text:
\begin{document}

\title{{\bf Automorphisms of $P_8$ singularities \\ and the complex crystallographic groups}}

\author{Victor Goryunov and Dmitry Kerner}
\date{}
\maketitle

\begin{abstract}
The paper completes the study of symmetries of parabolic function singularities
with relation to complex crystallographic groups that was started in \cite{GM,X9}.
We classify smoothable automorphisms of $P_8$ singularities
which split the kernel of the intersection form on the second homology. For such automorphisms,
the monodromy groups acting on the duals to the eigenspaces with degenerate intersection form are then identified as some of
complex affine reflection groups tabled in \cite{P}.
\end{abstract}

Singularity theory has been maintaining close relations with reflection groups since its early days, starting with the famous
works of Arnold and Brieskorn \cite{A,Br}. The first to emerge, as simple function singularities, were the
Weyl groups $A_k, D_k$ and $E_k$. They were followed by the $B_k$,  $C_k$ and $F_4$ as simple boundary
singularities or, equivalently, functions invariant under the involution \cite{Ab,Slo}. Since then appearance of Weyl groups
in a singularity problem became a kind of criterion of naturalness of the problem \cite{AGV2,AGLV1,AGLV2}.

The next to receive a singularity realisation was the list of all Coxeter groups, in the classification of
stable Lagrangian maps \cite{Gi}. Some of the
Shephard-Todd groups appeared in \cite{G1,G2,GB} in the context of simple functions equivariant with respect to a cyclic
group action. And finally, the first examples of complex crystallographic groups came out in \cite{GM,X9}
in connection with the symmetries of parabolic functions $J_{10}$ and $X_9$.
The affine reflection groups appeared there as monodromy groups, which
 is similar to the first realisations of other classes of reflection groups. This time it was the equivariant monodromy
corresponding to the symmetry eigenspaces $H_\chi$ in the vanishing second homology on which the intersection form
$\s$ has corank 1.

This paper completes the study of cyclically equivariant parabolic functions started in \cite{GM,X9} and considers the $P_8$ singularities.
The approach we are developing here is considerably shorter, with less calculations and more self-contained. This is allowed by
a preliminary observation that the modulus parameter may take on only exceptional values if corank$(\s|_{H_{\chi}}) = 1$: the
$j$-invariant of the underlying elliptic curves must be either 0 or 1728 (see Section \ref{Sker}).

\medskip
The structure of the paper is as follows.

Section 1 describes the crystallographic groups which will be involved.

Section 2 gives a classification of smoothable cyclic symmetries of $P_8$ singularities possessing eigenspaces $H_{\chi}$
with the property as already mentioned.

In Section 3, we obtain distinguished sets of generators in such eigenspaces and calculate the intersection numbers of the
elements in the sets. This allows us to describe the Picard-Lefschetz operators generating, as complex reflections,  the equivariant
monodromy action on the $H_{\chi}$.

Finally, in Section 4, we pass to the vanishing cohomology. In its subspaces dual to the $H_{\chi}$, we consider hyperplanes
formed by all the cocycles taking the same non-zero value on a fixed generator of the kernel of the hermitian intersection form
$\s|_{H_{\chi}}$. We show that the equivariant monodromy group acting on each of these hyperplanes is a complex
crystallographic group.

\medskip
Both authors gratefully acknowledge the support of the EC Marie Curie Training Site (LIMITS) on Algebraic Geometry,
Singularities and Knot Theory which funded the visits of the second author to Liverpool.
The research of the second author was also supported by the Skirball postdoctoral fellowship of the Center
of Advanced Studies in Mathematics, Be'er Sheva, Israel.

\section{The crystallographic groups}
We remind the description of the complex affine reflection groups
which will be relevant to our singularity constructions. In the notations of \cite{P}, the groups
are
$$
[K_3(3)], [K_3(4)], [K_3(6)], [G(3,1,2)]_2,  [K_5], [K_8], [K_{25}]\,.
$$
The linear parts of these groups are Shephard-Todd groups $L$  shown in Figure \ref{Fgr}, in the related notation of \cite{ST}.
There and in what follows $\w=exp(2\pi i/3)$.
The rank of $L$ is the number of the vertices in its diagram.
Each of the groups involved is a semi-direct product of its linear part and translation lattice.
The lattice of $[G(3,1,2)]_2$ is spanned over $\ZZ$ by the
$G(3,1,2)$-orbit of any order 2 root, while for any other group it is spanned the $L$-orbit of any root. For example, our list contains
all three one-dimensional groups, the $[K_3(m)]$, each generated by an order $m$ rotation in $\C$ and the lattice $\ZZ[1,i]$ if $m=4$ or
$\ZZ[1,\w]$ if $m=3,6$.

\begin{figure}[hbtp]
\begin{center}

\includegraphics[width=0.7\textwidth]{p8gr.eps}

\caption{Linear parts of the crystallographic groups. Each vertex represents a unit root. The order of the corresponding reflection
is written inside the vertex (order 2 omitted). An egde $a \to b$ is equipped with the hermitian product $\left< a, b \right>$
(no orientation is needed when the product is real). There are no edges between orthogonal roots.}
\label{Fgr}
\end{center}
\end{figure}

\section{The symmetries}

\subsection{Cubic curves}
Our first step in the study of linear automorphisms of the surfaces
$$x^3+y^3+z^3+3\a xyz =0\,, \qquad \a^3 \ne -1,$$
in $\C^3$, is consideration of the projective version of the question. This is a
description of automorphisms of projective curves $C_\a$ given by the same
equations in $\C P^2$. The question is classical, but, for the sake of the exposition,
we indicate a possible elementary approach.

A curve $C_\a$ has nine inflection points. There are four triplets of lines
in $\C P^2$ passing through these points, each line containing exactly three inflections
and each triplet containing all nine inflections. The lines are zero sets of the linear forms
$$
\begin{array}{lll}
x, & y, & z; \\
x+y+z, \quad & x+\w y + \ww z, \quad & x + \ww y + \w z; \\
x+y+\w z, \quad & x + \w y + z, \quad & \w x + y + z; \\
x+y+\ww z, \quad & x + \ww y + z, \quad & \ww x + y + z\,.
\end{array}
$$
A projective automorphism of $C_\a$ permutes the triplets and reorders the lines
within the triplets. It is sufficient to consider the images of the first triplet
only. A routine study of all the possibilities, followed by diagonalisation of each
projective transformation and rewriting the equation of an appropriate curve in the
projective eigencoordinates, give us

\begin{pr}\label{Pproj}
Any projective automorphism of a cubic curve can be reduced, by a choice of projective
coordinates, either to the diagonal projective transformation
$(x:y:z) \mapsto (\k_xx:\k_yy:\k_zz)$

of one of the following
cubics $f=0$ or to the inverse of such a transformation:
$$
\begin{array}{|l|l|l|l|}
\hline
&&& \vspace{-8pt} \\
 & \qquad\qquad\qquad  f & \k_x:\k_y:\k_z &
{\rm monomial \  basis \  of \  the \  local \  ring \  of \  } f

\\
&&& \vspace{-8pt} \\
 \hline &&& \vspace{-8pt} \\
1 & x^3 + y^3 + yz^2   & \,\, \w : 1 : -1 &
1, x, y, z, xy, xz, y^2\sim z^2, xy^2 \sim xz^2 \\ &&& \vspace{-8pt} \\
2 & x^3 + y^3 + yz^2   \simeq x^3 + y^3 + z^3 & \,\, \w : 1 : 1 &
1, x, y, z, xy, xz, y^2\sim z^2, xy^2 \simeq xyz \\ &&& \vspace{-8pt} \\
3 & x^2y + y^2z + z^2x & \,\, 1 : \w : \ww &
1,x,y,z,x^2 \sim yz, y^2 \sim xz, z^2 \sim xy, xyz
\\ &&&  \vspace{-8pt} \\
4 & x^2z + xy^2 + z^3 & \,\, 1 : i : -1 &
1,x,y,z,yz,x^2\sim z^2,xz\sim y^2,x^3 \\  &&& \vspace{-8pt} \\
5 & x^3 + y^3 + yz^2 + 3 \a xy^2\,, \,\, \a^3 \ne -\frac14 & \,\, 1:1:-1 &
1, x, y, z, xy, xz, y^2\sim z^2, xy^2 \sim xz^2 \\ &&& \vspace{-8pt} \\
6 & x^3+y^3+z^3+ 3 \a xyz\,, \,\, \a^3 \ne -1  & \,\, 1:\w:\ww  &
1,x,y,z,xy,xz,yz,xyz \\ &&& \vspace{-8pt} \\
7 & x^3+y^3+z^3+3\a xyz\,, \,\, \a^3 \ne -1 & \,\, 1:1:1 &
1,x,y,z,xy,xz,yz,xyz \vspace{-8pt} \\ &&& \\ \hline
\end{array}
$$
\end{pr}

\medskip
The contents of the last column of the table will be used in Section \ref{Ssmosym}.

\subsection{Kernel character constraints}\label{Sker}
We now lift a projective automorphism of a cubic curve $f=0$ to a linear
transformation $g$ of $\C^3$. It multiplies $f$ by a non-zero constant

and hence acts on the target $\C$ of the function $f$.
We consider deformations of the function-germ $f: (\C^3,0) \to (\C,0)$
as usual: choosing a small ball $B$ in $\C^3$ centred at the origin and deforming $f$ within the ball. However,
we restrict our attention to deformations which are $g$-equivariant with respect to the actions of $g$ on
$\C^3$ and $\C$.

\begin{dfn} {\em
If the function-germ $f$ admits $g$-equivariant deformations
with non-singular zero levels, we say that the symmetry $g$ of $f$
is} smoothable.
\end{dfn}

In this case, the smooth zero level $V \subset B$ of a small $g$-equivariant
deformation of $f$ is a Milnor fibre of $f$, and $g$ acts on $V$ and on its
second homology. Assuming from now on the order $r$ of $g$ finite, we obtain the
splitting into a direct sum of character subspaces
\begin{equation}
H_2(V,\C) = \oplus_{\chi^r=1} H_{2,\chi}
\label{Esum}
\end{equation}
so that $g$ acts on each summand as multiplication by the relevant
$r$th root $\chi$ of unity.

We extend the intersection form $\sigma$ from the lattice $H_2(V,\ZZ)$ to
the hermitian form $\sigma_h$ on $H_2(V,\C)$. The symmetric intersection form
of a parabolic function singularity in 3 variables is negative semi-definite, with a two-dimensional kernel.

\begin{dfn} {\em
We say that a smoothable symmetry $g$} splits {\em the kernel of
$\sigma_h$ if this kernel is shared by two different character subspaces
$H_{2,\chi}$. If the restriction of $\sigma_h$ to a $H_{2,\chi}$ is
degenerate, we call $\chi$ a} kernel character.
\end{dfn}

The aim of our construction, a complex crystallographic group, will be acting
on a hyperplane $\G$ in the cohomology subspace $H_{\chi}^2 = \{\gamma \in H^2(V,\C): g^*(\gamma) = \chi\gamma\}$ where
$\chi$ is a kernel character (cf. \cite{GM,X9}).
For this, our symmetry $g$ should split the kernel of $\sigma_h$, and the hyperplane $\G$ is defined as
the set of all elements of $H_{\chi}^2$ taking the same non-zero value
on a fixed generator of the line $Ker(\sigma_h) \cap H_{2,\chi}$.

\medskip
Our next step is to show that there is no kernel splitting in any of the modular
cases of Proposition \ref{Pproj}.

Assume $g$ is a lifting of one of the projective automorphisms from the table there.

Take a monomial basis $\f_0=1, \f_1, \dots, \f_7$ of the local ring
of $f$, with $\f_7$ of degree 3.
Consider the sub-unfolding of the $\R$-miniversal unfolding of $f$ in
which the modulus parameter does not participate:
$$
\F: \C^3 \times \C^{6} \to \C \times \C^{6}\,,
\quad
(x,y,z,\l_1,\dots,\l_{6}) \mapsto
(f+\l_1\f_1+\dots\l_{6}\f_{6},\l_1,\dots,\l_{6})\,.
$$
This map is defined globally. Its smooth fibres are Milnor fibres of $f$.
The action of $g$ naturally extends to the source and target of the map,
making $\F$ $g$-equivariant. We will denote the target of $\F$ by $\U$,
points there will be $u=(u_0,\dots,u_{6})$.

We simultaneously compactify all fibres $V_u$,
$u \in \U$, of $\F$ adding to each of them a copy
$C_u = \ov V_u \setminus V_u$ of the same projective curve $f=0$.
Each $\ov V_u \subset \C P^3$ is smooth near $C_u$.

