# Kannappan-Wilson and Van Vleck-Wilson functional equations on semigroups

Youssef Aserrar and Elhoucien Elqorachi  Ibn Zohr University, Faculty of
sciences, Department of mathematics,
Agadir, Morocco<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: January 1, 2020)

###### Abstract.

Let $S$ be a semigroup, $Z(S)$ the center of $S$ and $\sigma:S\rightarrow S$
is an involutive automorphism. Our main results is that we describe the
solutions of the Kannappan-Wilson functional equation

$\displaystyle\int_{S}f(xyt)d\mu(t)+\displaystyle\int_{S}f(\sigma(y)xt)d\mu(t)=2f(x)g(y),\
x,y\in S,$

and the Van Vleck-Wilson functional equation

$\displaystyle\int_{S}f(xyt)d\mu(t)-\displaystyle\int_{S}f(\sigma(y)xt)d\mu(t)=2f(x)g(y),\
x,y\in S,$

where $\mu$ is a measure that is a linear combination of Dirac measures
$(\delta_{z_{i}})_{i\in I}$, such that $z_{i}\in Z(S)$ for all $i\in I$.
Interesting consequences of these results are presented.

###### Key words and phrases:

Semigroup, automorphism, measure, Van Vleck’s functional equation, Wilson’s
functional equation.

###### 1991 Mathematics Subject Classification:

39B52, 39B32

##  1\. Set up, Notation and terminology

Throughout this paper $S$ denotes a semigroup, i.e., a set equipped with an
associative binary operation, $Z(S)$ denotes the center of $S$, i.e.,

$Z(S):=\left\\{s\in S\mid xs=sx\ \text{for all}\ x\in S\right\\}.$

######  Definition 1.1.

Let $f:S\rightarrow\mathbb{C}$.
$f$ is multiplicative, if $f(xy)=f(x)f(y)$ for all $x,y\in S$.
$f$ is additive, if $f(xy)=f(x)+f(y)$ for all $x,y\in S$.
$f$ is central, if $f(xy)=f(yx)$ for all $x,y\in S$.
$f$ is abelian, if $f$ is central and $f(xyz)=f(xzy)$ for all $x,y,z\in S$.

The map $\sigma:S\rightarrow S$ denotes an involutive automorphism of $S$.
That is $\sigma(xy)=\sigma(x)\sigma(y)$ and $\sigma(\sigma(x))=x$ for all
$x,y\in S$. For any function $f:S\rightarrow\mathbb{C}$ we define the function
$f^{*}:=f\circ\sigma$, and the functions $f^{e}:=\frac{f+f^{*}}{2}$,
$f^{\circ}:=\frac{f-f^{*}}{2}$. The function $f$ is said to be even if
$f=f^{*}$ and $f$ is odd if $f=-f^{*}$. Throughout the paper, we use without
explicit mentioning the fact that $\sigma(s)\in Z(S)$ for any $s\in Z(S)$.

If $\chi:S\rightarrow\mathbb{C}$ is a multiplicative function, we denote by
$\phi:S\rightarrow\mathbb{C}$ a non-zero solution of the special sine addition
law

$\phi(xy)=\phi(x)\chi(y)+\phi(y)\chi(x),\quad x,y\in S.$ (1.1)

##  2\. Introduction

The classical work of Wilson [20] was the first contribution to the theory of
Wilson’s functional equation

$f(x+y)+f(x-y)=2f(x)g(y),\quad x,y\in\mathbb{R},$

which is a generalization of the cosine functional equation

$g(x+y)+g(x-y)=2g(x)g(y),\quad x,y\in\mathbb{R}.$ (2.1)

Aczel’s monograph [1, Section 3.2.1 and Section 3.2.2] discusses the real
valued, continuous solutions of Wilson’s equation and contains references to
earlier works. These functional equations have been extended to abelian
groups, and they have been solved on that setting. An extension of Wilson’s
functional equation above is treated by Stetkær in [12, Corollary 5] who
derived the solution formulas for the functional equation

$f(x+y)-f(x+\sigma(y))=2g(x)h(y),\quad x,y\in G,$

on an abelian group $G$, where $\sigma:G\rightarrow G$ is an involution. The
d’Alembert’s functional equation

$f(xy)+f(x\sigma(y))=2f(x)f(y),\quad x,y\in S,$

where $\sigma:S\rightarrow S$ is an involutive anti-automorphism (i.e,
$\sigma(xy)=\sigma(y)\sigma(x)$ and $\sigma(\sigma(x))=x$ for all $x,y\in S$)
has been solved on groups by Davison [6] and others. See for example [14,
Chapter 9]. It was also studied on semigroups by Stetkær [18].

In [15], Stetkær solved the variant of d’Alembert’s functional equation

$f(xy)+f(\sigma(y)x)=2f(x)f(y),\quad x,y\in S,$

on a semigroup $S$, where $\sigma$ is an involutive automorphism. The
solutions are abelian and are of the form $f=\dfrac{\chi+\chi^{*}}{2}$, where
$\chi:S\rightarrow\mathbb{C}$ is a multiplicative function. Elqorachi and
Redouani [8] obtained the solutions of the variant of Wilson’s functional
equation

$f(xy)+\tau(y)f(\sigma(y)x)=2f(x)g(y),\quad x,y\in S,$ (2.2)

on groups, where $\sigma:S\rightarrow S$ is an involutive automorphism, and
$\tau:S\rightarrow\mathbb{C}$ is a multiplicative function such that
$\tau(x\sigma(x))=1$ for all $x\in S$. It was also solved on groups with
$\tau=1$ by Fadli et al. [9]. The results were extended to semigroups
generated by their squares by Ajebbar and Elqorachi [2]. Recently, Aserrar,
Chahbi and Elqorachi [3] obtained the solutions on semigroups. Theorem (3)
reproduces their main points in terms of multiplicative functions, and
solutions $\phi:S\rightarrow\mathbb{C}$ of Eq. (1.1) with $\tau=1$.

######  Theorem 2.1.

The solutions $f,g:S\rightarrow\mathbb{C}$ of Eq. (2.2) with $g\neq 0$ and
$\tau=1$, namely

$f(xy)+f(\sigma(y)x)=2f(x)g(y),\quad x,y\in S,$

are the following pairs

  1. (1)

$f=0$ and $g\neq 0$ arbitrary.

  2. (2)

$f=\alpha\chi+\beta\chi^{*}$ and $g=\dfrac{\chi+\chi^{*}}{2}$, where
$\alpha,\beta\in\mathbb{C}$ are constants such that $(\alpha,\beta)\neq(0,0)$
and $\chi:S\rightarrow\mathbb{C}$ is a non-zero multiplicative function.

  3. (3)

$f=\gamma_{1}\chi+\phi$ and $g=\chi$, where $\gamma_{1}\in\mathbb{C}$ is a
constant, $\chi:S\rightarrow\mathbb{C}$ is a non-zero multiplicative function,
$\phi$ is a non-zero solution of the sine addition law (1.1) such that
$\chi^{*}=\chi$ and $\phi^{*}=-\phi$.

Kannappan [10] generalized the cosine functional equation (2.1) to the
functional equation

$f(x+y+z_{0})+f(x-y+z_{0})=2f(x)f(y),\quad x,y\in\mathbb{R},$

where $z_{0}\neq 0$ is a real constant. He showed that any solution
$f:\mathbb{R}\rightarrow\mathbb{C}$ of this equation has the form
$f(x)=g(x-z_{0})$, where $g:\mathbb{R}\rightarrow\mathbb{C}$ is a periodic
solution of (2.1) with period $2z_{0}$.

In 2017, Stetkær [17] generalized the result of Kannappan by solving the
functional equation

$f(xyz_{0})+f(xy^{*}z_{0})=2f(x)f(y),\quad x,y\in S,$

on a semigroup $S$ such that $x\mapsto x^{*}$ is an involutive anti-
automorphism and $z_{0}\in Z(S)$ fixed. An other similar looking functional
equation is Van Vleck’s functional equation

$f(x-y+z_{0})-f(x+y+z_{0})=2f(x)f(y),\quad x,y\in\mathbb{R},$

which was studied by Van Vleck [19] with $z_{0}>0$ fixed. Stetkær [16]
extended Van Vleck’s result to general semigroups by solving the functional
equation

$f(xy^{*}z_{0})-f(xyz_{0})=2f(x)f(y),\quad x,y\in S,$

where $z_{0}\in Z(S)$ is fixed, $x\mapsto x^{*}$ is an involutive anti-
automorphism. He proved that the solutions are all abelian even in the setting
of semigroups. Elqorachi [7] described the solutions of the functional
equation

$\displaystyle\int_{S}f(xyt)d\mu(t)-\displaystyle\int_{S}f(x\sigma(y)t)d\mu(t)=2f(x)f(y),\
x,y\in S,$

where $S$ is a semigroup, $\sigma$ is an involutive automorphism and $\mu$ is
a measure that is a linear combination of Dirac measures
$(\delta_{z_{i}})_{i\in I}$ with $z_{i}\in Z(S)$ for all $i\in I$. Other
results about these functional equations can be found in [5] and [11].

The first purpose of the present paper is to solve the Kannappan-Wilson
functional equation

$\displaystyle\int_{S}f(xyt)d\mu(t)+\displaystyle\int_{S}f(\sigma(y)xt)d\mu(t)=2f(x)g(y),\
x,y\in S,$ (2.3)

where $S$ is a semigroup, $\mu$ is a measure that is a linear combination of
Dirac measures $(\delta_{z_{i}})_{i\in I}$ with $z_{i}\in Z(S)$ for all $i\in
I$ and $\sigma$ is an involutive automorphism. Equation (2.3) is a
generalization of (2.2), so here we show that the two equations are related
and consequently we find explicit formulas for the solutions expressed in
terms of multiplicative functions, and solutions of the special sine addition
law (1.1). As a consequence, we solve the Kannappan-Wilson functional equation

$f(xyz_{0})+f(\sigma(y)xz_{0})=2f(x)g(y),\quad x,y\in S,$ (2.4)

on semigroups, where $\sigma$ is an involutive automorphism and $z_{0}\in
Z(S)$ fixed. We also give the solutions of Jensen’s functional equation

$f(xyz_{0})+f(\sigma(y)xz_{0})=2f(x),\quad x,y\in S,$ (2.5)

and by taking $\sigma=id$, we derive the solutions of the functional equation

$f(xyz_{0})+f(yxz_{0})=2f(x)g(y),\quad x,y\in S,$ (2.6)

which is a generalization of the symmetrized multiplicative Cauchy equation

$f(xy)+f(yx)=2f(x)f(y),\quad x,y\in S,$

which was solved by Stetkær in [13].

The second aim of the paper is to solve the Van Velck-Wilson functional
equation

$\displaystyle\int_{S}f(xyt)d\mu(t)-\displaystyle\int_{S}f(\sigma(y)xt)d\mu(t)=2f(x)g(y),\
x,y\in S,$ (2.7)

and the Van Vleck functional equation

$f(xyz_{0})-f(\sigma(y)xz_{0})=2f(x)g(y),\quad x,y\in S,$ (2.8)

where $z_{0}\in Z(S)$. We also give some results that are not in the
literature about the functional equation

$f(xy)-f(\sigma(y)x)=2f(x)g(y),\quad x,y\in S,$ (2.9)

on a semigroup $S$, where $\sigma:S\rightarrow S$ is an involutive
automorphism. We show that the functional equations (2.7) and (2.8) are
related to (2.2), and we find explicit formulas for the solutions in terms of
multiplicative functions.

Until now, the functional equation (2.2) is the only one among all Wilson’s
type functional equations which is solved (with $g\neq 0$) on any semigroup,
and it is not difficult to see that the treatment of Eq. (2.3) on monoids or
groups compared with its treatment on semigroups is much easier thanks to the
existence of a neutral element which allows the equation to be related to Eq.
(2.2) easily. In the following remark, we give a proof of this fact.

######  Remark 2.2.

Let $f,g:S\rightarrow\mathbb{C}$ be a solution of Eq. (2.3). Suppose $S$ is a
monoid and $e$ its neutral element. If we put $y=e$ in Eq. (2.3) and taking
into account that $\sigma(e)=e$, we obtain

$\displaystyle\int_{S}f(xt)d\mu(t)+\displaystyle\int_{S}f(xt)d\mu(t)=2f(x)g(e),$

for all $x\in S$. That is $\displaystyle\int_{S}f(xt)d\mu(t)=f(x)g(e)$. Thus,
Eq. (2.3) becomes

$g(e)f(xy)+g(e)f(\sigma(y)x)=2f(x)g(y),\quad x,y\in S.$

Therefore, if $g(e)=0$ then $f=0$ or $g=0$. Now, if $g(e)\neq 0$, the last
equation above can be written as

$f(xy)+f(\sigma(y)x)=2f(x)\dfrac{g(y)}{g(e)},\quad x,y\in S,$

which means that the pair $\left(f,\dfrac{g}{g(e)}\right)$ is a solution of
Eq. (2.2).

The outline of the paper is as follows. The next section consists of two
subsections. In subsection 3.1, we solve the functional equation (2.3), and as
a consequence we find the solutions of (2.4), (2.5) and (2.6) in subsection
3.2. Subsection 4.1 contain the solutions of the functional equation (2.9),
and the solutions of (2.7) and (2.8) are given in subsection 4.2.

##  3\. Kannappan-Wilson functional equations

###  3.1. Solutions of Eq. (2.3)

The following lemma contain useful results about the solutions of the
functional equation (2.3).

######  Lemma 3.1.

Let $f,g:S\rightarrow\mathbb{C}$ be a solution of Eq. (2.3) such that $f\neq
0$ and $g\neq 0$. The following statements hold

  1. (1)

For all $x,y\in S$

$\displaystyle\begin{split}\displaystyle\int_{S}f(xt)d\mu(t)\left[\displaystyle\int_{S}g(ys)d\mu(s)+\displaystyle\int_{S}g(y\sigma(s))d\mu(s)\right]\\\
=2f(x)g(y)\displaystyle\int_{S}g(s)d\mu(s).\end{split}$ (3.1)

  2. (2)

Define $h(x):=\displaystyle\int_{S}f(xt)d\mu(t)$ for all $x\in S$. Then $h\neq
0$.

  3. (3)

Suppose $\displaystyle\int_{S}g(s)d\mu(s)\neq 0$, and define the functions

$F(x):=\dfrac{h(x)}{\displaystyle\int_{S}g(s)d\mu(s)}\quad\text{and}\quad
G(x):=\dfrac{\displaystyle\int_{S}g(xt)d\mu(t)+\displaystyle\int_{S}g(x\sigma(t))d\mu(t)}{2\displaystyle\int_{S}g(s)d\mu(s)}.$

Then $G\neq 0$, and

$F(xy)+F(\sigma(y)x)=2F(x)G(y),\quad x,y\in S.$ (3.2)

  4. (4)

If $\displaystyle\int_{S}g(s)d\mu(s)=0$, then $h(x)=\lambda_{1}f(x)$ for all
$x\in S$, where $\lambda_{1}\in\mathbb{C}\backslash\left\\{0\right\\}$ is a
constant.

