I've been playing a bit with ternary logic and arithmetic. The initial reason I started thinking about this is that it annoyed me how in binary a n-bit signed integer can represent the range [- 2^(n-1) ; 2^(n-1) - 1], which isn't symmetrical. Of course that's only a convention (although one that is very convenient for several reasons) and it could as well represent [- 2^(n-1) + 1/2 ; 2^(n-1) - 1/2], but then 0 wouldn't be represented.

In ternary, a n-"tit" signed integer can represent all the integers in [-(3^(n-1) - 1)/2 ; (3^(n-1) - 1)/2]. So a 3-tit signed integer can represent the 27 numbers between -13 and +13, for example.

Let's see how to construct such a numbering system. We could use 3's complement for negative numbers. For example, with 2-tit numbers:
12 = -4
20 = -3
21 = -2
22 = -1
00 = 0
01 = 1
02 = 2
10 = 3
11 = 4
Now that's very ugly, and there is no obvious way to determine whether a number is negative or positive from it's ternary representation. I don't like it.

So instead of assigning to each tit a value in {0, 1, 2}, we can instead assign them a value in {-1, 0, 1}. For convenience and legibility, I will note the different symbols as {1̃, 0, 1}, but remember that 1̃ has a value of -1. With those values, and a classic positional system, we can easily represent all natural numbers.
The positive numbers are:
0, 1, 11̃, 10, 11, 11̃1̃, 11̃0, ...
(0, 1, 2, 3, 4, 5, 6, ...)
The negative numbers are:
1̃, 1̃1, 1̃0, 1̃1̃, 1̃11, 1̃10, 1̃11̃, ...
(-1, -2, -3, -4, -5, -6, -7, ...)

This system has some nice properties:
1. All natural numbers have a unique representation
2. The range that can be represented by an n-tit number is symmetrical with regards to 0
3. The leftmost non-0 tit contains the sign information
4. The opposite of a number can be easily computed by replacing all 1s by 1̃s and all 1̃s by 1s
5. It's a positional system, so usual algorithms for addition and multiplication apply.
6. A n-tit variable can represent ~60% more values than a n-bit variable

We can also have a gray-like code. Here's how to generate a sequence where each consecutive term only differs by one tit:
1. Start with the sequence S ← {1̃, 0 1}
2. T and U both take the elements of S
3. S is reversed
4. Each element t of T has the tit 1̃ appended to its left
5. Each element s of S has the tit 0 appended to its left
6. Each element u of U has the tit 1 appended to its left
7. S ← T | S | U where "|" represents the concatenation operator.
8. Redo steps 2 to 7 until the sequence is long enough.

For example, with 2-tit numbers, we get:
1̃1̃ 1̃0 1̃1 01 00 01̃ 11̃ 10 11


Now let's do some logic!
We could easily emulate binary logic, for example by interpreting 0 as a boolean 0 and 1̃ and 1 as a boolean 1. That's not fun though, and wouldn't allow us to compute on ternary numbers, so let's invent toolean logic.
I tried coming up with a few logic gates, as extensions to the ones we're used to with booleans.

"and", it's a ternary multiplication. Notice how the lower-right quadrant is the same as with the boolean and. I'll note the operator as ⋅
 a⋅b | 1̃ 0 1
----+-------
1̃ | 1 0 1̃
0 | 0 0 0
1 | 1̃ 0 1
That one has some nice properties:
(a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) = a ⋅ b ⋅ c
a ⋅ a = abs(a) (that is, 1̃→1, 0→0 and 1→1)
1̃ ⋅ a = neg(a) (that is, 1̃→1, 0→0 and 1→1̃)
1 ⋅ a = a
0 ⋅ a = 0
a ⋅ b = neg(a) ⋅ neg(b)

"or", it's a ternary saturating addition. Again the lower-right quadrant is the same as with the boolean or. I'll note the operator as +
a+b | 1̃ 0 1
----+-------
1̃ | 1̃ 1̃ 0
0 | 1̃ 0 1
1 | 0 1 1
That one isn't as nice, so I would be happy if you have better suggestions. In particular, I hate that (a + b) + c ≠ a + (b + c). Still, a few interesting properties:
neg(a) + neg(b) = neg(a + b)
a + a = a
0 + a = a
a + neg(a) = 0

"xor", isn't an overflowing addition (half adder), because I wanted the lower-right cadrant to stay the same as the boolean xor. I'll note the operator as ⊕
 a⊕b | 1̃ 0 1
----+-------
1̃ | 0 1̃ 0
0 | 1̃ 0 1
1 | 0 1 0
Again, I hate that (a ⊕ b) ⊕ c ≠ a ⊕ (b ⊕ c)
Some properties:
a ⊕ b = (a + b) + ((a + b) ⋅ a ⋅ b) (can this expression be simplified?)
a ⊕ a = 0
0 ⊕ a = a
a ⊕ neg(a) = 0

There is no "not", because why would not(0) be 1 over 1̃? In any case both variant can be implemented as (a ⊕ 1) and (a ⊕ 1̃) respectively.