For non-critical values of $u$, take the intersection homomorphism
$\s:  H_2(V_u;\ZZ) \to H^2(V_u;\ZZ)$.

The Leray map sends $H_1(C_u; \ZZ)$ isomorphically onto the 2-dimensional
kernel of $\s$ \cite{L}.

Consider the 2-form $w = dx \we dy \we dz \we d\l_1 \we \dots \we d\l_{6}/
du_0 \we \dots \we du_{6}$ on the fibres
$V_u$. According to \cite{L}, its residue $\b$ along $C_u$ is a non-zero
holomorphic 1-form which does not depend on $u$.

The action of the automorphism $g$ on a $g$-symmetric fibre $V_u$
extends to $\ov V_u$.  The residue map
$Res: H^2(V_u;\C) \to H^1(C_u;\C)$ is $g$-equivariant: if $g^*$ multiplies a 2-form
by $\chi$, then it multiplies the residue of the form by $\chi$ too. Therefore, if $g$
splits $Ker(\s_h)$ on $V_u$, then its action on $H^1(C_u;\C)$ must also
have two distinct conjugate eigenvalues, with $\b$ and $\ov \b$ as eigenvectors.
Hence the curve $C_u=\{f=0\}$ must be either $\C/\ZZ[1,i]$ or $\C/\ZZ[1,\w]$.
Respectively, the kernel characters are either $\pm i$ or
$(\w,\ov \w)$ or $(-\w,-\ov\w)$.

Thus we have:

\begin{pr}
A smoothable automorphism $g$ splits the kernel of the intersection form
if and only if the section $w=dx \we dy \we dz /df$ of the vanishing cohomology
fibration of $f$ is a $g^*$-eigenvector with the eigenvalue from the set
$\{\pm i, \pm \w, \pm\ov\w\}$. The eigenvalue and its conjugate are the
kernel characters.
\end{pr}

If $g^*(w) = \chi w$ then for the hyperplane $\G$
mentioned above  one can take $Res^{-1}(Res(w)) \cap H_{\chi}^2$.

Since the degree of $f$ is 3, the eigenvalue of $g^*$ on $w=dx \we dy \we dz /df$
depends only on the projectivisation of $g$.

\begin{cor}\label{Clift}
All smoothable liftings of the first four automorphisms from Proposition
\ref{Pproj} split $Ker(\s_h)$. On the other hand, none of the liftings of the
last three cases does the same.
\end{cor}

\noindent
{\bf Remark.} The argument can be easily modified to the
symmetries of $X_9$ and $J_{10}$ function singularities, thus explaining
the absence of the moduli in the classification tables in \cite{GM,X9}.

\subsection{Smoothable symmetries}\label{Ssmosym}
Assume we have two actions of a finite cyclic group, on $(\C^k,0)$ and on $(\C,0)$.
Consider two function-germs $f_1,f_2:  (\C^k,0) \to (\C,0)$ equivariant with
respect to these actions: $f_i \circ \rho = \rho \circ f_i$ where $\rho$ is a generator of
the group. We say that the two functions are {\em $\R_\rho$-equivalent\/} if one
can be transformed into the other by a $\rho$-equivariant diffeomorphism-germ
of $(\C^k,0)$.

We apply this notion in the context of our cubic function $f$, its {\em diagonal\/}
symmetry $g$ and the induced action of $g$ on $\C$.  With a minor abuse of the
language, we will still be calling the corresponding equivalence the $\R_g$-equivalence.

For example, an $\R_g$-miniversal unfolding of $f$
is the restriction $F$ of the unfolding $\F$ of the previous subsection to
$\F^{-1}(U)$, where $U \subset \U$ is the set of fixed points of the natural
action of $g$ on $\U$.
Equivalently, in this case, for an $\R_g$-miniversal deformation of
$f$ one can take $f +$ arbitrary linear combinations of the elements $\psi_1, \dots, \psi_\tau$ of
a monomial basis of the local ring of $f$ that are multiplied by $g$ by the
same factor as $f$. The number $\tau$ appearing here will be called the
{\em $\R_g$-codimension\/} of $f$.

\begin{pr}\label{Proff0}
Assume a symmetry $g$ is smoothable. Then $g$ multiplies $f$ by the same
factor as it multiplies one of the monomials $1, x, y, z$.
\end{pr}

Indeed the conclusion is equivalent to generic fibres of an $\R_g$-miniversal unfolding
not being singular at $0 \in \C^3$.

With the help of Corollary \ref{Clift} and the last Proposition, we obtain after
straightforward calculations which we prefer to omit:

\begin{thm}\label{Tclass}
The complete list of  smoothable symmetries of $P_8$ function singularities
on $\C^3$ which split the kernel of the intersection form is given in Table \ref{Ta}.
Our classification is up to a choice of generators of the same cyclic group.
\end{thm}

\begin{table}[p]
\begin{center}
\caption{Smoothable symmetries of $P_8$ singularities splitting $Ker(\s)$}
\vspace{-15pt}
$$\begin{array}{|c|c|c|c|c|c|c|}
\hline
&&&&&&\\
f & g: x,y,z \mapsto & |g| &
\begin{array}{c}
{\rm versal} \\
{\rm monomials}
\end{array}
&
\begin{array}{c}
\!\!\!{\rm kernel}\!\!\! \\
\chi
\end{array}
&
\begin{array}{c}
{\rm affine}\\
{\rm group}
\end{array}
 &
{\rm notation} \\
&&&&&&\\
\hline
\hline
&&&&&&\\
x^3    &   \w x,y,-z   &   6   &   1, y, y^2    &   -\w, -\ww  &  \!\! [K_5] \!\!  &   C_3^{(3,3)} \\
+y^3 &&&&&& \\
+yz^2 &   x, \ww y, -\ww z    &   6   &      1,x        &   -\w, -\ww   &  [K_3(6)]    &   (P_8|\ZZ_6)'   \\
&&&&&& \\
& \ww x, \w y, -\w z  &  6  & 1,xy  &  -\w, -\ww & [K_3(6)] & (P_8|\ZZ_{6})'' \\
&&&&&& \\
& -x, -\ww y, \ww z &  6  &  x  &  -\w, -\ww  &  -  &  (P_8/\ZZ_6)' \\
&&&&&& \\
& i\w x, iy, -iz  &  12  &  z  &  -\w, -\ww  &  -  &  P_8/\ZZ_{12} \\
&&&&&& \\
&  \w x, y, z  &  3  &  1,y,y^2,z  & \w, \ww & [K_{25}] & D_4^{(3)} \\
&&&&&& \\
&  x, \ww y, \ww z  &  3  &  1,x  & \w,\ww  &  [K_3(6)] & (P_8|\ZZ_{3})' \\
&&&&&& \\
&  \ww x, \w y, \w z  &  3   &  1,xy,xz & \w, \ww & [G(3,1,2)]_2 & P_8|\ZZ_3 \\
&&&&&& \\
\hline
&&&&&& \\

x^3 &  -x,-\ww y, -\ww z  &  6  &  x  & \w, \ww & - & (P_8/\ZZ_6)'' \\
+y^3 &&&&&& \\
+z^3 & -\w x, -y, -z  &  6  &  y,z  &  \w, \ww  &  [K_3(3)]  & P_8/\ZZ_6 \\
&&&&&& \\
\hline
&&&&&& \\
\begin{array}{c}
x^2y \\ + y^2z \\ + z^2x
\end{array}
&  \ov\e_9 x, \w\ov\e_9 y, \ov{\w\e}_9 z  &  9   &  1 & \w, \ww & - & P_8|\ZZ_9 \\
&&&&&& \\
\hline
&&&&&& \\
x^2z   &  -x, -iy, z  &  4   &  1,z,z^2 & \pm i & [K_8] & C_3^{(2,4)} \\
+xy^2 &&&&&& \\
+z^3 & -\w x, -i\w y, \w z & 12 &  1  & \pm i  &  -  &  P_8|\ZZ_{12} \\
&&&&&& \\
& ix, -y, -iz  &  4  & x, yz  & \pm i & [K_3(4)]  &  P_8/\ZZ_4 \\
&&&&&& \\
& \ov\e_8 x , \e_8 y, -\ov\e_8 z  &  8  &  y  &  \pm i  &  -  &  P_8/\ZZ_8 \\
&&&&&& \\
\hline
\end{array}
$$
\label{Ta}
\end{center}
\end{table}

In the Table: the versal monomials are the monomials participating in $\R_g$-miniversal deformations
(these are selected from the last column of the
table in Proposition \ref{Pproj} as those multiplied by $g^*$ by the same factor as $f$),
the affine groups are those which will come out later as monodromy
groups acting on the hyperplanes in the character subspaces in the cohomology,
and $\e_k=exp(2\pi i/k)$.

\medskip\noindent
{\bf Remark.} During the proof of the Theorem, one obtains a few non-smoothable symmetries allowing linear terms in deformations:
$$
\begin{array}{|c|c|c|c|}
\hline
&&&   \vspace{-8pt} \\
f & g: x,y,z \mapsto & |g| &
{\rm versal \   monomials}    \vspace{-8pt}  \\
&&&\\
\hline
\hline
&&&   \vspace{-8pt} \\
x^3 + y^3 + yz^2 & -\w x, -y, z  &  6  &  y   \vspace{-8pt} \\
&&&\\
x^2y + y^2z + z^2x  &  \w x, \ww y, z  &  3  &  x, xz    \vspace{-8pt}  \\
&&&\\

&  -\w x, -\ww y, -z  &  6  &  x    \vspace{-8pt}  \\
&&&\\
x^2z + xy^2 + z^3  &  x, iy, -z & 4  &  z, xz    \vspace{-8pt}  \\
&&& \\

& -ix, y, iz  &  4  &  x   \vspace{-8pt} \\
&&& \\
\hline
\end{array}
$$
It is possible to relate complex crystallographic groups to the two singularities
with 2-parameter miniversal deformations, but this will be done in another paper.

\bigskip
Of course, one may consider a bit more general problem of finding all possible
smoothable symmetry {\em groups\/} $G$ of parabolic singularities, which are not necessarily cyclic.
In this situation the second homology splits into irreducible represenations of $G$, and we may be still looking
for cases when the kernel of the intersection form is shared by two of them.

For $P_8$, for example, this means that $G$ should contain one of the symmetries $g$ of the Table,
and hence the affine reflection group related to $G$ will be a lower rank subgroup of the one we are relating to $g$.
This does not leave too much dimensional room for further interesting crystallographic groups.

\section{Dynkin diagrams}
The equivariant monodromy of a $g$-equivariant function singularity $f$, that is, the one
within an $\R_g$-miniversal deformation of $f$ preserves the direct sum decomposition (\ref{Esum}).
Its action on an individual summand $H_{2,\chi}$ is generated by the Picard-Lefschetz operators $h_e$
corresponding to vanishing $\chi$-cycles $e$:
$$
h_e(c) = c - (1-\l_e) \left< c,e \right> e /\left< e, e \right>\,.
$$
Here $\l_e \ne 1$ is the eigenvalue of $h_e$: $h_e = \l_e e$.

In this Section we obtain all the necessary information describing the monodromy on
the kernel character subspaces of the singularities of Table \ref{Ta}. We choose
distinguished sets of generators of the subspaces $H_{2,\chi}$ (cf. \cite{Brmon,GZ, AGV2,AGLV1}), calculate
the intersection numbers of the elements
of the sets, and find the eigenvalues $\l_e$. The data will be collected into Dynkin diagrams of the singularities.

\begin{pr}\label{Pdim}
If $\chi$ is a kernel character of a symmetric singularity from Table \ref{Ta},
then the rank of $H^2_{\chi}$, equivalently the rank of $H_{2,\chi}$, coincides
with the $\R_g$-codimension $\tau$ of the singularity.
\end{pr}

{\em Proof.} If $f$ is $g$-invariant, then a basis of one of the two $H^2_{\chi}$
is formed, in the notations of Section \ref{Sker}, by the sections
$\f_i w$, where the $\f_i$ are all $g$-invariant monomials within
$\{\f_0, \dots, \f_7\}$. Their number is exactly $\tau$. (Cf. \cite{OS,W}.)

If $f$ is $g$-equivariant rather than invariant, the claim can be verified by
a direct case-by-case computation of the eigenvalues of $g^*$ on the
sections $\f_0 w, \dots, \f_7 w$. The observation needs a case-free understanding.
\hfill{$\Box$}

\medskip
Thus, if $\tau=1$ then the subspaces $H_{2,\chi}$ in the case of kernel $\chi$
are one-dimensional, hence contained in $Ker(\s_h)$, and therefore the monodromy
we are interested in is trivial. So, from now on we are considering only the cases
of at least two-parameter $\R_g$-miniversal deformations.

