From e6920d444f102ad5d315aaf4516196eb7d7e7207 Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Tue, 31 May 2022 16:35:46 +0200
Subject: [PATCH 01/18] Fix minus in SV B

---
 src/eko/scale_variations/expanded.py | 145 ++++++++++++++-------------
 1 file changed, 73 insertions(+), 72 deletions(-)

diff --git a/src/eko/scale_variations/expanded.py b/src/eko/scale_variations/expanded.py
index 179e511aa..bf1ca1dd3 100644
--- a/src/eko/scale_variations/expanded.py
+++ b/src/eko/scale_variations/expanded.py
@@ -1,6 +1,12 @@
 # -*- coding: utf-8 -*-
-r"""
-This module contains the scale variation operator in ``ModSV=expanded``
+r"""This module contains the scale variation operator for the expanded scheme (``ModSV=expanded``).
+
+The expressions can be obtained using Eqs. (3.33) and (3.38) of :cite:`AbdulKhalek:2019ihb`.
+Note, however, that our definition of the anomalous dimensions :math:`\gamma`
+includes a further minus sign with resepect to :cite:`AbdulKhalek:2019ihb`, as well as
+our definition of the coefficients of the beta function :math:`\beta_k`
+(compare Eq. (3.3) of :cite:`AbdulKhalek:2019ihb`).
+This effectively introduces a minus signs on terms which include a odd number of :math:`\gamma_j` and :math:`\beta_k` .
 """
 
 
@@ -12,104 +18,100 @@
 
 @nb.njit(cache=True)
 def gamma_1_variation(gamma, L):
-    r"""
-    Computes the |NLO| anomalous dimension variation.
+    r"""Computes the |NLO| anomalous dimension variation.
 
     Parameters
     ----------
-        gamma : numpy.ndarray
-            anomalous dimensions
-        L : float
-            logarithmic ratio of factorization and renormalization scale
+    gamma : numpy.ndarray
+        anomalous dimensions
+    L : float
+        logarithmic ratio of factorization and renormalization scale
 
     Returns
     -------
-        gamma_1 : complex
-            variation to :math:`\gamma^{(1)}`
+    gamma_1 : complex
+        variation to :math:`\gamma^{(1)}`
     """
-    return -L * gamma[0]
+    return L * gamma[0]
 
 
 @nb.njit(cache=True)
 def gamma_2_variation(gamma, L, beta0, g0e2):
-    r"""
-    Computes the |NNLO| anomalous dimension variation.
+    r"""Computes the |NNLO| anomalous dimension variation.
 
     Parameters
     ----------
-        gamma : numpy.ndarray
-            anomalous dimensions
-        L : float
-            logarithmic ratio of factorization and renormalization scale
-        beta0: float
-            :math:`\beta_0`
-        g0e2: complex
-            :math:`\gamma^{(0),2}`
+    gamma : numpy.ndarray
+        anomalous dimensions
+    L : float
+        logarithmic ratio of factorization and renormalization scale
+    beta0: float
+        :math:`\beta_0`
+    g0e2: complex
+        :math:`\left(\gamma^{(0)}\right)^2`
 
     Returns
     -------
-        gamma_2 : complex
-            variation to :math:`\gamma^{(2)}`
+    gamma_2 : complex
+        variation to :math:`\gamma^{(2)}`
     """
-    return -gamma[1] * L + 1 / 2 * (beta0 * gamma[0] + g0e2) * L**2
+    return gamma[1] * L + 1 / 2 * (beta0 * gamma[0] + g0e2) * L**2
 
 
 @nb.njit(cache=True)
 def gamma_3_variation(gamma, L, beta0, beta1, g0e2, g0e3, g1g0):
-    r"""
-    Computes the |N3LO| anomalous dimension variation.
+    r"""Computes the |N3LO| anomalous dimension variation.
 
     Parameters
     ----------
-        gamma : numpy.ndarray
-            anomalous dimensions
-        L : float
-            logarithmic ratio of factorization and renormalization scale
-        beta0: float
-            :math:`\beta_0`
-        beta0: float
-            :math:`\beta_1`
-        g0e2: complex
-            :math:`\gamma^{(0),2}`
-        g0e3: complex
-            :math:`\gamma^{(0),3}`
-        g1g0: complex
-            :math:`\gamma^{(1)} \gamma^{(0)}`
+    gamma : numpy.ndarray
+        anomalous dimensions
+    L : float
+        logarithmic ratio of factorization and renormalization scale
+    beta0: float
+        :math:`\beta_0`
+    beta0: float
+        :math:`\beta_1`
+    g0e2: complex
+        :math:`\left(\gamma^{(0)}\right)^2`
+    g0e3: complex
+        :math:`\left(\gamma^{(0)}\right)^3`
+    g1g0: complex
+        :math:`\gamma^{(1)} \gamma^{(0)}`
 
     Returns
     -------
-        gamma_3 : complex
-            variation to :math:`\gamma^{(3)}`
+    gamma_3 : complex
+        variation to :math:`\gamma^{(3)}`
     """
     return (
-        -gamma[2] * L
+        gamma[2] * L
         + (1 / 2) * (beta1 * gamma[0] + 2 * beta0 * gamma[1] + 2 * g1g0) * L**2
-        - (1 / 6) * (2 * beta0**2 * gamma[0] + 3 * beta0 * g0e2 + g0e3) * L**3
+        + (1 / 6) * (2 * beta0**2 * gamma[0] + 3 * beta0 * g0e2 + g0e3) * L**3
     )
 
 
 @nb.njit(cache=True)
 def non_singlet_variation(gamma, a_s, order, nf, L):
-    """
-    Scale Variation non-singlet dispatcher
+    """Non-singlet scale variation dispatcher.
 
     Parameters
     ----------
-        gamma : numpy.ndarray
-            anomalous dimensions
-        a_s :  float
-            target coupling value
-        order : int
-            perturbation order
-        nf : int
-            number of active flavors
-        L : float
-            logarithmic ratio of factorization and renormalization scale
+    gamma : numpy.ndarray
+        anomalous dimensions
+    a_s :  float
+        target coupling value
+    order : int
+        perturbation order
+    nf : int
+        number of active flavors
+    L : float
+        logarithmic ratio of factorization and renormalization scale
 
     Returns
     -------
-        sv_ker : numpy.ndarray
-            scale varion kernel
+    sv_ker : numpy.ndarray
+        scale varion kernel
     """
     sv_ker = 1.0
     if order >= 1:
@@ -127,26 +129,25 @@ def non_singlet_variation(gamma, a_s, order, nf, L):
 
 @nb.njit(cache=True)
 def singlet_variation(gamma, a_s, order, nf, L):
-    """
-    Scale Variation singlet dispatcher
+    """Singlet scale cariation dispatcher
 
     Parameters
     ----------
-        gamma : numpy.ndarray
-            anomalous dimensions
-        a_s :  float
-            target coupling value
-        order : int
-            perturbation order
-        nf : int
-            number of active flavors
-        L : float
-            logarithmic ratio of factorization and renormalization scale
+    gamma : numpy.ndarray
+        anomalous dimensions
+    a_s :  float
+        target coupling value
+    order : int
+        perturbation order
+    nf : int
+        number of active flavors
+    L : float
+        logarithmic ratio of factorization and renormalization scale
 
     Returns
     -------
-        sv_ker : numpy.ndarray
-            scale varion kernel
+    sv_ker : numpy.ndarray
+        scale varion kernel
     """
     sv_ker = np.eye(2, dtype=np.complex_)
     gamma = np.ascontiguousarray(gamma)

From 039beec06aec5f14be82b696817bd6b5b4ed2b6f Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Tue, 31 May 2022 16:57:39 +0200
Subject: [PATCH 02/18] Fix non-comm. in sv

---
 src/eko/scale_variations/expanded.py | 28 ++++++++++++++++++----------
 tests/eko/test_sv_expanded.py        |  2 +-
 2 files changed, 19 insertions(+), 11 deletions(-)

diff --git a/src/eko/scale_variations/expanded.py b/src/eko/scale_variations/expanded.py
index bf1ca1dd3..884a3427d 100644
--- a/src/eko/scale_variations/expanded.py
+++ b/src/eko/scale_variations/expanded.py
@@ -47,7 +47,7 @@ def gamma_2_variation(gamma, L, beta0, g0e2):
         logarithmic ratio of factorization and renormalization scale
     beta0: float
         :math:`\beta_0`
-    g0e2: complex
+    g0e2: complex or numpy.ndarray
         :math:`\left(\gamma^{(0)}\right)^2`
 
     Returns
@@ -55,11 +55,11 @@ def gamma_2_variation(gamma, L, beta0, g0e2):
     gamma_2 : complex
         variation to :math:`\gamma^{(2)}`
     """
-    return gamma[1] * L + 1 / 2 * (beta0 * gamma[0] + g0e2) * L**2
+    return gamma[1] * L + 1.0 / 2.0 * (beta0 * gamma[0] + g0e2) * L**2
 
 
 @nb.njit(cache=True)
-def gamma_3_variation(gamma, L, beta0, beta1, g0e2, g0e3, g1g0):
+def gamma_3_variation(gamma, L, beta0, beta1, g0e2, g0e3, g1g0, g0g1):
     r"""Computes the |N3LO| anomalous dimension variation.
 
     Parameters
@@ -72,12 +72,14 @@ def gamma_3_variation(gamma, L, beta0, beta1, g0e2, g0e3, g1g0):
         :math:`\beta_0`
     beta0: float
         :math:`\beta_1`
-    g0e2: complex
+    g0e2: complex or numpy.ndarray
         :math:`\left(\gamma^{(0)}\right)^2`
-    g0e3: complex
+    g0e3: complex or numpy.ndarray
         :math:`\left(\gamma^{(0)}\right)^3`
-    g1g0: complex
+    g1g0: complex or numpy.ndarray
         :math:`\gamma^{(1)} \gamma^{(0)}`
+    g0g1: complex or numpy.ndarray
+        :math:`\gamma^{(0)} \gamma^{(1)} `
 
     Returns
     -------
@@ -86,8 +88,12 @@ def gamma_3_variation(gamma, L, beta0, beta1, g0e2, g0e3, g1g0):
     """
     return (
         gamma[2] * L
-        + (1 / 2) * (beta1 * gamma[0] + 2 * beta0 * gamma[1] + 2 * g1g0) * L**2
-        + (1 / 6) * (2 * beta0**2 * gamma[0] + 3 * beta0 * g0e2 + g0e3) * L**3
+        + (1.0 / 2.0)
+        * (beta1 * gamma[0] + 2.0 * beta0 * gamma[1] + g1g0 + g0g1)
+        * L**2
+        + (1.0 / 6.0)
+        * (2.0 * beta0**2 * gamma[0] + 3.0 * beta0 * g0e2 + g0e3)
+        * L**3
     )
 
 
@@ -121,8 +127,9 @@ def non_singlet_variation(gamma, a_s, order, nf, L):
         sv_ker += a_s**2 * gamma_2_variation(gamma, L, beta0, gamma[0] ** 2)
     if order >= 3:
         beta1 = beta.beta(1, nf)
+        g0g1 = gamma[0] * gamma[1]
         sv_ker += a_s**3 * gamma_3_variation(
-            gamma, L, beta0, beta1, gamma[0] ** 2, gamma[0] ** 3, gamma[0] * gamma[1]
+            gamma, L, beta0, beta1, gamma[0] ** 2, gamma[0] ** 3, g0g1, g0g1
         )
     return sv_ker
 
@@ -162,7 +169,8 @@ def singlet_variation(gamma, a_s, order, nf, L):
         gamma0e3 = gamma0e2 @ gamma[0]
         # here the product is not commutative
         g1g0 = gamma[1] @ gamma[0]
+        g0g1 = gamma[0] @ gamma[1]
         sv_ker += a_s**3 * gamma_3_variation(
-            gamma, L, beta0, beta1, gamma0e2, gamma0e3, g1g0
+            gamma, L, beta0, beta1, gamma0e2, gamma0e3, g1g0, g0g1
         )
     return sv_ker
diff --git a/tests/eko/test_sv_expanded.py b/tests/eko/test_sv_expanded.py
index f0828f6f9..744fb5224 100644
--- a/tests/eko/test_sv_expanded.py
+++ b/tests/eko/test_sv_expanded.py
@@ -63,7 +63,7 @@ def scheme_diff(g, k, pto, is_singlet):
         """
         :math:`(ker_A - ker_B)/ker_{unv}` for truncated expansion
         Effects due to non commutativity are neglected thus,
-        he accuracy of singlet quantities is slightly worst.
+        the accuracy of singlet quantities is slightly worse.
         """
         if pto >= 1:
             diff = g[0] * k * a0

From 1189c790271fe109befbf06cd7f68614e2de293a Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Wed, 1 Jun 2022 14:37:29 +0200
Subject: [PATCH 03/18] Revert minus sign

---
 src/eko/scale_variations/expanded.py | 35 +++++++++++++---------------
 1 file changed, 16 insertions(+), 19 deletions(-)

diff --git a/src/eko/scale_variations/expanded.py b/src/eko/scale_variations/expanded.py
index 884a3427d..abbe21dd8 100644
--- a/src/eko/scale_variations/expanded.py
+++ b/src/eko/scale_variations/expanded.py
@@ -2,14 +2,11 @@
 r"""This module contains the scale variation operator for the expanded scheme (``ModSV=expanded``).
 
