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\begin{document}

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\begin{titlepage}
\title{
{\bf  Relation Between Holonomy}\\[.3cm]
{\bf Groups in Superstrings, M and F-theories}

\author{Adil Belhaj$^{1,2,3}$\thanks{\tt{belhaj@uniar.es}; GNPHE/0808}, Luis  J. Boya$^2$\thanks{\tt{luisjo@uniar.es}},
Antonio Segui$^2$\thanks{\tt{segui@uniar.es}}\\
{ \small $^1$ Centre National de l'Energie, des Sciences et des Techniques Nucl\'eaires,  CNESTEN} \\
{\small Rabat, Morocco} \\{ \small $^2$ Departamento de F\'{\i}sica
Te\'orica,
Universidad de  Zaragoza} \\
{ \small E-50009-Zaragoza, Spain}
\\
{\small $^3$  GNPHE, Point  focal:  Rabat, Morocco}
\\
 } } \maketitle
\thispagestyle{empty}
\begin{abstract}
We  consider   manifolds with special holonomy groups $SU(3)$, $G_2$ and $Spin(7)$
as suitable for  compactification  of superstrings, M-theory
and F-theory (with only one time)  respectively.    The  relations  of these groups with
 the octonions are discussed,  reinforcing their  role  in  the physics of string theory
  and duality. We also  exhibit three triple exact sequences   explaining the
 connections between the mentioned  special holonomy groups.
\end{abstract}
{\tt  KEYWORDS}:  Superstrings, M-theory, F-theory, Duality, Octonions,  Holonomy Groups.
\end{titlepage}

\newpage
\tableofcontents
\newpage
\section{Introduction}
 Once  one includes strings and/or other extended objects, extra dimensions  became  unavoidable,
 and thence the  necessity  of dimensional  reduction, introducing tiny compactifying  spaces, as
  we see only four extended dimensions. Over the few past years, there has been an increasing
   interest in studying duality in superstring theory  and supersymmetric  models, related to compactification.
   Interesting examples are mirror symmetry  in four dimensions between pairs of Calabi-Yau
threefolds in type II superstrings \cite{vafa} and  their strong/weak coupling dualities
    with  heterotic superstrings on $K3\times T^2$  \cite{KV}.
 The most important consequence
of the study of string duality is that all (five) superstring models are equivalent in the sense that
they correspond to different limits in the moduli space of the same theory, called M-theory  by Witten
\cite{witten,townsend,PT,adil1,adil2}. M-theory   is considered nowadays as the best candidate for the
unification
of the weak and strong coupling sectors of superstring models, and is described, at low energies,
 by eleven dimensional supergravity theory. In this paper, we shall also consider F-theory
 in 12 dimensions, initiated  by C. Vafa \cite{vafaft},  with metric of signature (2,10)
  (the original idea)  but also (1,11).

To preserve adequate supersymmetry (Susy) in our  $4$d  these intermediate spaces have to
 behave as special holonomy manifolds. In particular, to obtain ${\cal N}=1$ supersymmetry down
  to four dimensions, which is necessary to hold a phenomenologically  acceptable  chiral
    theory, we need a manifold with $G_2$ holonomy if coming from 11d M-theory
    \cite{witten,townsend,PT,Duff,AW,adil1,adil2,adil3,adil4}, or  a Calabi-Yau three-fold
     with SU(3)  holonomy if starting from heterotic superstrings  \cite{witten1}.
 The first case is due to the fact that the $G_2$  group is the maximal
  subgroup of $SO(7)$, for which the
eight dimensional spinor representation of $SO(7)$ can be decomposed as the fundamental representation
of $G_2$ and one singlet.  In the second case, as $Spin(6)=SU(4)$, the  $SU(3)$
subgroup shows again  the $4=1+3$ reducible  representation, guaranteeing ${\cal N}=1$  down to $4d$;
 the further break, e.g,  from one factor of  $E_8 \times  E_8$  to $E_6 \times SU(3)$  is well studied for example in
   \cite{witten1}.  Now the conventional F-theory of  Vafa \cite{vafaft} has signature $(2,10)$;
    one can consider however an F-theory with conventional metric $(1,11)$.  In this case, the holonomy
     of the compactifying  manifold might be $Spin(7)$, the largest of the exceptional holonomy
    groups. The alternative case  is $SU(4)$  corresponding to Calabi-Yau four-folds.
     Very recently, a GUT realization has  been given in the context of F-theory for such
     manifolds with elliptic K3 fibration
      \cite{BHV1,vafa08}.