We will also forget about the $(P_8|\ZZ_3)'$ singularity. Indeed its symmetry $g$
is the inverse of the square of the $(P_8|\ZZ_6)'$ symmetry and the two
equivariant miniversal deformations coincide.
Hence the character subspaces with degenerate intersection form are
the same (only the actual character assignments differ: $\chi_3 = \chi_6^{-2}$)
and the monodromy is the same.

\bigskip
For each remaining symmetry, we will choose a distinguished set of generators for $H_{2,\chi}$
in the way it has been done in \cite{G1,G2,GB}.
Namely, let $u_*$ be a point  in the complement $\C^{\tau} \setminus \Delta$ to the discriminant in the
base of the $\R_g$-miniversal deformation of our function. Set $V_{u_*}/g$ to be the quotient of the
fibre $V_{u_*}$ under the cyclic
$g$-action and $(V_{u_*}/g)'$ its part corresponding to irregular orbits.  Denote
by $\pi$ the factorisation map $V_{u_*} \to V_{u_*}/g$.
The homology group $H_2(V_{u_*}/g,(V_{u_*}/g)';\ZZ)$ is spanned by a
distinguished set of relative vanishing cycles (such a set is defined in the traditional singularity theory manner). Denote them $c_1, \dots, c_k$.
The inverse image $\pi^{-1}(c_j)$ is the cyclic orbit of one of its components,
let it be $\wt c_j$. For a character $\chi$, the chain
$$e_j=\sum_{i=0}^{order(g)-1} \chi^{-i}g^i(\wt c_j)$$
is an element of $H_{2,\chi}$.  The {\em vanishing $\chi$-cycles\/}
$e_1, \dots, e_k$ span $H_{2,\chi}=H_{2,\chi}(V_{u_*},\C)$, not necessarily as a basis.
They are defined up to multiplication by powers of $\chi$ and the sign change.

\medskip
\noindent
{\bf Examples.}  (a) The function $x^n+y^2+z^2$ invariant under the transformation $g: (x,y,z) \mapsto (\e_n x,y,z)$ was denoted $A_1^{(n)}$
in \cite{G1}.
The factorisation by the group action gives the boundary singularity $A_1: w + y^2 + z^2$, $w=x^n$, whose relative
vanishing homology is spanned by
one semi-cycle $\{ w + y^2 + z^2 = 1: w,y,z \in {\mathbf R}, w \ge 0\}$. The corresponding vanishing $\chi$-cycle, $\chi^n=1$, $\chi \ne 1$,
is formed by $n$ hemi-spheres with appropriate coefficients and has self-intersection $-n$ \cite{G1}, which is consistent
with the standard Morse vanishing cycle having self-intersection $-2$.
The Picard-Lefschetz operator $h$ is the classical monodromy of the ordinary $A_{n-1}$ singularity. Its eigenvalue
on $H_{2,\chi}$ is $\l=\chi$, since the quasi-homogeneous isotopy in the family $x^n + y^2 + z^2 = e^{it}$, $t \in [0,2\pi]$, finishes
with the transformation $(x,y,z) \mapsto (\e_n x,-y,-z)$ whose action on the homology coincides with that of $g$.

(b) The $A_2$-version of the previous singularity is $A_2^{(n)}: x^n+y^3+z^2$, with the same symmetry.
For it, each of the $H_{2,\chi}$ is spanned by two similarly defined
$\chi$-cycles which may be chosen so that their intersection number is $n/(1-\chi)$. The two Picard-Lefschetz operators $h_j$

satisfy the standard braiding relation $aba=bab$.

Diagrammatically, the $A_2^{(n)}$ singularity is represented by fusing the
rectangular $2 \times (n-1)$ Dynkin diagram to the $A_2$ diagram which may be equipped with the markings indicating the orders of the
vertices and the intersection numbers. Cf. the $P_8 \to D_4^{(3)}$ part of Figure \ref{Ffus}.

(c) The singularity $A_m/\ZZ_m$ is the function $x^{m+1}+yz$ with the equivariant symmetry $g: (x,y,z) \mapsto (\e_m x,\e_m y,z)$. As it has
been shown in \cite{G2}, its $\chi$-cycle has self-intersection $-m$, $\chi$ being any $m$th root of unity. The quasi-homogeneous
argument applied to the $\R_g$-miniversal deformation $x^{m+1}+yz + \a x$ demonstrates that the Picard-Lefschetz operator
on $H_{2,\chi}$ is multiplication by $\chi$.

\begin{thm}\label{Tdd}
For the symmetric $P_8$ singularities and their kernel characters $\chi$, there exist distinguished sets of vanishing
$\chi$-cycles described by Dynkin diagrams of Figure \ref{Fdi}.
\end{thm}

\begin{figure}[hbtp]
\begin{center}

\includegraphics[width=0.8\textwidth]{p8di.eps}

\caption{Dynkin diagrams of the symmetric $P_8$ singularities. The numbers inside vertices are the orders of the Picard-Lefschetz operators
(order 2 omitted). The number next to a vertex is the self-intersection of the $\chi$-cycle. As earlier, edges are marked with the intersection numbers
of the cycles. In the diagrams of all {\em invariant\/} singularities, the multiplicity of an edge indicates the length of the pair-wise braiding relation
inherited from the fundamental group of the complement to the discriminant:
$aba=bab$ for a simple edge, $(ab)^2=(ba)^2$ for a double and $(ab)^3=(ba)^3$ for a triple.}
\label{Fdi}
\end{center}
\end{figure}

{\em Proof\/}. We proceed on the case-by-case basis. We are considering only kernel characters of the $P_8$ singularities.

\medskip
$\mathbf{D_4^{(3)}}$. Factorisation by the action of $\ZZ_3$ provides the boundary $D_4$ singularity. Hence, according to the above
Examples, a Dynkin diagram of $D_4^{(3)}$ can be obtained by the modification of the standard $D_4$ diagram: the
roots should have squares $-3$ instead of $-2$, the Picard-Lefschetz operators $h_j$ become of order 3, and the intersection
numbers 1 of pairs of cycles change to $3/(1-\chi)$. Since $\chi=\w,\ov\w$, the latter may be reduced to $1-\w$
(for both values of $\chi$) using the ambiguities in the choice of the $\chi$-cycles.
The diagram may be obtained by the fusion of the cylindrical $P_8$ diagram as shown in Figure \ref{Ffus}.

\begin{figure}[hbtp]
\begin{center}

\includegraphics[width=0.48\textwidth]{p8fus.eps}

\caption{Obtaining some of the diagrams by fusion and folding.}
\label{Ffus}
\end{center}
\end{figure}

\medskip
$\mathbf{C_3^{(3,3)}}$. We extend the previous $\ZZ_3$-symmetry by $\ZZ_2$ acting by the sign change on $z$. This embeds the character
subspaces $H_{2,-\w}$ and $H_{2,-\ov\w}$ of the symmetric singularity $C_3^{(3,3)}$ into
$H_{2,\w}$ and $H_{2,\ov\w}$ of $D_4^{(3)}$ as subspaces anti-invariant under the involution.
This is absolutely similar to the relation between the $\ZZ_2$-symmetric singularity $C_3$ and the absolute $D_4$ singularity,cf.
\cite{Ab,AGV2,AGLV1}.
Hence the $C_3^{(3,3)}$ diagram of Figure \ref{Fdi} is provided by the folding of the $D_4^{(3)}$ shown in Figure \ref{Ffus}.

\medskip
$\mathbf{(P_8|\Z_6)'}$. The discriminant in this case is a semi-cubic cusp, hence the relation $aba=bab$ between
the two Picard-Lefschetz operators. A generic point of the discriminant corresponds to the $D_4|\ZZ_6$ singularity considered in
\cite{G2}, where the self-intersection number of its vanishing $\chi$-cycle was shown to be $-3$ for $\chi=-\w,-\ov\w$.
Since we are considering the kernel characters, the intersection form on the $H_{2,\chi}$ must be degenerate which implies that
the absolute value of the intersection number of the two $\chi$-cycles is 3. Since this number is in $\ZZ[1,\w]$, we can make it 3
using the $\chi$-cycle choice ambiguities once again.

Finally, the Picard-Lefschetz operator $h_{D_4|\ZZ_6}$ has order 6 as the classical monodromy operator of the absolute $D_4$ singularity.
Its non-trivial eigenvalue on $H_{2,\chi}$ is $\l=\ov\chi$ since the quasi-homogeneous isotopy $x^2+y^3+yz^2=e^{it}$, $t\in[0,2\pi]$,
finishes with the transformation $(x,y,z) \mapsto (-x, \w y, \w z)$ which coincides with $g^{-1}$ on the local homology.

Thus we have arrived at the $(P_8|\ZZ_6)'$ diagram of Figure \ref{Fdi}. Due to the generating reflections coming from the classical
monodromy of the $D_4$ singularities, it is natural to consider it as obtained by the fusion of Figure \ref{Ffus}.

\medskip
$\mathbf{(P_8|\Z_6)''}$. The discriminant of the versal family $x^3+y^3+yz^2 + \b xy + \a$ consist of two strata:
$$
A_5 = \{\a = 0\} \qquad {\rm and} \qquad 3A_1 = \{27\a+\b^3=0\}\,.
$$
The cubic tangency of the strata gives the braiding relation between the generators.
Of course, the self-intersection of the $3A_1$ vanishing $\chi$-cycle is $-6$ and the order of the reflection $h_{3A_1}$ is 2.
As for the $A_5$ $\chi$-cycle, it is possible to homotope it to the standard $A_1^{(6)}$ vanishing $\chi$-cycle, with the self-intersection
$-6$ according to Example (a). Now the argument similar to that for the previous singularity gives the intersection number $6$ of the
two cycles.  Quasi-homogeneous considerations of the $A_5$ local vanishing form $xy + z^6 = \e$ shows that the non-trivial
eigenvalue of its monodromy operator on $H_{2,\chi}$ is $\ov\chi$ since its local action coincides with $g^{-1}$. Hence the
operator $h_{A_5}$ is of order 6.

\medskip
$\mathbf{P_8|\Z_3}$. The discriminants of two real versions $x^3+y^3 \pm yz^2 + \a + \b xy + \g xz$ of a versal family are
the unions of the $A_2$ stratum $\a=0$ and $3A_1$ stratum $(54\a+\b^3 \pm 9\b\g^2)^2 = (\b^2 \mp 3\g^2)^3$. They are shown
in Figure \ref{Fnew}. The two strata are simply tangent to each other along their meeting lines.

Like for the previous singularity, it is possible to homotope the $A_2$ vanishing $\chi$-cycle to the standard
$A_1^{(3 )}$ vanishing $\chi$-cycle, with the self-intersection $-3$. An operator $h_{A_2}$ has the eigenvalue $\chi$.
The self-intersection of a $3A_1$ $\chi$-cycle is $-6$ and its monodromy reflection is an involution.

Assume the base point $u_*$ is chosen inside the front lips region of the left diagram of Figure \ref{Fnew}. A generic line through it is vertical.
Take on it a path system from $u_*$: two paths going straight to the $3A_1$ points and the third nearly straight to the $A_2$ point, bypassing
the upper $3A_1$ point on its way (the side is not important). Then the vanishing $\chi$-cycles may be chosen so that
$$
\left< e_{3A_1,lower},   e_{3A_1,upper} \right> = 3 \qquad {\rm and} \qquad
\left< e_{A_2},   e_{3A_1,upper} \right> = 3\,.
$$
Indeed the absolute values of these intersection numbers must be 3 following the braiding relations between the
two pairs of the Picard-Lefschetz operators
coming from the singularities of the discriminants. Moreover, both numbers are in $\ZZ[1,\w]$ and the $\chi$-cycles may be multiplied
by $-1$ and powers of $\w$.

Now we see that the last intersection number, $\left< e_{A_2},   e_{3A_1,lower} \right>$, is either $3\w$ or $3\ov\w$. This is guaranteed
by the degeneracy of the intersection form on $H_{2,\chi}$ and the number being in $\ZZ[1,\w]$. However, both options turn out to be
equivalent up to a braid transformation.

\begin{figure}[hbtp]
\begin{center}

\includegraphics[width=0.7\textwidth]{new.eps}

\caption{Two real versions of the $P_8|\ZZ_3$ discriminant, for $x^3+y^3+yz^2$ and $x^3+y^3-yz^2$.}
\label{Fnew}
\end{center}
\end{figure}

\medskip
$\mathbf{C_3^{(2,4)}}$. The discriminant coincides with the standard $C_3$ discriminant. Its smooth stratum is $2A_1$
and the cuspidal is $A_3$. The latter is nearly $A_1^{(4)}$ of Example (a), but with the order 4 symmetry changing the sign of one of the squared
variables, hence the self-intersection of a vanishing $\chi$-cycle $e_{A_3}$ is $-4$, while its operator $h_{A_3}$ has eigenvalue $-\ov\chi=\chi$.
Additionally, $\left<e_{2A_1},e_{2A_1}\right>$ is also $-4$ and $h_{2A_1}$ is an involution.