 The expressions can be obtained using Eqs. (3.33) and (3.38) of :cite:`AbdulKhalek:2019ihb`.
-Note, however, that our definition of the anomalous dimensions :math:`\gamma`
-includes a further minus sign with resepect to :cite:`AbdulKhalek:2019ihb`, as well as
-our definition of the coefficients of the beta function :math:`\beta_k`
-(compare Eq. (3.3) of :cite:`AbdulKhalek:2019ihb`).
-This effectively introduces a minus signs on terms which include a odd number of :math:`\gamma_j` and :math:`\beta_k` .
+Be aware that corresponding the signs of the ingredients there are a number of differences.
+However, the ultimate sign can be obtained by comparing to the exponentiated scheme in the
+trunctated solution.
 """
 
-
 import numba as nb
 import numpy as np
 
@@ -17,7 +14,7 @@
 
 
 @nb.njit(cache=True)
-def gamma_1_variation(gamma, L):
+def variation_as1(gamma, L):
     r"""Computes the |NLO| anomalous dimension variation.
 
     Parameters
@@ -32,11 +29,11 @@ def gamma_1_variation(gamma, L):
     gamma_1 : complex
         variation to :math:`\gamma^{(1)}`
     """
-    return L * gamma[0]
+    return -L * gamma[0]
 
 
 @nb.njit(cache=True)
-def gamma_2_variation(gamma, L, beta0, g0e2):
+def variation_as2(gamma, L, beta0, g0e2):
     r"""Computes the |NNLO| anomalous dimension variation.
 
     Parameters
@@ -55,11 +52,11 @@ def gamma_2_variation(gamma, L, beta0, g0e2):
     gamma_2 : complex
         variation to :math:`\gamma^{(2)}`
     """
-    return gamma[1] * L + 1.0 / 2.0 * (beta0 * gamma[0] + g0e2) * L**2
+    return -gamma[1] * L + 1.0 / 2.0 * (beta0 * gamma[0] + g0e2) * L**2
 
 
 @nb.njit(cache=True)
-def gamma_3_variation(gamma, L, beta0, beta1, g0e2, g0e3, g1g0, g0g1):
+def variation_as3(gamma, L, beta0, beta1, g0e2, g0e3, g1g0, g0g1):
     r"""Computes the |N3LO| anomalous dimension variation.
 
     Parameters
@@ -87,11 +84,11 @@ def gamma_3_variation(gamma, L, beta0, beta1, g0e2, g0e3, g1g0, g0g1):
         variation to :math:`\gamma^{(3)}`
     """
     return (
-        gamma[2] * L
+        -gamma[2] * L
         + (1.0 / 2.0)
         * (beta1 * gamma[0] + 2.0 * beta0 * gamma[1] + g1g0 + g0g1)
         * L**2
-        + (1.0 / 6.0)
+        - (1.0 / 6.0)
         * (2.0 * beta0**2 * gamma[0] + 3.0 * beta0 * g0e2 + g0e3)
         * L**3
     )
@@ -121,14 +118,14 @@ def non_singlet_variation(gamma, a_s, order, nf, L):
     """
     sv_ker = 1.0
     if order >= 1:
-        sv_ker += a_s * gamma_1_variation(gamma, L)
+        sv_ker += a_s * variation_as1(gamma, L)
     if order >= 2:
         beta0 = beta.beta_0(nf)
-        sv_ker += a_s**2 * gamma_2_variation(gamma, L, beta0, gamma[0] ** 2)
+        sv_ker += a_s**2 * variation_as2(gamma, L, beta0, gamma[0] ** 2)
     if order >= 3:
         beta1 = beta.beta(1, nf)
         g0g1 = gamma[0] * gamma[1]
-        sv_ker += a_s**3 * gamma_3_variation(
+        sv_ker += a_s**3 * variation_as3(
             gamma, L, beta0, beta1, gamma[0] ** 2, gamma[0] ** 3, g0g1, g0g1
         )
     return sv_ker
@@ -159,18 +156,18 @@ def singlet_variation(gamma, a_s, order, nf, L):
     sv_ker = np.eye(2, dtype=np.complex_)
     gamma = np.ascontiguousarray(gamma)
     if order >= 1:
-        sv_ker += a_s * gamma_1_variation(gamma, L)
+        sv_ker += a_s * variation_as1(gamma, L)
     if order >= 2:
         beta0 = beta.beta_0(nf)
         gamma0e2 = gamma[0] @ gamma[0]
-        sv_ker += a_s**2 * gamma_2_variation(gamma, L, beta0, gamma0e2)
+        sv_ker += a_s**2 * variation_as2(gamma, L, beta0, gamma0e2)
     if order >= 3:
         beta1 = beta.beta(1, nf)
         gamma0e3 = gamma0e2 @ gamma[0]
         # here the product is not commutative
         g1g0 = gamma[1] @ gamma[0]
         g0g1 = gamma[0] @ gamma[1]
-        sv_ker += a_s**3 * gamma_3_variation(
+        sv_ker += a_s**3 * variation_as3(
             gamma, L, beta0, beta1, gamma0e2, gamma0e3, g1g0, g0g1
         )
     return sv_ker

From e2c22e88087755e537d1b255afd32041fcac112a Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Wed, 1 Jun 2022 14:43:34 +0200
Subject: [PATCH 04/18] Update docs

---
 src/eko/scale_variations/expanded.py | 22 +++++++++++-----------
 1 file changed, 11 insertions(+), 11 deletions(-)

diff --git a/src/eko/scale_variations/expanded.py b/src/eko/scale_variations/expanded.py
index abbe21dd8..2ae7504b3 100644
--- a/src/eko/scale_variations/expanded.py
+++ b/src/eko/scale_variations/expanded.py
@@ -26,8 +26,8 @@ def variation_as1(gamma, L):
 
     Returns
     -------
-    gamma_1 : complex
-        variation to :math:`\gamma^{(1)}`
+    complex
+        variation at |NLO|
     """
     return -L * gamma[0]
 
@@ -49,8 +49,8 @@ def variation_as2(gamma, L, beta0, g0e2):
 
     Returns
     -------
-    gamma_2 : complex
-        variation to :math:`\gamma^{(2)}`
+    complex
+        variation at |NNLO|
     """
     return -gamma[1] * L + 1.0 / 2.0 * (beta0 * gamma[0] + g0e2) * L**2
 
@@ -80,8 +80,8 @@ def variation_as3(gamma, L, beta0, beta1, g0e2, g0e3, g1g0, g0g1):
 
     Returns
     -------
-    gamma_3 : complex
-        variation to :math:`\gamma^{(3)}`
+    complex
+        variation at |N3LO|
     """
     return (
         -gamma[2] * L
@@ -113,8 +113,8 @@ def non_singlet_variation(gamma, a_s, order, nf, L):
 
     Returns
     -------
-    sv_ker : numpy.ndarray
-        scale varion kernel
+    complex
+        scale variation kernel
     """
     sv_ker = 1.0
     if order >= 1:
@@ -133,7 +133,7 @@ def non_singlet_variation(gamma, a_s, order, nf, L):
 
 @nb.njit(cache=True)
 def singlet_variation(gamma, a_s, order, nf, L):
-    """Singlet scale cariation dispatcher
+    """Singlet scale variation dispatcher.
 
     Parameters
     ----------
@@ -150,8 +150,8 @@ def singlet_variation(gamma, a_s, order, nf, L):
 
     Returns
     -------
-    sv_ker : numpy.ndarray
-        scale varion kernel
+    numpy.ndarray
+        scale variation kernel
     """
     sv_ker = np.eye(2, dtype=np.complex_)
     gamma = np.ascontiguousarray(gamma)

From 0f0e7ebf0b97b2b1309d9af393a4632e7a5310e4 Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Wed, 1 Jun 2022 15:23:07 +0200
Subject: [PATCH 05/18] Update theory docs

---
 doc/source/theory/pQCD.rst           | 38 +++++++++++++---------------
 src/eko/scale_variations/expanded.py |  2 +-
 2 files changed, 19 insertions(+), 21 deletions(-)

diff --git a/doc/source/theory/pQCD.rst b/doc/source/theory/pQCD.rst
index 9409875d8..18a1b1547 100644
--- a/doc/source/theory/pQCD.rst
+++ b/doc/source/theory/pQCD.rst
@@ -146,11 +146,12 @@ This method provides many advantages:
     * it naturally incorporates renormalization group invariance,
       as the perturbative order increases, estimates of |MHOU| decrease;
     * the same procedure can be used for any perturbative process,
-      since the scale dependence of the strong coupling :math:`a_s(\mu^2)` and of PDFs is universal;
+      since the scale dependence of the strong coupling :math:`a_s(\mu^2)` and of |PDF| are universal;
 
 However, there is no unique prescription to determine the specific range of the scale variation,
-the most used prescription specify to vary the factor :math:`\mu_F/\mu_R` in the range:
-:math:`1/2 \le \mu_F/\mu_R \le 2`.
+the most common prescription specify to vary the factor :math:`\mu_F/\mu_R` in the range:
+:math:`1/2 \le \mu_F/\mu_R \le 2`. In the following we express this additional dependency as a function
+of :math:`k = \ln(\mu_F^2/\mu_R^2)`
 
 This variation can be performed at least at two different levels during the |PDF|
 evolution, always evaluating the strong coupling at :math:`\mu_R^2`.
@@ -159,30 +160,28 @@ evolution, always evaluating the strong coupling at :math:`\mu_R^2`.
       and the anomalous dimension are then modified using :cite:`Vogt:2004ns`:
 
         .. math ::
-            & \gamma^{(1)}(N) \to \gamma^{(1)}(N) - \beta_0 k \gamma^{(0)} \\
-            & \gamma^{(2)}(N) \to \gamma^{(2)}(N) - 2 \beta_0 k \gamma^{(1)} - ( \beta_1 k - \beta_0^2 k^2) \gamma^{(0)} \\
-            & \gamma^{(3)}(N) \to \gamma^{(3)}(N) - 3 \beta_0 k \gamma^{(2)} - ( 2 \beta_1 k - 3 \beta_0^2 k^2) \gamma^{(1)} - (\beta_2 k - \frac{5}{2} \beta_1 \beta_0 k^2 + \beta_0^3 k^3) \gamma^{(0)} \\
-            & k = \ln(\mu_F^2/\mu_R^2)
+            \gamma^{(1)}(N) &\to \gamma^{(1)}(N) - \beta_0 k \gamma^{(0)} \\
+            \gamma^{(2)}(N) &\to \gamma^{(2)}(N) - 2 \beta_0 k \gamma^{(1)} - ( \beta_1 k - \beta_0^2 k^2) \gamma^{(0)} \\
+            \gamma^{(3)}(N) &\to \gamma^{(3)}(N) - 3 \beta_0 k \gamma^{(2)} - ( 2 \beta_1 k - 3 \beta_0^2 k^2) \gamma^{(1)} - (\beta_2 k - \frac{5}{2} \beta_1 \beta_0 k^2 + \beta_0^3 k^3) \gamma^{(0)}
 
       This procedure corresponds to Eq. (3.32) of :cite:`AbdulKhalek:2019ihb`, and we recommend to use it along with
       ``ModEv='iterate-exact'`` in order to be in agreement with the treatment of the evolution integral expansion.
 