###### Proof.

(1) Let $x,y,t,s\in S$ be arbitrary. Applying Eq. (2.3) to the pair $(xyt,s)$
and then to $(\sigma(y)xt,s)$, we get respectively

$g(s)f(xyt)=\dfrac{1}{2}\left[\displaystyle\int_{S}f(xytsk)d\mu(k)+\displaystyle\int_{S}f(\sigma(s)xytk)d\mu(k)\right],$
(3.3)

and

$g(s)f(\sigma(y)xt)=\dfrac{1}{2}\left[\displaystyle\int_{S}f(\sigma(y)xtsk)d\mu(k)+\displaystyle\int_{S}f(\sigma(s)\sigma(y)xtk)d\mu(k)\right].$
(3.4)

On the other hand, if we apply Eq. (2.3) to the pair $(xt,ys)$ and then to
$(xt,y\sigma(s))$, and integrate the result with respect to $t$ and $s$, we
obtain respectively

$\displaystyle\begin{split}\dfrac{1}{2}\left[\displaystyle\int_{S}\int_{S}\left(\int_{S}f(xytsk)d\mu(k)+\displaystyle\int_{S}f(\sigma(s)\sigma(y)xtk)d\mu(k)\right)d\mu(t)d\mu(s)\right]\\\
=\displaystyle\int_{S}f(xt)d\mu(t)\displaystyle\int_{S}g(ys)d\mu(s),\end{split}$
(3.5)

and

$\displaystyle\begin{split}\dfrac{1}{2}\left[\displaystyle\int_{S}\int_{S}\left(\int_{S}f(\sigma(s)xytk)d\mu(k)+\displaystyle\int_{S}f(\sigma(y)xtsk)d\mu(k)\right)d\mu(t)d\mu(s)\right]\\\
=\displaystyle\int_{S}f(xt)d\mu(t)\displaystyle\int_{S}g(y\sigma(s))d\mu(s).\end{split}$
(3.6)

Thus, by integrating Eq. (3.3) and Eq. (3.4) with respect to $t$ and $s$, then
adding the two equations obtained and taking into account Eq. (3.5) and Eq.
(3.6), we obtain Eq. (3.1).
(2) Suppose $h=0$. Then Eq. (2.3) yields $2f(x)g(y)=0$ for all $x,y\in S$, so
$f=0$ or $g=0$. This is a contradiction since $f\neq 0$ and $g\neq 0$. Thus
$h\neq 0$. This occurs in (2).
(3) Suppose $\displaystyle\int_{S}g(s)d\mu(s)\neq 0$. Equation (3.2) follows
easily from (1) when we divide Eq. (3.1) by
$\left(\displaystyle\int_{S}g(s)d\mu(s)\right)^{2}$. In addition, suppose
$G=0$. It follows from Eq. (3.1) that

$\displaystyle\int_{S}g(s)d\mu(s)\left[\displaystyle\int_{S}f(xyt)d\mu(t)+\displaystyle\int_{S}f(\sigma(y)xt)d\mu(t)\right]=0,$

for all $x,y\in S$. That is $2f(x)g(y)\displaystyle\int_{S}g(s)d\mu(s)=0$,
which implies that $\displaystyle\int_{S}g(s)d\mu(s)=0$ since $f\neq 0$ and
$g\neq 0$. This is a contradiction. So $G\neq 0$. This is part (3).
(4) Assume that $\displaystyle\int_{S}g(s)d\mu(s)=0$. Replacing $(x,y)$ by
$(x,s)$ in the functional equation (2.3) and integrating the result obtained
with respect to $s$, we find that

$\displaystyle\int_{S}\int_{S}f(xst)d\mu(t)d\mu(s)+\displaystyle\int_{S}\int_{S}f(\sigma(s)xt)d\mu(t)d\mu(s)=0,$

for all $x\in S$. This implies

$\displaystyle\int_{S}\int_{S}f(xst)d\mu(t)d\mu(s)=-\displaystyle\int_{S}\int_{S}f(\sigma(s)xt)d\mu(t)d\mu(s).$
(3.7)

Now, if we replace $(x,y)$ by $(x,ys)$ in Eq. (2.3) and integrate the result
obtained with respect to $s$, we get

$\displaystyle\int_{S}\int_{S}f(xyst)d\mu(t)d\mu(s)+\displaystyle\int_{S}\int_{S}f(\sigma(ys)xt)d\mu(t)d\mu(s)=2f(x)\displaystyle\int_{S}g(ys)d\mu(s).$

That is, in view of Eq. (3.7)

$\displaystyle\int_{S}\int_{S}f(xyst)d\mu(t)d\mu(s)-\displaystyle\int_{S}\int_{S}f(s\sigma(y)xt)d\mu(t)d\mu(s)=2f(x)\displaystyle\int_{S}g(ys)d\mu(s).$
(3.8)

On the other hand, if we replace $(x,y)$ by $(xs,y)$ in Eq. (2.3) and
integrate the result obtained with respect to $s$, we obtain

$\displaystyle\int_{S}\int_{S}f(xsyt)d\mu(t)d\mu(s)+\displaystyle\int_{S}\int_{S}f(\sigma(y)xst)d\mu(t)d\mu(s)=2g(y)\displaystyle\int_{S}f(xs)d\mu(s).$
(3.9)

Thus, by adding Eq. (3.8) to Eq. (3.9), we get

$\displaystyle\int_{S}\int_{S}f(xsyt)d\mu(t)d\mu(s)=f(x)\displaystyle\int_{S}g(ys)d\mu(s)+g(y)\displaystyle\int_{S}f(xs)d\mu(s).$
(3.10)

Applying Eq. (3.10) to $(xk,y)$ and then to $(x,yk)$ and integrating the
result obtained with respect to $k$, we get respectively

$\displaystyle\begin{split}\displaystyle\int_{S}\int_{S}\int_{S}f(xksyt)d\mu(t)d\mu(s)d\mu(k)=\displaystyle\int_{S}f(xk)d\mu(k)\displaystyle\int_{S}g(ys)d\mu(s)\\\
+g(y)\displaystyle\int_{S}\int_{S}f(xks)d\mu(s)d\mu(k)\end{split}$

and

$\displaystyle\begin{split}\displaystyle\int_{S}\int_{S}\int_{S}f(xsykt)d\mu(t)d\mu(s)d\mu(k)=f(x)\displaystyle\int_{S}\int_{S}g(yks)d\mu(s)d\mu(k)\\\
+\displaystyle\int_{S}g(yk)d\mu(k)\displaystyle\int_{S}f(xs)d\mu(s)\end{split}$

Therefore, comparing the last two identities, we can see that

$f(x)\displaystyle\int_{S}\int_{S}g(yks)d\mu(s)d\mu(k)=g(y)\displaystyle\int_{S}\int_{S}f(xks)d\mu(s)d\mu(k).$

This implies, since $g\neq 0$ that there exists a constant
$\alpha\in\mathbb{C}$ such that

$\displaystyle\int_{S}\int_{S}f(xks)d\mu(s)d\mu(k)=\alpha f(x)\ \text{ for
all}\ x\in S.$

So, Eq. (3.10) becomes

$\alpha
f(xy)=f(x)\displaystyle\int_{S}g(ys)d\mu(s)+g(y)\displaystyle\int_{S}f(xs)d\mu(s).$
(3.11)

If $\alpha=0$, then replacing $x$ by $xk$ in Eq. (3.11) and integrating the
result obtained with respect to $k$, we find that

$\displaystyle\int_{S}f(xk)d\mu(k)\displaystyle\int_{S}g(ys)d\mu(s)=0,$

for all $x,y\in S$. This implies, since $h\neq 0$ that
$\displaystyle\int_{S}g(ys)d\mu(s)=0$ for all $y\in S$. Thus, Eq. (3.11)
becomes

$g(y)\displaystyle\int_{S}f(xs)d\mu(s)=0,$

for all $x,y\in S$, which is a contradiction since $h\neq 0$ and $g\neq 0$.
Therefore $\alpha\neq 0$, so dividing Eq. (3.11) by $\alpha$ then replacing
$y$ by $k$ and integrating the result obtained with respect to $k$, we obtain

$\displaystyle\int_{S}f(xk)d\mu(k)=\dfrac{1}{\alpha}f(x)\displaystyle\int_{S}\int_{S}g(ks)d\mu(s)d\mu(k),$

for all $x\in S$. This occurs in case (4) with
$\lambda_{1}:=\dfrac{1}{\alpha}\displaystyle\int_{S}\int_{S}g(ks)d\mu(s)d\mu(k)$.
In addition $h\neq 0$ implies $\lambda_{1}\neq 0$. This completes the proof of
Lemma 3.1. ∎

The first main result of the present paper is the following

######  Theorem 3.2.

The solutions $f,g:S\rightarrow\mathbb{C}$ of the functional equation (2.3)
with $g\neq 0$ can be listed as follows

  1. (1)

$f=0$ and $g\neq 0$ arbitrary.

  2. (2)

$f=\lambda_{2}m$ and $g=\displaystyle\int_{S}m(s)d\mu(s)\dfrac{m+m^{*}}{2}$,
where $m:S\rightarrow\mathbb{C}$ is a multiplicative function and
$\lambda_{2}\in\mathbb{C}\backslash\\{0\\}$ a constant such that
$\displaystyle\int_{S}m(s)d\mu(s)\neq 0$.

  3. (3)

$f=\alpha_{1}\chi+\beta_{1}\chi^{*}$ and
$g=\displaystyle\int_{S}\chi(s)d\mu(s)\dfrac{\chi+\chi^{*}}{2}$, where
$\chi:S\rightarrow\mathbb{C}$ is a multiplicative function and
$\alpha_{1},\beta_{1}\in\mathbb{C}$ are constants such that
$\left(\alpha_{1},\beta_{1}\right)\neq(0,0)$, $\chi\neq\chi^{*}$,
$\displaystyle\int_{S}\chi(s)d\mu(s)=\displaystyle\int_{S}\chi^{*}(s)d\mu(s)$
and $\displaystyle\int_{S}\chi(s)d\mu(s)\neq 0$.

  4. (4)

$f=\lambda(\gamma_{1}\chi+\phi)$ and
$g=\displaystyle\int_{S}\chi(s)d\mu(s)\chi$, where
$\chi:S\rightarrow\mathbb{C}$ is a non-zero multiplicative function, $\phi$ is
a non-zero solution of the sine addition law (1.1) and
$\lambda\in\mathbb{C}\backslash\\{0\\},\gamma_{1}\in\mathbb{C}$ are constants
such that $\chi^{*}=\chi$, $\phi^{*}=-\phi$,
$\displaystyle\int_{S}\chi(s)d\mu(s)\neq 0$ and
$\displaystyle\int_{S}\phi(s)d\mu(s)=0$.

Note that, of the exceptional case (1), $f$ and $g$ are abelian.

###### Proof.

Let $f,g:S\rightarrow\mathbb{C}$ be a solution of (2.3). If $f=0$, we can see
from Eq. (2.3) that $g\neq 0$ is arbitrary, so we are in family (1).
Henceforth we suppose that $f\neq 0$ and $g\neq 0$. We split the discussion
into the cases: $\displaystyle\int_{S}g(s)d\mu(s)=0$ and
$\displaystyle\int_{S}g(s)d\mu(s)\neq 0$.
Case 1: $\displaystyle\int_{S}g(s)d\mu(s)=0$. According to Lemma 3.1 (4),
there exists a constant $\lambda_{1}\in\mathbb{C}\backslash\\{0\\}$ such that
$\displaystyle\int_{S}f(xk)d\mu(k)=\lambda_{1}f(x)$ for all $x\in S$. So, Eq.
(2.3) can be written as

$f(xy)+f(\sigma(y)x)=2f(x)\dfrac{g(y)}{\lambda_{1}},\quad x,y\in S.$

Then, according to Theorem (3) and taking into account that $f\neq 0$, we have
two possibilities
(i) $f=\alpha\chi+\beta\chi^{*}$ and
$\dfrac{g}{\lambda_{1}}=\dfrac{\chi+\chi^{*}}{2}$, where
$\chi:S\rightarrow\mathbb{C}$ is a non-zero multiplicative function and
$(\alpha,\beta)\neq(0,0)$ are constants. Since
$\displaystyle\int_{S}g(s)d\mu(s)=0$, we can see that
$\displaystyle\int_{S}\chi(s)d\mu(s)=-\displaystyle\int_{S}\chi^{*}(s)d\mu(s)$.
Thus, inserting these forms in Eq. (2.3), we get after some rearrangements
that for all $x,y\in S$

$\displaystyle\begin{split}\left(\alpha\displaystyle\int_{S}\chi(s)d\mu(s)-\alpha\lambda_{1}\right)\chi(x)(\chi(y)+\chi^{*}(y))\\\
=\chi^{*}(x)\left(\beta\displaystyle\int_{S}\chi(s)d\mu(s)+\beta\lambda_{1}\right)(\chi(y)+\chi^{*}(y)).\end{split}$

If $\chi=\chi^{*}$, then $g=\lambda_{1}\chi$. So
$\displaystyle\int_{S}\chi(s)d\mu(s)=0$ and the identity above reduces to
$\alpha=-\beta$. This implies $f=0$, contradiction. So $\chi\neq\chi^{*}$.
Then by the help of [14, Theorem 3.18], we deduce that

$\left\\{\begin{array}[]{l}\alpha\displaystyle\int_{S}\chi(s)d\mu(s)-\alpha\lambda_{1}=0\\\
\beta\displaystyle\int_{S}\chi(s)d\mu(s)+\beta\lambda_{1}=0\end{array}\right.$
(3.12)

Since $f\neq 0$, then at least one of $\alpha$ and $\beta$ is non-zero.
(a) Assume $\alpha\neq 0$ and $\beta=0$. Then Eq.(3.12) implies
$\lambda_{1}=\displaystyle\int_{S}\chi(s)d\mu(s)$. This occurs in case (2)
with $\lambda_{2}:=\alpha$ and $m:=\chi$.
(b) Suppose $\alpha=0$ and $\beta\neq 0$. It follows from Eq. (3.12) that
$\lambda_{1}=-\displaystyle\int_{S}\chi(s)d\mu(s)$. So, we get family (2) with
$\lambda_{2}:=\beta$ and $m:=\chi^{*}$.
(c) If $\alpha\neq 0$ and $\beta\neq 0$. Then Eq. (3.12) implies
$\lambda_{1}=\displaystyle\int_{S}\chi(s)d\mu(s)=-\displaystyle\int_{S}\chi(s)d\mu(s)$.
Thus $\lambda_{1}=0$. This case does not occur.
(ii) $f=\gamma_{1}\chi+\phi$ and $\dfrac{g}{\lambda_{1}}=\chi$, where
$\chi:S\rightarrow\mathbb{C}$ is a non-zero multiplicative function, $\phi$ is
a non-zero solution of the sine addition law (1.1) and
$\gamma_{1}\in\mathbb{C}$ is a constant such that $\chi^{*}=\chi$ and
$\phi^{*}=-\phi$. In addition $\displaystyle\int_{S}g(s)d\mu(s)=0$ yields
$\displaystyle\int_{S}\chi(s)d\mu(s)=0$. Thus, inserting the forms of $f$ and
$g$ in Eq. (2.3), we get after some simplifications