A distinguished set of vanishing $\chi$-cycles consists of  $e_{2A_1,1},e_{2A_1,2}, e_{2A_1}$.
The singularities of the discriminant guarantee that the set may be chosen so that the braiding relations between the Picard-Lefschetz
operators are exactly those encoded in the diagram of Figure \ref{Fdi}. These imply
$$
\left<e_{2A_1,1}, e_{2A_1}\right> = 0 \qquad {\rm and} \qquad
|\left<e_{2A_1,1},e_{2A_1,2}\right>| = |\left<e_{2A_1,2}, e_{2A_1}\right>| = 2\sqrt2\,.
$$
Since the intersection numbers are in $\ZZ[1,i]$, we can make
$\left<e_{2A_1,1},e_{2A_1,2}\right> = \left<e_{2A_1,2}, e_{2A_1}\right> = 2(1-i)$ by possible multiplication of the cycles by powers of $i$.

{\em Remark.} We prefer calling this singularity $C$ rather than $B$ since it has only one vanishing cycle coming from more
than one critical point on one level (cf. \cite{Ab}).

\medskip
$\mathbf{P_8/\Z_6}$. The discriminant of the versal family $x^3+y^3+z^3 + \a y + \b z$ consists of 3 lines corresponding to the
three divisors $\a y + \b z$ of the cubic form $y^3 + z^3$. Each of the lines is a $2A_1^{(3)}$ stratum: each $A_1^{(3)}$ singularity
is off the origin and has symmetry $g^2: (x,y,z) \mapsto (\ov\w x,y,z)$. Hence all vanishing $\chi$-cycles have self-intersection $-6$.
Any Picard-Lefschetz operator has eigenvalue $\l = \ov\chi^2 = \chi$.

A distinguished set of $\chi$-cycles consists of 3 elements.
Since the rank of the intersection form on $H_{2,\chi}$ is 1, the absolute value of the intersection number of any pair is 6.
Since the number itself is in $\ZZ[1,\w]$, we can use the ambiguities in the choice of the cycles and make
$\left<e_1,e_2\right> = \left<e_2,e_3\right> = -6\w$. The rank condition implies that
$\left<e_3,e_1\right>$ is also $-6\w$.

{\em Remark.} It is possible to show that the relation between the cycles may be assumed to be $e_1+e_2+e_3=0$.

\medskip
$\mathbf{P_8/\Z_4}$. The discriminant of the versal family $x^2z + xy^2 + z^3 + \a x + \b yz$ has 3 strata:
$$
A_4: \a = 0\,, \qquad 2A_2: \b = 0 \,, \qquad 4A_1: 4\a + \b^2 =0\,.
$$
The $A_4$ degeneration reduces to the $A_4/\ZZ_4$ singularity of Example (c). Hence the self-intersection of $e_{A_4}$ is $-4$.
By the quasi-homogeneity, we see that the operator $h_{A_4}$ coincides locally with $g$ and thus has eigenvalue $\chi$ on $H_{2,\chi}$.
Similarly, each of the two $A_2$ degenerations is the $g^2$-symmetric singularity $A_2/\ZZ_2$.
Therefore, the self-intersection of $e_{2A_2}$ is also $-4$, while $h_{2A_2}$ is an involution. It is clear
that  a $\chi$-cycle vanishing at $4A_1$ has the self-intersection $-8$, and $h_{4A_1}$ is also an involution.

A distinguished set of vanishing $\chi$-cycles contains one cycle of each of the three kinds. Since the rank of the $H_{2,\chi}$ is
2, the $3 \times 3$ intersection matrix must have rank 1. Due to this, after possible multiplication of the cycles by powers of $i$,
we can assume that
$$
\left<e_{4A_1},e_{A_4}\right> = \left<e_{4A_1},e_{2A_2}\right> = 4(1+i)\,.
$$
This forces $\left<e_{A_4},e_{2A_2}\right> = -4$.

{\em Remark.} The relation between the cycles may be assumed to be either
$e_{A_4}+e_{4A_1}+ ie_{2A_2}=0$ or $ie_{A_4}+e_{4A_1}+ e_{2A_2}=0$.  \hfill{$\Box$}

\section{Affine monodromy}

We start with a reminder how a corank 1 semi-definite hermitian form $\wt q$ on $\wt V = \C^{n+1}$ defines an affine
reflection group on a hyperplane in the dual space \cite{Bou,GM,X9}.

First of all, choose a basis $e_0',e_1,\dots,e_n$ in $\wt V$ so that $e'_0$ is in the kernel $K$ of the form. We denote
the span of the $e_{j>0}$ by $V$ and write $v$ for the $V$-component of $\wt v \in \wt V$: $\wt v = v_0 e_0' + v$.
Set $Q$ to be the matrix of the restriction $q = \wt q|_V$: $Q=(\wt q (e_i,e_j))_{i,j>0}$.

In the dual space $\wt V^* \simeq K^* \oplus V$, we use coordinates $\a_0,\a_1,\dots,\a_n$ so that a linear functional
$\wt \a$ on $\wt V$ is written as
$$\wt \a (\wt v) = v_0\a_0 + v^T Q \a = v_0 \a_0 + q(v,\ov \a)\,.$$

Consider a reflection on $\wt V$ with a root $\wt u \notin K$ and the eigenvalue $\l$:
$$
A: \wt v \mapsto \wt v - (1-\l) \wt q (\wt v, \wt u) \wt u / \wt q (\wt u, \wt u)\,.
$$
Then its dual $A^*$ sends each hyperplane $\a_0=const$ in $\wt V^*$ into itself and on such a hyperplane it acts as
$$
\a \mapsto \a - (1 - \ov \l){\a_0u_0 + \ov q(\a,\ov u) \over \ov q(\ov u, \ov u)} \ov u\,,
$$
where $\ov q$ if the hermitian form on $V$ conjugate to $q$, with the matrix $\ov Q = Q^T$ in the chosen coordinates.
For $\a_0 \ne 0$, this is an affine reflection on the hyperplane $\G = \{\a_0 = const\} \simeq V$, with the root $\ov u$,
mirror $\wt a (\wt u) = \a_0 u_0 + \ov q (\a, \ov u) =0$ and eigenvalue $\ov\l \,.$ For $u_0=0$, the transformation is linear.

\bigskip
We are now ready to prove our main result

\begin{thm} Assume a symmetry of a function singularity $P_8$ on $\C^3$ splits the kernel of the intersection form $\s_h$
and $\chi$ is a kernel character. Assume the cohomology character subspace $H^2_{\chi}$ is at least of rank 2.
Let $\G \subset H^2_\chi$ be the hyperplane formed by all 2-cocycles taking a fixed non-zero value on a fixed generator
of the line $H_{2,\chi} \cap Ker(\s_h)$.
Then the equivariant monodromy acts on $\G$ as a complex crystallographic group. The correspondence between the symmetric
singularities and affine groups is given by Table \ref{Ta}.
\end{thm}

{\em Proof\/}.
Number the vanishing $\chi$-cycles of each diagram in Figure \ref{Fdi} from the left to the right
(the numbering of the last two vertices in $D_4^{(3)}$ is not important).
Omit the leftmost cycle $e_0$. In equivariant cases omit also $e_2$. It is easy to notice that
the remaining cycles may be multiplied by appropriate factors so that the encoded hermitian form becomes the negative
of the relevant form of Figure \ref{Fgr} while the orders of the related vertices coincide. Therefore,
taking the remaining vanishing cycles
as the basic vectors $e_{j>0}$ in $\wt V = H_{2,\chi}$ in the
introduction to this Section, we see that their Picard-Lefschetz operators $h_j$
generate the required Shephard-Todd group $L$ on $\G \subset H^2_{\chi} = \wt V^*$.

The next task is to obtain the translation lattices of the crystallographic groups.
The kernel of any intersection form in Figure \ref{Fdi} is spanned by the $e_0'=e_0 + {\mathbf a}$, where ${\mathbf a}$ is
a linear combination of the $e_{j>0}$. For example, in all $\tau =2$ cases, $\mathbf a$ is a non-zero multiple of $e_1$
(for equivariant singularities this is due to the two Remarks by the end of the previous Section).

Assume the order of a root of $L$
which generates the lattice of the required affine group coincides with the order of an operator $h_k$, $k>0$.
According to what has been said before the Theorem, its validity for such singularity will follow from
$\mathbf a$ being a multiple of an element of the $L$-orbit of $e_k$. Hence only $\tau > 2$ singularities are still to be checked.
And we have for them:
$$
\begin{array}{rcclcl}
C_3^{(3,3)}: & \mathbf a & = & (1-\ov\w, 1) & = & A_1A_2^2 e_1 \\

D_4^{(3)}:    & \mathbf a & = & (1-\ov\w,1,1)             & =  & A_1A_3^2A_1 e_2  \\
P_8|\ZZ_3:    & \mathbf a & = & (\ov\w - \w, -\w)    & =   & \ov\w A_2A_1^2 e_2 \\
C_3^{(2,4)}: & \mathbf a & = & (1+i,i) & = & iA_1^2e_2
\end{array}
$$
To make the calculations suitable for any of the two kernel characters, the $A_j$ here are either the Picard-Lefschetz
operators or their inverses, but always with the eigenvalues either $\w$ or $-1$ or $i$. The expressions obtained show that
the vectors $\mathbf a$ are maximal roots of the groups $L$ in the sense \cite{H}.
\hfill{$\Box$}

\bigskip
\noindent
{\bf Remark.} The multiplicities of vertices and edges in Figure \ref{Fdi} provide nearly complete presentations of all our rank $>1$
crystallographic groups as abstract groups. To obtain all defining relations, one should add:
\begin{itemize}
\item[(i)] $(h_1h_0h_2h_0)^2 = (h_0h_2h_0h_1)^2$ to the $P_8|\ZZ_3$ diagram of $[G(3,1,2)]_2$ (cf. \cite{M}), which corresponds
to one of the tangency lines in Figure \ref{Fnew}, right;
\item[(ii)] the condition that the classical monodromy (that is, the product of all the generators) of each
singularity has order 3. This is exactly the extra relation from \cite{M}.
\end{itemize}


TITLE: If and only if, which direction is which?
QUESTION [22 upvotes]: I can never figure out (because the English language is imprecise) which part of "if and only if" means which implication.
($A$ if and only if $B$) = $(A \iff B)$, but is the following correct:
($A$ only if $B$) = $(A \implies B)$
($A$ if $B$) = $(A \impliedby B)$
The trouble is, one never comes into contact with "$A$ if $B$" or "$A$ only if $B$" using those constructions in everyday common speech.