Can the logic operators do arithmetic? Yes they can!
Here's the truth table for a half adder
| 1̃ 0 1
----+-------
1̃ | 1 1̃ 0
0 | 1̃ 0 1
1 | 0 1 1̃
That is, (a ⊕ b) + ((a + b) ⋅ neg(a ⋅ b)). Can this expression be simplified?

And here's the truth table for a full adder
| 1̃1̃ 1̃0 1̃1 01 00 01̃ 11̃ 10 11
----+------------------------------
1̃ | 1̃0 1̃1 01̃ 00 01̃ 1̃1 01̃ 00 01
0 | 1̃1 01̃ 00 01 00 01̃ 00 01 11̃
1 | 01 00 01 11̃ 01 00 01 11̃ 10
Now I haven't got a clue as to where to start to get a toolean expression out of this.


What do you think about all this? Did I make any mistake?
Some more things I want to look into:
1. Other properties of toolean logic, especially to help find and simplify expressions
2. Find an expression for the full adder
3. Design ternary CMOS circuits
4. Check out what people smarter than me have written, but I want to play around a bit more before.

P91162
WTF
In boolean they are trivial, like
y(x1, x2, x3, x4) = !x1 & x2 & x3 & !x4 for y(x...)={1 if x==0110; 0 otherwise}

P91174
>Can't read french.
Lol, git gud. But yeah

P91176
Probably not, but I haven't ran an exhaustive search for those.
Thanks for the code, I see I made a dum-dum now. I assumed "Not(a+b) === Not(a)*Not(b)" and "Not(a*b) === Not(a)+Not(b)" were equivalent, which they are not, since not(not(a))≠a. What language is that btw? Looks convenient.


If I didn't make any other mistake, there is no logic that verify:
1. "or" and "and" are associative and commutative
2. None of the operators is a constant
3. The logic satisfies de Morgan's laws
4. "and" distributes over "or"
5. The set of operators is complete, in that it can be used to generate all binary operators by composition.

I can find some if I loosen any of those conditions though. Attached are the logics I found by removing the distributy constraints. Obviously, there are duplicates then.

P91183
> What language is that btw?
Why, it's Haskell.

> I can find some if I loosen any of those conditions though. Attached are the logics I found by removing the distributy constraints.
I would like to remove 3 (De Morgan) instead. Those look like being ad hoc for boolean logic.

P91229
>Why, it's Haskell.
I really should use more functional programming, it makes the code so much clearer.

>I would like to remove 3 (De Morgan) instead.
Fair enough. I'm running the search for those.

P91176
I've found 333 of them. I should have optimised my code better, because it took 12h.
Results and code attached. "ternary_ops.py" is a bunch of helper functions, "find_nands.py" is the script that runs the search.

>>P91261
> 12h
Dude, just use CVC.

henlo frens
wat r sum god places 2 git sick epic classic math books 4 cheap?
asking 4 fren of fren

https://annas-archive.org/
It has everything from libgen, z-library and others.

P90160
doesnt have asstr tbh

P90115
no u XD
P90131
heyo that,s kekkou neato desu
i didn,nt know aboutthat
P90132
P90141
P90160
P90166
sory frens i shoould have clarified i was looking 2 build a colection of physical (paper books

P90615
>i was looking 2 build a colection of physical (paper books
1. buy printer
2. download files from P90132 or P90160
3. print
4. punch holes in the paper
5. put in binder

First, try writing like a non-retard.

>A vector space just generalizes a field.
<No? Vectors are a whole separate set of objects you introduce. It's not just a generalization.
>But if you make the vector space the same field as the scalars it's the same.
<But vectors are different from scalars.
>But it's the same object.
<But they're different types. We write vectors in bold.
>Math doesn't have types.

>The next simplest example is the field F itself. Vector addition is just field addition, and scalar multiplication is just field multiplication. This property can be used to prove that a field is a vector space. Any non-zero element of F serves as a basis so F is a 1-dimensional vector space over itself.
...
Surely it causes problems if you can't assume vectors and scalars are distinct?

This is everything I hate about mathematics in one example. Are mathematicians not capable even of using tagged unions? Imagine if the vector space contained the objects used to construct the scalar field. It's insane and will make mathematics implode under special cases eventually. It's like math isn't made to find mathematical truths directly but so some "wiggers" (jewish) can go "UM ACKCHUALLY" and use intentionally hidden structure to make some important seeming abstractions in a way that somehow avoids helping anyone else understand. It's utterly disgusting.
Post other examples of this stuff itt, if you want.

op just refuses to accept the convention that "field" refers to the set itself (implying the existence of operations) instead of the tuple containg the set and its operations. this means when he hears someone say "a field is a vector space" he doesn't (or didn't) understand this is a statement about sets. now he understands but still insists he is right, saying this use of language is wrong, bad design and "obfuscation". op should calm down and try to understand instead of fighting.

P90057
>op should calm down and try to understand instead of fighting.
This is called being unteachable. A person who has an anger response to new information can't learn.

