I want to know how different crypto-currency protocols are from the Bitcoin protocol for addressing the pitches. You know that Split is when two miners mine a block of the same height. In the Bitcoin protocol, miners accept the first block they receive earlier, and dismantle it. Is this also the case for the protocols of other cryptocurrencies?

# Tag: theory

## Theory of Autometa – MathOverflow

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## Representation Theory – The Non-Degeneracy of Hyperbolic GCMs

To let $ A $ denote a generalized Cartan matrix (i.e., a matrix with zeros on the main diagonal and nonpositive integers in other places, such that $ A_ {ij} = 0 $ implied $ A_ {ji} = 0 $). We say that $ A $ is **hyperbolic type** if any right main sub-matrix of $ A $ is of finite or affine type (or equivalent to any correct subdiagram of the dynkin diagram associated with the dynkin diagram) $ A $ of the finite or affine type).

I want to prove the following sentence:

All hyperbolic generalized Cartan matrices are not degenerate.

It is well known that there are only a limited number of hyperbolic matrices, so I can do it directly. But there is a lot of work: There are more than 200 hyperbolic GCMs. Of course, some optimization is possible, but the problem is still extremely tedious.

So the question is: is there a more intellectual way to prove the above phrase? If there is one, can we say something about the sign of these determinants?

## elementary number theory – Show $ gcd (a, b) = 1 $ implies $ varphi (a cdot b) = phi (a) cdot phi (b) $

This means that if $ gcd (m, n) = 1 $, then $ φ (mn) = φ (m) φ (n) $, (Evidence: let $ A, B, C $ are the sets of nonnegative integers that are respectively too and less than coprime $ m, n $, and $ Mn $; then there is a bijection between $ A × B $ and $ C $, according to the Chinese remainder theorem.)

I also saw this on the wiki page of Euler's Totient function, but I had no idea$ dots $

My experiments:

After FTA we have:

$$ a = p_1 ^ { alpha_1} cdots p_n ^ { alpha_n} $$

$$ b = q_1 ^ { beta_1} cdots q_m ^ { beta_m} $$

Since $ gcd (a, b) = 1 $, to have $ p_i neq q_j $, Where $ i in (1, n), j in (1, m) $implies:

$$ a cdot b = p_1 ^ { alpha_1} cdots p_n ^ { alpha_n} cdot q_1 ^ { beta_1} cdots q_m ^ { beta_m} $$

From that:

begin {align}

& ~~~~~~ varphi (a cdot b) \

& = varphi (p_1 ^ { alpha_1} cdots p_n ^ { alpha_n} cdot q_1 ^ { beta_1} cdots q_m ^ { beta_m}) tag * {(1)} \

& = a cdot b (1 frac {1} {p_1}) cdots (1 frac {1} {p_n}) (1 frac {1} {q_1}) cdots (1 frac {1} {q_m}) tag * {(2)} \

& = a (1- frac {1} {p_1}) cdots (1- frac {1} {p_n}) cdot b (1- frac {1} {q_1}) cdots (1- frac {1} {q_m}) tag * {(3)} \

& = varphi (p_1 ^ { alpha_1} cdots p_n ^ { alpha_n}) cdot varphi (q_1 ^ { beta_1} cdots q_m ^ { beta_m}) tag * {(4)} \

& = varphi (a) cdot varphi (b) tag * {(5)}

end

Is this proof valid since I have seen the proof of Euler's product formula?$ ($used on step $ (2)) $ It seems like I would use this feature too. If I then use Euler's product formula to prove this property, it seems a bit circular, or are there other approaches $? $

## Homotopy Theory – What is an example of $ infty $ -topos that is not comparable to sheaves on a Grothendieck site?

My question is as in the title: Does anyone have an example (provided there is one) for one $ infty $-topos that are known not to correspond to the sheaves of a Grothendieck site.

One **$ infty $-topos** is like in Higher Topos Theory (HTT) 6.1.0.4: an $ infty $Category, which is an accessible left-accurate localization of presheaves on a small scale $ infty $-Category.

ONE **Location Grothendieck** is a small one $ infty $-Category $ mathcal {C} $ equipped with the $ infty $-Categorical variant of the classical concept of a Grothendieck topology $ mathcal {T} $as in HTT 6.2.2: a collection of **Siebe** (Sub-objects $ U to j (C) $ of representable presheaves $ mathcal {C} $) satisfy some axioms. **sheaves** on $ ( mathcal {C}, mathcal {T}) $ are presheaves of $ infty $-groupoids on $ mathcal {C} $ which ones are in for the strainers $ mathcal {T} $, Such form a complete subcategory $ mathrm {Shv} ( mathcal {C}, mathcal {T}) $ of $ infty $Category of presheaves.