As a whole,  there are many ways to get four dimensional models using different compactifications as intermediates.
These constructions could be related to each other  due to  different
  sort of dualities, which appear in the process. As an example, we point out here several
   equivalences in seven dimensions giving rise to a web of dualities (with F and M for F-theory,
   M-theory, \ldots) \cite{vafa}
\begin{equation}
F/K3 \times S^1\sim het/T^3\sim  IIB/S^2\times S^1\sim M/K3\sim IIA/S^2\times S^1.
\end{equation}
This   relation of special  holonomy manifolds and dualities  can be pursued, of course,
 to lower dimensions. In particular, one might naturally ask  for similar     relations
 in  four dimensions involving the special holonomy manifold compactifications previously
 mentioned. In this work, we  address  the problem of dualities  and semi-realistic compactifications,
 as regards the different holonomy groups.  We shall focus mainly in three groups, $SU(3)$,
 $G_2$ and $Spin(7)$ suitable for superstrings, M-theory and F-theory. We shall not be so
  much concerned with the manifolds themselves \cite{Joyce}. The relation of the above groups  with
   octonions should   be apparent; we shall devote some space to study it. We shall
    exhibit also some exact triple sequences, which we believe illuminate the relations between these groups and several subgroups.

The organization of this paper   is as follows. In section 2 we recall the classification
 of special holonomy manifolds (Berger 1955). In section 3, we review different ways of
  constructing    four dimensional models with minimal number of supercharges  from  higher dimensional
   supersymmetric theories.  The exact sequences mentioned above are explained in section 4.
       Background on toroidal  M-theory compactification,  division algebras and related group manifolds
        are  included in the two appendices.
\section{Special holonomy manifolds}
\subsection{Generalities on holonomy groups}
Extended objects, unification of forces and supersymmetry, all suggest extra dimensions
 of space-time. However we see only   $4= (1+3) $ dimensions: Some mechanism has to be advocated to
 prevent  the extra size  of the space to be visible. The compactification is the most
 accepted ingredient, namely making the extra  dimensions too small to be observable. In the original
 Kaluza-Klein type of theories, the observable  gauge group  in 4 dimensions came from
the {\it isometry} groups of the compactifying space (this is why
the U(1)  gauge group
  of electromagnetism came from circle compactification).  But when supersymmetry is present,
it was realized in the early 80s  that  the {\it holonomy} groups of
intermediate spaces  respond of the number of supercharges surviving
in four dimensions \cite{awada83}.

Here  we recall briefly the classification of  special holonomy groups and manifolds
 of Berger (1955) in a form suitable for all later physical
 consideration\footnote{ Note that
the book of Joyce \cite{Joyce} is the best modern reference  for
this subject.}.
 Let  $\cal M$ be any  $n$ dimensional differentiable manifold. The structure group of the tangent bundle
is a subgroup of the general linear group, $GL(n,R)$. Now the
maximal compact group of the linear group is $O(n)$. So the quotient
homogenous space $ \frac{GL(n,R)} {O(n)}$ is a contractible space;
hence, a manifold admits always a Riemannian metric $g$, whose  tangent
structure  group  is (a subgroup of) the orthogonal group.  The isometry
group $Iso(\cal M)$  is the set of diffeomorphisms  $\sigma$ leaving
$g$ invariant. For spheres we have $Iso(S^n)=O(n+1)$; for torii  $Iso(T^n)=U(1)^n$.

For an arbitrary $n$ dimensional Riemannian manifold $\cal M$, the
structure group of the tangent  bundle is, as said, a subgroup of
$O(n)$. Carrying a orthonormal frame $\epsilon$ of $n$ vectors in a
point $P$ through  a closed loop $\gamma$ in the manifold
\begin{equation}
\gamma: P \to P'\to P
\end{equation}
it becomes another frame $\epsilon'=o \cdot \epsilon $  which is  shifted by certain  element $o$ of $O(n)$.
This is the {\it holonomy element} of the loop. All elements of all possible loops on the manifold
 from $P$ make up
the holonomy group of the  manifold $Hol_P(\cal M)$,  which is  always a subgroup of $O(n)$, and is easily
 seen to be independent of the point $P$ for an arcwise-connected manifold.
 A generic Riemannian manifold  would have holonomy $O(n)$, or $SO(n)$ if it is orientable, whereas the isometry
 group generically is just the identity; in a way isometry and holonomy are complementary.

For any vector bundle with connection, the structure group reduces to the holonomy group (reduction theorem).
The corresponding Lie algebra of the holonomy group is generated by the curvature of the connection
(the Ambrose-Singer theorem)\cite{KN63}.

