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README.md: some corrections and cleanup #72

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29 changes: 14 additions & 15 deletions README.md
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Expand Up @@ -18,7 +18,7 @@

# GromovWitten

The package GromovWitten computes generating series for tropical Hurwitz numbers of elliptic curves via mirror symmetry and Feynman integrals, and thus, by a correspondence theorem, Hurwitz numbers in the sense algebraic geometry. Generalizations of the method also allow for the computation of Gromov-Witten invariants for ellptic curves, and are also implemented in the package. GromovWitten is based on the computeralgebra system OSCAR and is provided as a package for the Julia programming language.
The package GromovWitten computes generating series for tropical Hurwitz numbers of elliptic curves via mirror symmetry and Feynman integrals, and thus, by a correspondence theorem, Hurwitz numbers in the sense algebraic geometry. Generalizations of the method also allow for the computation of Gromov-Witten invariants for elliptic curves, and are also implemented in the package. GromovWitten is based on the computeralgebra system OSCAR and is provided as a package for the Julia programming language.

# Installation

Expand All @@ -28,7 +28,7 @@ We assume that Julia is installed in a recent enough version to run OSCAR. Navig
git clone https://github.com/singular-gpispace/GromovWitten.git
```

Navigate to GromovWitten folder execute the following command:
Navigate to the `GromovWitten` folder and execute the following command:

```bash
julia --project
Expand All @@ -43,7 +43,7 @@ import Pkg; Pkg.instantiate()
and load our package. On the first run this may take some time.

```bash
using GromovWitten
using GromovWitten
```


Expand All @@ -57,25 +57,24 @@ graph([(1, 3), (1, 2), (1, 2), (2, 4), (3, 4), (3, 4)])
```

```julia
julia> F=feynman_integral(G)
julia> F = feynman_integral(G);
```

```julia
julia> a = [0, 2, 1, 0, 0, 1];
```

```julia
julia> o=[1,3,4,2];
julia> o = [1,3,4,2];
```

```julia

julia> feynman_integral_branch_type_order(F,a,o)
julia> feynman_integral_branch_type_order(F, a, o)
128*q[2]^4*q[3]^2*q[6]^2
```

```julia
julia> feynman_integral_branch_type(F, a)
julia> feynman_integral_branch_type(F, a)
256*q[2]^4*q[3]^2*q[6]^2
```
also we can compute Feynman Integral of degree 3
Expand All @@ -87,29 +86,29 @@ julia> f = feynman_integral_degree(F, 3)
Finally we substitute all $q$ variables by $q_{1}$ after computing the sum of all Feynman Integral of degree up to 8.

```julia
julia> substitute(feynman_integral_degree_sum(F,8))
julia> substitute(feynman_integral_degree_sum(F,8))
10246144*q[1]^16 + 3294720*q[1]^14 + 886656*q[1]^12 + 182272*q[1]^10 + 25344*q[1]^8 + 1792*q[1]^6 + 32*q[1]^4
```


# Caching Feynman Integral

We can try to catch the previous result in a table.
We can try to cache the previous result in a table.
To do that, we define `feynman_integral_branch_type_cache` , `feynman_integral_degree_cache` and `feynman_integral_degree__sum_cache`. The last one returns the univariable polynomial of Feynman integral sum up to degree $d$.
We define first the Feynman graph,
We define first the Feynman graph,
```julia
julia> G = feynman_graph([(1, 3), (1, 2), (1, 2), (2, 4), (3, 4), (3,4)] )
graph([(1, 3), (1, 2), (1, 2), (2, 4), (3, 4), (3, 4)])
```

```julia
julia> F=feynman_integral(G)
julia> F = feynman_integral(G);
```
we cache the Feynman Integral.
```julia
julia> feynman_integral_degree_cache(F, 3);
```
To diplay the previous caching
To display the previous caching
```julia
julia> F.integral_cache[:degree]
Dict{Vector{Int64}, QQMPolyRingElem} with 4 entries:
Expand All @@ -127,12 +126,12 @@ We get the following table of comparison.
|degree|[10] => 5465008*q[1]^20 +…|5.453744 s(3.425 GiB)|0.000033 s ( 1.609 KiB)|


simillary for The univariable polynomial, we have
similarly for The univariable polynomial, we have
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this sentence still is wrong, but I couldn't figure it out what you even meant, so I can't suggest an improvement (well I can guess that "univariable" is supposed to mean "univariate" and "The" -> "the" but the entire sentence still doesn't make sense to me. No need to explain it to me either!! But perhaps think about it and whether you can rephrase it.

```julia
julia> feynman_integral_degree_sum_cache(F, 3)
1792*q[1]^6 + 32*q[1]^4
```
To diplay the previous caching
To display the previous caching
```julia
julia> F.integral_cache[:sum]
Dict{Vector{Int64}, QQMPolyRingElem} with 5 entries:
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