$\phi=\left(\displaystyle\int_{S}\phi(s)d\mu(s)-\lambda_{1}\gamma_{1}\right)\chi.$

This implies $\phi=0$, since $\chi^{*}=\chi$ and $\phi^{*}=-\phi$, which is
not possible because $\phi\neq 0$. This case does not occur.
Case 2: $\displaystyle\int_{S}g(s)d\mu(s)\neq 0$. By Lemma 3.1 (3), we have

$F(xy)+F(\sigma(y)x)=2F(x)G(y),\quad x,y\in S,$ (3.13)

where

$F(x):=\dfrac{h(x)}{\displaystyle\int_{S}g(s)d\mu(s)}\quad\text{and}\quad
G(x):=\dfrac{\displaystyle\int_{S}g(xt)d\mu(t)+\displaystyle\int_{S}g(x\sigma(t))d\mu(t)}{2\displaystyle\int_{S}g(s)d\mu(s)},$

and $h(x)=\displaystyle\int_{S}f(xt)d\mu(t)$ for all $x\in S$. In addition,
according to Lemma 3.1 (2) and (3), $F\neq 0$ and $G\neq 0$. Therefore,
applying Theorem (3) to Eq. (3.13), we have two cases to consider
(i) $F=\alpha\chi+\beta\chi^{*}$ and $G=\dfrac{\chi+\chi^{*}}{2}$, where
$\chi:S\rightarrow\mathbb{C}$ is a non-zero multiplicative function and
$(\alpha,\beta)\neq(0,0)$ are constants. On the other hand, since $(f,g)$ is a
solution of Eq. (2.3), we can see that for all $x,y\in S$

$F(xy)+F(\sigma(y)x)=\dfrac{2f(x)g(y)}{\displaystyle\int_{S}g(s)d\mu(s)}.$

So, Eq. (3.13) can be written as

$\dfrac{f(x)g(y)}{\displaystyle\int_{S}g(s)d\mu(s)}=F(x)G(y).$ (3.14)

Thus, replacing $(x,y)$ by $(x,s)$ in Eq. (3.14) and integrating the result
obtained with respect to $s$, we get that

$f(x)=F(x)\displaystyle\int_{S}G(s)d\mu(s),$

for all $x\in S$. That is

$f=\dfrac{\alpha\chi+\beta\chi^{*}}{2}\left(\displaystyle\int_{S}\chi(s)d\mu(s)+\displaystyle\int_{S}\chi^{*}(s)d\mu(s)\right).$

In addition, since $f\neq 0$, we get from Eq. (3.14) that $g=\gamma G$ for
some constant $\gamma\in\mathbb{C}\backslash\\{0\\}$. That is

$g=\dfrac{\gamma}{2}\left(\chi+\chi^{*}\right).$

So, $\chi+\chi^{*}\neq 0$ and
$\displaystyle\int_{S}\chi(s)d\mu(s)+\displaystyle\int_{S}\chi^{*}(s)d\mu(s)\neq
0$, since $g\neq 0$ and $f\neq 0$. Thus, inserting the forms of $f$ and $g$ in
Eq. (2.3), we obtain after some simplifications

$\alpha\left(\displaystyle\int_{S}\chi(s)d\mu(s)-\gamma\right)\chi+\beta\left(\displaystyle\int_{S}\chi^{*}(s)d\mu(s)-\gamma\right)\chi^{*}=0.$
(3.15)

If $\chi=\chi^{*}$, then
$f=\displaystyle\int_{S}\chi(s)d\mu(s)(\alpha+\beta)\chi$ and the identity
above yields $\gamma=\displaystyle\int_{S}\chi(s)d\mu(s)$ since
$\alpha+\beta\neq 0$ because $f\neq 0$. This occurs in case (2) with
$\lambda_{2}:=\displaystyle\int_{S}\chi(s)d\mu(s)(\alpha+\beta)$ and
$m:=\chi=\chi^{*}$.

Now, if $\chi\neq\chi^{*}$, then by applying [14, Theorem 3.18], Eq. (3.15)
yields

$\left\\{\begin{array}[]{l}\alpha\displaystyle\int_{S}\chi(s)d\mu(s)-\alpha\gamma=0\\\
\beta\displaystyle\int_{S}\chi^{*}(s)d\mu(s)-\beta\gamma=0\end{array}\right.$
(3.16)

So, we proceed as in case 1 (i) by discussing three cases.
(a) $\alpha\neq 0$ and $\beta=0$. We get
$\gamma=\displaystyle\int_{S}\chi(s)d\mu(s)$. This is case (2) with $m:=\chi$
and
$\lambda_{2}:=\dfrac{\alpha}{2}\left(\displaystyle\int_{S}\chi(s)d\mu(s)+\displaystyle\int_{S}\chi^{*}(s)d\mu(s)\right)$.
(b) $\alpha=0$ and $\beta\neq 0$. This implies
$\gamma=\displaystyle\int_{S}\chi^{*}(s)d\mu(s)$. This occurs in case (2) with
$m:=\chi^{*}$ and
$\lambda_{2}:=\dfrac{\beta}{2}\left(\displaystyle\int_{S}\chi(s)d\mu(s)+\displaystyle\int_{S}\chi^{*}(s)d\mu(s)\right)$.
(c) $\alpha\neq 0$ and $\beta\neq 0$. Eq. (3.16) yields
$\gamma=\displaystyle\int_{S}\chi(s)d\mu(s)=\displaystyle\int_{S}\chi^{*}(s)d\mu(s)$.
This is case (3) with $\alpha_{1}:=\alpha\displaystyle\int_{S}\chi(s)d\mu(s)$
and $\beta_{1}:=\beta\displaystyle\int_{S}\chi(s)d\mu(s)$.
(ii) $F=\gamma_{1}\chi+\phi$ and $G=\chi$, where $\chi:S\rightarrow\mathbb{C}$
is a non-zero multiplicative function, $\phi$ is a non-zero solution of the
sine addition law (1.1) and $\gamma_{1}\in\mathbb{C}$ is a constant such that
$\chi^{*}=\chi$ and $\phi^{*}=-\phi$. Since
$f=\displaystyle\int_{S}G(s)d\mu(s)F$ and $g=\gamma G$, where
$\gamma\in\mathbb{C}\backslash\\{0\\}$ is a constant, we get that

$f=\displaystyle\int_{S}\chi(s)d\mu(s)(\gamma_{1}\chi+\phi)\quad\text{and}\quad
g=\gamma\chi.$

Since $f\neq 0$, we can see that $\displaystyle\int_{S}\chi(s)d\mu(s)\neq 0$.
Thus, inserting the forms of $f$ and $g$ in Eq. (2.3), we get after some
rearrangements that

$\left(\gamma-\displaystyle\int_{S}\chi(s)d\mu(s)\right)\phi=\left(\gamma_{1}\left(\displaystyle\int_{S}\chi(s)d\mu(s)-\gamma\right)+\displaystyle\int_{S}\phi(s)d\mu(s)\right)\chi.$

This implies that, since $\chi^{*}=\chi$ and $\phi^{*}=-\phi$

$\gamma=\displaystyle\int_{S}\chi(s)d\mu(s)\quad\text{and}\quad\displaystyle\int_{S}\phi(s)d\mu(s)=0.$

This occurs in case (4) with $\lambda:=\displaystyle\int_{S}\chi(s)d\mu(s)$.

Conversely, if $f$ and $g$ are of the forms (1)–(4) in Theorem 3.2 we check by
elementary computations that $f$ and $g$ satisfy the functional equation
(2.3). This completes the proof of Theorem 3.2. ∎

###  3.2. Some consequences

The first consequence of our first main result reads as follows

######  Theorem 3.3.

The solutions $f,g:S\rightarrow\mathbb{C}$ of the functional equation (2.4)
with $g\neq 0$ can be listed as follows

  1. (1)

$f=0$ and $g\neq 0$ arbitrary.

  2. (2)

$f=\lambda_{2}m$ and $g=\dfrac{m(z_{0})}{2}(m+m^{*})$, where
$m:S\rightarrow\mathbb{C}$ is a multiplicative function and
$\lambda_{2}\in\mathbb{C}\backslash\\{0\\}$ a constant such that $m(z_{0})\neq
0$.

  3. (3)

$f=\alpha_{1}\chi+\beta_{1}\chi^{*}$ and
$g=\dfrac{\chi(z_{0})}{2}(\chi+\chi^{*})$, where $\chi:S\rightarrow\mathbb{C}$
is a multiplicative function and $\alpha_{1},\beta_{1}\in\mathbb{C}$ are
constants such that $\left(\alpha_{1},\beta_{1}\right)\neq(0,0)$,
$\chi\neq\chi^{*}$, $\chi(z_{0})=\chi^{*}(z_{0})$ and $\chi(z_{0})\neq 0$.

  4. (4)

$f=\lambda(\gamma_{1}\chi+\phi)$ and $g=\chi(z_{0})\chi$, where
$\chi:S\rightarrow\mathbb{C}$ is a non-zero multiplicative function, $\phi$ is
a non-zero solution of the sine addition law (1.1),
$\lambda\in\mathbb{C}\backslash\\{0\\},\gamma_{1}\in\mathbb{C}$ are constants
such that $\chi^{*}=\chi$, $\phi^{*}=-\phi$, $\chi(z_{0})\neq 0$ and
$\phi(z_{0})=0$.

Note that, of the exceptional case (1), $f$ and $g$ are abelian.

###### Proof.

Follows easily from Theorem 3.2 by taking $\mu=\delta_{z_{0}}$, where $z_{0}$
is a fixed element in $Z(S)$. ∎

######  Remark 3.4.

If $g=0$, the functional equation (2.4) can be written as

$f(xyz_{0})+f(\sigma(y)xz_{0})=0,\quad x,y\in S.$

Then, for all $x,y,z\in S$

$f(xyzz_{0})=-f(\sigma(yz)xz_{0})=f(\sigma(z)xyz_{0})=-f(xyzz_{0}).$

Thus, $f(xyzz_{0})=0$. So, if $S$ is a regular semigroup, we can see that
$f=0$ on $Sz_{0}:=\\{xz_{0}\ |\ x\in S\\}$, but $f$ does not vanish
necessarily on $S$. See [3, Remark 3.1].

The second consequence of Theorem 3.2 is the following result on the solution
of the variant of Jensen’s functional equation (2.5).

######  Corollary 3.5.

Let $f:S\rightarrow\mathbb{C}$ be a solution of Eq. (2.5). Then $f$ has the
form $f=\gamma_{1}+A$, where $A$ is a non-zero additive function and
$\gamma_{1}\in\mathbb{C}$ is a constant such that $A^{*}=-A$ and $A(z_{0})=0$.

Note that, $f$ is abelian.

###### Proof.

Let $f$ be a solution of Eq. (2.5). By applying Theorem 3.3 with $g=1$ we can
see that the only possibility for $g$ to be $1$ is in case (4) of Theorem 3.3.
That is for $\chi=\chi^{*}=1$. So, $f=\gamma_{1}+\phi$ where
$\gamma_{1}\in\mathbb{C}$ is a constant and $\phi$ is a non-zero solution of
the functional equation

$\phi(xy)=\phi(x)+\phi(y),\quad x,y\in S,$

such that $\phi^{*}=-\phi$ and $\phi(z_{0})=0$. That is $\phi$ additive which
occurs in Corollary 3.5 with $A:=\phi$.

Conversely, simple computations prove that the formula above for $f$ defines a
solution of Eq. (2.5). This completes the proof. ∎

An other interesting consequence is the following corollary about solutions of
Eq. (2.6).

######  Corollary 3.6.

The solutions $f,g:S\rightarrow\mathbb{C}$ be a solution of Eq. (2.6) with
$g\neq 0$ are the following

  1. (1)

$f=0$ and $g\neq 0$ arbitrary.

  2. (2)

$f=\lambda_{2}\chi$ and $g=\chi(z_{0})\chi$, where
$\chi:S\rightarrow\mathbb{C}$ is a non-zero multiplicative function and
$\lambda_{2}\in\mathbb{C}\backslash\\{0\\}$ a constant such that
$\chi(z_{0})\neq 0$.

Note that in case (2), $f$ and $g$ are abelian.

###### Proof.

The result follows directly from Theorem 3.3 by taking $\sigma=id$. ∎

##  4\. Van Vleck-Wilson functional equations

###  4.1. Solutions of Eq. (2.9)

The following lemma will be used later

######  Lemma 4.1.

Let $F,G:S\rightarrow\mathbb{C}$ be a solution of the functional equation

$F(xy)=F(x)G(y)-F(y)G(x)\quad x,y\in S.$ (4.1)

Then, $F$ and $G$ are linearly dependent.

###### Proof.

Follows from [4, Theorem 5.2] by taking $\sigma=id$. ∎

The next theorem gives the general solution of (2.9).

######  Theorem 4.2.

The solutions $f,g:S\rightarrow\mathbb{C}$ of Eq. (2.9) with $g\neq 0$, namely

$f(xy)-f(\sigma(y)x)=2f(x)g(y),\quad x,y\in S,$

are the following

  1. (1)

$f=0$ and $g\neq 0$ arbitrary.

  2. (2)

$f=\lambda\chi$ and $g=\dfrac{\chi-\chi^{*}}{2}$, where
$\chi:S\rightarrow\mathbb{C}$ is a multiplicative function and
$\lambda\in\mathbb{C}\backslash\\{0\\}$ is a constant such that
$\chi\neq\chi^{*}$.

Note that in case (2), $f$ and $g$ are abelian.