REPLY [0 votes]: IF-AND-ONLY IF statements have a double arrow symbol ($\Leftrightarrow$) indicating both direction of the statement is true no matter which direction you start from.
Eg: Former statement $\Leftrightarrow$ Latter statement.
IF is the Backward direction ($\Leftarrow$). That is, assume the Latter statement is true and prove the Former statement (often the easier part)
IF-AND-ONLY is the Forward direction ($\Rightarrow$). That is, assume the Former statement is true and prove the Latter statement (often the difficult part)


TITLE: For two commutative elements a,b of a finite order in a group G, why does there exist an order of order $LCM[|a|,|b|]$?
QUESTION [0 upvotes]: This is Q.43 of Chapter 4(Cyclic Groups) in Joseph A. Gallian's Contemporary Abstract Algebra.
I haven't been able to prove it myself but I also believe Gallian's solution is incorrect.
Joseph A. Gallian offered a proof as follows -

Let $|a|=m,|b|=b \ and \ gcd(m,n) = d$ Then $LCM[m,n] = \frac {mn}{d}, \ |{a}^d| = \frac {m}{d}, \ and \ |b| = n$ then by exercise 41 $|{a}^db| = LCM[m,n]$

Also, what we proved in exercise 41 was this -

$For \ a \ group \ G, \ a,b \in G$ such that $<a>\cap <b> = {e}$ where $e$ is the identity element of the group, then $|ab| = l.c.m[|a|,|b|]$

However, the Claim Gallian has used is certainly NOT true. We look at $G =\Bbb Z_{84}$
$$6,7 \in G$$
$$g.c.d(|6|,|7|) = g.c.d(14,12)= 2$$
Here, raising an element to the power of 2 would mean operating it with itself twice so if I am raising 7 to the power of 2 I just need to find $$2*7(mod 84)$$ which is 14.
However, the result doesn't hold at all because $$|14+6|=|20|=21$$ while it supposed to be $$L.C.M[|6|,|7|]=L.C.M[14,12] \ne 1$$
Have I actually found an error in Gallian?(No arrogance intended)
Edit : - I had accidentaly multiplied the elements in the add where I was supposed to add them. $$|14*6|=1|$$ has been changed to $$|14+6|=21$$

REPLY [2 votes]: Indeed, the argument you quoted is incorrect, for $d = \gcd(m,n)$, it does not follow that $\gcd\bigl(\frac{m}{d},n\bigr) = 1$. And we can't even heal it by instead taking $m$ and $\frac{n}{d}$ in case $\gcd\bigl(\frac{m}{d},n\bigr) > 1$, for an example consider $m = 12,\, n = 18$, then $\gcd\bigl(\frac{12}{6}, 18\bigr) = 2 > 1$ and $\gcd\bigl(12,\frac{18}{6}\bigr) = 3 > 1$.
Writing $m = d\mu$ and $n = d\nu$ with $\gcd(\mu,\nu) = 1$, we must split $d$ into two coprime factors $d_1,d_2$ such that $\gcd(d_1,\mu) = 1 = \gcd(d_2,\nu)$ and then look at $\frac{m}{d_1} = d_2\mu$ and $\frac{n}{d_2} = d_1\nu$. With such a factorisation, the order of $a^{d_1}b^{d_2}$ is $d_2\mu\cdot d_1\nu = \frac{mn}{d} = \operatorname{lcm}(m,n)$ since $\gcd(d_2\mu,d_1\nu) = 1$.


TITLE: how comes $s=4$ and $t=3$ for $4=7s-8t$
QUESTION [1 upvotes]: i am given these two problems:
$x\equiv 1 (\bmod 7 )$ and $x \equiv 5( \bmod 18)$
I tried this way:
$x\equiv 1 (\bmod 7 )$ is basically $x = 1 + 7s$ and $x\equiv 5 (\bmod 18 )$ is $x=5+18t$
then substituted x: $4 = 7s - 8t$
Now i am asked to use extended euclidean algorithm to solve this equation for $s$ and $t$ so that i can find $x$.
i tried this:

but i am getting $s=1$ and $t=-1$. what am i doing wrong? please help. the answer in the textbook is: $s=4$ and $t=3$

REPLY [1 votes]: You have solved $1=gcd(7,8)=7s-8t$ correctly with $s=1$ and $t=-1$, because $-7+8=1$.
Multiplying by $4$ we obtain $4=-4\cdot 7+4\cdot 8$. But this does not solve your simultaneous congruences for $x$ (and $s=4, t=3$ also do not solve it). Check the webpage for the Chinese remainder Theorem,
http://en.wikipedia.org/wiki/Chinese_remainder_theorem,
to see how the extended Euclidean algorithm yields the solution $x=113$, which is unique up to multiples of $7\cdot 18=126$. So $s=16$ and $t=6$.


TITLE: $\int_{0}^{\infty}\frac{\sqrt{x}}{1+x^2}\,dx$
QUESTION [0 upvotes]: I want to evaluate the integral $\displaystyle \int_{0}^{\infty}\frac{\sqrt{x}}{1+x^2}\,dx$ using complex analysis methods.
I know that I have to use a keyhole contour, but I don't know which function to integrate on the contour. For example if I had to integrate $\displaystyle \int_{0}^{\infty}\frac{\ln x}{1+x^2}\,dx$ then I would have chosen to integrate over a keyhole contour the function $\displaystyle f(z)=\frac{\ln^2 z}{1+z^2}\,dx$.
If i had known that function then the rest is routine, since if we declare the given integral as $I$ and the contour integral as $J$ , then there is a simple relationship of $I, \; J$. It holds that $J=2\pi i I $. Hence we have the result.
However, I did not understand  well the keyhole contour. If someone could show me how the other parts disappear I would be glad.

REPLY [2 votes]: Sometimes it is possible to remove branch cuts by using some clever (real) substitution.
In this case, by setting $x=y^2$ we have:
$$ I = \int_{0}^{+\infty}\frac{\sqrt{x}}{1+x^2}\,dx = \int_{\mathbb{R}}\frac{y^2}{1+y^4}\,dy$$
and the last integral can be computed thorugh the residue theorem by using a semicircle in the upper half-plane as a contour, leading to:
$$ I = 2\pi i\sum_{\xi\in\{\exp(\pi i/4),\exp(3\pi i/4)\}}\operatorname{Res}\left(\frac{1}{y^{-2}+y^2},y=\xi\right)=\color{red}{\frac{\pi}{\sqrt{2}}}.$$


TITLE: Is there a perfect Polish topology on the Baire space that is strictly finer than the usual topology?
QUESTION [3 upvotes]: The question is: is there a perfect Polish topology on the Baire space $\mathbb{N} ^ \mathbb{N}$ that is strictly finer than the usual topology on $\mathbb{N} ^ \mathbb{N}$? The usual topology is the product topology on $\mathbb{N} ^ \mathbb{N}$ with the discrete topology on $\mathbb{N}$.

REPLY [4 votes]: Yes, this is a corollary from Lemma 13.2 from “Classical Descriptive Set Theory” by Alexander Kechris.


text:
\begin{document}
\maketitle
\noindent
{\bf Abstract.}

\noindent
We prove the uniqueness of the viscosity solution to the Hamilton-Jacobi
equation associated with a Bolza problem of the Calculus of Variations,
assuming that the Lagrangian is autonomous, continuous, superlinear, and
satisfies the usual convexity hypothesis. Under the same assumptions we
prove also the uniqueness, in a class of lower semicontinuous functions,
of a slightly different notion of solution, where classical derivatives are
replaced only by subdifferentials. These results follow from a new comparison
theorem for lower semicontinuous viscosity supersolutions of the
Hamilton-Jacobi equation, that is proved in the general case of lower
semicontinuous Lagrangians.

\vspace{ 5 mm}

\noindent
{\bf Key words.} Discontinuous Lagrangians,
 Hamilton-Jacobi equations, viability theory, viscosity solutions.

\vspace{ 5 mm}

\noindent
{\bf AMS-MOS subject classification:} 49L20 (primary),
49L25 (secondary).

\section{Introduction}
Let us consider a {\em  Bolza problem\/} of the Calculus of
Variations
\begin{equation}\label{B}
\min \left\{ \int_{0}^{t} L (y (s),y' (s))ds \;+\;
\varphi (y(t)) : y \in W^{1,1} (0,t;\R^n),\; y(0)=x \right\},
\end{equation}
where the final cost $ \varphi \colon  \R^n \mapsto \R_+ \cup  \{ + \infty
\}$ is {\em  lower semicontinuous\/} and the Lagrangian
$L\colon  \R^n
\times \R^n \mapsto \R_+$  is {\em locally bounded\/} and {\em lower
semicontinuous\/}. We assume also that $L (x, \cdot )$
is {\em  convex\/} for every  $x \in \R^n$ and that the following
Tonelli type
{\em coercivity assumption\/} is satisfied: there exists
a function $\Theta \colon  \R^n \mapsto \R_+$ such that
\begin{equation} \label{T}
 \lim_{|u| \rightarrow \infty } \frac{ \Theta (u)}{|u|} =+
\infty \,, \qquad\forall \; (x,u)\in \R^n\times
\R^n,\;\; L (x,u) \geq \Theta ( u )\,.
\end{equation}
These assumptions guarantee the existence of absolutely continuous
minimizers (see, e.g., \cite{ces83}). The classical Lagrange problem (with the fixed final condition
$y (t)=z$) may be reduced to the form  (\ref{B}) by simply setting $
\varphi
(z):=0$ and $ \varphi := + \infty $ elsewhere.

The {\em value function\/}
$V \colon  \R_+ \times  \R^n \mapsto \R_+ \cup \{+ \infty \}$ for the Bolza
problem (\ref{B}) is
defined by
\begin{displaymath}
 V (t,x):= \min \left\{ \int_{0}^{t} L (y (s),y' (s))ds \;+\; \varphi
(y (t)) : y\in W^{1,1} (0,t;\R^n),\; y(0)=x \right\}.
\end{displaymath}
Under our assumptions on  $L$ and  $ \varphi $ the value function is  {\em
lower
semicontinuous\/} on  $\R_+ \times \R^n$ and
{\em
locally Lipschitz\/} on
$\R_+^{ \star } \times \R^n$, where $\R_+^{ \star }:=
\R_+ \setminus \{0\}$. Moreover it satisfies the  {\em initial condition\/}
\begin{displaymath}
 \forall \; x \in \R^n,\;\; \liminf_{h \rightarrow 0+,\; y \rightarrow x} V
(h,y)= \varphi  (x) \,.
\end{displaymath}

If  $V$ is smooth and  $L$ is continuous, then  $V$ is a classical solution
to the {\em  Hamilton-Jacobi equation\/}
\begin{equation}\label{H-J}
\cases{V_t+H(x,-V_x)=0& in $\R_+^{\star} \times \R^n$,
\cr\cr
V (0, \cdot )= \varphi& in $\R^n$,
\cr}
\end{equation}
with Hamiltonian $H\colon  \R^n
\times \R^n \mapsto \R$ defined by
\begin{equation}\label{H}
 H (x,p) := \sup_{u \in \scriptR^n} \left( \left\langle   p,u\right\rangle
 -L(x,u) \right),
\end{equation}
where  $ \left\langle   \cdot , \cdot \right\rangle $ denotes the scalar
product in  $\R^n$.
In other words, $H (x, \cdot )$ is the Legendre-Fenchel conjugate of $L (x,
\cdot )$.

It is  well known  that  (\ref{H-J}) may not have smooth solutions,
even if  $H$ and $\varphi $ are smooth. To overcome this difficulty,
different notions of
generalized
solutions have been proposed. The notion of viscosity
solution can be introduced by means of subdifferentials and
superdifferentials.

We recall that the
{\em
subdifferential\/}  $ \partial _-W (x)$ of
  a function $W\colon \R^n \mapsto \R \cup \{ +
\infty \}$ at a point  $x \in \mbox{\rm dom}(W)$ is defined by
\[
\partial_- W(x):=\left\{
p\in \R^n\;:\; \liminf_{y\to x}\frac{ W(y)-
W(x)-\langle p,y-x\rangle}{|y-x|} \geq 0\right\} .
\]
while the {\em superdifferential \/}  $ \partial _+ W (x)$ is defined by
$ \partial _+W (x):=- \partial _- (-W) (x)$.