 
-    * In ``ModSV='expanded'`` the |EKO| is multiplied by an additional kernel, such that
-      the scale variation is applied to the whole evolution operator:
+    * In ``ModSV='expanded'`` the full |EKO| is multiplied by an additional kernel:
 
         .. math ::
             \tilde{\mathbf{E}}(a_s \leftarrow a_s^0) & = \tilde{\mathbf{K}}(a_s) \tilde{\mathbf{E}}(a_s \leftarrow a_s^0) \\
             \tilde{\mathbf{K}}(a_s) & = 1 - k \gamma + \frac{1}{2} k^2 \left ( \gamma^{2} - \beta \frac{\partial \gamma}{\partial a_s} \right ) \\
-            & + \frac{1}{6} k^3 \left [ - \beta \frac{\partial}{\partial a_s} \left( \beta \frac{\partial \gamma}{\partial a_s} \right) + 3 \beta \frac{\partial \gamma}{\partial a_s} \gamma - \gamma^3 \right ] + \mathcal{O}(k^4)
+            & \hspace{10pt} + \frac{1}{6} k^3 \left [ - \beta \frac{\partial}{\partial a_s} \left( \beta \frac{\partial \gamma}{\partial a_s} \right) + 3 \beta \frac{\partial \gamma}{\partial a_s} \gamma - \gamma^3 \right ] + \mathcal{O}(k^4)
 
-      where scale variation kernel is expanded consistently order by order in :math:`a_s`,
+      where the scale variation kernel :math:`\tilde{\mathbf{K}}` is expanded consistently order by order in :math:`a_s`,
       leading to:
 
         .. math ::
-            \tilde{\mathbf{K}}(a_s) \approx & 1 - a_s k \gamma^{(0)} + a_s^2 \left [ - k \gamma^{(1)} + \frac{1}{2} k^2 \gamma^{(0)} (\beta_0 + \gamma^{(0)}) \right ] \\
-            & + a_s^3 \left [ -k \gamma^{(2)} + \frac{1}{2} k^2 \left(\beta_1 \gamma^{(0)} + 2 \gamma^{(1)} (\beta_0 + \gamma^{(0)} ) \right) \right. \\
-            & \left. - \frac{1}{6} k^3 \gamma^{(0)} \left(2 \beta_0^2 + 3 \beta_0 \gamma^{(0)}+\gamma^{(0),2} \right) \right] + \mathcal{O}(a^4)
+            \tilde{\mathbf{K}}(a_s) &\approx 1 - a_s k \gamma^{(0)} + a_s^2 \left [ - k \gamma^{(1)} + \frac{1}{2} k^2 \gamma^{(0)} (\beta_0 + \gamma^{(0)}) \right ] \\
+            & \hspace{10pt} + a_s^3 \left [ -k \gamma^{(2)} + \frac{1}{2} k^2 \left(\beta_1 \gamma^{(0)} + 2 \beta_0\gamma^{(1)}  + \gamma^{(1)}\gamma^{(0)} + \gamma^{(0)}\gamma^{(1)} \right) \right. \\
+            & \hspace{35pt} \left. - \frac{1}{6} k^3 \gamma^{(0)} \left(2 \beta_0^2 + 3 \beta_0 \gamma^{(0)} + \left(\gamma^{(0)}\right)^2 \right) \right] + \mathcal{O}(a_s^4)
 
 
       In this way the dependence of the |EKO| on :math:`k` is factorized outside the unvaried evolution kernel.
@@ -190,12 +189,11 @@ evolution, always evaluating the strong coupling at :math:`\mu_R^2`.
       It corresponds to Eq. (3.35) of :cite:`AbdulKhalek:2019ihb`, and we recommend to use it along with
       ``ModEv='truncated'`` in order to keep consistency with the evolution integral expansion.
 
-
-By construction, the corrections of the order :math:`\mathcal{O}(k^n)` will appear
-at the order :math:`n` in the expansion :math:`a_s`.
-This happens because :math:`\beta \approx \mathcal{O}(a_s^2)`, :math:`\gamma \approx \mathcal{O}(a_s)`
-and the contribution proportional to :math:`\mathcal{O}(k^n)` is originated
-by the `n-th` derivative in :math:`\gamma` :cite:`AbdulKhalek:2019ihb`.
+      By construction, the corrections of the order :math:`\mathcal{O}(k^n)` will appear
+      at the order :math:`n` in the expansion :math:`a_s`.
+      This happens because :math:`\beta \approx \mathcal{O}(a_s^2)`, :math:`\gamma \approx \mathcal{O}(a_s)`
+      and the contribution proportional to :math:`\mathcal{O}(k^n)` is originated
+      by the `n-th` derivative in :math:`\gamma` :cite:`AbdulKhalek:2019ihb`.
 
 Furthermore the distance between the varied |EKO| and the unvaried one will decrease while
 keeping higher order terms in :math:`a_s`
diff --git a/src/eko/scale_variations/expanded.py b/src/eko/scale_variations/expanded.py
index 2ae7504b3..bf2e30f42 100644
--- a/src/eko/scale_variations/expanded.py
+++ b/src/eko/scale_variations/expanded.py
@@ -76,7 +76,7 @@ def variation_as3(gamma, L, beta0, beta1, g0e2, g0e3, g1g0, g0g1):
     g1g0: complex or numpy.ndarray
         :math:`\gamma^{(1)} \gamma^{(0)}`
     g0g1: complex or numpy.ndarray
-        :math:`\gamma^{(0)} \gamma^{(1)} `
+        :math:`\gamma^{(0)} \gamma^{(1)}`
 
     Returns
     -------

From 2eb0536bbd7b73f0b040e58ea6ded25d79098b0a Mon Sep 17 00:00:00 2001
From: Alessandro Candido <candido.ale@gmail.com>
Date: Thu, 2 Jun 2022 15:56:33 +0200
Subject: [PATCH 06/18] Poetry update

---
 poetry.lock | 796 +++++++++++++++++++++++++++-------------------------
 1 file changed, 408 insertions(+), 388 deletions(-)

diff --git a/poetry.lock b/poetry.lock
index e107b38c6..8141afe7f 100644
--- a/poetry.lock
+++ b/poetry.lock
@@ -83,7 +83,7 @@ python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*"
 name = "attrs"
 version = "21.4.0"
 description = "Classes Without Boilerplate"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*, !=3.4.*"
 
@@ -139,7 +139,7 @@ docs = ["Sphinx (>=4.3.2,<5.0.0)", "sphinx-rtd-theme (>=1.0.0,<2.0.0)", "sphinxc
 name = "beautifulsoup4"
 version = "4.11.1"
 description = "Screen-scraping library"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.6.0"
 
@@ -154,7 +154,7 @@ lxml = ["lxml"]
 name = "bleach"
 version = "5.0.0"
 description = "An easy safelist-based HTML-sanitizing tool."
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -168,17 +168,17 @@ dev = ["pip-tools (==6.5.1)", "pytest (==7.1.1)", "flake8 (==4.0.1)", "tox (==3.
 
 [[package]]
 name = "certifi"
-version = "2021.10.8"
+version = "2022.5.18.1"
 description = "Python package for providing Mozilla's CA Bundle."
 category = "main"
 optional = false
-python-versions = "*"
+python-versions = ">=3.6"
 
 [[package]]
 name = "cffi"
 version = "1.15.0"
 description = "Foreign Function Interface for Python calling C code."
-category = "dev"
+category = "main"
 optional = false
 python-versions = "*"
 
@@ -236,14 +236,14 @@ test = ["flake8 (==3.7.8)", "hypothesis (==3.55.3)"]
 
 [[package]]
 name = "coverage"
-version = "6.3.3"
+version = "6.4.1"
 description = "Code coverage measurement for Python"
 category = "dev"
 optional = false
 python-versions = ">=3.7"
 
 [package.dependencies]
-tomli = {version = "*", optional = true, markers = "extra == \"toml\""}
+tomli = {version = "*", optional = true, markers = "python_full_version <= \"3.11.0a6\" and extra == \"toml\""}
 
 [package.extras]
 toml = ["tomli"]
@@ -276,17 +276,17 @@ python-versions = ">=3.5"
 name = "defusedxml"
 version = "0.7.1"
 description = "XML bomb protection for Python stdlib modules"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*, !=3.4.*"
 
 [[package]]
 name = "dill"
-version = "0.3.4"
+version = "0.3.5.1"
 description = "serialize all of python"
 category = "dev"
 optional = false
-python-versions = ">=2.7, !=3.0.*"
+python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*, !=3.4.*, !=3.5.*, !=3.6.*"
 
 [package.extras]
 graph = ["objgraph (>=1.7.2)"]
@@ -311,7 +311,7 @@ python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*, !=3.4.*"
 name = "entrypoints"
 version = "0.4"
 description = "Discover and load entry points from installed packages."
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.6"
 
@@ -339,7 +339,7 @@ pyrepl = ">=0.8.2"
 name = "fastjsonschema"
 version = "2.15.3"
 description = "Fastest Python implementation of JSON schema"
-category = "dev"
+category = "main"
 optional = false
 python-versions = "*"
 
@@ -348,7 +348,7 @@ devel = ["colorama", "jsonschema", "json-spec", "pylint", "pytest", "pytest-benc
 
 [[package]]
 name = "filelock"
-version = "3.6.0"
+version = "3.7.1"
 description = "A platform independent file lock."
 category = "dev"
 optional = false
@@ -393,7 +393,7 @@ docs = ["sphinx"]
 
 [[package]]
 name = "identify"
-version = "2.5.0"
+version = "2.5.1"
 description = "File identification library for Python"
 category = "dev"
 optional = false
@@ -420,7 +420,7 @@ python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*"
 
 [[package]]
 name = "importlib-metadata"
-version = "4.11.3"
+version = "4.11.4"
 description = "Read metadata from Python packages"
 category = "main"
 optional = false
@@ -438,7 +438,7 @@ testing = ["pytest (>=6)", "pytest-checkdocs (>=2.4)", "pytest-flake8", "pytest-
 name = "importlib-resources"
 version = "5.7.1"
 description = "Read resources from Python packages"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -482,7 +482,7 @@ test = ["pytest (>=6.0)", "pytest-cov", "flaky", "ipyparallel", "pre-commit", "p
 
 [[package]]
 name = "ipython"
-version = "8.3.0"
+version = "8.4.0"
 description = "IPython: Productive Interactive Computing"
 category = "main"
 optional = false
@@ -560,9 +560,9 @@ i18n = ["Babel (>=2.7)"]
 
 [[package]]
 name = "jsonschema"
-version = "4.5.1"
+version = "4.6.0"
 description = "An implementation of JSON Schema validation for Python"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -573,13 +573,13 @@ pyrsistent = ">=0.14.0,<0.17.0 || >0.17.0,<0.17.1 || >0.17.1,<0.17.2 || >0.17.2"
 
 [package.extras]
 format = ["fqdn", "idna", "isoduration", "jsonpointer (>1.13)", "rfc3339-validator", "rfc3987", "uri-template", "webcolors (>=1.11)"]
-format_nongpl = ["fqdn", "idna", "isoduration", "jsonpointer (>1.13)", "rfc3339-validator", "rfc3986-validator (>0.1.0)", "uri-template", "webcolors (>=1.11)"]
+format-nongpl = ["fqdn", "idna", "isoduration", "jsonpointer (>1.13)", "rfc3339-validator", "rfc3986-validator (>0.1.0)", "uri-template", "webcolors (>=1.11)"]
 
 [[package]]
 name = "jupyter-client"
 version = "7.3.1"
 description = "Jupyter protocol implementation and client libraries"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -600,7 +600,7 @@ test = ["codecov", "coverage", "ipykernel (>=6.5)", "ipython", "mypy", "pre-comm
 name = "jupyter-core"
 version = "4.10.0"
 description = "Jupyter core package. A base package on which Jupyter projects rely."
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -615,7 +615,7 @@ test = ["ipykernel", "pre-commit", "pytest", "pytest-cov", "pytest-timeout"]
 name = "jupyterlab-pygments"
 version = "0.2.2"
 description = "Pygments theme using JupyterLab CSS variables"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -648,7 +648,7 @@ python-versions = ">=3.6"
 
 [[package]]
 name = "llvmlite"
-version = "0.38.0"
+version = "0.38.1"
 description = "lightweight wrapper around basic LLVM functionality"
 category = "main"
 optional = false
@@ -717,15 +717,15 @@ python-versions = ">=3.6"
 name = "mistune"
 version = "0.8.4"
 description = "The fastest markdown parser in pure Python"
-category = "dev"
+category = "main"
 optional = false
 python-versions = "*"
 
 [[package]]
 name = "nbclient"
-version = "0.6.3"
+version = "0.6.4"
 description = "A client library for executing notebooks. Formerly nbconvert's ExecutePreprocessor."
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7.0"
 
@@ -733,7 +733,7 @@ python-versions = ">=3.7.0"
 jupyter-client = ">=6.1.5"
 nbformat = ">=5.0"
 nest-asyncio = "*"
-traitlets = ">=5.0.0"
+traitlets = ">=5.2.2"
 
 [package.extras]
 sphinx = ["autodoc-traits", "mock", "moto", "myst-parser", "Sphinx (>=1.7)", "sphinx-book-theme"]
@@ -743,7 +743,7 @@ test = ["black", "check-manifest", "flake8", "ipykernel", "ipython (<8.0.0)", "i
 name = "nbconvert"
 version = "6.5.0"
 description = "Converting Jupyter Notebooks"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -776,7 +776,7 @@ webpdf = ["pyppeteer (>=1,<1.1)"]
 name = "nbformat"
 version = "5.4.0"
 description = "The Jupyter Notebook format"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -793,7 +793,7 @@ test = ["check-manifest", "testpath", "pytest", "pre-commit"]
 name = "nbsphinx"
 version = "0.8.8"
 description = "Jupyter Notebook Tools for Sphinx"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.6"
 
