just endless proofs. Prove every step, based on first principles or based on things based on first principles. Airtight.

>>11169577
YOU CAN PICK ANY ELEMENT
ANY ELEMENT YOU WANT
WHATEVER ELEMENT YOU PICK
WHICHEVER YOU CHOOSE
IT STILL WORKS

>When I landed in college I took a class on logical proof and greatly enjoyed it. I had a bit of a knack for doing proofs quickly in my head.

>>11169581
>no matter what the value of x is
So x is just one number? What's this got to do with functions?

>>11169585
I... pick? Does this have to do with that "axiom of choice" thing? I kept trying to read the wiki article but I could never comprehend it. Can you state your thing without me having to pick? I like it better when there are simple relationships directly proven true, not when I have to... step in and do something. Like, there's something unmathematical about that concept, but I don't have the words to explain it yet.

>>11169600
stop reading wikipedia and start reading your pre-calculus book.

>>11169603
i finished highschool and i did up to calc 3 at the top of my class. I'm doing linear algebra now on my own. Anyway wikipedia is actually a valuable tool for understanding math concepts, usually. And where else can I expect to find a simple yet generally accurate explanation of a concept I'm unfamiliar with?

I have never seen a grown man be this confused about the concept of variables

>>11169609
this site is pretty good: www.lemonparty.org

>>11169614
It's okay, last year I spent about a week wondering why and how objects rotate when struck off their center of mass. I'm not stupid just really autistic. I think variables are reasonably confusing, because they invoke that word "any". I still don't really get what any means, without having to do an action myself. It seems less like a concept and more like a mechanism I personally have to undertake to find concepts.

>>11169620
Here i think you'd really like meatspin.site/

>>11169623
woah thanks bro!

What multiplication?????

Also, if R^r is isomorphic to the reals, isn't Hom(R^{r-q},R^q) extremely simple?

like won't the only homomorphism between them have to send every basis element of R^{r-q} to 1, and the other homomorphism is sending them all to zero?

>>11169688
>what is multiplication
[math]mult: R^{r-q} \rightarrow Hom_{\mathbb{R}} (R^q, R^r)[/math] given by [math]mult(a)(b)=ab[/math]
>isn't something something
I dunno, maybe. Probably not.
>like won't the only homomorphism between them have to send every basis element of R^{r-q} to 1, and the other homomorphism is sending them all to zero?
You completely misunderstood [math]\mathbb{R}[/math]'s algebraic structure.

>>11169621
Hello! I think your fundamental misunderstanding is that mathematics "means" anything at all! Every mathematical concept is merely a piece in a symbolic game, with some rules dictating how it might be transformed into another such piece. My friends and I would be happy to guide you along the path to mathematical enlightenment and research funding - you can join us on a cool little website we've put together at https://ncatlab.org.. Thanks, and remember to keep it categorical!

>>11169700
kys

>>11166717
Do Herstein then fill holes with Dummit and foote. After that you know basic algebra. Go open up Roman's Advanced Linear Algebra and you will learn linear algebra the right way that is easy and intuitive and will actually relate to math.

>>11169700
based

>>11169700

>>11169688
Source?

>>11169700
I think you're mocking me but I've come to similar conclusions before, and while I never researched category theory I bet it really is cool. That doesn't help me though, because even if math is just symbols, they have to fit together. The word "any" doesn't fit in the realm of math, no one seems to be able to show me a good definition of it other than "pick one". It unironically feels like collapsing a wavefunction.

Is there any way to parse the concept of a variable without the axiom of choice?

>>11169733

https://arxiv.org/pdf/1511.02888.pdf

>>11169698

So you be saying that wildeberger was right?

>>11169751

>>11169756
Not gonna lie I had one of the heartiest keks I've had in many months. But laughter is not too far from crying and I grow desperate and nihilistic in this confusion.

Its like math (and ontology, but that's not related to this convo) as I always knew it is crumbling beneath my feet and I am helpless. I spent several hours this morning just saying, what? What? You know the fucking Oomer meme? I'm becoming the Oomer. Well then I drank a coffee and proved some linealgebra and theorems and went to museums and felt kind of sane. But I am really confused. I guess confusion isn't really the worst feeling.