###### Proof.

Let $f,g:S\rightarrow\mathbb{C}$ be a solution of Eq. (2.9). If $f=0$, we can
see easily from (2.9) that $g\neq 0$ is arbitrary. This is case (1). From now
on, we assume that $f\neq 0$ and $g\neq 0$. Let $x,y\in S$ be arbitrary. By
applying Eq. (2.9) to the pair $(\sigma(y),x)$, we get that

$f(\sigma(y)x)-f^{*}(xy)=2f^{*}(y)g(x).$

By adding this to (2.9) we obtain

$f^{\circ}(xy)=f(x)g(y)+f^{*}(y)g(x).$ (4.2)

On the other hand, since $f=f^{e}+f^{\circ}$ and taking (4.2) into account, we
get from (2.9) that

$\displaystyle f^{e}(xy)-f^{e}(\sigma(y)x)$
$\displaystyle=2f(x)g(y)-f^{\circ}(xy)+f^{\circ}(\sigma(y)x)$
$\displaystyle=f(x)g(y)-f^{*}(y)g(x)+f^{\circ}(\sigma(y)x)$
$\displaystyle=f(x)g(y)+f^{*}(x)g^{*}(y).$

So, since $f^{*}=f^{e}-f^{\circ}$ we deduce that

$\displaystyle\begin{split}f^{e}(xy)-f^{e}(\sigma(y)x)=\left[f^{e}(x)+f^{\circ}(x)\right]\left[g^{e}(y)+g^{\circ}(y)\right]+\\\
\left[f^{e}(x)-f^{\circ}(x)\right]\left[g^{e}(y)-g^{\circ}(y)\right].\end{split}$

That is

$f^{e}(xy)-f^{e}(\sigma(y)x)=2f^{e}(x)g^{e}(y)+2f^{\circ}(x)g^{\circ}(y).$
(4.3)

Now, replacing $(x,y)$ by $(\sigma(y),x)$ in (4.3) and using that
$f^{e}\circ\sigma=f^{e}$ and $f^{\circ}\circ\sigma=-f^{\circ}$, we obtain

$f^{e}(\sigma(y)x)-f^{e}(xy)=2f^{e}(y)g^{e}(x)-2f^{\circ}(y)g^{\circ}(x).$
(4.4)

Thus, by comparing Eq. (4.3) with Eq. (4.4), we see that

$f^{e}(x)g^{e}(y)+f^{\circ}(x)g^{\circ}(y)=f^{\circ}(y)g^{\circ}(x)-f^{e}(y)g^{e}(x).$

This implies

$f^{e}(x)g^{e}(y)+f^{e}(y)g^{e}(x)=f^{\circ}(y)g^{\circ}(x)-f^{\circ}(x)g^{\circ}(y).$
(4.5)

In Eq. (4.5), the left hand side is an even function of $x$ while the right is
an odd function of $x$, so

$f^{e}(x)g^{e}(y)+f^{e}(y)g^{e}(x)=0,$ (4.6)

and

$f^{\circ}(y)g^{\circ}(x)-f^{\circ}(x)g^{\circ}(y)=0,$ (4.7)

for all $x,y\in S$. According to [14, Exercise 1.1 (b)] we deduce from (4.6)
that we have two possibilities: $f^{e}=0$ or $g^{e}=0$.
First case: $f^{e}=0$. That is $f=f^{\circ}$, so $f^{*}=-f$ and from Eq. (4.3)
we can see that $g^{\circ}=0$, i.e., $g=g^{e}$. In addition Eq. (4.2) becomes

$f(xy)=f(x)g(y)-f(y)g(x).$

According to Lemma 4.1, $f$ and $g$ are linearly dependent. Then $f=0$ or
$g=0$, since $f$ is odd and $g$ is even. This case does not occur since $f\neq
0$ and $g\neq 0$.
Second case: $g^{e}=0$. That is $g=g^{\circ}$. Since $g\neq 0$, we get from
Eq. (4.7) that $f^{\circ}=\lambda g$ for some constant $\lambda\in\mathbb{C}$.
Thus, equation (4.2) becomes

$\lambda g(xy)=f(x)g(y)+f^{*}(y)g(x).$ (4.8)

Now, replacing $(x,y)$ by $(x,\sigma(y))$ in (4.8) we obtain, since $g$ is odd

$\lambda g(x\sigma(y))=f(y)g(x)-f(x)g(y).$ (4.9)

If $\lambda=0$ then $f^{\circ}=0$, i.e., $f=f^{e}$. In addition, we get from
(4.9) that $f$ and $g$ are linearly dependent. Therefore $f=0$ or $g=0$, since
$f$ is even and $g$ is odd. This is a contradiction, since $f\neq 0$ and
$g\neq 0$. So $\lambda\neq 0$. Thus, equation (4.9) can be written as

$g(x\sigma(y))=\dfrac{1}{\lambda}f(y)g(x)-\dfrac{1}{\lambda}f(x)g(y).$

According to [4, Theorem 5.2] and taking into account that $g$ is a non-zero
odd function, we have two cases to consider
Case 1: $g=\alpha(\chi-\chi^{*})$ and
$\dfrac{1}{\lambda}f=\dfrac{\chi+\chi^{*}}{2}+\alpha_{1}\dfrac{\chi-\chi^{*}}{2}$,
where $\chi:S\rightarrow\mathbb{C}$ is a multiplicative function such that
$\chi\neq\chi^{*}$, $\alpha_{1}\in\mathbb{C}$ and
$\alpha\in\mathbb{C}\backslash\\{0\\}$. This implies that
$f=\dfrac{\lambda+\lambda\alpha_{1}}{2}\chi+\dfrac{\lambda-\lambda\alpha_{1}}{2}\chi^{*}$.
Thus, inserting these forms into (2.9), we obtain after some simplifications
that

$\dfrac{\lambda+\lambda\alpha_{1}}{2}\chi(x)+\dfrac{\lambda\alpha_{1}-\lambda}{2}\chi^{*}(x)=\lambda\alpha(1+\alpha_{1})\chi(x)+\lambda\alpha(1-\alpha_{1})\chi^{*}(x),$

for all $x\in S$. Therefore, since $\chi\neq\chi^{*}$, by the help of [14,
Theorem 3.18] we obtain, since $\lambda\neq 0$

$1+\alpha_{1}=2\alpha(1+\alpha_{1})\ \ \text{and}\ \ \
\alpha_{1}-1=2\alpha(1-\alpha_{1}).$

This implies $\alpha_{1}=2\alpha$. Then $\alpha_{1}^{2}=1$. That is
$\alpha_{1}=1$ or $\alpha_{1}=-1$. So, $\alpha=\dfrac{1}{2}$ or
$\alpha=\dfrac{-1}{2}$. Thus

$\left\\{\begin{array}[]{l}f=\lambda\chi\\\
g=\dfrac{\chi-\chi^{*}}{2}\end{array}\right.\quad\text{or}\quad\left\\{\begin{array}[]{l}f=\lambda\chi^{*}\\\
g=\dfrac{\chi^{*}-\chi}{2}\end{array}.\right.$

The pair $(f,g)$ on the left falls into case (2). So does the pair on the
right, except that there $\chi$ is replaced by $\chi^{*}$.
Case 2: $g=\phi$ and $\dfrac{1}{\lambda}f=\chi+\lambda_{2}\phi$, where
$\chi:S\rightarrow\mathbb{C}$ is a non-zero multiplicative function, $\phi$ is
a non-zero solution of the sine addition law (1.1) such that $\chi^{*}=\chi$,
$\phi^{*}=-\phi$ and $\lambda_{2}\in\mathbb{C}$ is a constant. So
$f=\lambda\chi+\lambda c_{2}\phi$. Therefore, inserting these forms into
(2.9), we get after some simplifications that

$c_{2}\chi(x)\phi(y)=(\chi(x)+\lambda_{2}\phi(x))\phi(y),$

for all $x,y\in S$. This implies, since $\phi\neq 0$ that
$(\lambda_{2}-1)\chi(x)=c_{2}\phi(x)$ for all $x\in S$. Thus $\lambda_{2}=1$
and $\lambda_{2}=0$ since $\chi$ is even and $\phi$ is add, which is
impossible. This case does not occur.

Conversely, we check by elementary computations that the pairs described in
Theorem 4.2 are solutions of Eq. (2.9). This completes the proof of Theorem
4.2. ∎

######  Remark 4.3.

The only solution $f:S\rightarrow\mathbb{C}$ of the functional equation

$f(xy)-f(\sigma(y)x)=2f(x)f(y),\quad x,y\in S,$

is $f\equiv 0$.

###  4.2. Solutions of Eq. (2.7) and Eq. (2.8)

In the next lemma we give some key properties of solutions of (2.7), namely

$\displaystyle\int_{S}f(xyt)d\mu(t)-\displaystyle\int_{S}f(\sigma(y)xt)d\mu(t)=2f(x)g(y),\
x,y\in S.$

######  Lemma 4.4.

Let $f,g:S\rightarrow\mathbb{C}$ be a solution of Eq. (2.7) such that $f\neq
0$ and $g\neq 0$. Let $h$ and $F$ be the functions defined in Lemma 3.1. The
following statements hold

  1. (1)

Suppose $\displaystyle\int_{S}g(s)d\mu(s)\neq 0$.Then $F\neq 0$, and

$F(xy)+F(\sigma(y)x)=2F(x)G_{1}(y),\quad x,y\in S,$ (4.10)

where
$G_{1}(x):=\dfrac{\displaystyle\int_{S}g(xt)d\mu(t)-\displaystyle\int_{S}g(x\sigma(t))d\mu(t)}{2\displaystyle\int_{S}g(s)d\mu(s)}$
and $G_{1}\neq 0$.

  2. (2)

If $\displaystyle\int_{S}g(s)d\mu(s)=0$, then $h(x)=\lambda_{1}f(x)$ for all
$x\in S$, where $\lambda_{1}\in\mathbb{C}\backslash\left\\{0\right\\}$ is a
constant.

###### Proof.

(1) follows by using similar computations to those of the proof of Lemma 3.1
(3).
(2) By using similar computations to those of the proof of Lemma 3.1 (4), we
show that

$\displaystyle\int_{S}\int_{S}f(xyst)d\mu(t)d\mu(s)-\displaystyle\int_{S}\int_{S}f(\sigma(y)sxt)d\mu(t)d\mu(s)=2f(x)\displaystyle\int_{S}g(ys)d\mu(s),$
(4.11)

and

$\displaystyle\int_{S}\int_{S}f(xyst)d\mu(t)d\mu(s)-\displaystyle\int_{S}\int_{S}f(\sigma(y)xst)d\mu(t)d\mu(s)=2g(y)\displaystyle\int_{S}f(xs)d\mu(s).$
(4.12)

Thus, comparing Eq. (4.11) with Eq. (4.12), we see that for all $x,y\in S$

$f(x)\displaystyle\int_{S}g(ys)d\mu(s)=g(y)\displaystyle\int_{S}f(xs)d\mu(s).$

Now, the result follows easily since $g\neq 0$. This completes the proof. ∎

Now, we are ready to solve the functional equation (2.7).

######  Theorem 4.5.

The solutions $f,g:S\rightarrow\mathbb{C}$ of Eq. (2.7) with $g\neq 0$ can be
listed as follows

  1. (1)

$f=0$ and $g\neq 0$ arbitrary.

  2. (2)

$f=\alpha_{1}\chi+\alpha_{2}\chi^{*}$ and
$g=\displaystyle\int_{S}\chi(s)d\mu(s)\dfrac{\chi-\chi^{*}}{2}$, where
$\chi:S\rightarrow\mathbb{C}$ is a multiplicative function and
$\alpha_{1},\alpha_{2}\in\mathbb{C}$ are constants such that
$(\alpha_{1},\alpha_{2})\neq(0,0)$, $\chi\neq\chi^{*}$ and
$\displaystyle\int_{S}\chi(s)d\mu(s)=-\displaystyle\int_{S}\chi^{*}(s)d\mu(s)\neq
0$.

Note that in case (2), $f$ and $g$ are abelian.

###### Proof.

Let $f,g:S\rightarrow\mathbb{C}$ be a solution of (2.7). If $f=0$, then we can
see easily from (2.7) that $g\neq 0$ is arbirary. This occurs in (1). Now, we
assume that $f\neq 0$ and $g\neq 0$. We discuss two cases :

$\displaystyle\int_{S}g(s)d\mu(s)=0\ \ \text{and}\ \
\displaystyle\int_{S}g(s)d\mu(s)\neq 0.$

Case 1: $\displaystyle\int_{S}g(s)d\mu(s)=0$. According to Lemma 4.4 (2),
there exists a constant $\lambda_{1}\in\mathbb{C}\backslash\\{0\\}$ such that
$\displaystyle\int_{S}f(xk)d\mu(k)=\lambda_{1}f(x)$ for all $x\in S$. Thus,
Eq. (2.7) can be written as

$f(xy)-f(\sigma(y)x)=2f(x)\dfrac{g(y)}{\lambda_{1}},\quad x,y\in S.$

Then, according to Theorem 4.2 and taking into account that $f\neq 0$, we have
one possibility

$f=\lambda\chi\ \ \text{and}\ \
\dfrac{g}{\lambda_{1}}=\dfrac{\chi-\chi^{*}}{2},$

where $\chi:S\rightarrow\mathbb{C}$ is a multiplicative function and
$\lambda\in\mathbb{C}\backslash\\{0\\}$ is a constant such that
$\chi\neq\chi^{*}$. Inserting the forms of $f$ and $g$ into Eq. (2.7), we
obtain after some simplifications that
$\lambda_{1}=\displaystyle\int_{S}\chi(s)d\mu(s)$. This is part (2) with
$\alpha:=\lambda$.
Case 2: $\displaystyle\int_{S}g(s)d\mu(s)\neq 0$. By Lemma 4.4 (1), we have
that

$F(xy)+F(\sigma(y)x)=2F(x)G_{1}(y),\quad x,y\in S,$

where

$F(x):=\dfrac{h(x)}{\displaystyle\int_{S}g(s)d\mu(s)}\quad\text{and}\quad
G_{1}(x):=\dfrac{\displaystyle\int_{S}g(xt)d\mu(t)-\displaystyle\int_{S}g(x\sigma(t))d\mu(t)}{2\displaystyle\int_{S}g(s)d\mu(s)},$

and $h(x):=\displaystyle\int_{S}f(xt)d\mu(t)$ for all $x\in S$. Thus,
according to Theorem (3) and taking into account that $F\neq 0$ and $G_{1}\neq
0$, we get

$F=\alpha\chi+\beta\chi^{*}\quad\text{and}\quad
G_{1}=\dfrac{\chi+\chi^{*}}{2},$

where $\alpha,\beta\in\mathbb{C}$ are constants such that
$(\alpha,\beta)\neq(0,0)$ and $\chi:S\rightarrow\mathbb{C}$ is a non-zero
multiplicative function. Therefore, proceeding similarly to the proof of
Theorem 3.2 case (2)(i), we find that

$f=\alpha_{1}\chi+\alpha_{2}\chi^{*}\ \ \text{and}\ \
g=\displaystyle\int_{S}\chi(s)d\mu(s)\dfrac{\chi-\chi^{*}}{2},$

where $\alpha_{1},\alpha_{2}\in\mathbb{C}$ are constants such that
$(\alpha_{1},\alpha_{2})\neq(0,0)$, $\chi\neq\chi^{*}$ and
$\displaystyle\int_{S}\chi(s)d\mu(s)=-\displaystyle\int_{S}\chi^{*}(s)d\mu(s)\neq
0$. This occurs in case (2).
The second possibility according to Theorem (3) is that $F=\gamma(c\chi+\phi)$
and $G_{1}=\chi$, where $\gamma\in\mathbb{C}\backslash\\{0\\}$ is a constant.
This leads to the contradiction $g=0$.

The converses are straightforward verifications. This completes the proof. ∎

The final result of the paper is the following corollary about the solutions
of the functional equation (2.8), namely

$f(xyz_{0})-f(\sigma(y)xz_{0})=2f(x)g(y),\quad x,y\in S.$

######  Corollary 4.6.

The solutions $f,g:S\rightarrow\mathbb{C}$ of Eq. (2.8) with $g\neq 0$ can be
listed as follows

  1. (1)

$f=0$ and $g\neq 0$ arbitrary.