@@ -809,7 +809,7 @@ traitlets = "*"
 name = "nest-asyncio"
 version = "1.5.5"
 description = "Patch asyncio to allow nested event loops"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.5"
 
@@ -823,7 +823,7 @@ python-versions = "*"
 
 [[package]]
 name = "numba"
-version = "0.55.1"
+version = "0.55.2"
 description = "compiling Python code using LLVM"
 category = "main"
 optional = false
@@ -831,15 +831,15 @@ python-versions = ">=3.7,<3.11"
 
 [package.dependencies]
 llvmlite = ">=0.38.0rc1,<0.39"
-numpy = ">=1.18,<1.22"
+numpy = ">=1.18,<1.23"
 
 [[package]]
 name = "numpy"
-version = "1.21.6"
+version = "1.22.4"
 description = "NumPy is the fundamental package for array computing with Python."
 category = "main"
 optional = false
-python-versions = ">=3.7,<3.11"
+python-versions = ">=3.8"
 
 [[package]]
 name = "packaging"
@@ -877,7 +877,7 @@ test = ["hypothesis (>=5.5.3)", "pytest (>=6.0)", "pytest-xdist (>=1.31)"]
 name = "pandocfilters"
 version = "1.5.0"
 description = "Utilities for writing pandoc filters in python"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*"
 
@@ -943,7 +943,7 @@ python-versions = "*"
 
 [[package]]
 name = "pillow"
-version = "9.1.0"
+version = "9.1.1"
 description = "Python Imaging Library (Fork)"
 category = "main"
 optional = true
@@ -1006,14 +1006,14 @@ wcwidth = "*"
 
 [[package]]
 name = "psutil"
-version = "5.9.0"
+version = "5.9.1"
 description = "Cross-platform lib for process and system monitoring in Python."
 category = "dev"
 optional = false
-python-versions = ">=2.6, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*"
+python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*"
 
 [package.extras]
-test = ["ipaddress", "mock", "unittest2", "enum34", "pywin32", "wmi"]
+test = ["ipaddress", "mock", "enum34", "pywin32", "wmi"]
 
 [[package]]
 name = "ptyprocess"
@@ -1038,7 +1038,7 @@ tests = ["pytest"]
 name = "py"
 version = "1.11.0"
 description = "library with cross-python path, ini-parsing, io, code, log facilities"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*, !=3.4.*"
 
@@ -1060,7 +1060,7 @@ test = ["pytest"]
 
 [[package]]
 name = "pybtex-docutils"
-version = "1.0.1"
+version = "1.0.2"
 description = "A docutils backend for pybtex."
 category = "main"
 optional = false
@@ -1074,7 +1074,7 @@ pybtex = ">=0.16"
 name = "pycparser"
 version = "2.21"
 description = "C parser in Python"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*"
 
@@ -1088,28 +1088,30 @@ python-versions = ">=3.6"
 
 [[package]]
 name = "pylint"
-version = "2.13.8"
+version = "2.14.0"
 description = "python code static checker"
 category = "dev"
 optional = false
-python-versions = ">=3.6.2"
+python-versions = ">=3.7.2"
 
 [package.dependencies]
-astroid = ">=2.11.3,<=2.12.0-dev0"
+astroid = ">=2.11.5,<=2.12.0-dev0"
 colorama = {version = "*", markers = "sys_platform == \"win32\""}
 dill = ">=0.2"
 isort = ">=4.2.5,<6"
 mccabe = ">=0.6,<0.8"
 platformdirs = ">=2.2.0"
 tomli = {version = ">=1.1.0", markers = "python_version < \"3.11\""}
+tomlkit = ">=0.10.1"
 typing-extensions = {version = ">=3.10.0", markers = "python_version < \"3.10\""}
 
 [package.extras]
-testutil = ["gitpython (>3)"]
+spelling = ["pyenchant (>=3.2,<4.0)"]
+testutils = ["gitpython (>3)"]
 
 [[package]]
 name = "pyparsing"
-version = "3.0.8"
+version = "3.0.9"
 description = "pyparsing module - Classes and methods to define and execute parsing grammars"
 category = "main"
 optional = false
@@ -1138,7 +1140,7 @@ python-versions = "*"
 name = "pyrsistent"
 version = "0.18.1"
 description = "Persistent/Functional/Immutable data structures"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.7"
 
@@ -1220,7 +1222,7 @@ python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*"
 name = "pywin32"
 version = "304"
 description = "Python for Window Extensions"
-category = "dev"
+category = "main"
 optional = false
 python-versions = "*"
 
@@ -1234,9 +1236,9 @@ python-versions = ">=3.6"
 
 [[package]]
 name = "pyzmq"
-version = "22.3.0"
+version = "23.1.0"
 description = "Python bindings for 0MQ"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.6"
 
@@ -1280,7 +1282,7 @@ jupyter = ["ipywidgets (>=7.5.1,<8.0.0)"]
 
 [[package]]
 name = "scipy"
-version = "1.8.0"
+version = "1.8.1"
 description = "SciPy: Scientific Library for Python"
 category = "main"
 optional = false
@@ -1325,7 +1327,7 @@ python-versions = "*"
 name = "soupsieve"
 version = "2.3.2.post1"
 description = "A modern CSS selector implementation for Beautiful Soup."
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.6"
 
@@ -1463,7 +1465,7 @@ test = ["pytest"]
 
 [[package]]
 name = "sqlalchemy"
-version = "1.4.36"
+version = "1.4.37"
 description = "Database Abstraction Library"
 category = "main"
 optional = true
@@ -1487,7 +1489,7 @@ mysql_connector = ["mysql-connector-python"]
 oracle = ["cx_oracle (>=7,<8)", "cx_oracle (>=7)"]
 postgresql = ["psycopg2 (>=2.7)"]
 postgresql_asyncpg = ["greenlet (!=0.4.17)", "asyncpg"]
-postgresql_pg8000 = ["pg8000 (>=1.16.6)"]
+postgresql_pg8000 = ["pg8000 (>=1.16.6,!=1.29.0)"]
 postgresql_psycopg2binary = ["psycopg2-binary"]
 postgresql_psycopg2cffi = ["psycopg2cffi"]
 pymysql = ["pymysql (<1)", "pymysql"]
@@ -1513,7 +1515,7 @@ tests = ["pytest", "typeguard", "pygments", "littleutils", "cython"]
 name = "tinycss2"
 version = "1.1.1"
 description = "A tiny CSS parser"
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">=3.6"
 
@@ -1540,24 +1542,32 @@ category = "main"
 optional = false
 python-versions = ">=3.7"
 
+[[package]]
+name = "tomlkit"
+version = "0.11.0"
+description = "Style preserving TOML library"
+category = "dev"
+optional = false
+python-versions = ">=3.6,<4.0"
+
 [[package]]
 name = "tornado"
 version = "6.1"
 description = "Tornado is a Python web framework and asynchronous networking library, originally developed at FriendFeed."
-category = "dev"
+category = "main"
 optional = false
 python-versions = ">= 3.5"
 
 [[package]]
 name = "traitlets"
-version = "5.1.1"
-description = "Traitlets Python configuration system"
+version = "5.2.2.post1"
+description = ""
 category = "main"
 optional = false
 python-versions = ">=3.7"
 
 [package.extras]
-test = ["pytest"]
+test = ["pre-commit", "pytest"]
 
 [[package]]
 name = "typing-extensions"
@@ -1610,7 +1620,7 @@ python-versions = "*"
 name = "webencodings"
 version = "0.5.1"
 description = "Character encoding aliases for legacy web content"
-category = "dev"
+category = "main"
 optional = false
 python-versions = "*"
 
@@ -1644,13 +1654,13 @@ testing = ["pytest (>=6)", "pytest-checkdocs (>=2.4)", "pytest-flake8", "pytest-
 
 [extras]
 box = ["banana-hep", "sqlalchemy", "pandas", "matplotlib"]
-docs = ["Sphinx", "sphinx-rtd-theme", "sphinxcontrib-bibtex"]
+docs = ["Sphinx", "sphinx-rtd-theme", "sphinxcontrib-bibtex", "nbsphinx"]
 mark = ["banana-hep", "sqlalchemy", "pandas", "matplotlib"]
 
 [metadata]
 lock-version = "1.1"
 python-versions = "^3.8,<3.11"
-content-hash = "23a035385bdb975a3d5bb7b809cdd4efda084a47b70d3ad600aafc9855e4a897"
+content-hash = "da35fd5a971b451f916bca8809d0fe2a36c31a3788d1dcdc430859ba5c90adb1"
 
 [metadata.files]
 a3b2bbc3ced97675ac3a71df45f55ba = [
@@ -1714,8 +1724,8 @@ bleach = [
     {file = "bleach-5.0.0.tar.gz", hash = "sha256:c6d6cc054bdc9c83b48b8083e236e5f00f238428666d2ce2e083eaa5fd568565"},
 ]
 certifi = [
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+    {file = "SQLAlchemy-1.4.37.tar.gz", hash = "sha256:3688f92c62db6c5df268e2264891078f17ecb91e3141b400f2e28d0f75796dea"},
 ]
 stack-data = [
     {file = "stack_data-0.2.0-py3-none-any.whl", hash = "sha256:999762f9c3132308789affa03e9271bbbe947bf78311851f4d485d8402ed858e"},
@@ -2814,6 +2830,10 @@ tomli = [
     {file = "tomli-2.0.1-py3-none-any.whl", hash = "sha256:939de3e7a6161af0c887ef91b7d41a53e7c5a1ca976325f429cb46ea9bc30ecc"},
     {file = "tomli-2.0.1.tar.gz", hash = "sha256:de526c12914f0c550d15924c62d72abc48d6fe7364aa87328337a31007fe8a4f"},
 ]
+tomlkit = [
+    {file = "tomlkit-0.11.0-py3-none-any.whl", hash = "sha256:0f4050db66fd445b885778900ce4dd9aea8c90c4721141fde0d6ade893820ef1"},
+    {file = "tomlkit-0.11.0.tar.gz", hash = "sha256:71ceb10c0eefd8b8f11fe34e8a51ad07812cb1dc3de23247425fbc9ddc47b9dd"},
+]
 tornado = [
     {file = "tornado-6.1-cp35-cp35m-macosx_10_9_x86_64.whl", hash = "sha256:d371e811d6b156d82aa5f9a4e08b58debf97c302a35714f6f45e35139c332e32"},
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@@ -2858,8 +2878,8 @@ tornado = [
     {file = "tornado-6.1.tar.gz", hash = "sha256:33c6e81d7bd55b468d2e793517c909b139960b6c790a60b7991b9b6b76fb9791"},
 ]
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-    {file = "traitlets-5.1.1.tar.gz", hash = "sha256:059f456c5a7c1c82b98c2e8c799f39c9b8128f6d0d46941ee118daace9eb70c7"},
+    {file = "traitlets-5.2.2.post1-py3-none-any.whl", hash = "sha256:1530d04badddc6a73d50b7ee34667d4b96914da352109117b4280cb56523a51b"},
+    {file = "traitlets-5.2.2.post1.tar.gz", hash = "sha256:74803a1baa59af70f023671d86d5c7a834c931186df26d50d362ee6a1ff021fd"},
 ]
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From 5954ace29a9981226e0e0b1597c48da372513133 Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Mon, 6 Jun 2022 18:04:33 +0200
Subject: [PATCH 07/18] Try to reduze sv kernel to 1 instance

---
 src/eko/evolution_operator/__init__.py        | 185 ++++++++++--------
 src/eko/evolution_operator/grid.py            |  50 ++---
 .../operator_matrix_element.py                |  70 +++----
 src/eko/scale_variations/__init__.py          |  31 ++-
 src/eko/strong_coupling.py                    |   6 +-
 5 files changed, 192 insertions(+), 150 deletions(-)

diff --git a/src/eko/evolution_operator/__init__.py b/src/eko/evolution_operator/__init__.py
index 23476a228..8ec01a25c 100644
--- a/src/eko/evolution_operator/__init__.py
+++ b/src/eko/evolution_operator/__init__.py
@@ -133,47 +133,49 @@ def quad_ker(
     ev_op_iterations,
     ev_op_max_order,
     sv_mode,
+    is_threshold,
 ):
-    """
-    Raw evolution kernel inside quad.
+    """Raw evolution kernel inside quad.
 