Ooo
>LOL

>>11169752
>wildberger was right
No, the reals are an extremely comfy analytic object, they just become absolute dogshit once you take away their ordering.

>>11169768
Does taking away ordering mean maintaining operation associations but merely removing the notion of greater or lesser? Like 1+1=2 but 2 and 1 have no "value" difference. What would be the point of this.

>>11169765
Young man, in mathematics you don't understand things. You just get used to them.

>>11169751
the concept of a variable comes from first order logic itself, you need to look even further back than the ZF(C) axioms

>>11169772
I guess so. I think I can come to accept the axiom of choice if I see how its truly necessary and naturally fits with the universe. It genuinely feels like quantum mechanics to me, even before I knew what the axiom of choice was, the notion of a variable felt like an undefined superposition and selecting an instantiation would be interacting with the system, collapsing the wavefunction.

There's this philosophical conundrum that's really bothering me though, even after I understand the AC, but I think I'm making progress on it. If I can't I'll have to repress it or ignore it so I don't kill myself, I wrote this little rhyme about it: "Death is scary but there's something worse, so I'll just live and avoid the hearse".

>>11169773
Now we're talking, can you elaborate? I know very little about formal logic.

>>11169765
Take your meds friend

>>11169783
>Now we're talking, can you elaborate? I know very little about formal logic.
Neither do I. The wikipedia page (https://en.wikipedia.org/wiki/First-order_logic)) goes through both the syntax and semantics of it, maybe that would be a good start if you want to really drill down into how these things work.

>>11169797
Yeah I really need to get on vyvanse

>>11169783
You seem to have a problem accepting the existence of an unknown. You do realise that we as humans accept that unknowns exist all the time, right? Like, what's behind the corner, or what will happen a second from now, or all the statements of logic and theorems of mathematics as yet unproven but which nonetheless form the basis for our thinking and the structure of the universe in which we live.

The unease you feel should be channeled, and I see that so far it has mostly been channeled, towards a yearning for new discovery. It's good to see you curious here, and you should be more confident to ask questions of people who might be able to help you understand better. But you must understand that no matter how much you understand, there will always, always be unknowns. I hope that's enticing.

>There's this philosophical conundrum that's really bothering me though
go on, spill the beans now you've taken over the general

>>11169801
>First-order logic uses quantified variables over non-logical objects
Ah shit, that's just variables as placeholders for objects, that's the simple stuff. The complex stuff is variables referring to non objects, to ranges of possibilities that have to be selected from seemingly by AC

>>11169733
>>11169698

I think I just got it.

you're supposed to think of elements of R^{r-q} as matrices. They send elements of R^q to R^r with multiplication (which is linear.)

Let's try not to think about the reals too much. It is best to leave that genie behind the curtain.

>>11169817
>unknown
It's not quite exactly that. It's a specific subset of the unknowns. Its not that I'm particularly afraid of not knowing the solution to a diff eq, its that accepting entirely new systems of patterns causality etc is really jarring to me. You know, growing up as a monkey for about 15 years with notions of determinism and material realism and blabla, then you start to learn about quantum mechanics and infinitesimal numbers and you have to think in terms of probabilities and limits. Its just hard going from "reality is one way, why dont other things fit into it", to "various tools are needed to describe reality, including tools that dont provide definite answers (at least in the traditional sense)"

That, I'll cope with. The philosophy thing is really just maddening and drives me nuts, I posted a lot of thoughts about it in sqtddtot earlier today, but here's the fundamental thing:

Let object A have no traits.... now try to think about object A. Any analysis of it fails

>>11169771
Yes.
The point is that, when you work over things like [math]Hom(A, \mathbb{R} )[/math] that's the R you're working with. If you take a and b transcedental you might even have some isomorphism from R to itself with f(a)=b and f(b)=a.
It's behavior is extremely unintuitive and shitty.

>>11169838
This might not sound very appealing on /sci/, but the Dao De Jing deals directly with your problems.

>>11169841

> If you take a and b transcedental you might even have some isomorphism from R to itself with f(a)=b and f(b)=a.