  2. (2)

$f=\alpha_{1}\chi+\alpha_{2}\chi^{*}$ and
$g=\chi(z_{0})\dfrac{\chi-\chi^{*}}{2}$, where $\chi:S\rightarrow\mathbb{C}$
is a multiplicative function and $\alpha_{1},\alpha_{2}\in\mathbb{C}$ are
constants such that $(\alpha_{1},\alpha_{2})\neq(0,0)$, $\chi\neq\chi^{*}$ and
$\chi(z_{0})=-\chi^{*}(z_{0})\neq 0$.

Note that in case (2), $f$ and $g$ are abelian.

###### Proof.

The result follows from Theorem 4.5 by taking $\mu=\delta_{z_{0}}$. ∎

### Acknowledgement

Our sincere regards and gratitude go to Professor Henrik Stetkær for many
valuable comments on our papers. We would also like to express our thanks to
the referees for useful comments.

## References

  * [1] J. Aczél, Lectures on Functional Equations and their Applications, Academic Press (New York, 1966).
  * [2] O. Ajebbar and E. Elqorachi, Variants of Wilson’s functional equation on semigroups, Commun. Korean Math., 35 (2020), 711–722.
  * [3] Y. Aserrar, A. Chahbi and E. Elqorachi, A variant of Wilson’s functional equation on semigroups, Commun. Korean Math., 38 (2023), 1063–1074.
  * [4] Y. Aserrar and E. Elqorachi, Cosine and sine addition and subtraction law with an automorphism Annal. Math Silesianae (2023). https://doi.org/10.2478/amsil-2023-0021
  * [5] B. Bouikhalene and E. Elqorachi, An extension of Van Vleck’s functional equation for the sine, Acta Math. Hungar., 150 (2016), no. 1, 258–267.
  * [6] T.M.K. Davison, D’Alembert’s functional equation on topological monoids, Publ. Math. Debrecen., 75 (2009)(1/2), 41–66.
  * [7] E. Elqorachi, Integral Van Vleck’s and Kannappan’s functional equations on semigroups, Aequationes Math., 91 (2017), 83–98.
  * [8] E. Elqorachi and A. Redouani, Solutions and stability of a variant of Wilson’s functional equation, Proyecciones(Antofagasta, On line)., vol. 37 (2018), no. 2, pp. 317-344.
  * [9] B. Fadli, D. Zeglami and S. Kabbaj, A variant of Wilson’s functional equation, Publ Math. Debrecen., 87 (2015), no. 3-4, 415–427.
  * [10] Pl. Kannappan, A functional equation for the cosine, Can. Math. Bull., 2 (1968), 495-498. https://doi.org/10.4153/CMB-1968-059-8
  * [11] A.M. Perkins and P.K. Sahoo, On two functional equations with involution on groups related to sine and cosine functions, Aequationes Math., 89 (2015)(5), 1251–1263.
  * [12] H. Stetkær, Functional equations on abelian groups with involution, Aequationes. Math., 54 (1997), 144–172.
  * [13] H. Stetkær, On multiplicative maps, Semigroup Forum., 63 (2001), no. 3, 466–468.
  * [14] H. Stetkær, Functional equations on groups, World scientific Publishing CO, (Singapore, 2013).
  * [15] H. Stetkær, A variant of d’Alembert’s functional equation, Aequationes Math., no 3, 89 (2015), 657–662.
  * [16] H. Stetkær, Van Vleck’s functional equation for the sine, Aequationes Math., 90 (2016), 25–34.
  * [17] H. Stetkær, Kannappan’s functional equation on semigroups with involution, Semigroup Forum., 94 (2017), 17–30.
  * [18] H. Stetkær, The small dimension lemma and d’Alembert’s equation on semigroups, Aequationes Math., 95 (2021), 281–299.
  * [19] E.B.Van. Vleck, A functional equation for the sine, Ann. Math. Second Ser., 11 (1910)(4), 161–165.
  * [20] W. H. Wilson, On certain related functional equations, Bull. Amer. Math. Soc., 26 (1919-20), no 7., 33–312.


envname-P envname#1 tcb@breakable

# biquality-learn: a Python library for Biquality Learning

Pierre Nodet<EMAIL_ADDRESS>Orange Innovation, Université Paris-
SaclayChâtillonFrance92320 ,  Vincent Lemaire<EMAIL_ADDRESS>Orange InnovationLannionFrance22300 ,  Alexis Bondu<EMAIL_ADDRESS>Orange InnovationChâtillonFrance92320 and  Antoine Cornuéjols
<EMAIL_ADDRESS>AgroParisTech, INRAe, Université Paris-
SaclayPalaiseauFrance91120

###### Abstract.

The democratization of Data Mining has been widely successful thanks in part
to powerful and easy-to-use Machine Learning libraries. These libraries have
been particularly tailored to tackle Supervised Learning. However, strong
supervision signals are scarce in practice, and practitioners must resort to
weak supervision. In addition to weaknesses of supervision, dataset shifts are
another kind of phenomenon that occurs when deploying machine learning models
in the real world. That is why Biquality Learning has been proposed as a
machine learning framework to design algorithms capable of handling multiple
weaknesses of supervision and dataset shifts without assumptions on their
nature and level by relying on the availability of a small trusted dataset
composed of cleanly labeled and representative samples. Thus we propose
biquality-learn: a Python library for Biquality Learning with an intuitive and
consistent API to learn machine learning models from biquality data, with
well-proven algorithms, accessible and easy to use for everyone, and enabling
researchers to experiment in a reproducible way on biquality data.

Python, Biquality Learning, Weakly Supervised Learning, Dataset Shift

##  1\. Introduction

The democratization of Data Mining has been widely successful thanks in part
to powerful and easy to use Machine Learning libraries such as scikit-learn
(Pedregosa et al., 2011), weka (Witten and Frank, 2002), or caret (Kuhn,
2008). These libraries have been at the core of enforcing good practices in
Machine Learning and providing efficient solutions to complex problems. These
libraries have been particularly tailored to tackle Supervised Learning and
occasionally Semi-Supervised Learning and Unsupervised Learning. However,
strong supervision signals are scarce in practice, and practitioners must
resort to weak supervision. Learning with weak supervisions, or Weakly-
Supervised Learning (Zhou, 2017), is a diverse field, as diverse as the
identified weaknesses of supervision. Usually, weaknesses of supervision are
divided into three groups, namely inaccurate supervision when samples are
mislabeled, inexact supervision when labels are not adapted to the
classification task, or incomplete supervision when labels are missing which
reflects the inadequacy of the available labels in the real world (Kuhn,
2008). For each weakness of supervision, algorithms have to be specifically
hand designed to alleviate them. In addition to weaknesses of supervision,
dataset shifts are another kind of phenomenon that occurs when deploying
machine learning models in the real world (Quinonero-Candela et al., 2008).
Dataset shifts happen when the data distribution observed at training time
differs from what is expected from the data distribution at testing time
(Moreno-Torres et al., 2012). Shifts in the joint distribution of features and
targets can be further divided into four subgroups of shifts, covariate shift
for shifts in the feature distribution, prior shift for shifts in the target
distribution, concept drift for shifts in the decision boundary, and class-
conditional shift for shifts in the feature distribution for a given target.
Again, designing algorithms to handle dataset shifts usually requires
assumptions on the nature of the shift (David et al., 2010). Because of the
diverse nature of possible weaknesses of supervision and dataset shifts, and
robust algorithms’ associated assumptions, it is impossible for practitioners
to choose the suited approach to their problem.

Biquality Learning is a machine learning framework that has been proposed to
design algorithms capable of handling multiple weaknesses of supervision, and
dataset shifts without assumptions on their nature (Nodet et al., 2021a). It
relies on the availability of a small trusted dataset composed of cleanly
labeled and representative samples for the targeted classification task, in
addition to the usual untrusted dataset composed of potentially corrupted and
biased samples. Even though the trusted dataset is not big or rich enough to
properly learn the targeted classification task, it is sufficient to learn a
mapping function from the untrusted distribution to the trusted distribution
to train machine learning models on corrected untrusted samples.

Leveraging trusted data has proven to be particularly efficient to combat
distribution shifts (Hendrycks et al., 2018; Fang et al., 2020) especially on
the most engaging corruptions such as instance-dependant label noise (Nodet et
al., 2021b). In many real-world scenarios, these trusted data are available or
can easily be made available to use Biquality Learning algorithms to train
robust machine learning models. One occurrence is when annotating an entire
dataset is expansive to the point of being prohibitive, but labeling a small
part of the dataset is doable. In Fraud Detection and Cyber Security, labeling
samples require complex forensics from domain experts, limiting the number of
clean samples. However, the rest of the dataset can be labeled by hand-
engineered rules (Ratner et al., 2020) with labels that cannot properly be
trusted. Another scenario happens where data shifts happen during the labeling
process over time. It arises in MLOps (Kreuzberger et al., 2022), when a model
is first learned on clean data and then deployed in production, or when past
predictions are used to learn an updated model (Veeramachaneni et al., 2016).
Finally, when multiple annotators are responsible for dataset labeling, which
happens in NLP, the annotators’ efficiency in following these guidelines may
vary. However, suppose one annotator can be trusted. In that case, all the
other annotators can be considered untrusted, and associating each untrusted
annotator against the trusted annotator can be viewed as a Biquality Learning
task (Yuen et al., 2011).

Multiple libraries have been developed recently for the purpose of handling
covariate shift, especially for Domain Adaptation (de Mathelin et al., 2021)
or for dealing with weak supervisions (Campagner et al., 2022). However,
Biquality Learning lacks an accessible library with an intuitive and
consistent API to learn machine learning models from Biquality Data, with
well-proven algorithms. Thus we propose biquality-learn: a Python library for
Biquality Learning.

##  2\. biquality-learn

We designed the biquality-learn library following the general design
principles of scikit-learn, meaning that it provides a consistent interface
for training and using biquality learning algorithms with an easy way to
compose building blocks provided by the library with other blocks from
libraries sharing these design principles (Buitinck et al., 2013). It includes
various reweighting algorithms, plugin correctors, and functions for
simulating label noise and generating sample data to benchmark biquality
learning algorithms.

biquality-learn and its dependencies can be easily installed through pip:

    pip install biquality-learn

Overall, the goal of biquality-learn is to make well-known and proven
biquality learning algorithms accessible and easy to use for everyone and to
enable researchers to experiment in a reproducible way on biquality data.

  * •

Source Code: https://github.com/biquality-learn/biquality-learn

  * •

Documentation: https://biquality-learn.readthedocs.io/

  * •

License: BSD 3-Clause

##  3\. Design of the API

Scikit-learn (Pedregosa et al., 2011) is a machine learning library for Python
with a design philosophy emphasizing consistency, simplicity, and performance.
The library provides a consistent interface for various algorithms, making it
easy for users to switch between models. It also aims to make machine learning
easy to get started with through user-friendly API and precise documentation.
Additionally, it is built on top of efficient numerical libraries (numpy
(Harris et al., 2020), and SciPy (Virtanen et al., 2020)) to ensure that
models can be trained and used on large datasets in a reasonable amount of
time.

In biquality-learn, we followed the same principle, implementing a similar API
with fit, transform, and predict methods. In addition to passing the input
features $X$ and the labels $Y$ as in scikit-learn, in biquality-learn, we
need to provide information regarding whether each sample comes from the
trusted or trusted or untrusted dataset. We require an additional
sample_quality untrusted dataset: the additional sample_quality parameter
serves to specify property to specify from which dataset the sample
originates. Especially from which dataset the sample originates where a value
of 0 indicates an untrusted a value of 0 indicates an untrusted sample, and 1
indicates a trusted sample, and 1 a trusted one.

##  4\. Algorithms Implemented

In biquality-learn, we purposely implemented only a specific class of
algorithms centered on approaches for tabular data and classifiers, thus
restricting approaches that are genuinely classifier agnostic or implementable
within scikit-learn’s API. We did so not to break the design principles shared
with scikit-learn and not impose a particular deep learning library such as
PyTorch (Paszke et al., 2019), or TensorFlow (Abadi et al., 2015) on the user.

We summarized all implemented algorithms and what kind of corruption they can
handle in the following Table.

Algorithms | Dataset | Weaknesses
---|---|---
Shifts | of Supervision
EasyAdapt (Daumé III, 2009) | $\checkmark$ | $\times$
TrAdaBoost (Dai et al., 2007) | $\checkmark$ | $\times$
Unhinged (Linear/Kernel) (van Rooyen et al., 2015) | $\times$ | $\checkmark$
Backward (Natarajan et al., 2017; Patrini et al., 2017) (with GLC (Hendrycks et al., 2018)) | $\times$ | $\checkmark$
IRLNL (Liu and Tao, 2015; Wang et al., 2017) (with GLC (Hendrycks et al., 2018)) | $\times$ | $\checkmark$
Plugin (Zhang et al., 2021) (with GLC (Hendrycks et al., 2018)) | $\times$ | $\checkmark$
$K$-KMM (Fang et al., 2020) | $\checkmark$ | $\checkmark$
IRBL (Nodet et al., 2021b) | $\times$ | $\checkmark$
$K$-PDR | $\checkmark$ | $\checkmark$

Table 1. Algorithms Implemented in biquality-learn

##  5\. Training Biquality Classifiers

Training a biquality learning algorithm using biquality-learn is the same
procedure as training a supervised algorithm with scikit-learn thanks to the
design presented in Section 3. The features $X$ and the targets $Y$ of samples
belonging to the trusted dataset $D_{T}$ and untrusted dataset $D_{U}$ must be
provided as one global dataset $D$. Additionally, the indicator representing
if a sample is trusted or not has to be provided:
$\textit{sample\\_quality}=\mathbbm{1}_{X\in D_{T}}$.

Here is an example of how to train a biquality classifier using the $K$-KMM
($K$-Kernel Mean Matching) (Fang et al., 2020) algorithm from biquality-learn:

[] from sklearn.linear_models import LogisticRegression from bqlearn.kdr
import KKMM

kkmm = KKMM(kernel=”rbf”, LogisticRegression()) kkmm.fit(X, y,
sample_quality=sample_quality) kkmm.predict(X_new)

##  6\. scikit-learn’s metadata routing

scikit-learn’s metadata routing is a Scikit Learn Enhancement Proposal
(SLEP006) describing a system that can be used to seamlessly incorporate
various metadata in addition to the required features and targets in scikit-
learn estimators, scorers and transformers. biquality-learn uses this design
to integrate the sample_quality property into the training and prediction
process of biquality learning algorithms. It allows one to use biquality-
learn’s algorithms in a similar way to scikit-learn’s algorithms by passing
the sample_quality property as an additional argument to the fit, predict, and
other methods.

Currently, the main components provided by scikit-learn support this design
and is already usable for cross-validators. However, it will be extended to
all components in the future, and biquality-learn will significantly benefit
from many “free” features. When https://github.com/scikit-learn/scikit-
learn/pull/24250 will be merged, it will be possible to make a bagging
ensemble of biquality classifiers thanks to the BaggingClassifier implemented
in scikit-learn without overriding its behavior on biquality data.

[]

from sklearn.ensemble import BaggingClassifier

bag = BaggingClassifier(kkmm).fit(X, y, sample_quality=sample_quality)

##  7\. Cross-Validating Biquality Classifiers

Any cross-validators working for usual Supervised Learning can work in the
case of Biquality Learning. However, when splitting the data into a train and
test set, untrusted samples need to be removed from the test set to avoid
computing supervised metrics on corrupted labels. That is why
make_biquality_cv is provided by biquality-learn to post-process any scikit-
learn compatible cross-validators.