Vector space? You mean module over a field?

Isn't a vector that can be a scalar called tensor?

P90602
no

Can someone explain where the following mathy things are used?
I'm interested in where it is used mainly, and also how it's used.

>Fourier Series
I know it's used to solve heat conduction equation and that's why the autist Fourier came up with all this but I'm sure there's more uses to this
>Partial Differential Equations
>Analytic Functions
>Complex Interpretations
>Vector Calculus

>Fourier Series
Spectral analysis
>Analytic Functions
>Complex Interpretations
>Vector Calculus
Almost everywhere. It's a basis of applied maths.
>Partial Differential Equations
Same. They are more obscure though, like string/membrane oscillations or gas flow.

>Fourier Series
Aren't the used all the time in signal processing? Anything to do with a superposition of waves really. Visible light, radio, sound, etc. Heard there are some more niche uses as well.
>PDE
I think these are used to death in finance. Don't ask me why. Something something modelling unknown functions.
>Analytic Functions
Aren't they related to doing approximations? If so, comp sci is a large field of use.
>Complex Interpretations
AFAIK they are used for getting really weird integrals, quantum mechanics, and solving polynomial equations. Also fractals.
>Vector Calculus
Fields. Gravity, electromagnetism, different flows. Makes sense since in these fields you are working with 3d spaces, movement, intensities, etc.

P89351
P89353
I'm actually looking for where they're used in Physics, also Physics that does not involve Electricity, Magnetism, Optics and all those electron or light related topics but things like force, heat, etc.

>Gravity
>different flows
>string/membrane oscillations or gas flow
Good, this is fine. I'll read more into this.

P89453
Those topics are ubiquitous. F(vector)=m d2/dt2 x(vector), well, the whole theoretical mechanics is a bunch of vectors and their integrals. Heat dissipation may be modelled by PDE. What else? I don't know.

Cute fact I learned recently: Start with a triangle ABC and extend sides AB, BC, and AC out to points D, E, and F, respectively. Pick an arbitrary point M on side AB. Draw a line from M to E, and let M' be where it crosses side AC. Continuing, use a line through M' and D to make a point M'' on side BC, then use a line through M'' and F to make a point M''' on side AB. After two circuits around the triangle in this manner, we always come back to the original starting point M.

This is a consequence of Pappus's hexagon theorem. Can you see why?

Have you been studying anything interesting lately? For myself, I've been trying to learn a bit more group theory.

I taught myself some basics of model theory: compactness theorem, quantifier elimination, the Löwenheim-Skolem theorems. The results are pretty damn fascinating but I don't see myself using any of it in my work going further...

I am learning Pseudoholomorphic curve but I am not familiar with Gromov–Witten invariants so, I am working on that first.

bit of analysis
pretty confusing desu

P77503
What do you mean by interesting ?
Recently, I have been interested in all the other integration theories besides Riemann's definition. I find it interesting to compare them.
>For myself, I've been trying to learn a bit more group theory.
Silly, a group is just a groupoid with a single object.

P77945
so cool!! do you have any books to recommend ? Do you usually study on your own or in a group ? I want to learn more about applications of topology in mathematical logic.
I learned a bit about Stone's duality, which associates a logic with a class of topological spaces, a theory with a space of this type, and formulae with an open where they are true. From this, the points of this space are the models of your theory, and the continuous functions between such spaces are model morphisms. It's like in algebraic geometry: a polynomial is sent on the closed set where it is equal to 0, and the corresponding algebra of functions is equivalent to the associated space. For classical logic, these spaces (of Stone) are compact, which corresponds to the compactness theorem in logic (even if there's still something unclear for me ... The formulas also have to be closed sets, I think).

learning a bit about projective geometry

I was wondering if anyone would be down to figure out π and φ again, just because we can and I'm schizo enough to not trust the fact that π is 3.14... and φ is 1.61...

[bold:The π question]
we know that π is used in the calculation of the lenght of the circle:
L=2*π*R
L=D*π

L=Lenght of the Circle
R=Radius of the Circle
D=Diameter of the Circle

In the Bible (KJV here), at verse 1 Kings 7:23 it is said that:
"And he made a molten sea, [bold:ten] cubits from the one brim to the other: it was [bold:round all about], and his height was five cubits: and a line of [bold:thirty] cubits did compass it round about."

it is also repeated in 2 Chronicles 4:2:
"Also he made a molten sea of [bold:ten] cubits from brim to brim, [bold:round in compass], and five cubits the height thereof; and a line of [bold:thirty] cubits did compass it round about."

giving us the values of:
D=10 cubits => R=5 cubits
L=30 cubits
L=D*π <=> 30=10*π => π=3
[bold:Note]:The lenght of the circle is 30 with the diameter being 10; if π was truly the 3.14... value it is given today, then either the value in bible is rounded down from 31.4/31.5 to 30 or the currently recognised value of π is false

Some mathematicians throughout their life attempted to calculate π via polygons, by using the Radius connected from the centre of the Polygon to one of its corners or perpendicular to one of its sides, and the higher the Polygon side count is, the closer to reality the value comes, in theory

The only usable solution to calculating π would be through calculations as attempting IRL experiments would result in false values due to too many physical factors that would throw us off, even slightly


[bold:The φ question]
We know that φ is the value of the golden ratio following this ecuation:
A/B=(A+B)/A as long as A>B>0
A=Lenght of a rectangle
B=Width/Height of a rectangle

The value of φ fluctuated between different values, with it being today at the value of 1.61...