[citation needed]

>>11169843
Okay I'll read it

>>11169851
It's only a few pages long. Enjoy!

>>11169850
Oh, right, that was C. My bad.

>>11169708
thank you for the advice

>ywn be a part of the mathematics community talking about Serre's Epsilon conjecture over a cup of coffee
wish I worked harder during undergrad and got a higher GPA

>>11169316
thanks, I'll give it a try. I would like to be able to continue doing math in some way, I think.

>>11169225
Reflection basically says that if something is true about sets then there is a stage of the cumulative hierarchy where that property is true. A very useful theorem of the reflection principle says that for any finite list of axioms of ZFC there is a stage in the cumulative hierarchy for which each formula is absolute. This is done is one formalization in forcing. You say, pick some stage where the following finite list of formulas hold and force over that set.

>>11169700
t.constructionist

>>11169751
>Is there any way to parse the concept of a variable without the axiom of choice?
Not that we know of.

>>11170116
if you look in sqtddtot we figured it out, with alternate dimensions

also pic related is the funniest picture ive ever seen and i usualyl hate nerd memes

>>11169765
>I always knew it is crumbling beneath my feet and I am helpless
LMFAO. Weakling detected. Please proceed to kill your own self immediately after reading this message.

>>11169765
>I grow desperate and nihilistic in this confusion.
that's great, hopefully you reach its natural conclusion and an hero so nobody interested in math irl would ever have to deal with your narcissistic whiny retarded ass.

>>11169969
What is wrong with having your maths be choice-free? Aside from the inability to rigorously define the notion of a variable, that is.

>>11169783
>here's this philosophical conundrum that's really bothering me though
You're just experiencing one of the many natural plebfilters math has created throughout its lifespan against subhuman plebs who can only conceive of things in very limited and crippled ways, masking it as """rigour""". And it appears like you're on the "getting filtered real hard" side of things.

>>11170141
Why are you so rude? It doesn't hurt my feelings at all its just really unpleasant to witness and kind of annoying. I've gained the ire of many people over the years, I never really understood it. You remind me of this time I was biking to work and this chained up pitbull wouldnt stop howling at me, it wouldn't change demeanor in the slightest when I tried talking to it. Like a nerve snapped in its mind and all it could barf out was nonsensical rage. Its just kind of disgusting and pathetic.

>>11170149
>its just really unpleasant to witness and kind of annoying
Describes your posts quite well.
>I never really understood it
Might have something to do with your debilitating mental and social retardation you pass of as "le epic quirky autism" to come off as special and definitely not retarded. I swear guys I just have autism! I am not mentally handicapped!

Didn't even bother reading the rest, kiddo. Hehe... I am going to go read some cool maths now to hone my intuitons.

>>11170153
>that softening of demeanor
Bullyanon a CUTE!

>>11170149
An hero

Who is Bill Dubuque? I am assuming that's not his real name cause I can't find anything in Google of him.

>>11170455
Who is asking? Who told you that name?

> haven't visited /mg/ in a month
> see people arguing about a "philosophical conundrum"
First thing that came to my mind is that they arguing about axiom of choice.
> scrolls up.
> kek

>>11170462
Answers a lot of questions in M.S.E. in a pretty nice format.

>>11170462
I think he might be onto us. Usual place?

>>11170149
You sound nice anon, why do you post on this site where everyone is a complete asshole?

>>11170465
We're actually arguing about variables

>>11169700
kek

>>11170669
Also, what issue did you take with the anon offering first order logic as an explanation for variables? Going back to the difference of functions case, the principal of functional extensionality states that functions are equal if they're equal at each argument. This means that X - X = 0 <=> \all x \in R x - x = 0, which is a statement on quantified variables which you seemed OK with.

>>11169743
>>11169751
look at fucking first order logic. leave us alone. this is shit only foundations morons care about.

>>11170149
it's like we're reading a bunch of reddit posts just copied into the quick reply box and posted without thought. you're shitting up our thread. please, stop. we're not here to discuss incredibly simple ideas over and over again and to entertain someone who can't seem to grasp them. we're here to talk about math with one another.