Note: The question examples of $ ( infty, 1) $ – topoi, which are not given as sheaves on a Grothendieck topology, seems superficially to correspond to this. In practice it is not exactly the same. As the answers to this question show, many are interesting $ infty $topoi that can be described without reference to any Grothendieck site. However, it is still conceivable that a suitable location is available.

Also note: each one $ infty $-topos $ mathcal {X} $ can be obtained as an accessible left-accurate localization of some $ mathrm {Shv} ( mathcal {C}, mathcal {T}) $ in terms of a suitable class of $ infty $-linked morphisms (HTT 6.2.2, 6.5.3.14), e.g. For example, the class of hyperbias. However, this does not exclude immediate $ mathcal {X} $ equivalent to $ mathrm {Shv} ( mathcal {C} & # 39 ;, mathcal {T} & # 39;) $ for another Grothendieck location $ ( mathcal {C} & # 39 ;, mathcal {T} & # 39;) $,

## mp.mathematical physics – What do physicists understand by a topological quantum gravity theory?

This is a jargon-like question.

The fact that this is more likely to be posted in a physics forum indicates two things

- I know too little physics.
- An explanation with more math taste is more appreciated.

### background

I should first explain what a gravitational theory is in my imagination: it seems to be a theory governing the relationship between space and matter. For example, Hilbert addressed this issue by introducing a function for the metrics space. As a consequence, Einstein's field equations were introduced, relating the curvature of the space-time and mass-energy momentum tensor.

A quantum field theory seems (to me) to be a field theory in which possibly every field could occur. In our case, the fields are the metrics whose amplitudes can be calculated by a certain "quantized" action weight

### question

This leads to the confusing part: A topological theory (seems) means a theory that does not depend on the geometry (especially the metric)! What then does a topological quantum gravity theory mean?

## nt.number Theory – Why is the Congruent Number Problem open?

So I read about the Congruent Number Problem.

One of the sentences on the subject says how the two things are equivalent: a positive integer $ n $ be a congruent number and elliptic curve $ y ^ 2 = x ^ 3-n ^ 2 x $ have a non-trivial rational solution. Later it is said in the notes that the problem of the congruent numbers is still open.

My question now is: does the above result not give a criterion to determine whether a certain positive integer is a congruent number or not? The reason could be that we have no way to know when such elliptic curves have a non-trivial solution and therefore no direct way to know if $ n $ is a congruent number?

And wait, when we say we're looking for a criterion, what exactly do we mean by that?

P .: I am new to the theory of elliptic curves, so I apologize if the above question about MO's standards seems a bit ignorant or not.

## Complexity theory – First attempt to convert context-free grammar into Chomsky normal form

This is my first attempt to convert a context-free grammar into a normal Chomsky form. I think I have the right answer, would appreciate any feedback if I made a mistake somewhere.

**Context free grammar**

```
V = {M, N, O}
Σ = {+, *, (, ), x}
R = {
M ---> M + N
M ---> N
N ---> O * N
N ---> O
O ---> ( M )
O ---> x
```

**Chomsky normal form conversion**

**Step one**

M will appear on the right side, so I'll create a new state called A:

```
A ---> M
M ---> M + N
M ---> N
N ---> O * N
N ---> O
O ---> ( M )
O ---> x
```

**Step two**

Ignored because there are no epsilon symbols

**Step three**

Give terminals next to non-terminals their own rule:

```
A ---> M Y N
M ---> M Y N
M ---> O Z N
N ---> O Z N
N ---> Q M P
O ---> Q M P
O ---> x
Y ---> +
Z ---> *
Q ---> (
P ---> )
```

**Step four**

Remove productions

```
A ---> B N
M ---> B N
M ---> C N
N ---> C N
N ---> D P
O ---> D P
O ---> x
Y ---> +
Z ---> *
Q ---> (
P ---> )
B ---> M Y
C ---> O Z
D ---> Q M
```

**The context-free grammar is so …**

```
A ---> B N
M ---> B N
M ---> C N
N ---> C N
N ---> D P
O ---> D P
O ---> x
Y ---> +
Z ---> *
Q ---> (
P ---> )
B ---> M Y
C ---> O Z
D ---> Q M
```

## Representation Theory – Representations of Chevalley algebras over arbitrary fields

Professor Humphrey's "Introduction to Lie Algebra and Representation Theory" explains how we can reduce a semi-simple complex Lie algebra (and its representations) to any field.