     Parameters
     ----------
-        u : float
-            quad argument
-        order : int
-            perturbation order
-        method : str
-            method
-        mode0: int
-            pid for first sector element
-        mode1 : int
-            pid for second sector element
-        is_log : boolean
-            is a logarithmic interpolation
-        logx : float
-            Mellin inversion point
-        areas : tuple
-            basis function configuration
-        a1 : float
-            target coupling value
-        a0 : float
-            initial coupling value
-        nf : int
-            number of active flavors
-        L : float
-            logarithm of the squared ratio of factorization and renormalization scale
-        ev_op_iterations : int
-            number of evolution steps
-        ev_op_max_order : int
-            perturbative expansion order of U
-        sv_mode: int, `enum.IntEnum`
-            scale variation mode, see `eko.scale_variations.Modes`
+    u : float
+        quad argument
+    order : int
+        perturbation order
+    method : str
+        method
+    mode0: int
+        pid for first sector element
+    mode1 : int
+        pid for second sector element
+    is_log : boolean
+        is a logarithmic interpolation
+    logx : float
+        Mellin inversion point
+    areas : tuple
+        basis function configuration
+    a1 : float
+        target coupling value
+    a0 : float
+        initial coupling value
+    nf : int
+        number of active flavors
+    L : float
+        logarithm of the squared ratio of factorization and renormalization scale
+    ev_op_iterations : int
+        number of evolution steps
+    ev_op_max_order : int
+        perturbative expansion order of U
+    sv_mode: int, `enum.IntEnum`
+        scale variation mode, see `eko.scale_variations.Modes`
+    is_threshold : boolean
+        is this an itermediate threshold operator?
 
     Returns
     -------
-        ker : float
-            evaluated integration kernel
+    float
+        evaluated integration kernel
     """
     ker_base = QuadKerBase(u, is_log, logx, mode0)
     integrand = ker_base.integrand(areas)
@@ -192,7 +194,7 @@ def quad_ker(
             order, method, gamma_singlet, a1, a0, nf, ev_op_iterations, ev_op_max_order
         )
         # scale var expanded is applied on the kernel
-        if sv_mode == sv.Modes.expanded:
+        if sv_mode == sv.Modes.expanded and not is_threshold:
             ker = np.ascontiguousarray(ker) @ np.ascontiguousarray(
                 sv.expanded.singlet_variation(gamma_singlet, a1, order, nf, L)
             )
@@ -210,40 +212,43 @@ def quad_ker(
             nf,
             ev_op_iterations,
         )
-        if sv_mode == sv.Modes.expanded:
+        if sv_mode == sv.Modes.expanded and not is_threshold:
             ker = ker * sv.expanded.non_singlet_variation(gamma_ns, a1, order, nf, L)
 
     # recombine everthing
     return np.real(ker * integrand)
 
 
-class Operator:
-    """
-    Internal representation of a single EKO.
+class Operator(sv.ModeMixin):
+    """Internal representation of a single EKO.
 
     The actual matrices are computed upon calling :meth:`compute`.
 
     Parameters
     ----------
-        config : dict
-            configuration
-        managers : dict
-            managers
-        nf : int
-            number of active flavors
-        q2_from : float
-            evolution source
-        q2_to : float
-            evolution target
-        mellin_cut : float
-            cut to the upper limit in the mellin inversion
+    config : dict
+        configuration
+    managers : dict
+        managers
+    nf : int
+        number of active flavors
+    q2_from : float
+        evolution source
+    q2_to : float
+        evolution target
+    mellin_cut : float
+        cut to the upper limit in the mellin inversion
+    is_threshold : bool
+        is this an itermediate threshold operator?
     """
 
     log_label = "Evolution"
     # complete list of possible evolution operators labels
     full_labels = br.full_labels
 
-    def __init__(self, config, managers, nf, q2_from, q2_to=None, mellin_cut=5e-2):
+    def __init__(
+        self, config, managers, nf, q2_from, q2_to, mellin_cut=5e-2, is_threshold=False
+    ):
         self.config = config
         self.managers = managers
         self.nf = nf
@@ -251,6 +256,7 @@ def __init__(self, config, managers, nf, q2_from, q2_to=None, mellin_cut=5e-2):
         self.q2_to = q2_to
         # TODO make 'cut' external parameter?
         self._mellin_cut = mellin_cut
+        self.is_threshold = is_threshold
         self.op_members = {}
 
     @property
@@ -258,20 +264,14 @@ def n_pools(self):
         n_pools = self.config["n_integration_cores"]
         if n_pools > 0:
             return n_pools
-        return os.cpu_count() + n_pools
+        # so we subtract from the maximum number
+        return max(os.cpu_count() + n_pools, 1)
 
     @property
     def fact_to_ren(self):
         r"""Returns the factor :math:`(\mu_F/\mu_R)^2`"""
         return self.config["fact_to_ren"]
 
-    @property
-    def sv_mode(self):
-        """Returns the scale variation mode"""
-        if self.config["ModSV"] is not None:
-            return sv.Modes[self.config["ModSV"]]
-        return sv.Modes.unvaried
-
     @property
     def int_disp(self):
         """Returns the interpolation dispatcher"""
@@ -282,25 +282,41 @@ def grid_size(self):
         """Returns the grid size"""
         return self.int_disp.xgrid.size
 
+    def mur2_shift(self, q2):
+        """Computes shifted renormalization scale.
+
+        Parameters
+        ----------
+        q2 : float
+            factorization scale
+
+        Returns
+        -------
+        float
+            renormalization scale
+        """
+        if self.sv_mode == sv.exponentiated:
+            return q2 / self.fact_to_ren
+        return q2
+
     @property
     def a_s(self):
         """Returns the computed values for :math:`a_s`"""
         sc = self.managers["strong_coupling"]
         a0 = sc.a_s(
-            self.q2_from / self.fact_to_ren, fact_scale=self.q2_from, nf_to=self.nf
+            self.mur2_shift(self.q2_from), fact_scale=self.q2_from, nf_to=self.nf
         )
-        a1 = sc.a_s(self.q2_to / self.fact_to_ren, fact_scale=self.q2_to, nf_to=self.nf)
+        a1 = sc.a_s(self.mur2_shift(self.q2_to), fact_scale=self.q2_to, nf_to=self.nf)
         return (a0, a1)
 
     @property
     def labels(self):
-        """
-        Compute necessary sector labels to compute.
+        """Compute necessary sector labels to compute.
 
         Returns
         -------
-            labels : list(str)
-                sector labels
+        list(str)
+            sector labels
         """
         order = self.config["order"]
         labels = []
@@ -322,22 +338,21 @@ def labels(self):
         return labels
 
     def quad_ker(self, label, logx, areas):
-        """
-        Partially initialized integrand function
+        """Partially initialized integrand function.
 
         Parameters
         ----------
-            label: tuple
-                operator element pids
-            logx: float
-                Mellin inversion point
-            areas : tuple
-                basis function configuration
+        label: tuple
+            operator element pids
+        logx: float
+            Mellin inversion point
+        areas : tuple
+            basis function configuration
 
         Returns
         -------
-            quad_ker : functools.partial
-                partially initialized intration kernel
+        functools.partial
+            partially initialized intration kernel
 
         """
         return functools.partial(
@@ -357,6 +372,7 @@ def quad_ker(self, label, logx, areas):
             ev_op_iterations=self.config["ev_op_iterations"],
             ev_op_max_order=self.config["ev_op_max_order"],
             sv_mode=self.sv_mode,
+            is_threshold=self.is_threshold,
         )
 
     def initialize_op_members(self):
@@ -379,18 +395,17 @@ def run_op_integration(
         self,
         log_grid,
     ):
-        """
-        Run the integration for each grid point
+        """Run the integration for each grid point
 
         Parameters
         ----------
-            log_grid : tuple(k, logx)
-                log grid point with relative index
+        log_grid : tuple(k, logx)
+            log grid point with relative index
 
         Returns
         -------
-            column : list
-                computed operators at the give grid point
+        list
+            computed operators at the give grid point
 
         """
         column = []
@@ -425,7 +440,7 @@ def run_op_integration(
         return column
 
     def compute(self):
-        """compute the actual operators (i.e. run the integrations)"""
+        """Compute the actual operators (i.e. run the integrations)."""
         self.initialize_op_members()
 
         # skip computation ?
@@ -446,8 +461,8 @@ def compute(self):
         logger.info(
             "%s: µ_R^2 distance: %e -> %e",
             self.log_label,
-            self.q2_from / self.fact_to_ren,
-            self.q2_to / self.fact_to_ren,
+            self.mur2_shift(self.q2_from),
+            self.mur2_shift(self.q2_to),
         )
         if self.sv_mode.name != "unvaried":
             logger.info(
diff --git a/src/eko/evolution_operator/grid.py b/src/eko/evolution_operator/grid.py
index 0a28802f7..50221a0b4 100644
--- a/src/eko/evolution_operator/grid.py
+++ b/src/eko/evolution_operator/grid.py
@@ -1,6 +1,5 @@
 # -*- coding: utf-8 -*-
-"""
-This module contains the :class:`OperatorGrid` class.
+"""This module contains the :class:`OperatorGrid` class.
 
 The first is the driver class of eko as it is the one that collects all the
 previously instantiated information and does the actual computation of the Q2s.
@@ -15,13 +14,14 @@
 
 from .. import basis_rotation as br
 from .. import matching_conditions, member
+from .. import scale_variations as sv
 from ..matching_conditions.operator_matrix_element import OperatorMatrixElement
 from . import Operator, flavors, physical
 
 logger = logging.getLogger(__name__)
 
 
-class OperatorGrid:
+class OperatorGrid(sv.ModeMixin):
     """
     The operator grid is the driver class of the evolution.
 
@@ -186,7 +186,12 @@ def get_threshold_operators(self, path):
                 # Compute the operator and store it
                 logger.info("Prepare threshold operator")
                 op_th = Operator(
-                    self.config, self.managers, seg.nf, seg.q2_from, seg.q2_to
+                    self.config,
+                    self.managers,
+                    seg.nf,
+                    seg.q2_from,
+                    seg.q2_to,
+                    is_threshold=True,
                 )
                 op_th.compute()
                 self._threshold_operators[new_op_key] = op_th
@@ -199,18 +204,17 @@ def get_threshold_operators(self, path):
         return thr_ops
 
     def compute(self, q2grid=None):
-        """
-        Computes all ekos for the q2grid.
+        """Computes all ekos for the `q2grid`.
 
         Parameters
         ----------
-            q2grid: list(float)
-                List of q^2
+        q2grid: list(float)
+            List of :math:`Q^2`
 
         Returns
         -------
-            grid_return: list(dict)
-                List of ekos for each value of q^2
+        list(dict)
+            List of ekos for each value of :math:`Q^2`
         """
         # use input?
         if q2grid is None:
@@ -220,23 +224,28 @@ def compute(self, q2grid=None):
             q2grid = [q2grid]
         # And now return the grid
         grid_return = {}
+        # shift path for expanded scheme
+        shift = (
+            lambda q2, f2r=self.config["fact_to_ren"]: q2 * f2r
+            if self.sv_mode == sv.Modes.expanded
+            else lambda q2: q2
+        )
         for q2 in q2grid:
-            grid_return[q2] = self.generate(q2)
+            grid_return[q2] = self.generate(shift(q2))
         return grid_return
 
     def generate(self, q2):
-        """
-        Computes a single EKO.
+        r"""Computes a single EKO.
 
         Parameters
         ----------
-            q2: float
-                Target value of q^2
+        q2: float
+            Target value of :math:`Q^2`
 
         Returns
         -------
-            final_op: dict
-                eko E(q^2 <- q_0^2) in flavor basis as numpy array
+        dict
+            eko :math:`\mathbf E(Q^2 \leftarrow Q_0^2)` in flavor basis as numpy array
         """
         # The lists of areas as produced by the thresholds
         path = self.managers["thresholds_config"].path(q2)
@@ -288,13 +297,8 @@ def generate(self, q2):
                 final_op = final_op @ rot @ matching @ phys_op
 
         values, errors = final_op.to_flavor_basis_tensor()
-        fact_to_ren = self.config["fact_to_ren"]
         return {
             "operators": values,
             "operator_errors": errors,
-            "alphas": self.managers["strong_coupling"].a_s(
-                q2 / fact_to_ren, fact_scale=q2, nf_to=path[-1].nf
-            )
-            * 4.0
-            * np.pi,
+            "alphas": 0.0,  # TODO drop this completely
         }
diff --git a/src/eko/matching_conditions/operator_matrix_element.py b/src/eko/matching_conditions/operator_matrix_element.py
index bbf62c15c..9577d04e9 100644
--- a/src/eko/matching_conditions/operator_matrix_element.py
+++ b/src/eko/matching_conditions/operator_matrix_element.py
@@ -285,27 +285,26 @@ def quad_ker(
 
 
 class OperatorMatrixElement(Operator):
-    """
-    Internal representation of a single |OME|.
+    """Internal representation of a single |OME|.
 
     The actual matrices are computed upon calling :meth:`compute`.
 