>>11169700
>today: what is a variable, really?
>in one month: pic related

>>11170806
>burned out mathematical beauty receptors
haha that sounds horrible glad I'm not just chasing highs like that guy

>>11170806
>can't solve a basic integral

always kek at that

>>11170806
>burned out mathematical beauty receptors
How do you even burn those out with category theory?
Lemme fix that real quick.
Also rephrased the third topic to make it more concrete and added some stuff.

meta question:

sometimes, I get stuck on a problem, and after peaking at the solution it's obvious, and was usually some elaborate symbol crunching I missed. I'm not sure how best to synthesize information in these scenarios. In these cases, the problems solution isn't fundementally important to the topic, and redoing the problem a few times spaced out over a couple weeks to a month usually keeps the solution (often more like a "trick") in my memory for long enough (forever?).

how do you deal with problems like that? do I just go "oh", and move on? are they worth revisiting? I feel bad if I leave problems behind that I couldn't solve again if I tried, so I frequently flip back to problems to make sure I can solve everything in the book. FWIW, I normally do this kind of work when I'm tired and/or not operating at 100% capacity, it seems like a good use of fogbrain time.

>>11170941
>>11170941
so, in other words, when you get BTFO'd by a problem, how do you recover? ggwp and move on, or do you request a rematch?

>>11170970
I NEVER look at the solutions or the proofs of problems, if I am unable to prove them myself I just skip them and go on and then return to them after some time.

>>11171007
seriously? I'll try that, sounds badass, but I worry that so much crud or holes in my knowledge will pile up that continuing will be infeasible.

>>11171007
do you check the solutions to ensure your proof was correct, at least?

>>11167235
The identity looked "linear" so I just guessed it was an affine function.

>>11167235
I don't know because I didn't try but my immediate instinct was to plug in 0 and 1 and get some more identities to work with. So maybe.

>>11170864
actually not horrible changes, but not enough to make me save over my file.

>can imagine very complex structures
>can make them traslade, rotate and such
>when it comes to drawing I can't even do a cube with perspective

>>11170970
I spend a bit of time identifying what specifically it was I needed to think of, how I would have thought of it, and what was preventing me from thinking of it. Once I understand that, I move on. Then if something similar comes up again I'm sure to get it.
I always try to go through this process meticulously, even if very quickly. This is in part because I rarely give up on a problem, and in part because sometimes when I think I understand what I was doing wrong I actually didn't quite get it and then I'll get caught up in the same way again much later. The best way to get through a problem WITHOUT looking is to start by identifying as specifically as you can what's blocking you, and if you did that already for a similar problem you should figure it out quickly.

>applied maths class
>talking about a nonlinear operator that sends a function to a solution of a DE
>professor asks if we've ever calculated this kind of Fréchet differential by hand before
>no
>he does it
>it's some of the most beautiful autism I've ever seen in my life
>>11171226
If you're drawing for your own intuition, shitty drawings are perfectly fine.
I used to make drawings of just about everything in contact geometry, helped a lot.
>>11171215
I'm particularly proud of the Lurie joke.

>>11171226
What does that have to do with porn addiction?

>>11171237
could you replicate this beauty here for us?

>>11170470
I've made the dropoff.

>>11171237
>If you're drawing for your own intuition, shitty drawings are perfectly fine.
I don't agree with this line. They don't have to be frameable works of art, but you can't take any reliable geometric intuition from a drawing that's so shit it doesn't represent what it's supposed to be a drawing of.

>>11171324
No, you don't keep the drawing, that's a trap. Drawing it out is just to help make the visualization concrete, you absolutely shouldn't use it as a reference.

>>11171324
Where did you get that I said you should keep drawings?
The point is that you cannot "make the visualization concrete" if your drawing skills are so shit that what ends up on the paper isn't what you were trying to visualize.

>>11171377
No, but it's "correct in your head while you draw it". Drawing is just forcing your visualization to concretizing itself, there's no issue if the drawing is bad, since you perform the appropriate corrections in your head, such as "these should be parallel", "this should be jutting out of the paper" or even highly convoluted stuff like "these two paralellograms span a four-dimensional subspace of the tangent space".