Many mathematicians also figured out that the Pentagon and the Pentagram has φ all over it, so those would be some sources of inspiration as well, I suppose


I'll come from time to time over here to share and give my opinion, as well as the results of my calculations on the size of π & φ, have a wonderful day and good luck at your calculations, cheers! ^^

very based thread, this is how we rewrite un\*x without wiggers

Pi day is coming up. Did you ever get around to calculating pi?

Here's a useful place to start. I assume you remember sine, cosine, and tangent from school; if not, you can quickly look them up online.

If you took trigonometry, you'll remember that there are formulas to get
sin(A+B)
and
cos(A+B)
in terms of sin(A), sin(B), cos(A), and cos(B). Try seeing if you can derive these formulas. Pic related should help get you started. (Hint: Find formulas for whatever sides and angles you can. It may also help you to draw a rectangle around the whole thing.)

Once you have those, it's easy to work out the formula for
tan(A+B).
You can get it in terms of only tan(A) and tan(B).

Another key insight is that if you measure the angle in radians, then for small angles sin(θ) and tan(θ) are approximately equal to θ. Draw some pictures and see if you can see why.

P83800
Also I forgot to mention another important thing: On a circle of radius 1, the length of the arc inside an angle is equal to the measure of the angle in radians. That's how all this ties in with calculating pi. If we can calculate angles in radians, we are calculating the lengths of pieces of a circle.

P83809

ever attempted to remember the full pi?

P83854
I can calculate it farther than I have it memorized.

Someone on 4/sci/ just made a video attempting to explain Uncle Ted's PhD thesis at the level of "a general audience with some calculus familiarity." Did he succeed? How much of it do you understand?

/watch?v=wD4xrnzKN1Y

P80990
laugh all you want
its almost finished, should be done by the end of march

P80778
So, what's it about?

P80998
>its almost finished, should be done by the end of march
Is it about the age of consent?

P81005
nothing

As part of studying group theory I've been playing around with various concrete groups, among them the rotational symmetry groups of the regular polyhedra. And I noticed something cool, although I'm far from the first to notice it.

In a regular dodecahedron, you can connect certain vertices together to form an inscribed cube. There are five different ways of doing this. If you draw out the edges of the five cubes, each cube in a different color, then on each face of the dodecahedron you end up drawing a pentagram with each of the five line segments in a different color. And on each face of the dodecahedron the colors in the pentagram are arranged differently; in other words, each face displays a different circular permutation of the five colors. There are twenty-four possible circular permutations of five objects, and the twelve permutations seen on the dodecahedron are specifically the even permutations, permutations you can reach with an even number of swaps.

This is an easy way to see that the rotational symmetries of the dodecahedron are isomorphic to the even permutations of five objects. Each rotation in the group of symmetries permutes the five colored cubes, and given an even permutation of the five colors, you can figure out the rotation it corresponds to by picking a face, identifying which face will need to rotate to that face's location to get the desired permutation of colors up to rotation, then seeing how that face will need to be rotated about its center to make the permutation match exactly.

Files related are a picture of the cubes and a foldable version from
https://www.chiark.greenend.org.uk/~sgtatham/polypics/dodec-cubes.html

Some more nice pictures and links.

Here's a picture showing making a dodecahedron by adding "hats" or "roofs" to the faces of one of the inscribed cubes:
https://www.cutoutfoldup.com/813-dodecahedron-as-a-cube-with-hats.php

This construction is apparently in Euclid's Elements (Book XIII, Proposition 17), so it's a very old observation indeed.

This page has an animation showing how the hats can be folded one way to make a dodecahedron and another way to make a cube:
https://cage.ugent.be/~hs/polyhedra/dodeca.html

(In either configuration, there's an empty space in the middle, so you should not incorrectly take this as implying the two solids have the same volume.)

Related: forming a tetrahedron by connecting vertices of a cube.

P79672
What program do u use for this?

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P79683
I didn't make any of these pictures; I just found them on the web. The OP image were apparently generated by the code discussed here:
https://www.chiark.greenend.org.uk/~sgtatham/polyhedra/

When I was confirming it for myself I just drew everything out on a Schlegel diagram (a diagram like the ones in pic related). When I get the time it might be fun to print out a foldable version like the PDF in the OP so I can see it in three dimensions.

Given a triangle with one obtuse angle, is it possible to cut the triangle into smaller triangles, all of them acute? (An acute triangle is a triangle with three acute angles. A right angle is of course neither acute nor obtuse.) If this cannot be done, give a proof of impossibility. If it can be done, what is the smallest number of acute triangles into which any obtuse triangle can be dissected?