     Parameters
     ----------
-        config : dict
-            configuration
-        managers : dict
-            managers
-        is_backward: bool
-            True for backward evolution
-        q2: float
-            matching scale
-        nf: int
-            number of active flavor below threshold
-        L: float
-            log of K threshold squared
-        is_msbar: bool
-            add the |MSbar| contribution
+    config : dict
+        configuration
+    managers : dict
+        managers
+    nf: int
+        number of active flavor below threshold
+    q2: float
+        matching scale
+    is_backward: bool
+        True for backward evolution
+    L: float
+        log of K threshold squared
+    is_msbar: bool
+        add the |MSbar| contribution
     """
 
     log_label = "Matching"
@@ -324,7 +323,7 @@ class OperatorMatrixElement(Operator):
     ]
 
     def __init__(self, config, managers, nf, q2, is_backward, L, is_msbar):
-        super().__init__(config, managers, nf, q2)
+        super().__init__(config, managers, nf, q2, None)
         self.backward_method = config["backward_inversion"] if is_backward else ""
         if is_backward:
             self.is_intrinsic = True
@@ -335,13 +334,12 @@ def __init__(self, config, managers, nf, q2, is_backward, L, is_msbar):
 
     @property
     def labels(self):
-        """
-        Compute necessary sector labels to compute.
+        """Computes the necessary sector labels to compute.
 
         Returns
         -------
-            labels : list(str)
-                sector labels
+        list(str)
+            sector labels
         """
 
         labels = []
@@ -377,23 +375,21 @@ def labels(self):
         return labels
 
     def quad_ker(self, label, logx, areas):
-        """
-        Partially initialized integrand function
+        """Partially initialized integrand function.
 
         Parameters
         ----------
-            label: tuple
-                operator element pids
-            logx: float
-                Mellin inversion point
-            areas : tuple
-                basis function configuration
+        label: tuple
+            operator element pids
+        logx: float
+            Mellin inversion point
+        areas : tuple
+            basis function configuration
 
         Returns
         -------
-            quad_ker : functools.partial
-                partially initialized intration kernel
-
+        functools.partial
+            partially initialized integration kernel
         """
         return functools.partial(
             quad_ker,
@@ -412,17 +408,15 @@ def quad_ker(self, label, logx, areas):
 
     @property
     def a_s(self):
-        """
-        Returns the computed values for :math:`a_s`.
+        """Returns the computed values for :math:`a_s`.
+
         Note that here you need to use :math:`a_s^{n_f+1}`
         """
         sc = self.managers["strong_coupling"]
-        return sc.a_s(self.q2_from / self.fact_to_ren, self.q2_from, nf_to=self.nf + 1)
+        return sc.a_s(self.mur2_shift(self.q2_from), self.q2_from, nf_to=self.nf + 1)
 
     def compute(self):
-        """
-        compute the actual operators (i.e. run the integrations)
-        """
+        """Compute the actual operators (i.e. run the integrations)"""
         self.initialize_op_members()
 
         # At LO you don't need anything else
diff --git a/src/eko/scale_variations/__init__.py b/src/eko/scale_variations/__init__.py
index 853d95ce7..b16fac3e4 100644
--- a/src/eko/scale_variations/__init__.py
+++ b/src/eko/scale_variations/__init__.py
@@ -1,7 +1,5 @@
 # -*- coding: utf-8 -*-
-"""
-This module contain the possible scale variations integrals.
-"""
+"""This module contain the possible scale variations schemes."""
 import enum
 
 from . import expanded, exponentiated
@@ -13,3 +11,30 @@ class Modes(enum.IntEnum):
     unvaried = enum.auto()
     exponentiated = enum.auto()
     expanded = enum.auto()
+
+
+def sv_mode(s):
+    """Returns the scale variation mode.
+
+    Parameters
+    ----------
+    s : str
+        string representation
+
+    Returns
+    -------
+    enum.IntEnum
+        enum representation
+    """
+    if s is not None:
+        return Modes[s]
+    return Modes.unvaried
+
+
+class ModeMixin:
+    """Mixin to cast scale variation mode."""
+
+    @property
+    def sv_mode(self):
+        """Returns the scale variation mode"""
+        return sv_mode(self.config["ModSV"])
diff --git a/src/eko/strong_coupling.py b/src/eko/strong_coupling.py
index 29a56675d..96e313543 100644
--- a/src/eko/strong_coupling.py
+++ b/src/eko/strong_coupling.py
@@ -12,7 +12,9 @@
 import numpy as np
 import scipy
 
-from . import constants, thresholds
+from . import constants
+from . import scale_variations as sv
+from . import thresholds
 from .beta import b as beta_b
 from .beta import beta
 
@@ -242,6 +244,8 @@ def from_dict(cls, theory_card, masses=None):
             raise ValueError(f"{hqm_scheme} is not implemented, choose POLE or MSBAR")
         # adjust factorization scale / renormalization scale
         fact_to_ren = theory_card["fact_to_ren_scale_ratio"]
+        if sv.sv_mode(theory_card["ModSV"]) is not sv.Modes.exponentiated:
+            fact_to_ren = 1.0
         heavy_flavors = "cbt"
         if masses is None:
             masses = np.power(

From 2ab68c2875affd46f491aaf4505fc44d5333ab57 Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Wed, 8 Jun 2022 14:26:57 +0200
Subject: [PATCH 08/18] Move sv b path shifting

---
 src/eko/evolution_operator/grid.py | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/src/eko/evolution_operator/grid.py b/src/eko/evolution_operator/grid.py
index 50221a0b4..192dd544a 100644
--- a/src/eko/evolution_operator/grid.py
+++ b/src/eko/evolution_operator/grid.py
@@ -224,14 +224,14 @@ def compute(self, q2grid=None):
             q2grid = [q2grid]
         # And now return the grid
         grid_return = {}
-        # shift path for expanded scheme
-        shift = (
-            lambda q2, f2r=self.config["fact_to_ren"]: q2 * f2r
-            if self.sv_mode == sv.Modes.expanded
-            else lambda q2: q2
-        )
         for q2 in q2grid:
-            grid_return[q2] = self.generate(shift(q2))
+            # shift path for expanded scheme
+            q2_gen = (
+                q2 * self.config["fact_to_ren"]
+                if self.sv_mode == sv.Modes.expanded
+                else q2
+            )
+            grid_return[q2] = self.generate(q2_gen)
         return grid_return
 
     def generate(self, q2):

From 1f9abefbc8547c67a37ea5e87c2388f51309703e Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Wed, 8 Jun 2022 16:52:05 +0200
Subject: [PATCH 09/18] Revert sv kernel order

---
 src/eko/evolution_operator/__init__.py | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/eko/evolution_operator/__init__.py b/src/eko/evolution_operator/__init__.py
index 8ec01a25c..5f91210dd 100644
--- a/src/eko/evolution_operator/__init__.py
+++ b/src/eko/evolution_operator/__init__.py
@@ -195,9 +195,9 @@ def quad_ker(
         )
         # scale var expanded is applied on the kernel
         if sv_mode == sv.Modes.expanded and not is_threshold:
-            ker = np.ascontiguousarray(ker) @ np.ascontiguousarray(
+            ker = np.ascontiguousarray(
                 sv.expanded.singlet_variation(gamma_singlet, a1, order, nf, L)
-            )
+            ) @ np.ascontiguousarray(ker)
         ker = select_singlet_element(ker, mode0, mode1)
     else:
         gamma_ns = ad.gamma_ns(order, mode0, ker_base.n, nf)
@@ -213,7 +213,7 @@ def quad_ker(
             ev_op_iterations,
         )
         if sv_mode == sv.Modes.expanded and not is_threshold:
-            ker = ker * sv.expanded.non_singlet_variation(gamma_ns, a1, order, nf, L)
+            ker = sv.expanded.non_singlet_variation(gamma_ns, a1, order, nf, L) * ker
 
     # recombine everthing
     return np.real(ker * integrand)

From 2cbf981f2778f7725031234ed15b4937fb3064d4 Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Mon, 13 Jun 2022 11:54:27 +0200
Subject: [PATCH 10/18] Fix typo in docstring

---
 src/eko/evolution_operator/__init__.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/eko/evolution_operator/__init__.py b/src/eko/evolution_operator/__init__.py
index 5f91210dd..45871fa7d 100644
--- a/src/eko/evolution_operator/__init__.py
+++ b/src/eko/evolution_operator/__init__.py
@@ -352,7 +352,7 @@ def quad_ker(self, label, logx, areas):
         Returns
         -------
         functools.partial
-            partially initialized intration kernel
+            partially initialized integration kernel
 
         """
         return functools.partial(

From 57e21b070c7a47624d63587a76a003c370a94afc Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Wed, 15 Jun 2022 12:40:46 +0200
Subject: [PATCH 11/18] Fix LHA doc strings

---
 benchmarks/lha_paper_bench.py | 100 +++++++++++++++++-----------------
 1 file changed, 49 insertions(+), 51 deletions(-)

diff --git a/benchmarks/lha_paper_bench.py b/benchmarks/lha_paper_bench.py
index 27573b721..0f1e08451 100644
--- a/benchmarks/lha_paper_bench.py
+++ b/benchmarks/lha_paper_bench.py
@@ -1,6 +1,6 @@
 # -*- coding: utf-8 -*-
 """
-Benchmark to :cite:`Giele:2002hx` (LO + NLO) and :cite:`Dittmar:2005ed` (NNLO)
+Benchmark to :cite:`Giele:2002hx` (LO + NLO) and :cite:`Dittmar:2005ed` (NNLO).
 """
 import numpy as np
 from banana import register
@@ -33,9 +33,7 @@
 
 
 class LHABenchmark(Runner):
-    """
-    Globally set the external program to LHA
-    """
+    """Globally set the external program to LHA."""
 
     def __init__(self):
         super().__init__()
@@ -44,36 +42,34 @@ def __init__(self):
         self.rotate_to_evolution_basis = True
 
     def plain_theory(self, pto):
-        """
-        Plain theories at given PTO.
+        """Generate plain theories at given PTO.
 
         Parameters
         ----------
-            pto : int
-                perturbation order
+        pto : int
+            perturbation order
 
         Returns
         -------
-            list(dict)
-                theory updates
+        list(dict)
+            theory updates
         """
         th = self.theory.copy()
         th.update({"PTO": pto})
         return [th]
 
     def sv_theories(self, pto):
-        """
-        Scale variation theories.
+        """Generate scale variation theories.
 
         Parameters
         ----------
-            pto : int
-                perturbation order
+        pto : int
+            perturbation order
 
         Returns
         -------
-            list(dict)
-                theory updates
+        list(dict)
+            theory updates
         """
         low = self.theory.copy()
         low["PTO"] = pto
@@ -87,29 +83,27 @@ def sv_theories(self, pto):
 
     @staticmethod
     def skip_pdfs(_theory):
-        """
-        Adjust skip_pdf by the used theory
+        """Adjust skip_pdf by the used theory.
 
         Parameters
         ----------
-            theory_updates : list(dict)
-                theory updates
+        theory : dict
+            theory update
 
         Returns
         -------
-            list :
-                current skip_pdf
+        list
+            allowed PDFs in LHA
         """
         return default_skip_pdfs
 
     def run_lha(self, theory_updates):
-        """
-        Enforce operators and PDF
+        """Enforce operator grid and PDF.
 
         Parameters
         ----------
-            theory_updates : list(dict)
-                theory updates
+        theory_updates : list(dict)
+            theory updates
         """
         self.run(
             theory_updates,
@@ -124,16 +118,16 @@ def run_lha(self, theory_updates):
         )
 
     def benchmark_plain(self, pto):
-        """Plain configuration"""
+        """Run plain configuration."""
         self.run_lha(self.plain_theory(pto))
 
     def benchmark_sv(self, pto):
-        """Scale variations"""
+        """Run scale variations."""
         self.run_lha(self.sv_theories(pto))
 
 
 class BenchmarkVFNS(LHABenchmark):
-    """Variable Flavor Number Scheme"""
+    """Provide |VFNS| settings."""
 
     def __init__(self):
         super().__init__()
@@ -151,7 +145,7 @@ def __init__(self):
 
 
 class BenchmarkFFNS(LHABenchmark):
-    """Fixed Flavor Number Scheme"""
+    """Provide |FFNS| settings."""
 
     def __init__(self):
         super().__init__()
@@ -169,19 +163,29 @@ def __init__(self):
 
     @staticmethod
     def skip_pdfs(theory):
+        """Adjust skip_pdf by the used theory.
+
+        Parameters
+        ----------
+        theory : dict
+            theory update
+
+        Returns
+        -------
+        list
+            allowed PDFs in FFNS LHA
+        """
         ffns_skip_pdfs = default_skip_pdfs.copy()
         # remove bottom
         ffns_skip_pdfs.extend([-5, 5, "T24"])
-        # in NNLO V8 becomes available
+        # in NNLO also V8 gets removed
         if theory["PTO"] >= 2:
             ffns_skip_pdfs.remove("V8")
         return ffns_skip_pdfs
 
 
 class BenchmarkRunner(BenchmarkVFNS):
-    """
-    Generic benchmark runner using the LHA VFNS settings
-    """
+    """Generic benchmark runner using the LHA |VFNS| settings."""
 
     def __init__(self, external):
         super().__init__()
@@ -189,18 +193,12 @@ def __init__(self, external):
         self.sandbox = True
 
     def benchmark_sv(self, pto):
-        """
-        Scale variations
+        """Run scale variations.
 