>>11170149
>this is the only board dumb enough to fall for this for an entire day

>>11170941
you realize you're a brainlet and that you've learned nothing by peaking at the solution lol

Okay I'm really tired, how can I show the integrability of [eqn]x\mapsto |\ln(x)|^kx^{a-1}e^{-x}\mathbf{1}_{[0,1]}(x)+|\ln(x)|^kx^{b-1}e^{-x}\mathbf{1}_{[1,+\infty)}(x)[/eqn] on [math]\mathbb{R}_+^*[/math] for every [math]k\in\mathbb{N},(a,b)\in\mathbb{R}_+^{*}[/math]. The big 1 is the indicator function. I'm trying to compare this function with another function at 0 and infinity but I can't find one... Can someone help me?

>>11171725
>how can I show the integrability of
mathematica

>>11171035
yes, I do. 99% of the time it coincides with the proof in the book though

>>11171725
Split it up into an integral from 0 to 1 and another from 1 to plus infinity.
For the first one, discard e^-x, since it's bounded by [1/e, 1]. Then, if a>1, you can bound it by [0, 1], and this reduces to the integral of ln(x)^k, and you can solve for the integral of ln(x) analytically and bound the powers the usual way. Or use integration by parts. If a<1, you bound it by a=0, and integrate by parts.
For the other part of the integral, bound ln(x) by x, and bound x^{b-1} by some x^c, where c is some integer larger than b-1. You should have x up to something times e^-x, which is easy and you can just integrate by parts.

Any mistakes are not the responsibility of the author.

>>11171773
Thank you, I only had a hard time finding the functions for comparison. It seems like this function is [math]o(x^{a/2-1})[/math] at [math]0[/math] and [math]o(\frac{1}{x^2})[/math] at infinity. I don't need the exact value as this this function is just here to prove that I can safely differentiate the gamma function inside the integral sign.

>>11171758
how do i get to this point of mastery

also where's the guy a few weeks ago who talked about having a chaotic organization (or lack of) rather than being autistic about it, i want his advice

>>11171834
practice

Two guys will travel to a place 20 miles away from their house. They both will start at the same time, and will walk at a constant velocity of 4 miles/hour.
However, they own a horse that can carry one person at a time and can run at 10 miles/hour, regardless of whether someone is riding or not. Find the minimum time required for both of them to reach their destination.

>>11172991
Your problem is really ambiguous.
Can the horse come back alone to search the other guy?
What does it mean for "both of them to reach their destination"? Should they be simultaneously at the destination together? Are they starting from the same house (are they fags)?

>>11173028
They will start from the same place
The horse can come back alone
If guy1 takes t1 hours, and guy2 takes t2 hours, the answer would be max(t1, t2)

>>11166717
Read Artin's algebra. It covers group theory and linear algebra simultaneously, which are very related.

>>11173378
>>11173378

>>11172991
the chad answer is to have the horse switch riders every epsilon seconds then have epsilon approach zero and have both guys moving at an increased average speed.

>>11172773
I'm practicing daily, just have to practice well and smart. thanks for the help, I've had a good morning trying to overcome my urgers to check the answer book.

>>11173426
Fairly sure the answer is independent of when and where they ride or walk, the time should be the same.
Hard to formally say that, tho.

>>11173491
Wait, anon's answer actually fucking works.
We consider walking speed to be stationary. Then, the horse moves one of them epsilon forward, goes back epsilon, and takes the second one epsilon forwards. So it moves the collective forward epsilon at 6/3=2 miles per hour, and each traveller has average inertial velocity 6 miles per hour by adding the four back.
So they take 20/6=3,33333333 hours.

>>11173502
Wait, I forgot to properly account for the way back in the adjusted reference, since it moves backwards at 14 miler per hour.
Someone else do it properly.

Is discrete math a meme? Since all CS brainlets study it so I think it's a meme.

>>11173507
the class "discrete math" is typically about basic logic, basic set theory, basic proof techniques, etc. "Discrete math" in general refers to subjects like combinatorics and graph theory.

>>11169420
>69420
>69
>420
>69 and 420

based digits confirm

rate my AA book

>>11173782
I don't see any Anti-Aircraft content in there anon. Did you attach the right picture?