A  {\em continuous function\/}
$W\colon  \R_+^{ \star } \times \R^n \mapsto \R$ is said to be a  {\em viscosity
solution to the Hamilton-Jacobi equation  (\ref{H-J})\/} if the following
conditions are satisfied:
\begin{equation}
 \label{hj1}
\forall \;  (t,x)\in\R_+^{ \star } \times \R^n, \; \forall \; (p_t,p_x) \in
\partial _-W (t,x),\;\; p_t + H (x,-p_x) \geq 0\,,
\end{equation}
\begin{equation}
 \label{hj2a}
\forall \; (t,x) \in\R_+^{ \star } \times \R^n, \; \forall \; (p_t,p_x) \in
\partial _+W (t,x),\;\; p_t + H (x,-p_x) \leq 0\,.
\end{equation}
In \cite{cl2} and \cite{cl2a} the uniqueness of a
bounded uniformly continuous viscosity solution of (\ref{H-J}) is proved
under some assumptions on  $H$, which imply in particular that
$H$ is continuous and  $H ( \cdot ,p)$ is uniformly continuous for
every $p \in\R^n$ . In
\cite[Theorem 4.5]{dm-fra} we proved the following result, that can be
applied also to unbounded solutions.
\begin{Theorem}
\label{dm1.1}
Assume that $L$ is continuous. Let  $W\colon\R_+^{ \star } \times \R^n
\mapsto \R_+$ be a locally Lipschitz viscosity solution of  (\ref{H-J})
which satisfies the initial condition
\begin{equation}\label{hj2}
\forall \; x \in \R^n,\;\; \liminf_{h \rightarrow 0+,y \rightarrow x} W
(h,y)= \varphi  (x) \,.
\end{equation}
Then  $W=V$ on  $\R_+^{ \star } \times \R^n $.
\end{Theorem}

To describe the new uniqueness results, we introduce the notion of {\em
contingent directional derivatives\/} of a function $W\colon \R^n \mapsto \R
\cup \{+ \infty \}$. These are defined, for every   $x \in \mbox{\rm
dom}(W)$ and for every  $u \in \R^n$, by
\begin{equation}\label{dirder}
D _{\uparrow } W (x) (u) :=
\liminf_{ \textstyle {h \rightarrow 0+ \atop v \rightarrow
u}} \frac{W (x+hv) - W (x)}{h} \;\;.
\end{equation}

The main result of this paper is the following theorem, which shows that
the value function  $V$ is the unique viscosity solution in the larger
class of continuous functions whose contingent derivatives satisfy the
following very weak assumptions:
\begin{equation}
 \label{new}
\forall \;  (t,x) \in \mbox{\rm dom}(W),\;\; t >0,\;\;D _{\uparrow }W (t,x)
(0,0) =0 \,,
\end{equation}
\begin{equation}
 \label{new1}
\forall \;  (t,x) \in \mbox{\rm dom}(W),\;\; t >0,\;\;\exists \;
u\in\R^{n}, \;\;D _{\uparrow }W (t,x) (-1,u) <+\infty\,.
\end{equation}
\begin{Theorem} \label{visc}
Assume that $L$ is continuous.
 Let $W \colon  \R_+^{ \star }
\times \R^n \mapsto \R_+ $ be a continuous viscosity solution of
(\ref{H-J}) which satisfies  (\ref{new}),  (\ref{new1}), and the initial
condition (\ref{hj2}). Then $W=V$ on  $\R_+^{ \star }
\times \R^n$.
\end{Theorem}

In  \cite[Theorem 4.4]{dm-fra} we considered also a different notion of
generalized solution and we proved the following uniqueness result in the
class of locally Lipschitz functions.
\begin{Theorem}
Assume that $L$ is continuous.
 Let $W \colon  \R_+^{ \star }
\times \R^n \mapsto \R_+ $ be a locally Lipschitz function which satisfies
the initial condition  (\ref{hj2}) and solves the Hamilton-Jacobi equation
(\ref{H-J}) in the following sense :
\begin{equation}
\label{dm-fra3}
 \forall \; (t,x) \in\R_+^{ \star } \times \R^n, \; \forall \; (p_t,p_x)
\in \partial _-W (t,x),\;\; p_t + H (x,-p_x) = 0 \,.
\end{equation}
 Then $W=V$ on $\R_+^{ \star } \times \R^n$.
\end{Theorem}

In this paper we shall prove the following result, which provides
uniqueness in the larger class of lower semicontinuous functions $W$
satisfying  (\ref{new}) and  (\ref{new1}).
\begin{Theorem} \label{dm-fra4}
Assume that $L$ is continuous.
 Let $W \colon  \R_+^{ \star }
\times \R^n \mapsto \R_+ \cup \{+ \infty \}$ be a lower semicontinuous function which
satisfies the initial condition  (\ref{hj2}), the technical conditions
(\ref{new}) and  (\ref{new1}), and
solves the Hamilton-Jacobi equation  (\ref{H-J}) in the sense of
(\ref{dm-fra3}).
Then $W=V$ on $\R_+^{ \star } \times \R^n$.
\end{Theorem}

In the proof of Theorem \ref{dm-fra4} we use the following comparison
result, which follows immediately from  \cite[Theorem 4.1, Remark 4.2 and
Proposition  4.3]{dm-fra}.
\begin{Theorem} \label{dm-fra5}
Assume that $L$ is continuous.
 Let $W \colon  \R_+^{ \star }
\times \R^n \mapsto \R_+ \cup \{ +
\infty \}$ be a lower semicontinuous function which
satisfies the initial condition  (\ref{hj2}). Suppose that  $W$ is a
subsolution of the Hamilton-Jacobi equation   (\ref{H-J}) in the following
sense:
\begin{equation} \label{viscsubsol}
\forall \; (t,x) \in\R_+^{ \star } \times \R^n, \; \forall \; (p_t,p_x)
\in \partial _-W (t,x),\;\; p_t + H (x,-p_x) \leq  0
\end{equation}
Then $W \leq V$ on $\R_+^{ \star } \times \R^n$.
\end{Theorem}

In the proof of Theorems \ref{visc} and \ref{dm-fra5} we use also a very
general comparison result (Theorem \ref{dm-fra1.6})
 for lower semicontinuous viscosity
supersolutions of the Hamilton-Jacobi equation  (\ref{H-J}). To our
knowledge the strongest result in this direction, dealing with possibly
discontinuous Lagrangians, is the following theorem proved in
\cite[Theorem
5.1]{dm-fra}, where the notion of supersolution is given by using
contingent inequalities.
\begin{Theorem}
\label{dm-fra1.5}
 Let $W \colon  \R_+
\times \R^n \mapsto \R_+ \cup \{ +
\infty \} $ be a lower semicontinuous function which
satisfies the initial condition  $W (0, \cdot )= \varphi $. Suppose that
$W$ is a supersolution of (\ref{H-J}) in the following sense :
$$ \forall \; (t,x) \in \mbox{\rm dom}(W),\;t>0, \; \exists \; u \in
\R^n,\;\; D _{\uparrow }W (t,x) (-1,u) \leq  -L (x,u) \,. $$
Then $W \geq V$ on $\R_+ \times \R^n$.
\end{Theorem}

The comparison result for viscosity supersolutions we are going to prove
is the following theorem, where we need the additional assumptions
(\ref{new}) and  (\ref{new1}).

\begin{Theorem}
\label{dm-fra1.6}
 Let $W \colon  \R_+
\times \R^n \mapsto \R_+ \cup \{ +
\infty \}$ be a lower semicontinuous function which
satisfies (\ref{new}) and  (\ref{new1}).
Suppose that $W (0, \cdot )= \varphi $ and  that  $W$ is a viscosity
supersolution of the Hamilton-Jacobi equation (\ref{H-J}), i.e., $W$
satisfies  (\ref{hj1}).
Then $W \geq V$ on $\R_+ \times \R^n$.
\end{Theorem}

\section{Preliminaries}

Let $K\subset \R^n$ be a nonempty subset and $x\in K$. The
contingent cone $T_K(x)$ to $K$ at $x$ is defined by
\[
v\in T_K(x) \; \Longleftrightarrow  \;\liminf_{h\to 0+}
\frac{\mbox{\rm dist}(x+hv,K)}{h}=0 \, .
\]
The negative polar cone $T^-$ to a subset $T\subset \R^n$ is given by
\[
T^- :=\{v\in \R^n\,:\;\forall \; w\in T\,,\;\langle v,w \rangle\leq 0\} \, .
\]
When $K$ is convex , then $\left[T_K (x) \right]^-$
coincides with the the usual normal cone $N_{K} (x)$ of convex analysis.

The epigraph ${\cal E}pi(W)$ of a function
$W\colon \R^n \mapsto \R \cup \{ +
\infty \}$ is defined by
\begin{displaymath}
 {\cal E}pi(W) := \left\{ (x,r)\in \R^n\times\R : r \geq W (x) \right\} .
\end{displaymath}
We shall

need the following
version of Rockafellar's result (see \cite{Rockafellar}).
\begin{Lemma}\label{Rockafellar}
Let $x\in \mbox{\rm dom}(W)$ and let
$(p,0)\in\left[T_{{\cal E}pi(W)}(x,W(x))\right]^-$ be such that
$p\neq 0$. Then there exist
$x_\varepsilon $ converging to $x$ (as $ \varepsilon  \rightarrow 0+$) and
\begin{displaymath}
 (p_\varepsilon,q_\varepsilon)\in\left[T
_{{\cal E}pi(W)}(x_\varepsilon,W(x_\varepsilon ))\right]^-
\end{displaymath}
converging to $ (p,0)$ as  $ \varepsilon  \rightarrow 0+$
such that $q_{ \varepsilon }<0$ for every $\varepsilon>0$.
\end{Lemma}

A closed subset $K$ of $\R^n$ is called a viability domain
of a set-valued map $G\colon K \leadsto  \R^n$ if for every $x\in K$
\[
G(x)\cap T_K(x)\neq\emptyset\,.
\]
The following statement summarizes several versions of the viability
theorem (see~\cite{Aubin}).

\begin{Theorem}[Viability]\label{viability}
Let $K\subset \R^n$ be a closed set and let $G\colon K\leadsto \R^n$ be
 an upper semicontinuous
set-valued map with compact convex values.
The following conditions are equivalent:
\begin{description}
\item[\mbox{\rm (a)}] $K$ is a viability domain of $G$;
\item[\mbox{\rm (b)}]  $G(x)\cap\overline {co}\,T_K(x)\neq\emptyset$
for every $x\in K$;
\item[\mbox{\rm (c)}]  for every $x\in K$ there exist
$ \varepsilon >0$ and a solution $y\colon {[0,\varepsilon[}\mapsto K$
to the
Cauchy problem
\end{description}
\begin{equation}\label{diG}
\cases{
y'(t)\in G(y(t)) \,,
\cr
y(0)=x \,.&
\cr}
\end{equation}
\end{Theorem}

The equivalence (a) $\Longleftrightarrow$ (b)
was proved in \cite{Ushakov}. This proof was simplified in
\cite[page~85]{Aubin}. The fact that (a)
$\Longleftrightarrow$ (c) was first proved by
Bebernes and Schuur in \cite{beb}. A proof can be found
in \cite{AubinCellina} or \cite{Aubin}.

\vspace{ 5 mm}
The next theorem allows to deal with some unbounded
set-valued maps with closed convex values. As usual,  $B_R$ denotes
the closed ball with centre $0$ and radius $R$.

\begin{Theorem}\label{viability0}
Let $K \subset \R^n$ be a closed set and let
$G\colon K\leadsto \R^n$ be an upper semicontinuous set-valued map
with closed convex values. We assume that for every $x \in K$ there
exists $R>0$ such that
for all small $h>0$
\begin{displaymath}
 \mbox{\rm dist}  (x+h G(x),K)= \mbox{\rm dist} (x+h (G (x) \cap
 B_{R}),K) \,.
\end{displaymath}
Then the following statements are equivalent:
\begin{description}
\item[\mbox{\rm (a)}]  $K$ is a viability domain of $G$;
\item[\mbox{\rm (b)}] for every $x\in K$ and
for every
$p \in \left[   T_K (x)\right]^-$
we have $\displaystyle
\inf_{u \in G(x)}\left\langle  p,u \right\rangle \leq 0$.
\end{description}
\end{Theorem}

Theorem~\ref{viability0} is a direct consequence of the following more
technical result, which will be crucial in the proof of
Theorem~\ref{dm-fra1.6}.

\begin{Theorem}\label{viability1}
Let $K \subset \R^n$ be a closed
set, let $x\in K$, and let $G\colon K\leadsto \R^n$ be a set-valued map
with non-empty closed convex values.
Assume that there exists $R>0$ such that
for all small $h>0$
\begin{equation}\label{dist}
 \mbox{\rm dist}  (x+h G(x),K)= \mbox{\rm dist} (x+h (G (x) \cap B_R),K)
 \,.
\end{equation}
Assume also that the support function, defined by
$$
 \sigma (y,p) := \sup_{u \in G (y)}
 \left\langle   p,u\right\rangle \,,
$$
satisfies the following upper semicontinuity condition at $x$:
for every $\varepsilon>0$ there exists a
neighbourhood $U$ of $x$ such that
\begin{equation}\label{usc}
\sigma(x,p)+\varepsilon|p|> \sigma(y,p)
\end{equation}
for every $y\in U\cap K$ and for every $p$ in the set
$$
N_{G(x)}(B_{R}):=\{p\in\R^{n}:
\exists\; u\in G(x)\cap B_{R},\  p\in N_{G(x)}(u)\}\,.
$$
Finally, assume that for every $y\in K$ in a suitable neighbourhood of
$x$ we have
\begin{equation}\label{geq0}
\sigma (y,-p) \geq 0
\end{equation}
for every
$p \in \left[   T_K (y)\right]^-$ such that
$ -p \in N_{G (x)} (B_{R}) $.
Then $G(x)\cap T_K(x) \cap B_R\neq\emptyset$.
\end{Theorem}

{}From the proof given below it follows that the same result
holds if $\left[   T_K (y)\right]^-$ is replaced by the set of
proximal normals to $K$ at~$y$.