The illustration shows a typical attempt that leads nowhere. The triangle has been divided into three acute triangles, but the fourth is obtuse, so nothing has been gained by the preceding cuts.

fix tex

It isn't possible. In fact, no triangle can be cut into smaller triangles, all of them acute.

1. At some point or another, you will have to bisect the obtuse angle in your initial triangle. Otherwise it will stay obtuse.
2. By cutting a triangle into two smaller triangle by drawing a line from one corner to the opposite vertex, at least one of the resulting triangles will be right or obtuse. (because 180°≥2×90°)
2.1. There is another way of cutting a triangle into smaller triangles, which is to draw three lines between each corner and a central point. In this case too, at least one of the resulting triangles will be obtuse, because 360°>3×90°.

P76283
It is possible.

Everyone knows how to prove the pigeonhole principle by induction. Can you prove induction from the pigeonhole principle?

yes

P7469
Natural numbers are defined inductively [tex: n, m ::= 0 | S_n] which means that [tex: 0 \in \mathbb{N}] and that if [tex: n \in \mathbb{N}] then [tex: S_n \in \mathbb{N}] (where the constructor S is the successor operator, i.e., the function from [tex: n \mapsto n+1]
An inductive definition is a pair [tex: (C, \alpha)] with [tex: \alpha : C \rightarrow \mathbb{N}]. We call the arity of constructor c, the number [tex: \alpha(c)]
Let [tex: (C, \alpha)] be an inductive definition. To that definition we associate :
- the set [tex: I_0 = \{ c \in C | \alpha(c) = 0 \}
- for a set [tex: I_n], the set [tex: I_{n+1} = \{ (c, x) \in C \times I_{n}^{\alpha(c)} | \alpha(c) > 0 \} \cup I_n]
We can now define the set $I = \cup_{n \in \mathbb{N}} I_n$ to be the associated inductive structure.

This gives us the following lemma :
>Let E be an inductive set and [tex: e \in E]. Then either e is a constant (constructor with arity 0) or there exists c,x such that [tex: e = (c,x)] where c is a constructor of arity k and x is a k-tuple of elements in E. These cases are all mutually exclusive, in that the elements c and x are unique.
The proof is given by definition. Notice however that we can consider a constructor c like a function from its inductive set to itself (in the form of [tex: c : E^{\alpha(c)} \rightarrow E]). In general, this distinction is not so important, since all constructors are injective by definition.
That allows us to define structural induction.
>Let P be a property dependent on x, on an inductive set E generated by [tex: (C, \alpha)]. Then P is true on E for all natural numbers iff P(c) is true for all constants [tex: c \in C] and if c has arity k, for all [tex: x_1, ..., x_k \in E], [tex: P(x_1)] and [tex: P(x_2)] and ... and [tex: P(x_k)] implies [tex: P(c, (x_1, ..., x_k))].
That way we can define a function using constructors:
>Let E be an inductive set generated by [tex: (C, \alpha)] and F an arbitrary set. Suppose that for each constant [tex: c_0], we have an element [tex: f_{c_0} \in F], and for each constructor c of arity a we have a function [tex: f_c : F^a \rightarrow F] Then there exists a unique function f such that [tex: f(c_0) = f_{c_0}] et for each constructor c, [tex: f((c, (x_1, ..., x_a))) = f_c(f(x_1), ..., f(x_a))].
Notice that with our definition, if we define [tex: \mathbb{N}] with the constructor 0 of arity 0 and the constructor s of arity 1, then the principle of induction to define a function of the form of [tex: \mathbb{N} \rightarrow F] is about giving an element [tex: c_0 \in F] and a bijection [tex: F \rightarrow F]. You need to show that this principle of inductive definition is enough to construct functions defined by an element [tex: c_0 \in F] and a function [tex: \mathbb{N} \times F \rightarrow F].

You have a rectangular chocolate bar marked into m x n squares, and you wish to break up the bar into its constituent squares. At each step, you may pick up one piece and break it along any of its marked vertical or horizontal lines.

Prove that every method finishes in the same number of steps.

Every time you break a piece, you increase the total number of pieces by one. You start with 1 piece, you end up with n*m pieces, so you need n*m - 1 steps.

when you try to add 2 large binary numbers on the computer, they overflow and the computers returns a negative number
when you try to add all natural numbers, you get -1/12
proof we live in a computer simulation?

fixed sized ints are the dumbest wigger shit on earth
you dont use zero static allocations, stupid LARPers
literally every time they write anything too they have a 30 hour debate over whether some number can overflow or is theoretically impossible (a distinction they of course cannot correctly pin down being code monkeys), and then they still say arbitrary size is impractical

P69307
I'd like to see floats replaced too.

So apparently Google used AI to discover new ways to troll people in an 8-dimensional Set game, among other mathematical discoveries:

https://github.com/google-deepmind/funsearch

Is this a big step forward in AI or is it just a speedup on problems where they could have used a search algorithm not based on language models to discover similar results? How do you think generative AI will influence mathematics in the coming years?

