  In previous papers we investigated so-called sum of translates functions
$F({\mathbf{x}},t):=J(t)+\sum_{j=1}^n \nu_j K(t-x_j)$, where $J:[0,1]\to
\underline{\mathbb{R}}:={\mathbb{R}}\cup\{-\infty\}$ is a "sufficiently
nondegenerate" and upper-bounded "field function", and $K:[-1,1]\to
\underline{\mathbb{R}}$ is a fixed "kernel function", concave both on $(-1,0)$
and $(0,1)$, and also satisfying the singularity condition $K(0)=\lim_{t\to 0}
K(t)=-\infty$. For node systems ${\mathbf{x}}:=(x_1,\ldots,x_n)$ with
$x_0:=0\le x_1\le\dots\le x_n\le 1=:x_{n+1}$, we analyzed the behavior of the
local maxima vector ${\mathbf{m}}:=(m_0,m_1,\ldots,m_n)$, where
$m_j:=m_j({\mathbf{x}}):=\sup_{x_j\le t\le x_{j+1}} F({\mathbf{x}},t)$. Among
other results we proved a strong intertwining property: if the kernels are also
decreasing on $(-1,0)$ and increasing on $(0,1)$, and the field function is
upper semicontinuous, then for any two different node systems there are $i,j$
such that $m_i({\mathbf{x}})<m_i({\mathbf{y}})$ and
$m_j({\mathbf{x}})>m_j({\mathbf{y}})$. Here we partially succeed to extend this
even to nonsingular kernels.