Here is an example of how to use scikit-learn’s RandomizedSearchCV function to
perform hyperparameter validation for a biquality learning algorithm in
biquality-learn:

[]

from sklearn.model_selection import RandomizedSearchCV from
sklearn.utils.fixes import loguniform from bqlearn.model_selection import
make_biquality_cv

param_dist = ”final_estimator__C”: loguniform(1e3, 1e5) n_iter=20

random_search = RandomizedSearchCV( kkmm, param_distributions=param_dist,
n_iter=n_iter, cv=make_biquality_cv(X, sample_quality, cv=3) )
random_search.fit(X, y, sample_quality=sample_quality)

##  8\. Simulating Corruptions with the Corruption API

The corruption module in biquality-learn provides several functions to
artificially create biquality datasets by introducing synthetic corruption.
These functions can be used to simulate various types of label noise or
imbalances in the dataset. We hope to ease the benchmark of biquality learning
algorithms thanks to the corruption API, with a special touch on the
reproducibility and standardization of these benchmarks for researchers.

Here is a brief overview of the functions available in the corruption module:

  * •

make_weak_labels: Adds weak labels to a dataset by learning a classifier on a
subset of the dataset and using its predictions as a new label.

  * •

make_label_noise: Adds noisy labels to a dataset by randomly corrupting a
specified fraction of the samples thanks to a given noise matrix.

  * •

make_instance_dependent_label_noise: Adds instance-dependent noisy labels by
corrupting samples with a probability depending on the sample and a given
noise matrix.

  * •

uncertainty_noise_probability: Computes the probability of corrupting a sample
based on the prediction uncertainty of a given classifier (Nodet et al.,
2021b).

  * •

make_feature_dependent_label_noise: Adds instance-dependent noisy labels by
corrupting a specified fraction of the labels with a probability depending on
a random linear map between the features space and the labels space (Xia et
al., 2020).

  * •

make_imbalance: Creates an imbalanced dataset by oversampling or undersampling
the minority classes (Buda et al., 2018).

  * •

make_sampling_biais: Creates a sampling biais by sampling not at random a
subset of the dataset from the original dataset. The sampling scheme follows a
Gaussian distribution with a shifted mean and scaled variance computed from
the first principal component of a PCA learned from the dataset (Gretton et
al., 2009).

##  9\. Conclusion

We presented biquality-learn, a Python library for Biquality Learning. We
exposed the design behind its API to make it easy to use and consistent with
scikit-learn. We notably showed the future-proof of our design by showing how
well it integrates with the future design of scikit-learn. In the future,
biquality-learn could be supported with deep learning capabilities with a twin
library with a principled design, committing to a deep learning library.
Finally, the capacity of biquality-learn could be extended to particularly
needed capabilities in real-world scenarios, such as evaluating machine
learning models on untrusted data.

## References

  * (1)
  * Abadi et al. (2015) Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Geoffrey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dandelion Mané, Rajat Monga, Sherry Moore, Derek Murray, Chris Olah, Mike Schuster, Jonathon Shlens, Benoit Steiner, Ilya Sutskever, Kunal Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viégas, Oriol Vinyals, Pete Warden, Martin Wattenberg, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng. 2015\.  TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems.  https://www.tensorflow.org/ Software available from tensorflow.org.
  * Buda et al. (2018) Mateusz Buda, Atsuto Maki, and Maciej A Mazurowski. 2018\.  A systematic study of the class imbalance problem in convolutional neural networks.  _Neural Networks_ 106 (2018), 249–259.
  * Buitinck et al. (2013) Lars Buitinck, Gilles Louppe, Mathieu Blondel, Fabian Pedregosa, Andreas Mueller, Olivier Grisel, Vlad Niculae, Peter Prettenhofer, Alexandre Gramfort, Jaques Grobler, et al. 2013\.  API design for machine learning software: experiences from the scikit-learn project.  _arXiv preprint arXiv:1309.0238_ (2013).
  * Campagner et al. (2022) Andrea Campagner, Julian Lienen, Eyke Hüllermeier, and Davide Ciucci. 2022.  Scikit-Weak: A Python Library for Weakly Supervised Machine Learning. In _Rough Sets_ , JingTao Yao, Hamido Fujita, Xiaodong Yue, Duoqian Miao, Jerzy Grzymala-Busse, and Fanzhang Li (Eds.). Springer Nature Switzerland, Cham, 57–70.
  * Dai et al. (2007) Wenyuan Dai, Qiang Yang, Gui-Rong Xue, and Yong Yu. 2007\.  Boosting for transfer learning. In _International Conference on Machine Learning_. 193–200.
  * Daumé III (2009) Hal Daumé III. 2009\.  Frustratingly easy domain adaptation.  _arXiv preprint arXiv:0907.1815_ (2009).
  * David et al. (2010) Shai Ben David, Tyler Lu, Teresa Luu, and Dávid Pál. 2010\.  Impossibility theorems for domain adaptation. In _Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics_. JMLR Workshop and Conference Proceedings, 129–136.
  * de Mathelin et al. (2021) Antoine de Mathelin, François Deheeger, Guillaume Richard, Mathilde Mougeot, and Nicolas Vayatis. 2021.  ADAPT: Awesome Domain Adaptation Python Toolbox.  _arXiv preprint arXiv:2107.03049_ (2021).
  * Fang et al. (2020) Tongtong Fang, Nan Lu, Gang Niu, and Masashi Sugiyama. 2020\.  Rethinking Importance Weighting for Deep Learning under Distribution Shift, In Neural Information Processing Systems.  _Advances in Neural Information Processing Systems_ 33 (2020), 11996–12007.
  * Gretton et al. (2009) Arthur Gretton, Alex Smola, Jiayuan Huang, Marcel Schmittfull, Karsten Borgwardt, and Bernhard Schölkopf. 2009.  Covariate shift by kernel mean matching.  _Dataset shift in machine learning_ 3, 4 (2009), 5\.
  * Harris et al. (2020) Charles R. Harris, K. Jarrod Millman, Stéfan J van der Walt, Ralf Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J. Smith, Robert Kern, Matti Picus, Stephan Hoyer, Marten H. van Kerkwijk, Matthew Brett, Allan Haldane, Jaime Fernández del Río, Mark Wiebe, Pearu Peterson, Pierre Gérard-Marchant, Kevin Sheppard, Tyler Reddy, Warren Weckesser, Hameer Abbasi, Christoph Gohlke, and Travis E. Oliphant. 2020.  Array programming with NumPy.  _Nature_ 585 (2020), 357–362.  https://doi.org/10.1038/s41586-020-2649-2
  * Hendrycks et al. (2018) Dan Hendrycks, Mantas Mazeika, Duncan Wilson, and Kevin Gimpel. 2018.  Using Trusted Data to Train Deep Networks on Labels Corrupted by Severe Noise. In _Advances in Neural Information Processing Systems_ , Vol. 31. 10456–10465.
  * Kreuzberger et al. (2022) Dominik Kreuzberger, Niklas Kühl, and Sebastian Hirschl. 2022.  Machine Learning Operations (MLOps): Overview, Definition, and Architecture.  _arXiv preprint arXiv:2205.02302_ (2022).
  * Kuhn (2008) Max Kuhn. 2008.  Building Predictive Models in R Using the caret Package.  _Journal of Statistical Software, Articles_ 28, 5 (2008), 1–26.  https://doi.org/10.18637/jss.v028.i05
  * Liu and Tao (2015) Tongliang Liu and Dacheng Tao. 2015.  Classification with noisy labels by importance reweighting.  _IEEE Transactions on pattern analysis and machine intelligence_ 38, 3 (2015), 447–461.
  * Moreno-Torres et al. (2012) Jose G Moreno-Torres, Troy Raeder, Rocío Alaiz-Rodríguez, Nitesh V Chawla, and Francisco Herrera. 2012.  A unifying view on dataset shift in classification.  _Pattern recognition_ 45, 1 (2012), 521–530.
  * Natarajan et al. (2017) Nagarajan Natarajan, Inderjit S Dhillon, Pradeep Ravikumar, and Ambuj Tewari. 2017.  Cost-Sensitive Learning with Noisy Labels.  _J. Mach. Learn. Res._ 18, 1 (2017), 5666–5698.
  * Nodet et al. (2021a) Pierre Nodet, Vincent Lemaire, Alexis Bondu, Antoine Cornuéjols, and Adam Ouorou. 2021a.  From Weakly Supervised Learning to Biquality Learning: an Introduction. In _International Joint Conference on Neural Networks (IJCNN)_. IEEE.
  * Nodet et al. (2021b) Pierre Nodet, Vincent Lemaire, Alexis Bondu, Antoine Cornuejols, and Adam Ouorou. 2021b.  Importance Reweighting for Biquality Learning. In _International Joint Conference on Neural Networks (IJCNN)_. IEEE.
  * Paszke et al. (2019) Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. 2019.  PyTorch: An Imperative Style, High-Performance Deep Learning Library.  In _Advances in Neural Information Processing Systems 32_. 8024–8035.
  * Patrini et al. (2017) Giorgio Patrini, Alessandro Rozza, Aditya Menon, Richard Nock, and Lizhen Qu. 2017\.  Making Deep Neural Networks Robust to Label Noise: a Loss Correction Approach. In _IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_.
  * Pedregosa et al. (2011) Fabian Pedregosa, Gaël Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, et al. 2011\.  Scikit-learn: Machine learning in Python.  _Journal of machine learning research_ 12 (2011), 2825–2830.
  * Quinonero-Candela et al. (2008) Joaquin Quinonero-Candela, Masashi Sugiyama, Anton Schwaighofer, and Neil D Lawrence. 2008.  _Dataset shift in machine learning_.  Mit Press.
  * Ratner et al. (2020) Alexander Ratner, Stephen H Bach, Henry Ehrenberg, Jason Fries, Sen Wu, and Christopher Ré. 2020.  Snorkel: Rapid training data creation with weak supervision.  _The VLDB Journal_ 29, 2 (2020), 709–730.
  * van Rooyen et al. (2015) Brendan van Rooyen, Aditya Menon, and Robert C Williamson. 2015\.  Learning with Symmetric Label Noise: The Importance of Being Unhinged.  In _Neural Information Processing Systems_ , C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama, and R. Garnett (Eds.). 10–18.
  * Veeramachaneni et al. (2016) Kalyan Veeramachaneni, Ignacio Arnaldo, Vamsi Korrapati, Constantinos Bassias, and Ke Li. 2016\.  AI2: Training a Big Data Machine to Defend. In _2016 IEEE 2nd International Conference on Big Data Security on Cloud (BigDataSecurity)_. 49–54.
  * Virtanen et al. (2020) Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, Stéfan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric Larson, C J Carey, İlhan Polat, Yu Feng, Eric W. Moore, Jake VanderPlas, Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Henriksen, E. A. Quintero, Charles R. Harris, Anne M. Archibald, Antônio H. Ribeiro, Fabian Pedregosa, Paul van Mulbregt, and SciPy 1.0 Contributors. 2020.  SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python.  _Nature Methods_ 17 (2020), 261–272.  https://doi.org/10.1038/s41592-019-0686-2
  * Wang et al. (2017) Ruxin Wang, Tongliang Liu, and Dacheng Tao. 2017.  Multiclass learning with partially corrupted labels.  _IEEE transactions on neural networks and learning systems_ 29, 6 (2017), 2568–2580.
  * Witten and Frank (2002) Ian H Witten and Eibe Frank. 2002.  Data mining: practical machine learning tools and techniques with Java implementations.  _Acm Sigmod Record_ 31, 1 (2002), 76–77.
  * Xia et al. (2020) Xiaobo Xia, Tongliang Liu, Bo Han, Nannan Wang, Mingming Gong, Haifeng Liu, Gang Niu, Dacheng Tao, and Masashi Sugiyama. 2020.  Part-dependent label noise: Towards instance-dependent label noise.  _Advances in Neural Information Processing Systems_ 33 (2020), 7597–7610.
  * Yuen et al. (2011) Man-Ching Yuen, Irwin King, and Kwong-Sak Leung. 2011\.  A survey of crowdsourcing systems. In _2011 IEEE third international conference on privacy, security, risk and trust and 2011 IEEE third international conference on social computing_. IEEE, 766–773.
  * Zhang et al. (2021) Mingyuan Zhang, Jane Lee, and Shivani Agarwal. 2021.  Learning from noisy labels with no change to the training process. In _International Conference on Machine Learning_. PMLR, 12468–12478.
  * Zhou (2017) Zhi-Hua Zhou. 2017\.  A brief introduction to weakly supervised learning.  _National Science Review_ 5, 1 (08 2017), 44–53.