         Parameters
         ----------
-            pto : int
-                perturbation order
-
-        Returns
-        -------
-            list(dict)
-                theory updates
+        pto : int
+            perturbation order
         """
         high, low = self.sv_theories(pto)
 
@@ -221,15 +219,15 @@ def benchmark_sv(self, pto):
 if __name__ == "__main__":
 
     # Benchmark to LHA
-    # obj = BenchmarkVFNS()
-    obj = BenchmarkFFNS()
+    obj = BenchmarkVFNS()
+    # obj = BenchmarkFFNS()
 
     # obj.benchmark_plain(0)
-    # obj.benchmark_sv(2)
+    obj.benchmark_sv(1)
 
     # # VFNS benchmarks with LHA settings
-    programs = ["LHA", "pegasus", "apfel"]
-    for p in programs:
-        obj = BenchmarkRunner(p)
-        # obj.benchmark_plain(2)
-        obj.benchmark_sv(2)
+    # programs = ["LHA", "pegasus", "apfel"]
+    # for p in programs:
+    #     obj = BenchmarkRunner(p)
+    #     # obj.benchmark_plain(2)
+    #     obj.benchmark_sv(2)

From 8fadae31f1b62a97ba9893e9eb2c41b2a7b83804 Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Wed, 15 Jun 2022 14:09:34 +0200
Subject: [PATCH 12/18] Recover scheme A

---
 benchmarks/lha_paper_bench.py          | 2 +-
 src/eko/evolution_operator/__init__.py | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/benchmarks/lha_paper_bench.py b/benchmarks/lha_paper_bench.py
index 0f1e08451..b4dbe9aa2 100644
--- a/benchmarks/lha_paper_bench.py
+++ b/benchmarks/lha_paper_bench.py
@@ -223,7 +223,7 @@ def benchmark_sv(self, pto):
     # obj = BenchmarkFFNS()
 
     # obj.benchmark_plain(0)
-    obj.benchmark_sv(1)
+    obj.benchmark_sv(2)
 
     # # VFNS benchmarks with LHA settings
     # programs = ["LHA", "pegasus", "apfel"]
diff --git a/src/eko/evolution_operator/__init__.py b/src/eko/evolution_operator/__init__.py
index 45871fa7d..2b43cc64d 100644
--- a/src/eko/evolution_operator/__init__.py
+++ b/src/eko/evolution_operator/__init__.py
@@ -295,7 +295,7 @@ def mur2_shift(self, q2):
         float
             renormalization scale
         """
-        if self.sv_mode == sv.exponentiated:
+        if self.sv_mode == sv.Modes.exponentiated:
             return q2 / self.fact_to_ren
         return q2
 

From d14ef8e0f672f9a5be8f4ea54c176fd480f00dd1 Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Mon, 11 Jul 2022 14:53:48 +0200
Subject: [PATCH 13/18] Recover tests

---
 src/eko/evolution_operator/grid.py |  1 -
 tests/eko/test_ev_op_grid.py       | 18 ++----------------
 tests/eko/test_ev_operator.py      |  7 +++++++
 tests/eko/test_msbar_masses.py     |  1 +
 tests/eko/test_strong_coupling.py  |  1 +
 5 files changed, 11 insertions(+), 17 deletions(-)

diff --git a/src/eko/evolution_operator/grid.py b/src/eko/evolution_operator/grid.py
index 192dd544a..e773b3025 100644
--- a/src/eko/evolution_operator/grid.py
+++ b/src/eko/evolution_operator/grid.py
@@ -300,5 +300,4 @@ def generate(self, q2):
         return {
             "operators": values,
             "operator_errors": errors,
-            "alphas": 0.0,  # TODO drop this completely
         }
diff --git a/tests/eko/test_ev_op_grid.py b/tests/eko/test_ev_op_grid.py
index d505bb033..af6baa74c 100644
--- a/tests/eko/test_ev_op_grid.py
+++ b/tests/eko/test_ev_op_grid.py
@@ -108,16 +108,12 @@ def test_compute_q2grid(self):
         opg = opgrid.compute()
         assert len(opg) == 2
         assert all(
-            k in op
-            for k in ["operators", "operator_errors", "alphas"]
-            for op in opg.values()
+            k in op for k in ["operators", "operator_errors"] for op in opg.values()
         )
         opg = opgrid.compute(3)
         assert len(opg) == 1
         assert all(
-            k in op
-            for k in ["operators", "operator_errors", "alphas"]
-            for op in opg.values()
+            k in op for k in ["operators", "operator_errors"] for op in opg.values()
         )
 
     def test_grid_computation_VFNS(self):
@@ -127,16 +123,6 @@ def test_grid_computation_VFNS(self):
         operators = opgrid.compute(qgrid_check)
         assert len(operators) == len(qgrid_check)
 
-    def test_alphas(self):
-        opgrid = self._get_operator_grid()
-        # q2 has not be precomputed - but should work nevertheless
-        opg = opgrid.compute(3)
-        sv_opgrid = self._get_operator_grid(
-            theory_update={"fact_to_ren_scale_ratio": 2.0}
-        )
-        sv_opg = sv_opgrid.compute(3)
-        assert opg[3]["alphas"] < sv_opg[3]["alphas"]
-
     def test_mod_expanded(self):
         theory_update = {
             "PTO": 1,
diff --git a/tests/eko/test_ev_operator.py b/tests/eko/test_ev_operator.py
index a085442d3..2cc0ae30b 100644
--- a/tests/eko/test_ev_operator.py
+++ b/tests/eko/test_ev_operator.py
@@ -45,6 +45,7 @@ def test_quad_ker(monkeypatch):
             ev_op_iterations=0,
             ev_op_max_order=0,
             sv_mode=1,
+            is_threshold=False,
         )
         np.testing.assert_allclose(res_ns, 0.0)
         res_s = quad_ker(
@@ -63,6 +64,7 @@ def test_quad_ker(monkeypatch):
             ev_op_iterations=0,
             ev_op_max_order=0,
             sv_mode=1,
+            is_threshold=False,
         )
         np.testing.assert_allclose(res_s, 1.0)
         res_s = quad_ker(
@@ -81,6 +83,7 @@ def test_quad_ker(monkeypatch):
             ev_op_iterations=0,
             ev_op_max_order=0,
             sv_mode=1,
+            is_threshold=False,
         )
         np.testing.assert_allclose(res_s, 0.0)
     for label in [(br.non_singlet_pids_map["ns+"], 0), (100, 100)]:
@@ -101,6 +104,7 @@ def test_quad_ker(monkeypatch):
                 ev_op_iterations=0,
                 ev_op_max_order=0,
                 sv_mode=sv,
+                is_threshold=False,
             )
             np.testing.assert_allclose(res_sv, 1.0)
 
@@ -121,6 +125,7 @@ def test_quad_ker(monkeypatch):
         ev_op_iterations=0,
         ev_op_max_order=0,
         sv_mode=1,
+        is_threshold=False,
     )
     np.testing.assert_allclose(res_ns, 0.0)
 
@@ -204,6 +209,7 @@ def test_n_pools(self):
             {},
             3,
             1,
+            10,
         )
         assert o.n_pools == os.cpu_count() - excluded_cores
 
@@ -336,6 +342,7 @@ def quad_ker_pegasus(
                     ev_op_iterations,
                     10,
                     0,
+                    False,
                 ),
                 epsabs=1e-12,
                 epsrel=1e-5,
diff --git a/tests/eko/test_msbar_masses.py b/tests/eko/test_msbar_masses.py
index beed1d40e..42452cccd 100644
--- a/tests/eko/test_msbar_masses.py
+++ b/tests/eko/test_msbar_masses.py
@@ -29,6 +29,7 @@
     "Qmt": 174.9,
     "PTO": 2,
     "ModEv": "TRN",
+    "ModSV": None,
 }
 
 
diff --git a/tests/eko/test_strong_coupling.py b/tests/eko/test_strong_coupling.py
index 26751138d..675885747 100644
--- a/tests/eko/test_strong_coupling.py
+++ b/tests/eko/test_strong_coupling.py
@@ -66,6 +66,7 @@ def test_init(self):
                     ktThr=1.0,
                     MaxNfAs=6,
                     HQ="POLE",
+                    ModSV=None,
                 )
                 sc2 = StrongCoupling.from_dict(setup)
                 assert sc2.q2_ref == scale_ref

From 8aaecee50eb60d95d2f97659608f928cc2be74e8 Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Mon, 11 Jul 2022 15:10:39 +0200
Subject: [PATCH 14/18] Increase test coverage

---
 tests/eko/test_ev_operator.py | 18 ++++++++++++++++++
 1 file changed, 18 insertions(+)

diff --git a/tests/eko/test_ev_operator.py b/tests/eko/test_ev_operator.py
index 2cc0ae30b..ad2384d68 100644
--- a/tests/eko/test_ev_operator.py
+++ b/tests/eko/test_ev_operator.py
@@ -213,6 +213,24 @@ def test_n_pools(self):
         )
         assert o.n_pools == os.cpu_count() - excluded_cores
 
+    def test_exponentiated(self):
+        tcard = copy.deepcopy(theory_card)
+        tcard["fact_to_ren_scale_ratio"] = 2.0
+        tcard["ModSV"] = "exponentiated"
+        ocard = copy.deepcopy(operators_card)
+        g = OperatorGrid.from_dict(
+            tcard,
+            ocard,
+            ThresholdsAtlas.from_dict(tcard),
+            StrongCoupling.from_dict(tcard),
+            InterpolatorDispatcher.from_dict(ocard),
+        )
+        # setup objs
+        o = Operator(g.config, g.managers, 3, 2.0, 10.0)
+        np.testing.assert_allclose(o.mur2_shift(40.0), 10.0)
+        o.compute()
+        self.check_lo(o)
+
     def test_compute_parallel(self, monkeypatch):
         tcard = copy.deepcopy(theory_card)
         ocard = copy.deepcopy(operators_card)

From 2fb4cdd4ff58bb11d6585e859358c599fb86e258 Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Mon, 11 Jul 2022 15:14:17 +0200
Subject: [PATCH 15/18] Fix parallel test

---
 tests/eko/test_ev_operator.py | 3 +++
 1 file changed, 3 insertions(+)

diff --git a/tests/eko/test_ev_operator.py b/tests/eko/test_ev_operator.py
index ad2384d68..95c00cad6 100644
--- a/tests/eko/test_ev_operator.py
+++ b/tests/eko/test_ev_operator.py
@@ -199,6 +199,9 @@ def test_labels(self):
 
     def test_n_pools(self):
         excluded_cores = 3
+        # make sure we actually have more the those cores (e.g. on github we don't)
+        if os.cpu_count() <= excluded_cores:
+            return
         o = Operator(
             dict(
                 order=1,

From 97aa9f8458a0f0e134f97e4dbc87857e0374d3e7 Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Mon, 11 Jul 2022 15:32:58 +0200
Subject: [PATCH 16/18] Fix updated tests

---
 src/eko/evolution_operator/__init__.py | 8 ++++----
 tests/eko/test_ev_operator.py          | 2 +-
 2 files changed, 5 insertions(+), 5 deletions(-)

diff --git a/src/eko/evolution_operator/__init__.py b/src/eko/evolution_operator/__init__.py
index 8f43c9903..5fa8def51 100644
--- a/src/eko/evolution_operator/__init__.py
+++ b/src/eko/evolution_operator/__init__.py
@@ -143,12 +143,12 @@ def quad_ker(
         quad argument
     order : int
         perturbation order
-    method : str
-        method
     mode0: int
         pid for first sector element
     mode1 : int
         pid for second sector element
+    method : str
+        method
     is_log : boolean
         is a logarithmic interpolation
     logx : float
@@ -203,7 +203,7 @@ def quad_ker(
         # scale var expanded is applied on the kernel
         if sv_mode == sv.Modes.expanded and not is_threshold:
             ker = np.ascontiguousarray(
-                sv.expanded.singlet_variation(gamma_singlet, a1, order, nf, L)
+                sv.expanded.singlet_variation(gamma_singlet, as1, order, nf, L)
             ) @ np.ascontiguousarray(ker)
         ker = select_singlet_element(ker, mode0, mode1)
     else:
@@ -220,7 +220,7 @@ def quad_ker(
             ev_op_iterations,
         )
         if sv_mode == sv.Modes.expanded and not is_threshold:
-            ker = sv.expanded.non_singlet_variation(gamma_ns, a1, order, nf, L) * ker
+            ker = sv.expanded.non_singlet_variation(gamma_ns, as1, order, nf, L) * ker
 