 \subsection{ Classification of special holonomy manifolds}
Only special groups can act as holonomy groups; the classification
of possible holonomy groups was  carried by M. Berger in 1955. If
the manifold is irreducible, $ Hol ({\cal M})$ should be in $ O(n)$.
Its Lie algebra, as we said, is generated by the curvature. For the
irreducible non symmetric case, there are three double series
 corresponding to the numbers $\mathbf{R},\mathbf{C}$ and $\mathbf{H}$, and two isolated cases
  related to the octonion numbers $\mathbf{O}$.  For  each number domain there are the  generic
 case and the unimodular subgroup restriction.  The classification is given in the following Table \cite{Boya1}
\begin{center}
\begin{tabular}{lll}
Numbers & Group & Unimodular Form \\
\hline \\
$\mathbf{R}$ & $O(n)$ & $SO(n)$ \\
& generic case & orientable, $w_{1}=0$ \\
$\mathbf{C}
$ & $U(n)$ & $SU(n)$ \\
& K\"{a}hler, $d\omega =0$ & Calabi-Yau, $c_{1}=0$ \\
$
{\mathbf{H}}
$ & $Sp(1) \times/_2 Sp(n)$ & $Sp(n)$ \\
& Quaternionic &  Hyperk\"{a}hler \\
$\mathbf{O}$ & $Spin(7)$ in $8d$ spaces & $G_{2}$ in $7d$ spaces \\
& $Oct(1)$ & Aut$\mathbf{O}$
\end{tabular}
\end{center}

We provide some explanations.  An arbitrary $n$-dimensional Riemannian manifold
$ \cal{M}$  has $O(n)$  as the maximal holonomy group. The obstruction
   to   orientability is measured  by the first  Stiefel-Witney class of the tangent bundle,
   $w_1({\cal M})=w_1(T{\cal M}) \in H^1( {\cal M} ,Z_2)$.\\
A  $n$-dimensional complex K\"{a}hler  manifold parameterized  by $z_i, i=1,...,n$  has
 a closed  regular real K\"{a}hler  two form  $\omega$ given in a local chart by
\begin{equation}
\omega=iw_{i{\bar j}}dz_i\wedge d{\bar z_{\bar j}}, \quad d\omega=0
\end{equation}
and its holonomy group is $U(n)$.  Now as $ \frac{U(n)}{SU(n)}=U(1)$, we have the diagram

\begin{equation}
\begin{tabular}{lllll}
$\;$$\;$SU(n) &  &  &  &  \\
$\quad$ $\;$ $\downarrow $ &  &  &  &  \\
$\;$ U(n) & $\longrightarrow $ & B & $\longrightarrow $ & M \\
det $\downarrow $ &  &  &  &  \\
$\;$$\;$U(1) & $\longrightarrow $ & B

\'{}

& $\longrightarrow $ & M
\end{tabular}
\end{equation}
where   the middle line  is the   frame bundle: $B$ is  the principal bundle of orthogonal unitary frames.
 The last bundle  is mapped  to an element of $H^2({\cal M}, Z)$; hence, the determinant map defines the
  first Chern class of the bundle as $c_1({\cal M}) \in H^2({\cal M},Z)$.
It turns out that when $c_1=0$, the K\"{a}hler manifold becomes a Calabi-Yau manifold  with
$SU(n)$  holonomy group and it is Ricci flat, $Ric=0$. Note that one-dimensional Calabi-Yau manifold  is
nothing but a complex elliptic curve. Then its Hodge diagram is given by
\def\m#1{\makebox[10pt]{$#1$}}
\begin{equation}
  {\arraycolsep=2pt
  \begin{array}{*{5}{c}}
    &&\m{h^{0,0}}&& \\ &\m{h^{1,0}}&&\m{h^{0,1}}& \\
    &&\m{h^{1,1}}&&
  \end{array}} \;=\;
  {\arraycolsep=2pt
  \begin{array}{*{5}{c}}
    &&\m1&& \\ &\m1&&\m1& \\ &&\m{1}&&
  \end{array}}
\end{equation}
The second example of   such geometries  is the  K3 surface with  $SU(2)$ as holonomy group.
Its  Hodge diagram reads \cite{Boyafest} as
\def\m#1{\makebox[10pt]{$#1$}}
\begin{equation}
  {\arraycolsep=2pt
  \begin{array}{*{5}{c}}
    &&\m{h^{0,0}}&& \\ &\m{h^{1,0}}&&\m{h^{0,1}}& \\
    \m{h^{2,0}}&&\m{h^{1,1}}&&\m{h^{0,2}} \\
    &\m{h^{2,1}}&&\m{h^{1,2}}& \\ &&\m{h^{2,2}}&&
  \end{array}} \;=\;
  {\arraycolsep=2pt
  \begin{array}{*{5}{c}}
    &&\m1&& \\ &\m0&&\m0& \\ \m1&&\m{20}&&\m{1.} \\
    &\m0&&\m0& \\ &&\m1&&
  \end{array}}
\end{equation}
Since $SU(2)=Sp(1)$, the K3  surface is also a hyperk\"{a}hler (Calabi-Yau) manifold.
 Notice that Hyperk\"{a}hler manifolds are also Calabi-Yau, but the quaternionic
 manifolds in general are not. Quaternionic manifolds have for holonomy
 $Sp(1) \times Sp(n)/Z_2$, abreviated as $Sp(1) \times/_2 Sp(n)$ in the Table.