P69071
when is someone gonna take /All and make a Lambda+ bot model?

how do i get god at math
i,m get filtered just trying to prove basic properties of coomplex arithmeitc-

Junior (cont):
• Algebraic geometry of schemes, schemes over a ring, projective spectra, derivatives of a function, Serre duality, coherent sheaves, base change. Proper and separable schemes, a valuation criterion for properness and separability (Hartshorne). Functors, representability, moduli spaces. Direct and inverse images of sheaves, higher direct images. With proper mapping, higher direct images are coherent.
• Cohomological methods in algebraic geometry, semicontinuity of cohomology, Zariski's connectedness theorem, Stein factorization.
• Kähler manifolds, Lefschetz's theorem, Hodge theory, Kodaira's relations, properties of the Laplace operator (chapter zero of Griffiths-Harris, is clearly presented in the book by André Weil, "Kähler manifolds"). Hermitian bundles. Line bundles and their curvature. Line bundles with positive curvature. Kodaira-Nakano's theorem on the vanishing of cohomology (Griffiths-Harris).
• Holonomy, the Ambrose-Singer theorem, special holonomies, the classification of holonomies, Calabi-Yau manifolds, Hyperkähler manifolds, the Calabi-Yau theorem.
• Spinors on manifolds, Dirac operator, Ricci curvature, Weizenbeck-Lichnerovich formula, Bochner's theorem. Bogomolov's theorem on the decomposition of manifolds with zero canonical class (Arthur Besse, "Einstein varieties").
• Tate cohomology and class field theory (Cassels-Fröhlich, blue book). Calculation of the quotient group of a Galois group of a number field by the commutator. The Brauer Group and its applications.
• Ergodic theory. Ergodicity of billiards.
• Complex curves, pseudoconformal mappings, Teichmüller spaces, Ahlfors-Bers theory (according to Ahlfors's thin book).

Senior:
• Rational and profinite homotopy type. The nerve of the etale covering of the cellular space is homotopically equivalent to its profinite type. Topological definition of etale cohomology. Action of the Galois group on the profinite homotopy type (Sullivan, "Geometric topology").
• Etale cohomology in algebraic geometry, comparison functor, Henselian rings, geometric points. Base change. Any smooth manifold over a field locally in the etale topology is isomorphic to A^n. The etale fundamental group (Milne, Danilov's review from VINITI and SGA 4 1/2, Deligne's first article).
• Elliptic curves, j-invariant, automorphic forms, Taniyama-Weil conjecture and its applications to number theory (Fermat's theorem).
• Rational homotopies (according to the last chapter of Gel'fand-Manin's book or Griffiths-Morgan-Long-Sullivan's article). Massey operations and rational homotopy type. Vanishing Massey operations on a Kahler manifold.
• Chevalley groups, their generators and relations (according to Steinberg's book). Calculation of the group K_2 from the field (Milnor, Algebraic K-Theory).
• Quillen's algebraic K-theory, BGL^+ and Q-construction (Suslin's review in the 25th volume of VINITI, Quillen's lectures - Lecture Notes in Math. 341).
• Complex analytic manifolds, coherent sheaves, Oka's coherence theorem, Hilbert's nullstellensatz for ideals in a sheaf of holomorphic functions. Noetherian ring of germs of holomorphic functions, Weierstrass's theorem on division, Weierstrass's preparation theorem. The Branched Cover Theorem. The Grauert-Remmert theorem (the image of a compact analytic space under a holomorphic morphism is analytic). Hartogs' theorem on the extension of an analytic function. The multidimensional Cauchy formula and its applications (the uniform limit of holomorphic functions is holomorphic).

you don't get god at maths by doing proofs. you get bog standard mathematicians from doing proofs. you get good at maths by solving lots of tricky problems.

Specialist: (Fifth year of College):
• The Kodaira-Spencer theory. Deformations of the manifold and solutions of the Maurer-Cartan equation. Maurer-Cartan solvability and Massey operations on the DG-Lie algebra of the cohomology of vector fields. The moduli spaces and their finite dimensionality (see Kontsevich's lectures, or Kodaira's collected works). Bogomolov-Tian-Todorov theorem on deformations of Calabi-Yau.
• Symplectic reduction. The momentum map. The Kempf-Ness theorem.
• Deformations of coherent sheaves and fiber bundles in algebraic geometry. Geometric theory of invariants. The moduli space of bundles on a curve. Stability. The compactifications of Uhlenbeck, Gieseker and Maruyama. The geometric theory of invariants is symplectic reduction (the third edition of Mumford's Geometric Invariant Theory, applications of Francis Kirwan).
• Instantons in four-dimensional geometry. Donaldson's theory. Donaldson's Invariants. Instantons on Kähler surfaces.
• Geometry of complex surfaces. Classification of Kodaira, Kähler and non-Kähler surfaces, Hilbert scheme of points on a surface. The criterion of Castelnuovo-Enriques, the Riemann-Roch formula, the Bogomolov-Miyaoka-Yau inequality. Relations between the numerical invariants of the surface. Elliptic surfaces, Kummer surface, surfaces of type K3 and Enriques.
• Elements of the Mori program: the Kawamata-Viehweg vanishing theorem, theorems on base point freeness, Mori's Cone Theorem (Clemens-Kollar-Mori, "Higher dimensional complex geometry" plus the not translated Kollar-Mori and Kawamata-Matsuki-Masuda).
• Stable bundles as instantons. Yang-Mills equation on a Kahler manifold. The Donaldson-Uhlenbeck-Yau theorem on Yang-Mills metrics on a stable bundle. Its interpretation in terms of symplectic reduction. Stable bundles and instantons on hyper-Kähler manifolds; An explicit solution of the Maurer-Cartan equation in terms of the Green operator.