>>11173837
nice catch, here's the right picture

>>11173782
Weak but works.
Honestly, I shouldn't talk about it being weak, my college takes two semesters to cover that.

>>11173889
10/10

wot of vector spaces were a grill?

>>11173957
on that topic, which is the cutest mathematical structure?

>>11173957
L^2 claimed.
>>11174037
Contact metric structure.

>>11173782
what book?

>>11170864
Menance Category theorist coming.

>>11174088
Saracino, Abstract Algebra: A First Course

>>11174156
so dumb. holy shit

Daily reminder that Terry Tao is an otaku.http://shanghikid.50megs.com/Otakudom/ContReal/contReal.htm

>>11166644
This is always the best edition of /mg/. Infinity cucks btfo. Praise the lord Wild burger.

>>11174289
It's always the analysts.

>>11174376
Based

>>11174289
but he has a family

>>11174156
not sure what to think of this - the forced "applied" meme around CT that came with Scala trannies bothers me a bit desu

>>11173782
Anonymous Alcoholics?

>>11174837
are his kids good at math? must suck to have a literal child prodigy as your dad

[eqn]
\text{ }^{\color{#571da2}{\displaystyle\text{W}}}\text{ }^{^{^{^{\color{#462eb9}{\displaystyle\text{h}}}}}}\text{ }^{^{^{^{^{^{^{\color{#3f47c8}{\displaystyle\text{y}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{\color{#3f62cf}{\displaystyle\text{ }}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#437ccc}{\displaystyle\text{i}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#4b90bf}{\displaystyle\text{s}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#56a0ae}{\displaystyle\text{ }}}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#62ab99}{\displaystyle\text{t}}}}}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#71b484}{\displaystyle\text{h}}}}}}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#82ba70}{\displaystyle\text{i}}}}}}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#96bc5f}{\displaystyle\text{s}}}}}}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#a9bd52}{\displaystyle\text{ }}}}}}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#bcbb48}{\displaystyle\text{o}}}}}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#ceb541}{\displaystyle\text{n}}}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#dcab3c}{\displaystyle\text{ }}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#e39938}{\displaystyle\text{/}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{\color{#e68033}{\displaystyle\text{s}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{\color{#e3632d}{\displaystyle\text{c}}}}}}}}}\text{ }^{^{^{^{\color{#de4227}{\displaystyle\text{i}}}}}}\text{ }^{\color{#da2121}{\displaystyle\text{/}}}
[/eqn]

Is anyone aware of where is it possible to find second printing of Aluffi?

>>11175307
no it's literally never been done

>>11175346
its not rainbows

I will kill myself

im gay lol

>>11175763
Ok trannier

>>11175307
Any new copy you order from anywhere should be the second printing. The AMS will 100% send you the newest print run, Amazon very likely will but if they don't you can always return it. If you want to buy from a smaller bookstore or an individual just send them a message and ask them to check for you.

>answer a question in stackexchange
>insult the person who asked it
>give him a shitty answer that's fundamentally correct
>people get high school flashbacks where I'm their fat bald jewish teacher who doesn't explain shit and insults everyone who isn't also jewish and then shower me in dislikes
That place is absolutely hillarious.
Anyhow, love you guys.

>>11173382
i can't bear looking at actual numbers, so i cant help you

>>11173028
its not ambiguous its a related rates problem that 15 year olds have to solve for their ap class nigger

>>11176582
new
>>11176582

I want to be a pure mathematician i'm currently studying Calc 2 and finding it quite manageable and enjoyable. I'll easily be done before the end of the year.

How much harder do things get? What should I do next? Iv'e heard that reading a book on proofs should be by next step. What book should I read and should I do it before or after Calc 3 and DiffyQ?

Is Brilliant sufficient for Calc 3 practice problems? Since Khan doesn't offer any/enough problem sets for Calc 3 and differential equations.

>>11176600
ill make an analogous statement to yours
>i want to be a full time weaboo
>im currently watching the simpsons and find it quite manageable and enjoyable

>>11174997
His kids are probably going to get into Princeton, let's be serious. Ed Thorp's grandkids are all at MIT