To let $ L $ Be a complex semisimple Lie algebra. To let $ {e_ alpha, alpha in Phi; h_i, 1 leq i leq l } $ to be a Chevalley base of $ L $, Where $ l $ is the rank of $ L $ and $ Phi $ his root system. Then we can take $ L_ mathbb {Z} $ to be $ mathbb {Z} $-Span of this Chevalley base. For any field $ k $, we have that $ L (k) = L_ mathbb {Z} otimes k $ defines a Lie algebra over $ k $, We call $ L (k) $ a Chevalley algebra.

Now let it go $ rho: L to mathfrak {gl} (V) $ be a representation of $ L $, Then we can take a suitable (ie permissible) $ mathbb {Z} $grids $ V_ mathbb {Z} $ in the $ V $, Now let it go $ L_V $ be that $ mathbb {Z} $Grid inside $ L $ that stabilizes $ V_ mathbb {Z} $, Then we can prove that $ L_V $ must be closed under the lying clip and therefore $ L_V (k) = L_V otimes k $ defines a Lie algebra over $ k $, Furthermore $$ rho_k: L_V otimes k to mathfrak {gl} (V_ mathbb {Z} otimes k): ell otimes 1 mapsto rho ( ell) otimes 1 $$ is a representation of $ L_V (k) $, There is also a morphism $ varphi: L_V (k) to L (k) $ from Lie algebras. Consequently $ rho_k circ varphi $ defines a representation of $ L (k) $,

This construction depends on the choice of the grid $ V_ mathbb {Z} $, Therefore, I wonder if we can also reduce the following situations $ mathbb {C} $ to $ k $,

- Suppose we have a morphism $ f: V to W $ from $ L $presentations for. Select allowed grids $ V_ mathbb {Z} $ and $ W_ mathbb {Z} $ to the $ V $ and $ W $, Do we have a corresponding morphism? $ V_ mathbb {Z} k to W_ mathbb {Z} k $ from $ L (k) $-repräsentationen? Do we need any conditions for the allowed grids for this to be true?
- To let $ L ' leq L $ be a reductive subalgebra of $ L $ that contains the Cartan subalgebra $ H = langle h_i mid 1 leq i leq l rangle $ from $ L $, Then $ L '$ must be of the form $$ H oplus langle e_ alpha mid alpha in S rangle $$ Where $ S $ is a closed subsystem of the root system $ Phi $, is $ L '(k) $ a subalgebra of $ L (k) $?
- Look again at the situation of 2. Let $ V $ be a representation of $ L $, Consider the limitation of $ V $ to $ L '$ and suppose $ V = V_1 oplus V_2 oplus dots oplus V_n $ is a decomposition of $ V $ in $ L '$presentations for. Can we decompose? $ V_ mathbb {Z} $ as $ (V_ mathbb {Z}) _ i oplus dots oplus (V_ mathbb {Z}) _ n $ so that $ (V_ mathbb {Z}) _ i otimes mathbb {C} cong V_i $ for all $ i $? If not in general, what if $ V = V_1 oplus dots oplus V_n $ is the decomposition $ V $ in $ L '$-isotypic components and thus unique? Moreover, if there is such decomposition, $ ((V_ mathbb {Z}) _ 1 otimes k) oplus dots oplus ((V_ mathbb {Z}) _ n otimes k) $ a decomposition of $ V_k $ in $ L '(k) $-repräsentationen?

Help or hints are welcome.

## elementary number theory – What is an indexed set?

I held on to something. I try to understand what is said in the image linked above.

**"A feature that makes me out of a lot of Λ on a lot one should index the quantity a with Λ. The set Λ is called index and one is the indexed amount. If I (λ) = one, we will write oneλ for I (λ).**

There are no examples and further explanations in the textbook. As far as I understand, Λ can be a set {1,2,3,4 …} or say {a, b, c, d ..} and the set *one* can be {tree, fox, grass}. So, after set a has been indexed by set by = {1,2,3}, we get a = {tree (1), fox (2), grass (3)} (**We are sorry**I do not know how to type so that "1" appears under the word "tree.")

I'm right? And by the way, what does that mean? **"If I (λ) = one, we will write oneλ for I (λ).** I swear, no λ is mentioned before this line in the textbook. Is λ a variable that could represent one of these elements of {1,2,3 …}? Suppose I could say that λ = 1 or λ = 15, right?

Explain what it is about. Thank you in advance.