Specialist (cont):
• Pseudoholomorphic curves on a symplectic manifold. Gromov-Witten invariants. Quantum cohomology. Mirror hypothesis and its interpretation. The structure of the symplectomorphism group (according to the article of Kontsevich-Manin, Polterovich's book "Symplectic geometry", the green book on pseudoholomorphic curves and lecture notes by McDuff and Salamon)
• Complex spinors, the Seiberg-Witten equation, Seiberg-Witten invariants. Why the Seiberg-Witten invariants are equal to the Gromov-Witten invariants.
• Hyperkähler reduction. Flat bundles and the Yang-Mills equation. Hyperkähler structure on the moduli space of flat bundles (Hitchin-Simpson).
• Mixed Hodge structures. Mixed Hodge structures on the cohomology of an algebraic variety. Mixed Hodge structures on the Maltsev completion of the fundamental group. Variations of mixed Hodge structures. The nilpotent orbit theorem. The SL(2)-orbit theorem. Closed and vanishing cycles. The exact sequence of Clemens-Schmid (Griffiths red book "Transcendental methods in algebraic geometry").
• Non-Abelian Hodge theory. Variations of Hodge structures as fixed points of C^*-actions on the moduli space of Higgs bundles (Simpson's thesis).
• Weil conjectures and their proof. l-adic sheaves, perverse sheaves, Frobenius automorphism, weights, the purity theorem (Beilinson, Bernstein, Deligne, plus Deligne, Weil conjectures II)
• The quantitative algebraic topology of Gromov, (Gromov "Metric structures for Riemannian and non-Riemannian spaces"). Gromov-Hausdorff metric, the precompactness of a set of metric spaces, hyperbolic manifolds and hyperbolic groups, harmonic mappings into hyperbolic spaces, the proof of Mostow's rigidity theorem (two compact Kählerian manifolds covered by the same symmetric space X of negative curvature are isometric if their fundamental groups are isomorphic, and dim X> 2).
• Varieties of general type, Kobayashi and Bergman metrics, analytic rigidity (Siu)

You should be able to solve this.

P60706
It lags a lot, but it does looks clean, even when zooming in.

Anyway, here's a proper answer:
We know from their Taylor expansion that (not writing everything because fuck it)
[tex:sin(x) =_{x→0} x - \frac{x³}{6} + \frac{x^5}{120} + ...]
[tex:tan(x) =_{x→0} x + \frac{x³}{6} + \frac{x^5}{120} + ...]
[tex:atan(x) =_{x→0} x - \frac{x³}{3} + \frac{x^5}{5} + ...]
[tex:asin(x) =_{x→0} x + \frac{x³}{6} + \frac{3 x^5}{40} + ...]
Therefore,
[tex:sin(tan(x)) =_{x→0} x + \frac{x^3}{6} - \frac{x^5}{40} - 275 \frac{x^7}{5040} + o(x^8)]
[tex:tan(sin(x)) =_{x→0} x + \frac{x^3}{6} - \frac{x^5}{40} - 107 \frac{x^7}{5040} + o(x^8)]
[tex:atan(asin(x)) =_{x→0} x - \frac{x^3}{6} + \frac{13 x^5}{120} - \frac{173 x^7}{5040} + o(x^8)]
[tex:asin(atan(x)) =_{x→0} x - \frac{x^3}{6} + \frac{13 x^5}{120} - \frac{341 x^7}{5040} + o(x^8)]

Which means that [tex:sin(tan(x))-tan(sin(x)) =_{x→0} \frac{-168}{5040} x^7 + o(x^8)]
and [tex:asin(atan(x))-atan(asin(x)) =_{x→0} \frac{-168}{5040} x^7 + o(x^8)]

This means that [tex:\frac{sin(tan(x))-tan(sin(x))}{asin(atan(x))-atan(asin(x))} =_{x→0} \frac{1 + o(x)}{1 + o(x)} = 1+o(x)], so [tex:lim_{x→0} \frac{sin(tan(x))-tan(sin(x))}{asin(atan(x))-atan(asin(x))} = 1]

P60706
Yep, indeed. Wrong post quote, meant for P60680

P60734
>It lags a lot, but it does looks clean, even when zooming in.
What software? On most things I try the rounding errors create garbage in that region. Does it sample new points as you zoom in or just enlarge what it's already drawn?