     # recombine everthing
     return np.real(ker * integrand)
diff --git a/tests/eko/test_ev_operator.py b/tests/eko/test_ev_operator.py
index ae4fa8e0f..795712c6c 100644
--- a/tests/eko/test_ev_operator.py
+++ b/tests/eko/test_ev_operator.py
@@ -226,7 +226,7 @@ def test_exponentiated(self):
             tcard,
             ocard,
             ThresholdsAtlas.from_dict(tcard),
-            StrongCoupling.from_dict(tcard),
+            Couplings.from_dict(tcard),
             InterpolatorDispatcher.from_dict(ocard),
         )
         # setup objs

From 2d92104333972bcd900defd0e980dd8432500879 Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Mon, 11 Jul 2022 15:45:36 +0200
Subject: [PATCH 17/18] Fix ekos_product

---
 src/ekobox/utils.py | 3 ---
 1 file changed, 3 deletions(-)

diff --git a/src/ekobox/utils.py b/src/ekobox/utils.py
index 858525f0b..e228d7cc4 100644
--- a/src/ekobox/utils.py
+++ b/src/ekobox/utils.py
@@ -40,7 +40,6 @@ def ekos_product(eko_ini, eko_fin, in_place=True):
 
     final_op_dict = {}
     final_op_error_dict = {}
-    final_alphas_dict = {}
     final_dict = {}
     for q2, op2 in ope2_dict.items():
         final_op_dict[q2] = np.einsum("ajbk,bkcl -> ajcl", ope1, op2)
@@ -49,11 +48,9 @@ def ekos_product(eko_ini, eko_fin, in_place=True):
             "ajbk,bkcl -> ajcl", ope1, ope2_error_dict[q2]
         ) + np.einsum("ajbk,bkcl -> ajcl", ope1_error, op2)
 
-        final_alphas_dict[q2] = eko_fin["Q2grid"][q2]["alphas"]
         final_dict[q2] = {
             "operators": final_op_dict[q2],
             "operator_errors": final_op_error_dict[q2],
-            "alphas": final_alphas_dict[q2],
         }
 
     final_eko = None

From 1cfcf8b301f48b37b2107d3d7f743dff7f4c3db5 Mon Sep 17 00:00:00 2001
From: Felix Hekhorn <felix.hekhorn@uni-tuebingen.de>
Date: Mon, 11 Jul 2022 15:58:53 +0200
Subject: [PATCH 18/18] Do some pylint fixes

---
 src/eko/anomalous_dimensions/as4/ggg.py     | 6 +++---
 src/eko/anomalous_dimensions/as4/gps.py     | 4 ++--
 src/eko/anomalous_dimensions/as4/gqg.py     | 6 +++---
 src/eko/harmonics/f_functions/f11.py        | 1 +
 src/eko/harmonics/f_functions/f13.py        | 1 +
 src/eko/harmonics/f_functions/f14_f12.py    | 2 +-
 src/eko/harmonics/f_functions/f16.py        | 2 +-
 src/eko/harmonics/f_functions/f17.py        | 1 +
 src/eko/harmonics/f_functions/f18.py        | 2 +-
 src/eko/harmonics/f_functions/f19.py        | 1 +
 src/eko/harmonics/f_functions/f20.py        | 1 +
 src/eko/harmonics/f_functions/f21.py        | 1 +
 src/eko/harmonics/f_functions/f9.py         | 1 +
 src/eko/matching_conditions/as3/aHg.py      | 2 +-
 src/eko/matching_conditions/as3/aHgstfac.py | 1 +
 src/eko/matching_conditions/as3/agg.py      | 2 +-
 16 files changed, 21 insertions(+), 13 deletions(-)

diff --git a/src/eko/anomalous_dimensions/as4/ggg.py b/src/eko/anomalous_dimensions/as4/ggg.py
index 8e83a1d71..939cda60d 100644
--- a/src/eko/anomalous_dimensions/as4/ggg.py
+++ b/src/eko/anomalous_dimensions/as4/ggg.py
@@ -137,17 +137,17 @@ def gamma_gg_nf3(n, sx):
 
 
 @nb.njit(cache=True)
-def gamma_gg_nf2(n, sx):
+def gamma_gg_nf2(_n, _sx):
     return 0
 
 
 @nb.njit(cache=True)
-def gamma_gg_nf1(n, sx):
+def gamma_gg_nf1(_n, _sx):
     return 0
 
 
 @nb.njit(cache=True)
-def gamma_gg_nf0(n, sx):
+def gamma_gg_nf0(_n, _sx):
     return 0
 
 
diff --git a/src/eko/anomalous_dimensions/as4/gps.py b/src/eko/anomalous_dimensions/as4/gps.py
index 31b34f9e4..20878f063 100644
--- a/src/eko/anomalous_dimensions/as4/gps.py
+++ b/src/eko/anomalous_dimensions/as4/gps.py
@@ -79,12 +79,12 @@ def gamma_ps_nf3(n, sx):
 
 
 @nb.njit(cache=True)
-def gamma_ps_nf2(n, sx):
+def gamma_ps_nf2(_n, _sx):
     return 0
 
 
 @nb.njit(cache=True)
-def gamma_ps_nf1(n, sx):
+def gamma_ps_nf1(_n, _sx):
     return 0
 
 
diff --git a/src/eko/anomalous_dimensions/as4/gqg.py b/src/eko/anomalous_dimensions/as4/gqg.py
index 028875186..76047a249 100644
--- a/src/eko/anomalous_dimensions/as4/gqg.py
+++ b/src/eko/anomalous_dimensions/as4/gqg.py
@@ -329,17 +329,17 @@ def gamma_qg_nf3(n, sx):
 
 
 @nb.njit(cache=True)
-def gamma_qg_nf2(n, sx):
+def gamma_qg_nf2(_n, _sx):
     return 0
 
 
 @nb.njit(cache=True)
-def gamma_qg_nf1(n, sx):
+def gamma_qg_nf1(_n, _sx):
     return 0
 
 
 @nb.njit(cache=True)
-def gamma_qg_nf0(n, sx):
+def gamma_qg_nf0(_n, _sx):
     return 0
 
 
diff --git a/src/eko/harmonics/f_functions/f11.py b/src/eko/harmonics/f_functions/f11.py
index 5ec1984e4..b0e46365e 100644
--- a/src/eko/harmonics/f_functions/f11.py
+++ b/src/eko/harmonics/f_functions/f11.py
@@ -1,4 +1,5 @@
 # -*- coding: utf-8 -*-
+# pylint: skip-file
 """This module contains implemtation of F11.
 
 When using it, please cite :cite:`Blumlein:2009ta`.
diff --git a/src/eko/harmonics/f_functions/f13.py b/src/eko/harmonics/f_functions/f13.py
index f540a834e..38046f88e 100644
--- a/src/eko/harmonics/f_functions/f13.py
+++ b/src/eko/harmonics/f_functions/f13.py
@@ -1,4 +1,5 @@
 # -*- coding: utf-8 -*-
+# pylint: skip-file
 """This module contains implemtation of F13.
 
 When using it, please cite :cite:`Blumlein:2009ta`.
diff --git a/src/eko/harmonics/f_functions/f14_f12.py b/src/eko/harmonics/f_functions/f14_f12.py
index de0745bb7..48c8b1168 100644
--- a/src/eko/harmonics/f_functions/f14_f12.py
+++ b/src/eko/harmonics/f_functions/f14_f12.py
@@ -1,5 +1,5 @@
 # -*- coding: utf-8 -*-
-# pylint: disable=too-many-lines
+# pylint: skip-file
 """This module contains implemtation of F14 - F12.
 
 When using it, please cite :cite:`Blumlein:2009ta`.
diff --git a/src/eko/harmonics/f_functions/f16.py b/src/eko/harmonics/f_functions/f16.py
index db9ecd017..26caa9353 100644
--- a/src/eko/harmonics/f_functions/f16.py
+++ b/src/eko/harmonics/f_functions/f16.py
@@ -1,5 +1,5 @@
 # -*- coding: utf-8 -*-
-# pylint: disable=too-many-lines
+# pylint: skip-file
 """This module contains implemtation of F16.
 
 When using it, please cite :cite:`Blumlein:2009ta`.
diff --git a/src/eko/harmonics/f_functions/f17.py b/src/eko/harmonics/f_functions/f17.py
index cc0f0a4cd..be9709821 100644
--- a/src/eko/harmonics/f_functions/f17.py
+++ b/src/eko/harmonics/f_functions/f17.py
@@ -1,4 +1,5 @@
 # -*- coding: utf-8 -*-
+# pylint: skip-file
 """This module contains implemtation of F17.
 
 When using it, please cite :cite:`Blumlein:2009ta`.
diff --git a/src/eko/harmonics/f_functions/f18.py b/src/eko/harmonics/f_functions/f18.py
index 344a2fe84..0a5d55948 100644
--- a/src/eko/harmonics/f_functions/f18.py
+++ b/src/eko/harmonics/f_functions/f18.py
@@ -1,5 +1,5 @@
 # -*- coding: utf-8 -*-
-# pylint: disable=too-many-lines
+# pylint: skip-file
 """This module contains implemtation of F18.
 
 When using it, please cite :cite:`Blumlein:2009ta`.
diff --git a/src/eko/harmonics/f_functions/f19.py b/src/eko/harmonics/f_functions/f19.py
index f372b89d9..76a180194 100644
--- a/src/eko/harmonics/f_functions/f19.py
+++ b/src/eko/harmonics/f_functions/f19.py
@@ -1,4 +1,5 @@
 # -*- coding: utf-8 -*-
+# pylint: skip-file
 """This module contains implemtation of F19.
 
 When using it, please cite :cite:`Blumlein:2009ta`.
diff --git a/src/eko/harmonics/f_functions/f20.py b/src/eko/harmonics/f_functions/f20.py
index 2db510935..23149d287 100644
--- a/src/eko/harmonics/f_functions/f20.py
+++ b/src/eko/harmonics/f_functions/f20.py
@@ -1,4 +1,5 @@
 # -*- coding: utf-8 -*-
+# pylint: skip-file
 """This module contains implemtation of F20.
 
 When using it, please cite :cite:`Blumlein:2009ta`.
diff --git a/src/eko/harmonics/f_functions/f21.py b/src/eko/harmonics/f_functions/f21.py
index a2aba7b90..81895821b 100644
--- a/src/eko/harmonics/f_functions/f21.py
+++ b/src/eko/harmonics/f_functions/f21.py
@@ -1,4 +1,5 @@
 # -*- coding: utf-8 -*-
+# pylint: skip-file
 """This module contains implemtation of F21.
 
 When using it, please cite :cite:`Blumlein:2009ta`.
diff --git a/src/eko/harmonics/f_functions/f9.py b/src/eko/harmonics/f_functions/f9.py
index 94d41fb2e..3099e9679 100644
--- a/src/eko/harmonics/f_functions/f9.py
+++ b/src/eko/harmonics/f_functions/f9.py
@@ -1,4 +1,5 @@
 # -*- coding: utf-8 -*-
+# pylint: skip-file
 """This module contains implemtation of F9.
 
 When using it, please cite :cite:`Blumlein:2009ta`.
diff --git a/src/eko/matching_conditions/as3/aHg.py b/src/eko/matching_conditions/as3/aHg.py
index 7482accd3..a7ea6dbbb 100644
--- a/src/eko/matching_conditions/as3/aHg.py
+++ b/src/eko/matching_conditions/as3/aHg.py
@@ -1,5 +1,5 @@
 # -*- coding: utf-8 -*-
-# pylint: disable=too-many-lines
+# pylint: skip-file
 import numba as nb
 import numpy as np
 
diff --git a/src/eko/matching_conditions/as3/aHgstfac.py b/src/eko/matching_conditions/as3/aHgstfac.py
index 2ddd57d70..8651be75c 100644
--- a/src/eko/matching_conditions/as3/aHgstfac.py
+++ b/src/eko/matching_conditions/as3/aHgstfac.py
@@ -1,4 +1,5 @@
 # -*- coding: utf-8 -*-
+# pylint: skip-file
 import numba as nb
 import numpy as np
 
diff --git a/src/eko/matching_conditions/as3/agg.py b/src/eko/matching_conditions/as3/agg.py
index e0bf47284..e5fd62576 100644
--- a/src/eko/matching_conditions/as3/agg.py
+++ b/src/eko/matching_conditions/as3/agg.py
@@ -1,5 +1,5 @@
 # -*- coding: utf-8 -*-
-# pylint: disable=too-many-lines
+# pylint: skip-file
 import numba as nb
 import numpy as np