\vspace{ 5 mm}

{\bf Proof} --- \hspace{ 2 mm} Let us
define the  function $g \colon  \R_{+}\mapsto \R_+$ by
\begin{displaymath}
 g (h) := \frac{1}{2}\mbox{\rm dist} (x+hG (x),K)^2.
\end{displaymath}
Observe that  $g$ is locally Lipschitz around zero and $g (0)=0$.
For all small $h >0$ let us consider
$u_{h} \in G (x) \cap B _R$ and $x_h \in K$ such that $\mbox{\rm dist}
(x+hG
(x),K)=
|x+hu_h -x_h|$.
Then $x_h\rightarrow x$ when $h \rightarrow 0+$.

Consider $h>0$ such that
$g' (h)$ does exist and fix any $u \in G (x)$.
Since
$G(x)$ is convex, for all nonnegative $h, \;k$ we have $ (h+k) G (x)=h G
(x)
+k G (x)$.
 Therefore
\begin{displaymath}
 g' (h) \leq \frac{1}{2}\lim_{k \rightarrow 0+} \frac{|x+ hu_{h}+ku
-x_h|^2
- |x+ hu_h -x_h|^2 }{k} = \left\langle p_h, u
\right\rangle ,
\end{displaymath}
where $p_h :=x+ hu_h -x_h$.
Consequently
\begin{equation}\label{gprime}
 g' (h) \leq \inf_{u \in G (x)}\left\langle   p_h, u\right\rangle =
-\sup_{u \in G (x)} \left\langle   -p_h,u\right\rangle  = -\sigma
(x,-p_h)\, .
\end{equation}

As $x_h$ is a point of $K$ with minimum distance from $x+hu_h$, the
vector $p_h$ is a proximal normal to $K$, therefore it
belongs to $\left[   T_K (x_h)\right]^-$.
On the other hand, $u_h$ is the
point of $G (x)$ with minimum distance from  $(x_{h}-x)/h$. Thus
$-p_h \in N_{G (x)} (u_h)\subset N_{G (x)} (B_{R})$. By (\ref{geq0})
we have
\begin{equation}\label{geq00}
-\sigma(x_{h},-p_{h})\le 0\,.
\end{equation}

By the uniform upper semicontinuity (\ref{usc}) of $\sigma$
for every $\varepsilon>0$ there exists $h_{\varepsilon}>0$
such that for $0<h<h_{\varepsilon}$
\begin{equation}\label{gpr}
-\sigma  \left( x, -p_h\right) \leq -\sigma \left(
x_h,-p_h\right) + |p_h| \varepsilon\leq
-\sigma \left( x_h,-p_h\right) + hR\varepsilon\, ,
\end{equation}
where the last inequality follows from the
fact that $|p_h| \leq hR$.
{}From (\ref{gprime}), (\ref{geq00}), and (\ref{gpr}) we obtain that
$
 g' (h) \leq hR \varepsilon
$
for every $h\in(0,h_{\varepsilon})$ at which the derivative $ g' (h)$
exists.

Integrating $g'$ we deduce that for $0<h<h_{\varepsilon}$
\begin{displaymath}
 g (h) \leq  \frac{Rh^2}{2}\varepsilon\,.
\end{displaymath}
This implies that for $0<h<h_{\varepsilon}$
\begin{displaymath}
\frac{\mbox{\rm dist} (x+hu_{h},K) }h=
\frac{\mbox{\rm dist} (x+hG (x),K)}h  \leq \sqrt{R\varepsilon}\, .
\end{displaymath}
Let $h_i \rightarrow 0+$ be a sequence such that  $u_{h_i}$ converges to
some $u \in G (x)$. From the very definition of contingent cone we deduce
that $u \in T_K(x)$.
$\; \; \Box$

\section{Proof of the comparison theorem}
This section is devoted to the proof of the new comparison theorem
for viscosity supersolutions.

\vspace{ 5 mm}

{\bf Proof of Theorem \ref{dm-fra1.6}} --- \hspace{ 2 mm}
We first claim that $H$ is locally bounded. Indeed for all $x\in \R^{n}$
we have $H (x,p) \geq -L (x,0)$, thus $H$ is locally bounded from below.
On the other hand,
$H (x,p) \leq \Theta^{*} (p)$, where  $\Theta^{*}$ denotes the
Legendre-Fenchel conjugate of  $\Theta$. Since the function  $\Theta$ has
a superlinear
growth,  the convex function $\Theta^{ * }$ takes
only finite values, so it is locally bounded. This shows that
$H$ is also locally bounded from above.

Consequently,
the function $H(x,\cdot)$ is
locally Lipschitz with respect to $p$, locally uniformly with respect to
$x$.
By (\ref{T}) $H(\cdot,p)$ is upper semicontinuous with respect to $x$.
These two properties together imply that for every $x\in \R^{n}$, for every
$M>0$, and for every $\varepsilon>0$ there exists a neighbourhood $U$
of $x$ such that
\begin{equation}\label{uscH}
\forall\; y\in U\,,\ \forall\; p\in B_{M}\,, \quad
H(x,p)+\varepsilon > H(y,p) \,.
\end{equation}

Let us define the set-valued map $G\colon  \R^n
\leadsto  \R \times \R^n \times \R$ with closed convex values by
\begin{displaymath}
  G (x):= \left\{  (-1,u,-L (x,u) - \rho ) :  \rho  \geq 0,\; u \in
\R^n \right\} .
\end{displaymath}
Let $K:= {\cal E}pi(W)$, let $ (t,x) \in \mbox{\rm dom}(W)$  with
$t>0$,
and let $z:=(t,x,W(t,x))$. We want
to show that all assumptions of Theorem \ref{viability1}
are satisfied (here
$z$ plays the role of $x$). To prove (\ref{dist})
we show that there exists $R>0$
such that for all small $h>0$
\begin{eqnarray*}
 &\exists \; u \in \R^n,\;\; | (-1,u,-L (x,u))| \leq R\, ,&
\\
 &\mbox{\rm dist}\left( (t,x,W(t,x)) + h (-1,u,-L (x,u)),K \right)=
 \mbox{\rm dist}\left(  (t,x,W(t,x))+h G (x),K \right)\,. &
\end{eqnarray*}
As $W\ge0$, it is easy to see that for every $h>0$
there exist $ u_h \in \R^n$ and
$z_h=(t_h,x_h, r_h)\in
\R \times \R^n \times \R$,
with $r_h\ge W(t_h,x_h)$,
such that
\begin{eqnarray*}
&|(t,x,W(t,x)) + h (-1,u_h, -L (x,u_h) ) - (t_h,x_h, r_h)| &
\\
&= \mbox{\rm dist} ((t,x,W(t,x))+hG (x), K)
\leq h (L (x,0)+1)\,. &
\end{eqnarray*}
If $W(t,x)-hL (x,u_h)-r_h>0$, by increasing $r_h$ we could make
$W(t,x)-hL (x,u_h)-r_h=0$, which would contradict the definition of
distance.
Therefore $W(t,x)-hL (x,u_h)-r_h\le 0$, hence
\begin{displaymath}
hL (x,u_h) +W (t_h,x_h) -W (t,x) \le
hL (x,u_h) +r_h -W (t,x) \leq h(L (x,0)+1)\,.
\end{displaymath}

We claim that $u_h$ is bounded for all $h>0$ small enough.
Assume by contradiction that there exist $h_i  \rightarrow  0+$ such that
$|u_{h_i}| \rightarrow \infty $.
\vspace{ 3 mm}

{\it Case 1.\/} Assume first that for a subsequence, still
 denoted by  $h_{i}$, we have $h_i|u_{h_i}| \geq c$ for some $c>0$.
Since $W(t_h,x_h)  \ge0$, we have
\begin{displaymath}
 h_iL (x,u_{h_i}) -W (t,x) \leq h_i (L (x,0) +1)\,;
\end{displaymath}
dividing by $h_i|u_{h_i}|$ and taking the limit we get
\begin{displaymath}
 \limsup_{i \rightarrow \infty } \frac{L (x,u_{h_i})}{|u_{h_i}|} <
 +\infty \, ,
\end{displaymath}
which contradicts  (\ref{T}).

\vspace{ 3 mm}

{\it Case 2.\/} It remains to consider the case $h_i |u_{h_i}|
\rightarrow 0$. Since
$$ \max \left\{ |t_h +h -t | ,\;\;  |x_h - x - hu_h| \right\} \leq h(L
(x,0)+1),$$
 we deduce that for some $v \in \R^n$ and for a subsequence, still
 denoted by  $h_i$,
 we have
\begin{displaymath}
\lim_{i\rightarrow  \infty } \frac{t_{h_i} - t}{h_i |u_{h_i}|} =0
\qquad\mbox{and}\qquad \lim_{i \rightarrow \infty } \frac{x_{h_i} -x}{h_i
|u_{h_i}|}=
v\,.
\end{displaymath}
Furthermore,
\begin{displaymath}
h_iL (x,u_{h_i}) +W (t_{h_i},x_{h_i})  -W (t,x) \leq h_i(L (x,0)+1)\,;
\end{displaymath}
dividing by $h_i|u_{h_i}|$ and taking the limit yields
\begin{displaymath}
 \lim_{i\rightarrow \infty } \frac{W (t_{h_i},x_{h_i})  -W (t,x)}{h_i
|u_{h_i}|} = - \infty \,.
\end{displaymath}
Hence $D _{\uparrow }W (t,x)  (0,v)=- \infty $, which implies
$D _{\uparrow }W (t,x)  (0,0)=- \infty $ (see \cite{af90sva}).
This contradicts (\ref{new}) and completes the proof of our
claim.

\vspace{ 3 mm}

Thus $|u_h| $ is uniformly bounded when $h>0$ is small.
As $L$ is locally bounded, there exists $R>0$ such that
$| (-1,u_h, -L (x,u_h))| \leq R$ for all small $h>0$.

Observe next that, if  $(p_t,p_x,q) \in N_{G (x)} (B _R)$, then $q\ge
0$. Moreover, if $q=0$, then $p_{x}=0$; if $q>0$,
then there exists $u\in  B _R$ such that
$ p_x/q  \in \partial_u L (x,u)$, the subdifferential of
$L(x,\cdot)$ at $u$. As $L$ is locally bounded, this implies that
there exists a
constant $M$ such that $p_x/q\in B_M$
for every
$(p_t,p_x,q) \in N_{G (x)} (B _R)$ with
$q> 0$.

To prove (\ref{usc}) it is enough to show that
for every $\varepsilon>0$ there exists a neighbourhood $U$ of $x$
such that
\begin{equation}\label{usc1}
\sup_{u\in\scriptR^{n}} (-p_t + \left\langle   p_x,u\right\rangle -qL (x,u)) +
\varepsilon |(p_t,p_x,q)|
\end{equation}
$$
> \sup_{u\in\scriptR^{n}} (-p_t + \left\langle   p_x,u\right\rangle -qL (y,u))
$$
for every $y\in U$ and for every $(p_t,p_x,q)\in N_{G (x)} (B _R)$.
If $q=0$, then $p_{x}=0$ and (\ref{usc1}) is trivial. If $q>0$, then
(\ref{usc1})
can be written as
$$
-p_t + qH(x,\frac{p_x}{q}) +
\varepsilon |(p_t,p_x,q)|>
-p_t + qH(y,\frac{p_x}{q})\,,
$$
which follows easily from (\ref{uscH}).

Let us check that (\ref{geq0}) holds true. Fix
$(s,y,r) \in {\cal E}pi (W)$, with $s>0$, and
$ (p_t,p_x,q) \in \left[  T_{ {\cal E}pi (W)} (s,y,r)\right]^- $.
Since $(0,0,1)\in T_{ {\cal E}pi (W)} (s,y,r)$, we have $ q \leq 0$.
Therefore (\ref{geq0}) is equivalent to
\begin{equation}\label{geq01}
\sup_{u\in\scriptR^{n}} (p_t + \left\langle
 - p_x,u\right\rangle +qL (y,u))\ge 0 \,.
\end{equation}
If $q<0$, then $(p_t/|q|,p_x/|q|,-1)\in \left[  T_{ {\cal E}pi (W)}
(s,y,r)\right]^-$, hence
$ (p_t/|q|,p_x/|q|) \in \partial_-W (s,y)$ (see \cite[page~249]{af90sva})
and we deduce (\ref{geq01}) from  (\ref{hj1}).
If $q=0$ and $p_x\neq0$, then the supremum in (\ref{geq01}) is
$+\infty$. If $q=0$ and $p_x=0$, then
$(p_t,0,0)\in \left[  T_{ {\cal E}pi (W)}
(s,y,r)\right]^-$. By (\ref{new1}) there exists $u\in \R^{n}$ such
that $z:=D _{\uparrow }W (s,y) (-1,u) <+\infty$.
As ${\cal E}pi (D _{\uparrow } W (s,y) ( \cdot,\cdot ) ) =
 T_{{\cal E}pi (W)} (s,y,W (s,y)) $, the vector
 $(-1,u, z)$ belongs to
 $T_{{\cal E}pi (W)} (s,y,W (s,y))$, which is contained in
 $T_{ {\cal E}pi (W)}(s,y,r)$. By the definition of $\left[  T_{ {\cal E}pi
(W)}
(s,y,r)\right]^-$ we obtain $-p_t\le 0$, which yields (\ref{geq01})
when $q=0$ and $p_x=0$.