P60397
>picrel
<Man I really need create more neuroplasticity in my brain by doing more advanced math.

Why is there alphabet in the math problem is this some kind of cipher and i being serious been long time since doing this types of math that I forgot how to do it.

these codes arent rendering for me
using links 2

OH NO!!!!
Autistic Madoka has trapped Rena in the center of a perfectly circular pond. Autistic Madoka is waiting on the perimeter of the pond and can run exactly 4x faster than Rena can swim, and will always move to the part of the shore that is closest to Rena as possible

if Rena can touch land for even a split second before Autistic Madoka catches her she can escape, but if Madoka is waiting for her when she arrives then she'll kill Rena!

HOW TF CAN RENA ESCAPE TO SAVE US???

note: if Autistic Madoka can take two paths of the same distance to the location she'll flip a coin and choose one

Autistic madoka can change directions instantly!

P64523
As Rena starts fleeing from the pond Autistic Madoka yells after her
>Not so fast, you think you've escaped but first you must tell me what is the maximum speed difference between your swimming and my running that would still allow you to escape?

if Rena doesn't answer correctly then Autistic Madoka will snap her fingers causing Rena to die instantly

Hint: The ratio is [bold: not] [tex: \pi+1].

The problem I'm trying to figure out is how to drown both of them quietly and hide the bodies so that no one will know before they decompose.

Given: alligator, duct-tape, ball gag

ball gag + (duct-tape) * alligator = win

P64541
THIS, fuck tranime

P64541
55% basted jd
touching cancerime is cring aids
555 gallon 5B + elektron + M4 buster = win
>drowning
in flames powerful enough to melt calcium

P64540
Using a mix of P64523's and P64521's strategies, I manage to escape as long as Autistic Madoka's speed is less than 4.373. It's only a numerical solution though, and Rena's trajectory is suboptimal anyway.

Suppose the cost of shipping a rectangular box is proportional to the sum of its length, width, and height. Is it possible to save money by putting the box you want to deliver inside another box that costs less to ship?

If the proportionality constant is negative.

I don't think so. At least not for a "very long" package ((a,b,c)=(1,0,0), so you can fit it in the diagonal of a cube [tex:=> 1 < 1/\sqrt{3}*3=1.73]) nor a cubic package.

In dimension 1, it isn't possible to fit a segment of length [tex:a] inside a segment of length [tex:b<a]. That's pretty much a tautology.

In dimension 2, fitting a rectangle of size [tex:(a_1, a_2)] inside a rectangle [tex:(b_1, b_2)] requires [tex:2*(a_1+a_2) \le 2*(b_1+b_2)]. This can be proved by applying the triangle inequality on each of the triangles in the corners.

The same rule should apply in dimension 3 and above, but I can't draw it

You should be able to solve this.

hmmm
>cartoon picture
>primary school tier math problem
>insinuating tone
That's my Yuki. Easy. Next problem.

Given:
- the diameter of the small circle is 2*sqrt(2)
- the radius of the large circle is 2

Goal: compute the area of the crescent.

- area_crescent = area_smallc/2 + area_triangle - area_largec/4
- area_smallc = pi*sqrt(2)^2 = 2pi
- area_largec = pi*2^2 = 4pi
- area_triangle = 2^2/2 = 2
- area_crescent = 2pi/2 + 2 - 4pi/4 = pi + 2 - pi = 2

Answer: 2 square meters

On average, how many times do you need to roll a die before all six different numbers have turned up?

>>P61582
python3 lol cringe

P61602
Can someone decrypt this please? I don't want malware on my phone if I can help it.

P61659
>https://watch.goodluckgabe.life/videos/watch/ca022f95-dcc5-4dcc-98bc-b2d03722d766
lol wtf its a peertube video of a guy doing asmr cum tribute to 911 haha

P61659

It's a zipbomb, run!

P61299
for a perfectly weighted cube die, as long as you keep your roll constant (constant newtons [bold:I hate that guy] of force), at the same angle and at the same everything, except for the side that is up of the die, you should be able to get each and every side each in 6 throws

yet, IRL, there are multiple variables that come to get calculated, such as: the centre of gravity for the die, how close to symetrical perfection is the die, on inpact, how is the die affected elastically and plastically, as well as the surface it interacts with, what's the temperature of the die, surface, gas/liquid the die is sorounded by that separates the surface and the die etc etc

There are a lot of variables to calculate and each and every die roll is unique due to all of these factors, and even more that mathematicaly would take quite a bit to calculate even with a computer

Best method is either IRL analysis or Simulation with changing variables and getting the average from there, yet the IRL analysis would take days, weeks, months or even years while the Simulations would take an hours, maybe two, maybe ten maybe a full day or even a full week before there are enough results to satisfy the analist, at which point the average shall be calculated