HD189245 | 98470 | F7V | 0.490 | 1.2  |  $150_{-50}^{+150}$ | 53.3 |  $1.88\pm 0.01$\alphalph\alphalph __Desidera et al. (2015) | -/-
HD191089 | 99273 | F5V | 0.440 | 1.3  |  $24_{-5}^{+5}$ | 26.4 |  $0.488\pm 0.005$ | y/y\alphalph\alphalph __Churcher et al. (2011)
HD197481 | 102409 | M1V | 1.423 | 0.6  |  $21_{-5}^{+7}$ | 7.6 |  $4.84\pm 0.02$ | y\alphalph\alphalph __Rameau et al. (2013)/y\alphalph\alphalph __Kalas et al. (2004)
HD197890† | 102626 | K3V | 1.053 | 1.0  |  $45_{-35}^{+55}$ | 107.6 | 0.3804  | y/-
HD202917 | 105388 | G7V | 0.650 | 0.9  |  $45_{-10}^{+5}$ | 10.6 |  $3.36\pm 0.01$ | y/y
HD206860 | 107350 | G0V | 0.580 | 1.1  |  $300_{-100}^{+700}$ | 7.5 | 4.86  | y\alphalph\alphalph __Beichman et al. (2006b)\alphalph\alphalph __Moro-Martín et al. (2015)/-
HD206893 | 107412 | F5V | 0.440 | 1.24\alphalph\alphalph __Milli et al. (2017) |  $250_{-200}^{+450}$\alphalph\alphalph __Delorme et al. (2017) | 23.2 | - | y/y
HD207575 | 107947 | F6V | 0.490 | 1.24  |  $45_{-15}^{+5}$ | 22.7 | - | y/-
HD213845 | 111449 | F7V | 0.434 | 1.4  |  $250_{-50}^{+750}$ | 24.2 | - | y\alphalph\alphalph __Beichman et al. (2006a)/-
HD215641 | 112491 | G8V | 0.760 | 1.00  |  $440_{-40}^{+40}$ | 3.9 | - | -/-
HD216956 | 113368 | A3V | 0.090 |  $1.92^{+0.02}_{-0.02}$\alphalph\alphalph __Mamajek (2012) |  $440_{-40}^{+40}$ | 90.0 | - | y\alphalph\alphalph __Backman & Gillett (1987)/y\alphalph\alphalph __Holland et al. (1998)
HD217343 | 113579 | G5V | 0.640 | 1.05  |  $70$ | 9.1 | - | -/-
HD217987 | 114046 | M2V | 1.440 | 0.47  |  $100-10000$\alphalph\alphalph __Delorme et al. (2012) | 2.3 | - | -/-
HD218396 | 114189 | A5V | 0.257 | 1.46  |  $42$ | 26.6 | - | y\alphalph\alphalph __Zuckerman & Song (2004)/y\alphalph\alphalph __Su et al. (2009)
HD218860 | 114530 | G8V | 0.738 | 1.0  |  $149_{-49}^{+31}$ | 5.5 |  $5.17\pm 0.02$ | y/-
HD221575 | 116258 | K2V | 0.930 | 0.90  |  $250_{-100}^{+150}$ | 4.1 | - | -/-
HD223340 | - | K1V | 0.820 | 0.91  |  $149_{-49}^{+31}$ | 4.6 | - | -/-
HD224228 | 118008 | K2V | 0.985 | 0.86  |  $149_{-49}^{+31}$ | 3.6 | - | y/-
- | 6276 | G9V | 0.800 | 0.9  |  $149_{-49}^{+31}$ | 4.4 | 6.40  | y/-
- | 116384 | K7V | 1.347 | 0.729\alphalph\alphalph __Newton et al. (2017) |  $289_{-260}^{+971}$ | 4.0 | - | -/-
- | 17157 | K7V | 0.913 | 0.74  |  $100_{-30}^{+50}$ | 3.7 | - | -/-
- | 23309 | M0V | 1.383 | 0.55  |  $10_{-3}^{+3}$ | 5.5 |  $8.60\pm 0.07$ | n/-
- | 31878 | K7V | 1.297 | 0.643\alphalph\alphalph __Lindgren & Heiter (2017) |  $149_{-49}^{+31}$ | 4.0 |  $9.06\pm 0.08$ | -/-
- | 36985 | M2V | 1.476 | 0.621\alphalph\alphalph __Passegger et al. (2018) |  $260_{-260}^{+420}$\alphalph\alphalph __Poveda et al. (2009) | 3.6 | 12.16  | -/-
- | 44722 | K7V | 1.450 | 0.638  |  $600_{-500}^{+200}$ | 3.5 | - | -/-
- | 46634 | G5V | 1.180 | 0.93  |  $300_{-150}^{+300}$ | 4.6 |  $3.05\pm 0.03$ | -/-
- | 51317 | M2V | 1.501 | 0.44  |  $130_{-20}^{+40}$ | 1.7 | - | -/-
BD$+$20 2465† | - | M5V | 1.300 |  $0.42^{+0.01}_{-0.01}$\alphalph\alphalph __Morin et al. (2008) |  $50_{-30}^{+150}$ | 2.8 | 2.60  | -/-
CD$-$46 1064 | - | K3V | 1.048 | 0.8  |  $45_{-10}^{+5}$ | 8.1 |  $3.74\pm 0.04$ | -/-
Table 6: continued. Table 7: Same as LABEL:tab_result_sco for the $89$ stars in the initial sample of our HARPS YNS RV survey.  Stellar characteristics | Survey results. |  |
---|---|---|---
Name | HIP | ST | TBL | $N_{\rm m}$ | RV | BVS | RV- |  $<$FWHM$>$ |  $<$log$R^{\prime}_{\rm HK}$$>$ | V | T | B
HD/BD/CD |  |  |  |  | $\overbrace{\hskip 76.82234pt}$ | $\overbrace{\hskip 76.82234pt}$ | BVS |  | ($\sigma_{logR^{\prime}_{\rm HK}}$) |  |  |
|  |  |  |  | A | rms |  $<$U$>$ | A | rms |  $<$U$>$ | corr. |  |  |  |  |
|  |  |  |  | $\underbrace{\hskip 76.82234pt}$ | $\underbrace{\hskip 76.82234pt}$ |  |  |  |  |  |
|  | (day) |  |  | ($\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$) | ($\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$) |  | ($\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$) |  |  |  |
HD105 | 490 | G0V | 4606 | 36 | 236.8 | 61.1 | 4.2 | 310.7 | 9.8 | 72.9 | -0.67 | 22.6 | -4.32(0.02) | A | - | -
HD984 | 1134 | F7V | 867 | 21 | 301.9 | 84.6 | 16.4 | 571.8 | 38.2 | 137.4 | -0.46 | 59.7 | -4.37(0.02) | A | - | -
HD987 | 1113 | G8V | 2621 | 19 | 502.6 | 116.8 | 2.5 | 393.6 | 6.4 | 103.1 | -1.08 | 13.0 | -4.09(0.04) | A | - | -
HD1466 | 1481 | F8V | 4400 | 19 | 135.8 | 39.6 | 7.4 | 189.8 | 17.6 | 55.9 | -0.66 | 32.3 | -4.34(0.02) | A | - | -
HD3221 | 2729 | K4V | 4014 | 5 | 4793.4 | 1794.3 | 68.6 | 3115.1 | 172.5 | 1447.8 | -0.54 | 203.6 | -4.47(0.01) | - | - | -
HD6569 | 5191 | K1V | 8 | 4 | 24.0 | 11.4 | 1.5 | 17.8 | 3.8 | 7.5 | -1.49 | 9.6 | -4.229(0.003) | A | - | -
HD7661 | 5938 | K0V | 1525 | 29 | 96.0 | 28.1 | 1.2 | 50.5 | 3.3 | 12.5 | -1.42 | 9.2 | -4.33(0.05) | A | - | -
HD10008 | 7576 | K0V | 4021 | 17 | 20.5 | 6.2 | 1.0 | 17.5 | 2.7 | 5.2 | 0.44 | 7.5 | -4.36(0.03) | A | - | -
HD16765 | 12530 | F71V | 926 | 27 | 173.0 | 47.2 | 11.0 | 402.9 | 27.9 | 93.1 | -0.42 | 48.5 | -4.48(0.02) | A | - | -
HD17925 | 13402 | K1V | 1470 | 40 | 113.6 | 30.5 | 1.1 | 58.7 | 3.1 | 17.4 | -0.87 | 10.0 | -4.31(0.02) | A | - | -
HD18599 | 13754 | K2V | 1055 | 16 | 115.9 | 38.9 | 1.5 | 54.3 | 3.9 | 17.6 | -1.46 | 9.8 | -4.31(0.02) | A | - | -
HD19668 | 14684 | K0V | 4402 | 20 | 143.1 | 33.2 | 1.8 | 91.9 | 4.7 | 26.0 | -0.95 | 11.8 | -4.28(0.03) | A | - | -
HD24916 | 18512 | K4V | 792 | 22 | 24.3 | 7.4 | 0.8 | 36.7 | 2.2 | 11.6 | 0.46 | 7.8 | -4.54(0.04) | A | - | -
HD25457 | 18859 | F6V | 4089 | 78 | 187.2 | 49.5 | 3.3 | 345.4 | 7.7 | 60.2 | -0.66 | 28.2 | -4.36(0.04) | A | - | -
HD26923 | 19859 | G0IV | 4010 | 47 | 50.2 | 12.7 | 1.1 | 24.7 | 3.1 | 5.6 | 1.12 | 8.0 | -4.49(0.05) | A | - | -
HD29391 | 21547 | F0IV | 3996 | 81 | 1476.1 | 326.2 | 28.4 | 2564.3 | 72.1 | 487.8 | - | 101.4 | -4.36(0.01) | P | - | -
HD30447 | 22226 | F3V | 792 | 19 | 194.1 | 47.2 | 47.7 | 7048.7 | 106.6 | 1728.3 | - | 120.7 | -4.45(0.02) | P | - | -
HD35650 | 25283 | K6V | 1195 | 13 | 48.3 | 15.3 | 1.2 | 43.6 | 3.0 | 12.3 | -0.67 | 9.4 | -4.36(0.04) | A | - | -
HD37572 | 26373 | K0V | 2826 | 34 | 347.1 | 75.4 | 1.7 | 223.6 | 4.4 | 52.8 | -1.28 | 15.6 | -4.17(0.02) | A | - | -
HD39060 | 27321 | A6V | 3702 | 5108 | 755.5 | 341.4 | 58.9 | - | - | - | - | - | - | P | - | -
HD41593 | 28954 | K0V | 1194 | 15 | 69.5 | 19.8 | 1.1 | 59.4 | 2.9 | 16.7 | -0.66 | 9.1 | -4.37(0.03) | A | - | -
HD43989 | 30030 | G0V | 4433 | 17 | 819.0 | 238.6 | 15.6 | 743.7 | 34.3 | 197.6 | -0.83 | 64.2 | -4.21(0.06) | A | - | -
HD44627 | 30034 | K1V | 4892 | 23 | 670.6 | 177.7 | 3.2 | 428.6 | 8.1 | 110.9 | - | 19.1 | -4.02(0.03) | A | - | -
HD45081 | 29964 | K4V | 4020 | 17 | 852.4 | 252.1 | 5.0 | 911.0 | 13.0 | 246.9 | - | 24.8 | -3.89(0.07) | A | - | -
HD45270 | 30314 | G1V | 2827 | 19 | 137.4 | 51.8 | 3.1 | 193.3 | 8.4 | 59.2 | -0.76 | 27.6 | -4.34(0.03) | A | - | -
HD59967 | 36515 | G3V | 1065 | 25 | 65.3 | 19.7 | 1.4 | 41.6 | 3.6 | 10.5 | -0.99 | 9.0 | -4.33(0.03) | A | - | -
HD61005 | 36948 | G8V | 2369 | 33 | 186.5 | 48.8 | 2.5 | 127.4 | 6.3 | 32.7 | -1.16 | 14.5 | -4.26(0.04) | A | - | -
HD63608 | 37923 | K0V | 1069 | 36 | 80.7 | 22.2 | 1.1 | 49.1 | 3.0 | 11.7 | -0.63 | 8.0 | -4.32(0.05) | A | - | -
HD77825 | 44526 | K2V | 283 | 6 | 47.0 | 18.3 | 1.4 | 46.5 | 3.4 | 14.9 | -1.09 | 9.6 | -4.31(0.03) | A | - | -
HD82558 | 46816 | K1V | 1092 | 24 | 296.0 | 79.7 | 7.1 | 475.6 | 17.5 | 124.1 | -0.43 | 42.0 | -3.99(0.06) | A | - | -
HD89449† | 50564 | F6IV | 1227 | 28 | 129.9 | 41.4 | 4.3 | 351.6 | 9.1 | 113.6 | -0.26 | 25.3 | -4.97(0.11) | A | - | -
HD90905 | 51386 | F5V | 2551 | 24 | 125.6 | 40.8 | 1.9 | 187.7 | 5.0 | 49.0 | -0.74 | 15.0 | -4.35(0.04) | A | - | -
HD92945 | 52462 | K1V | 4533 | 38 | 182.3 | 40.5 | 1.5 | 86.2 | 3.9 | 23.8 | -1.37 | 10.3 | -4.26(0.03) | A | - | -
HD95086 | 53524 | A8III | 1532 | 103 | 1279.0 | 268.8 | 14.8 | 2594.0 | 36.3 | 511.1 | - | 45.4 | -4.47(0.03) | P | - | -
HD95650 | 53985 | M2V | 4046 | 12 | 23.8 | 7.6 | 3.1 | 46.1 | 7.3 | 13.6 | - | 8.0 | -4.33(0.04) | - | - | -
HD99211 | 55705 | A7V | 3068 | 112 | 818.0 | 149.4 | 86.8 | 49704.0 | 214.7 | 5222.3 | - | 183.4 | -4.243(0.008) | P | - | -
HD102458 | 57524 | G4V | 2964 | 26 | 988.5 | 307.7 | 10.0 | 1358.4 | 23.8 | 431.7 | -0.79 | 43.6 | -3.99(0.02) | A | - | -
HD103743 | 58241 | G4V | 1065 | 30 | 102.2 | 32.1 | 2.6 | 141.6 | 6.6 | 35.1 | -0.79 | 14.9 | -4.31(0.03) | A | - | -
HD105690 | 59315 | G5V | 2975 | 133 | 269.4 | 61.3 | 2.3 | 193.7 | 5.4 | 44.0 | -0.91 | 15.1 | -4.28(0.03) | A | - | -
HD106906† | 59960 | F5V | 1230 | 46 | 5290.4 | 1285.7 | 32.9 | 9949.8 | 83.3 | 2272.4 | - | 74.0 | -4.57(0.02) | A | - | SB1
HD108767 | 60965 | K0V | 1201 | 18 | 47.9 | 12.4 | 0.7 | 25.3 | 2.0 | 7.7 | 0.21 | 8.2 | -4.38(0.01) | A | - | -
HD116434† | 65426 | A2V | 439 | 58 | 6273.2 | 1223.2 | 1235.6 | - | - | - | - | - | - | P | - | -
HD118100 | 66252 | K5V | 720 | 10 | 534.7 | 155.6 | 6.4 | 139.5 | 15.6 | 43.0 | -3.48 | 16.5 | -4.11(0.03) | A | - | -
HD131399† | 72940 | A1V | 189 | 87 | 19478.2 | 6381.7 | 16.4 | 1287.6 | 39.3 | 237.0 | - | 43.1 | -3.85(0.01) | - | - | SB1
HD141943 | - | G2V | 2648 | 58 | 861.4 | 222.8 | 10.3 | 1169.7 | 25.3 | 269.6 | -0.66 | 53.0 | -4.03(0.02) | A | - | -
HD146464 | 79958 | K3V | 51 | 5 | 872.1 | 352.1 | 5.0 | 546.2 | 12.3 | 220.6 | -1.55 | 25.8 | -4.16(0.1) | A | - | -
HD146624 | 79881 | A0V | 4762 | 335 | 232.7 | 39.8 | 29.2 | 70472.6 | 72.9 | 6163.4 | - | 65.1 | -3.429(0.004) | P | - | -
HD152555 | 82688 | F8V | 1135 | 22 | 225.0 | 64.8 | 6.1 | 283.2 | 15.9 | 69.9 | -0.74 | 26.3 | -4.29(0.02) | A | - | -
HD159492 | 86305 | A5IV | 4751 | 90 | 612.4 | 124.9 | 20.8 | 2413.2 | 52.1 | 359.7 | - | 79.4 | -4.45(0.01) | P | - | -
HD164249 | 88399 | F6V | 1113 | 25 | 56.9 | 14.6 | 6.9 | 134.2 | 17.0 | 36.7 | -0.24 | 29.8 | -4.77(0.07) | A | - | -
HD169178 | - | K0V | 1123 | 19 | 354.7 | 135.7 | 1.8 | 85.6 | 4.7 | 25.0 | - | 10.7 | -4.23(0.03) | A | - | -
HD171488 | 91043 | G2V | 1111 | 18 | 2190.4 | 717.7 | 13.1 | 2249.5 | 33.2 | 680.4 | -1.0 | 58.2 | -3.99(0.03) | A | - | -
HD172555 | 92024 | A7V | 2975 | 262 | 1883.8 | 329.6 | 70.6 | 94415.3 | 185.8 | 6148.5 | - | 172.4 | -4.167(0.007) | P | - | -
HD174429 | 92680 | G9IV | 4572 | 42 | 3362.3 | 970.2 | 26.0 | 1167.7 | 60.0 | 320.7 | -1.77 | 109.9 | - | A | - | -
HD177171† | 93815 | F6V | 144 | 21 | 21233.2 | 5931.1 | 53.4 | - | - | - | - | - | -4.05(0.04) | - | - | SB1
HD181321† | 95149 | G2V | 3757 | 28 | 2610.2 | 683.2 | 3.9 | 244.0 | 9.6 | 59.1 | - | 19.5 | -4.24(0.02) | A | - | SB1
HD181327 | 95270 | F6V | 3496 | 56 | 63.1 | 16.3 | 3.8 | 156.8 | 9.0 | 30.6 | -0.26 | 28.8 | -4.58(0.03) | A | - | -
HD183414 | 96334 | G3V | 3097 | 68 | 247.9 | 63.6 | 2.4 | 214.0 | 6.2 | 58.3 | -1.01 | 16.0 | -4.24(0.02) | A | - | -
HD186704 | 97255 | G0V | 451 | 4 | 345.8 | 168.8 | 4.7 | 77.1 | 11.9 | 29.3 | - | 22.3 | -4.309(0.008) | - | T | SB1
HD188228 | 98495 | A0V | 4315 | 194 | 1398.7 | 319.2 | 150.2 | 211414.9 | 368.1 | 18294.0 | - | 148.0 | -2.928(0.003) | P | - | -
HD189245 | 98470 | F7V | 4285 | 46 | 646.2 | 119.4 | 21.0 | 5149.4 | 43.1 | 757.5 | - | 104.2 | -4.3(0.02) | P | - | -
HD191089 | 99273 | F5V | 1025 | 26 | 107.1 | 22.8 | 15.7 | 300.0 | 37.3 | 78.4 | - | 57.0 | -4.53(0.01) | P | - | -
HD197481 | 102409 | M1V | 5619 | 55 | 667.6 | 144.0 | 3.2 | 470.1 | 8.0 | 101.2 | -1.37 | 16.2 | -4.29(0.23) | A | - | -
HD197890† | 102626 | K3V | 13 | 3 | 922.1 | 376.6 | 85.2 | 3975.6 | 217.1 | 1713.8 | - | 216.2 | -4.28(0.03) | - | - | -
HD202917 | 105388 | G7V | 4438 | 20 | 436.1 | 124.6 | 5.0 | 438.1 | 12.4 | 134.8 | -0.84 | 23.0 | -3.98(0.04) | A | - | -
HD206860 | 107350 | G0V | 1332 | 22 | 122.1 | 35.8 | 2.8 | 180.9 | 6.7 | 51.4 | - | 16.1 | -4.39(0.03) | A | - | -
HD206893 | 107412 | F5V | 941 | 78 | 297.4 | 87.7 | 10.0 | 357.5 | 27.6 | 85.4 | - | 50.7 | -4.63(0.02) | P | T | C\alphalph\alphalph __Grandjean et al. (2019)
HD207575 | 107947 | F6V | 2391 | 39 | 227.6 | 49.3 | 9.5 | 389.5 | 19.9 | 97.6 | - | 49.1 | -4.33(0.02) | P | - | -
HD213845 | 111449 | F7V | 4436 | 79 | 220.4 | 44.2 | 9.6 | 454.3 | 25.8 | 82.6 | - | 52.3 | -4.7(0.02) | P | - | -
HD215641 | 112491 | G8V | 1639 | 78 | 101.5 | 20.5 | 1.2 | 84.4 | 3.4 | 12.2 | 0.22 | 8.3 | -4.46(0.03) | A | - | -
HD216956 | 113368 | A3V | 4929 | 834 | 382.4 | 44.3 | 27.0 | - | - | - | - | - | -3.965 | P | - | -
HD217343 | 113579 | G5V | 2727 | 26 | 276.6 | 96.9 | 2.7 | 298.7 | 6.6 | 85.6 | -1.08 | 19.3 | -4.22(0.03) | A | - | -
HD217987 | 114046 | M2V | 5109 | 130 | 63.8 | 18.5 | 0.9 | 39.5 | 2.3 | 8.2 | 0.95 | 5.8 | -4.9(0.06) | A | - | -
HD218396 | 114189 | A5V | 2727 | 124 | 3616.3 | 924.5 | 38.5 | 7376.0 | 95.5 | 2191.3 | - | 62.3 | -4.3(0.02) | P | - | -
HD218860 | 114530 | G8V | 1137 | 18 | 137.1 | 36.2 | 2.1 | 82.8 | 5.4 | 18.6 | -1.3 | 11.7 | -4.24(0.03) | A | - | -
HD221575 | 116258 | K2V | 869 | 16 | 45.7 | 12.2 | 1.4 | 33.4 | 3.7 | 12.0 | -0.54 | 8.8 | -4.44(0.02) | A | - | -
HD223340 | - | K1V | 868 | 10 | 54.4 | 17.3 | 1.7 | 42.2 | 4.4 | 10.7 | -1.34 | 9.8 | -4.34(0.03) | A | - | -
HD224228 | 118008 | K2V | 2815 | 31 | 35.4 | 8.8 | 1.0 | 31.5 | 2.8 | 7.5 | 0.32 | 7.8 | -4.36(0.03) | A | - | -
- | 6276 | G9V | 1201 | 20 | 95.6 | 27.3 | 1.6 | 91.0 | 4.2 | 27.6 | -0.65 | 9.4 | -4.32(0.03) | A | - | -
- | 116384 | K7V | 733 | 8 | 104.3 | 41.2 | 1.7 | 60.2 | 4.5 | 19.2 | -1.18 | 8.6 | -4.46(0.18) | A | - | -
- | 17157 | K7V | 1457 | 6 | 67.5 | 22.3 | 1.4 | 37.1 | 3.6 | 14.9 | - | 8.1 | -4.58(0.03) | A | - | -
- | 23309 | M0V | 4030 | 16 | 245.4 | 64.1 | 2.3 | 116.1 | 5.9 | 33.0 | -1.83 | 11.7 | -3.82(0.03) | A | - | -
- | 31878 | K7V | 435 | 11 | 84.3 | 31.6 | 1.6 | 48.3 | 4.1 | 16.1 | -1.86 | 8.5 | -4.28(0.02) | A | - | -
- | 36985 | M2V | 789 | 20 | 670.1 | 247.4 | 2.1 | 58.8 | 5.4 | 14.7 | - | 8.1 | -4.32(0.03) | A | - | SB1
- | 44722 | K7V | 90 | 4 | 8.0 | 3.4 | 1.5 | 10.7 | 3.8 | 4.0 | -0.49 | 7.6 | -4.55(0.04) | A | - | -
- | 46634 | G5V | 281 | 3 | 38.2 | 16.1 | 1.4 | 32.6 | 3.5 | 13.9 | -1.0 | 9.8 | -4.32(0.02) | A | - | -
- | 51317 | M2V | 4541 | 139 | 17.7 | 3.6 | 1.2 | 24.5 | 3.2 | 3.5 | -0.38 | 4.8 | -5.03(0.06) | A | - | -
BD$+$20 2465† | - | M5V | 3983 | 40 | 63.5 | 16.1 | 1.2 | 33.6 | 3.1 | 6.5 | -0.61 | 6.7 | -4.03(0.05) | A | - | -
CD$-$46 1064 | - | K3V | 1185 | 12 | 477.8 | 134.9 | 8.7 | 211.2 | 21.6 | 68.7 | -1.53 | 17.4 | -4.2(0.03) | A | - | -
Table 7: continued.