{}From Theorem \ref{viability1} we deduce that
\begin{displaymath}
 G (x) \cap \left(  T_{ {\cal E}pi (W)} (t,x,W (t,x)) \right) \ne
 \emptyset\,.
\end{displaymath}
As ${\cal E}pi (D _{\uparrow } W (t,x) ( \cdot,\cdot ) ) =
 T_{{\cal E}pi (W)} (t,x,W (t,x))$, we obtain that
\begin{displaymath}
 \exists \; u \in \R^n,\;  D _{\uparrow }W (t,x) (-1,u) \leq -L (x,u) .
\end{displaymath}
This and Theorem  \ref{dm-fra1.5} imply that $W \geq V$ on  $ \R_+ \times
\R^n$. $\; \; \Box$

\begin{Remark}\label{rem}
{ \rm {}From the proof of Theorem \ref{dm-fra1.6} we see that condition
(\ref{new1}) yields, for $t>0$,
$$
 (p_t,0,0) \in \left[  T_{ {\cal E}pi (W)} (t,x,W(t,x))\right]^-
 \; \Longrightarrow  \;
 p_t  \ge 0  \,.
$$
If, in addition, the subsolution inequality
(\ref{viscsubsol}) is satisfied, and
$$(p_t,0,0) \in \left[  T_{ {\cal E}pi (W)} (t,x,W(t,x))\right]^-,$$
then by Rockafellar's Lemma \ref{Rockafellar} there exist
$ (t_i,x_i) \rightarrow (t,x)$
and $(p^i_{t},p^i_{x},q_i) \in
\left[   T_{{\cal E}pi (W)} (t_i,x_i, W(t_i,x_i)) \right]^-$,
with $q_i <0$, such that $(p^i_{t},p^i_{x},q_i) \rightarrow (p_t,0,0)$.
Then $(-p^i_t/q_i,-p^i_x/q_i) \in \partial_- W (t_i,x_i)$ and from
(\ref{viscsubsol}) we obtain $-p^i_t/q_i - L(x_i,0)\le 0$, hence
$p^i_t + q_i L(x_i,0)\le 0$. Taking the limit we get
$p_t \le 0$. This shows that (\ref{new1}) and (\ref{viscsubsol}) together imply
the following geometric condition for $t>0$:
$$
 (p_t,0,0) \in \left[  T_{ {\cal E}pi (W)} (t,x,W(t,x))\right]^-
 \; \Longrightarrow  \;
 p_t  = 0  \,.
$$}
\end{Remark}

\section{Proofs of the uniqueness results}
We begin with the proof of the uniqueness theorem for viscsity solutions.

\vspace{ 5 mm}

{\bf Proof of Theorem \ref{visc}} --- \hspace{ 2 mm} By Theorem
\ref{dm-fra1.6}
we have
 $W \geq V$. Recall
that the hypograph of $W$ is defined by
\begin{displaymath}
 {\cal H}yp (W):=\{ (t,x,r) \in \R_+^{\star} \times \R^n \times
\R : r \leq W (t,x)\}\, .
\end{displaymath}
Define the closed set
$$
K := {\cal H}yp (W) \cup ( \R_- \times \R^n \times \R )\, .
$$
We claim that for all $(t,x,r) \in K$
\begin{equation} \label{cd12}
\forall \; u \in \R^n,\;\; (-1,u, -L (x,u) ) \in \overline {co}
\;T_{K} (t,x,r)\, .
\end{equation}
It is enough to prove it in the case $t>0$ and $r=W (t,x)$, since the other
cases are evident.
Fix $u \in \R^n$. Then by (\ref{hj2a})
\begin{equation} \label{cd13}
 \forall \; (p_t,p_x) \in \partial_+ W (t,x),\; p_t + \left\langle
-p_x,u\right\rangle -L (x,u)  \leq 0 \,.
\end{equation}

We want to prove that
\begin{equation} \label{cd14}
\forall\; (-p_t,-p_x,q) \in \left[   T_{ {\cal H}yp (W)} (t,x,W
(t,x))\right]^- \;\;
p_t + \left\langle   -p_x,u\right\rangle -qL (x,u)  \leq 0\, .
\end{equation}
When $q >0$ we have
$(p_t/q,p_x/q) \in \partial_+ W (t,x)$, thus (\ref{cd14})
follows from (\ref{cd13}).
By Lemma  \ref{Rockafellar}, applied to $-W$, if
$(0,0,0)\ne (-p_t,-p_x,0) \in
\left[ T_{{\cal H}yp (W)}
(t,x, W (t,x)) \right] ^-$, then for some $ (t_i,x_i) \rightarrow (t,x)$
and $(-p^i_{t},-p^i_{x},q_i) \in
\left[   T_{{\cal H}yp (W)} (t_i,x_i, W(t_i,x_i)) \right]^-$,
with $q_i >0$, we have $(p^i_{t},p^i_{x},q_i) \rightarrow (p_t,p_x,0)$.
So
\begin{displaymath}
 p^i_{t} + \langle   -p^i_{x},u\rangle -q_iL (x_i,u)  \leq 0 .
\end{displaymath}
Taking the limit we get
$ p_t + \left\langle   -p_x,u\right\rangle  \leq 0$, which concludes
the proof of (\ref{cd14}).

By the separation theorem, (\ref{cd12}) follows from
(\ref{cd14}).  Since the lower set-valued  limit of contingent cones is
equal
to Clarke's tangent cone  (see for instance  \cite{af90sva}),
from the continuity of $L$ we deduce that, for all
$(t,x) \in \R_+^{ \star } \times \R^n,$
\begin{displaymath}
 (-1,u,-L (x,u)) \in C_{ {\cal H}yp(W)}  (t,x,W (t,x)) .
\end{displaymath}
Fix $\varepsilon >0$.
Then it is not difficult to check that
\begin{displaymath}
\forall \; u \in \R^n,\;\;  (-1,u,-L (x,u) -\varepsilon )
\in \mbox{\rm Int} \left(
C_{ {\cal H}yp(W)}
 (t,x,W (t,x)) \right) .
\end{displaymath}
By \cite[Proposition 13, p. 425]{AE} this yields
\begin{equation} \label{B++}
 \liminf_{h \rightarrow 0+, \,v \rightarrow u} \frac{W (t -h, x+hv)-W
(t,x)}{h}  \geq -L (x,u) -\varepsilon  .
\end{equation}

We  have to show that for all $ t>0, \; x \in \R^n$, $V (t,x) \geq
W (t,x)$. Let $y$  be a minimizer of the Bolza
problem (\ref{B}). Since $L$ is
continuous, $y' \in L^{ \infty } (0,t;\R^{n})$ by \cite{ambr}.
Consider a sequence of continuous functions $u_i\colon
[ 0,t] \mapsto \R^n$ which is bounded in $L^{ \infty } (0,t;\R^{n})$
and converges to $y'$ almost
everywhere in $[0,t]$, and let $t_i \rightarrow 0+$ and $x_i \rightarrow y
(t)$
be such that $W (t_i,x_i)\rightarrow \varphi (y (t))$.
Define
\begin{displaymath}
 y_i (s) :=  x_i - \int_{s}^{t-t_i} u_i ( \tau )d \tau,\;\; \forall \; s \in
[0,t-t_i] .
\end{displaymath}
and $y_i (s):=x_i$ for $s > t-t_i$.
Then $y_i$ converges to $y$ uniformly in $[0,t]$.
Fix  $i$ and set $ \psi (s):=W (t-s,y_i (s))$ for $s \in {[ 0,t-t_i]}$. By
(\ref{B++})
for every $s\in {[ 0,t-t_i[}$ we have
\begin{equation}
 \label{coine+}
\limsup_{h\to 0+ }  \frac{\psi  (s+h) -\psi  (s)}h \geq -L (y_i (s),  u_i (s))-\varepsilon .
\end{equation}
Consider  the system
\begin{displaymath}
\cases{
(\alpha ' (s),   z' (s) ) = (1, -L (  y_i (s),  u_i (s))
-\varepsilon), \;\; s  \geq 0\, &
\cr\cr
(\alpha  (0), z (0))  = (0, W (t, y_i (0)))\, .&
\cr}
\end{displaymath}
where we have set $u_i (s):=u_i (t)$ for all  $s \geq t-t_i$.
It has the unique solution
$$( \alpha  (s),z (s)) :=  \left( s, W (t, y _i (0)) -
\int_{0}^{s} L (y_i (
\tau ), u_i ( \tau )) d \tau - \varepsilon s \right) .$$
According to Theorem \ref{viability} and  (\ref{coine+}),
this solution is
viable in ${\cal H}yp ( \psi ) \cup ({[t-t_i, +\infty [} \times \R)$, i.e.,
for
all $s \in {[0,t-t_i]}$ we have
 $ ( \alpha  (s),z (s)) \in {\cal H}yp ( \psi )$. Thus for all $s \in
{[0,t-t_i]}$
$$W (t-s,y_i (s)) \geq  W (t,y_i (0)) - \int_{0}^{s} L ( y_i (
\tau ),   u_i ( \tau ))d \tau  - \varepsilon s\,.$$
In particular
\begin{displaymath}
 W (t, y_i (0)) \leq W (t_i,x_i) + \int_{0}^{t-t_i}  L ( y_i (
\tau ),   u_i ( \tau ))d \tau  + \varepsilon (t-t_i)\,.
\end{displaymath}
Since the functions $L$ and $W$ are continuous,
taking the limit  we
get
\begin{displaymath}
 W (t,x)- \varepsilon  t \leq \varphi (y (t)) + \int_{0}^{t} L ( y ( \tau
),   y' ( \tau ))d \tau V (t,x) \,.
\end{displaymath}
Finally, as $ \varepsilon  \rightarrow 0+$ we obtain $W (t,x) \leq V
(t,x)$.
$\; \; \Box$

\vspace{ 5 mm}

{\bf Proof of Theorem \ref{dm-fra4}} --- \hspace{ 2 mm}
The inequality $W\ge V$ is proved in Theorem \ref{dm-fra1.6},
while the inequality $W\le V$ follows from Theorem \ref{dm-fra5}.
$\; \; \Box$

\vspace{ 5 mm}

config: arxiv


TITLE: General solutions for a functional equation
QUESTION [1 upvotes]: I wonder if we can show the existence of solutions other than $f(x)=x$ for the functional equation
$$f(x+1)=f(x)+1,\text{    }f(x^3)=(f(x))^3$$
where f is a real-valued function on the real line.

REPLY [1 votes]: For $x=0,\pm 1$ we have $x^3=x$, so that the second equation implies $f(x)=f(x)^3$ and thus $f(x)=0,\pm 1$.  In order for $f(1)=f(0)+1 = f(-1)+2$, we must have $f(-1)=-1$, $f(0)=0$, and $f(1)=1$.  Thus by $f(x+1) = f(x)+1$ we must have $f(x)=x$ for all integers.
The second equation can be transformed to $f(x)=f(x^{1/3})^3$.  Combined with the above observation, this implies that for any $x$ which is a cube root of an integer, $f(x)=x$.  The using the first equation again, any $x$ which is a cube root of an integer mod 1 (i.e. $x=n^{1/3}-\lfloor n^{1/3}\rfloor$ for some integer $n$) must also have $f(x)=x$.
Now cube roots of integers mod 1 are dense in the interval $[0,1]$, and thus $f(x)=x$ on a dense set of $[0,1]$, and by the first equation the same holds on a dense set of $\mathbb{R}$.  So if you require continuity for $f$, then $f(x)=x$ is the only solution.
If you don't require continuity, then there will be other solutions.  This is analogous to how the Cauchy equation has uncountably many solutions if you don't impose some condition like continuity: by iterating the two equations defining $f$ (i.e. adding/subtracting 1 and taking cubes or cube roots), you can only reach a countable number of points on the real line.  Without imposing continuity, all the other points are unconstrained.  So there are uncountably many solutions, but giving a closed-form description of any of the non-trivial ones is difficult.