##  Appendix B Age and mass estimations

First, the ages and masses of the targets of the HARPS YNS, the SOPHIE YNS,
and the HARPS Sco-Cen surveys were taken from the literature when available.
We estimated the ages and mass of the remaining ones. Whenever possible, we
used the methods described in Desidera et al. (2015) and in Desidera et al.
(2021), then in a homogeneous scale with the above works. Briefly, we
considered membership to groups and associations (adopting the group ages from
Bonavita et al. (2016), also discussed in the upcoming paper), indirect
indicators such as rotation, chromospheric and coronal activity, and
$6708\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ equivalent width, complemented
by isochrone fitting. Preference was given to the moving-group criterion
whenever possible (confirmed members). For field objects, the weights assigned
to the various methods depend on colors or spectral types and age range (e.g.,
saturation of chromospheric activity and coronal emission versus age below
$100-$150\text{\,}\mathrm{Myr}$$; high sensitivity of lithium to age for K
dwarfs younger than $300-$500\text{\,}\mathrm{Myr}$$, with limits only at
older ages). Masses were derived using the PARAM interpolation
code\alphalph\alphalph __http://stev.oapd.inaf.it/cgi-bin/param_1.3 (da Silva
et al. 2006) as in Desidera et al. (2015). For the remaining targets, masses
were estimated from an empirical $M_{*}=f($B-V$)$ relation. The spectral types
were first estimated from the $B-V$ using the relation from Lang (1999) (page
564), and the masses were thereafter deduced from the spectral types by using
the relation from Cox (2000) (page 209).

##  Appendix C Star with companion data

We present in this appendix the data for the stars with a companion in the
HARPS Sco-Cen survey (_cf._ Section 3): RV time series, BVS versus RV diagram,
residuals of the companion fit, and bisectors or CCF.

Figure 23: Summary of binaries for which orbits could not be derived. _First
column_ : RV time variations. _Second column_ : BVS vs. RV diagram (black) and
its best linear model (red dashed line).

(a)

(b)

(c)

Figure 24: CCFs of HD 129590 (24(a)), of HD 137057 (24(b)), and of HD 143811
(24(c)). For these three stars, each curve corresponds to a CCF derived from
one of their spectra.

(a)

(b)

Figure 25: Bisectors of HD 111102 (25(a)) and HD 149790 (25(b)). Figure 26: RV
of the characterized binaries. _First column_ : RV time variations (_top_) and
Keplerian fit residuals. The fits are presented in Figure 4 for HD 108857, in
Figure 5 for HD 108904, and in Figure 9 for HD 149790. _Second column_ : BVS
vs. RV diagram (black) and its best linear model (dashed red line). _Third
column_ : BVS vs. RV residuals diagram and its best linear model (dashed red
line). Figure 27: Stars with RV trend for which a linear regression was
performed. _First column_ : _top_ : RV time variations (black) with the model
of its linear regression (red line). _Bottom_ : Residuals of the linear
regression. _Second column_ : BVS vs. RV diagram (black) and its best linear
model (dashed red line). _Third column_ : BVS vs. RV residuals diagram and its
best linear model (dashed red line).  Figure 28: Posterior distributions of
the MCMC fit performed on the HD 108904 RV and HD 108904 B relative astrometry
simultaneously. _From top to bottom:_ RV offset to take into account that the
RVs are relative to a reference ($V_{0}$), RV jitter amplitude ($jitter$),
semimajor axis ($sma_{b}$), eccentricity ($ecc_{b}$), argument of periastron
($parg_{b}$), period ($per_{b}$), mass ($m_{b}$), inclination ($I_{b}$), and
longitude of the ascending node ($O_{b}$) of HD 108904 B.

##  Appendix D Statistical tests

We present the two statistical tests we used to determine whether the
populations of companions studied by different surveys are statistically
different (and therefore whether their occurrence rates are statistically
different): the p-value test, and the pooled version of the two-proportion
Z-test.

These tests are evaluated under a null hypothesis; if a test is validated,
then the null hypothesis can be rejected. For these tests, we used the
following null hypothesis: the occurrence rate of the two compared companion
populations in the studied ($P$, $M_{c}\sin{i}$) box are identical. Thus, if a
test is successful and therefore the null hypothesis is rejected, we can
deduce that it is unlikely that the occurrence rates of the two compared
populations are identical.

For these tests, we assimilated the surveys performed to constrain the
companion occurrence rates to Bernoulli tests $\mathcal{B}(n,p)$, with $n$ the
number of targets for a given survey and $p$ the actual companion occurrence
rate.

###  D.1 p-value test

The $p_{value}$ test is a statistical test that evaluates the probability of
obtaining the number of observed successes by chance for a given null
hypothesis. If this probability is below a validation criterion $\alpha$, the
test is validated.

In our case, we compared two Bernoulli tests ($\mathcal{B}_{1}(n_{1},p_{1})$
and $\mathcal{B}_{2}(n_{2},p_{2})$) under the following null hypothesis:
$p_{1}=p_{2}$. To perform this test, we computed the probability of obtaining
the results of the first test ($k_{1}$ success among $n_{1}$ observations),
knowing the probability $p_{1}=p_{2}$:

$p_{value}=\dbinom{n}{k}p_{2}^{k_{1}}(1-p_{2})^{n_{1}-k_{1}}$

with $\dbinom{n}{k}$ the binomial coefficient of $k$ among $n$.

If $p_{value}<\alpha$, the probability of observing the results of the first
Bernouilli test under the null hypothesis by chance is below $\alpha,$ and the
latter can therefore be rejected with a confidence level of $1-\alpha$.

###  D.2 Pooled version of the two-proportion Z-test

The pooled version of the two-proportion Z-test is a parametric test that
allows testing a difference in proportion of successes between two Bernoulli
process ($\mathcal{B}_{1}(n_{1},p_{1})$ and $\mathcal{B}_{2}(n_{2},p_{2})$,
respectively).

This test can be used only if the two compared samples are large enough for
their probability of success to follow a normal law (as the binomial
distribution tends toward a normal distribution when $n$ tends toward
infinity). Many criteria are used to ensure it. The most widely used is
$n_{1,2}>30$, but stricter criteria are also used, for example,
$n_{1,2}p_{1,2}>5;n_{1,2}(1-p_{1,2})>5$, or $n_{1,2}p_{1,2}(1-p_{1,2})>9$. In
our case, only the more relaxed criterion, $n_{1,2}>30$, is validated.

Then, if these criteria are validated, the difference in the proportion of
successes between two Bernoulli process also follows a normal distribution
($\mathcal{N}(\mu,\sigma^{2})$). Moreover, in the case of our null hypothesis
$p_{1}=p_{2}$, the mean of the normal distribution equals $0$. Thus, to test
this null hypothesis, we computed the actual difference of the proportion of
success between the two compared Bernoulli processes, and we tested whether it
actually followed a centered normal distribution
($\mathcal{N}(0,\sigma^{2})$). This is the case when

$\frac{|p_{1}-p_{2}|}{\sqrt{p(1-p)(\frac{1}{n_{1}}+\frac{1}{n_{2}}})}>\mathcal{U}_{1-\frac{\alpha}{2}},$
(1)

with $p=\frac{n_{1}f_{1}+n_{2}f_{2}}{n_{1}+n_{2}}$ and with
$\mathcal{U}_{1-\frac{\alpha}{2}}$ the quantile of order $1-\frac{\alpha}{2}$
of the reduced centered normal distribution.

If Equation 1 is verified, the test is validated and the null hypothesis
($p_{1}=p_{2}$) can be rejected with a confidence level of $1-\alpha$.