Finally, the  two cases related to the octonions are  $G_2=Aut (Octonions)$  and $Spin(7)$.
The former is well known  and we shall elaborate on it later; as for the $"Oct(1)"$
label for the later, this  will also be justified in the Appendix $2$.

Note that, in general,   a manifold  with a specific holonomy  group $ Hol( {\cal M})=G$
 implies  the manifold carries an additional structure, preserved by the group $G$. For
 example, an orientable  manifold, with holonomy  within $SO(n)$,   has an invariant volume element, indeed
$SO(n)=O(n)\bigcap SL(n,R)$.  A  K\"{a}hler  manifold, with holonomy inside $U(n)$
has an invariant closed 2-form as $U(n)=O(2n)\bigcap Sp(n)$. A manifold with $G_2$ holonomy
 will carry an invariant $3$-form, etc.
\section{  Semi-realistic compactifications}
Consider first superstring  theory, which lives in ten dimensions
\cite{Schwarz}; down to four dimensions we want only ${\cal N} =1$,
i.e. four supercharges, as to allow for parity violation. We know
that compactification on  a $SU(n)$-holonomy manifold would reduce
the supercharges by a factor of $1/2^{n-1}$, so $SU(3)$-holonomy
(1.e., a Calabi-Yau $3$-fold) would be just right to descend from
the heterotic string ($16$ supercharges) to a four  dimensional model with only four superchages.  Indeed, the search for
CY$_3$ manifolds was a big industry in the $80$s \cite{Candelas}.
This choice is also natural, as  $SU(3) \subset SU(4)=Spin(6)
\rightarrow SO(6)$, and obviously $4=3+1$ leaves one surviving
spinor.

In M-Theory with $11$ dimensions the preferable compactifying manifold would be one with $G_2$ holonomy:
now the inclusions are $G_2 \subset SO(7) \leftarrow Spin(7)$, and $8=1+7$, as $2^{(7-1)/2}=8$, type real.
$G_2$-holonomy manifolds are also Ricci flat; they were first proposed for the M-Teory in \cite{PT}.
Indeed, $G_2$ holonomy manifolds would preserve $1/2^3$ supercharges, and in $11$d there are $2^{(11-1)/2}=32$,
type real again as $10-1=9 \equiv 1$ mod $8$.

We can also consider $8$d compactifying manifolds in at least two context: $1$) Descend $11 \rightarrow 3$ just
for illustrative purposes, and $2$) F-Theory with metric ($1,11$); the original sugestion of Vafa was $12=(2,10)$,
see \cite{vafaft, Boya2}. Here the manifolds of choice would be either CY$_4$, that is, $SU(4)$-holonomy manifolds,
preserving $4$ charges out of $32$ (which is what we want), or $Spin(7)$, the last of the exceptional
holonomy groups; $Spin(7)$ does the job as it has an irreducible $8$d representation, same as $Spin(8)$ and
$Spin(7) \subset Spin(8)$.

Finally, starting from the conventional F-Theory with signature $(2,10)$ it is necessary to compactify in
 a manifold with signature $(1,6)$. The following Table sums up the situation:

 \begin{center}
\begin{tabular}{lll}
Theory & \quad Dim Change &  \qquad Holonomy \\
\hline \\
Heter. String & $\quad 10d\longrightarrow 4d$ & $\qquad SU(3)$ (CY$_3$ manifold) \\
M-theory  & $\quad 11d\longrightarrow 4d$ & $\qquad G_2$ (Ricci flat) \\
M-theory  & $\quad 11d\longrightarrow 3d$ & $\qquad Spin(7)$ (Ricci flat) \\
F-theory $(1,11)$  & $\quad 12d\longrightarrow 4d$ & $\qquad Spin(7), SU(4) ($CY$_4)$ \\
F-theory $(2,10)$ & $\quad 12d\longrightarrow 4d$ & \qquad Indefinite form of $Spin(7)$
\end{tabular}
\end{center}
\bigskip

Let us  comment this table. There is clearly a double  inclusion
\begin{equation}
SU(3) \mbox{(dim 8, rep. 6)} \subset G_2 \mbox{(dim 14, rep. 7)} \subset Spin(7) \mbox{(dim 21, rep. 8)}
\end{equation}
linked with the dimensions $(10,11$ and $12)$ of the physical (!) theories.
It is remarkable  that these three groups have a neat relation with octonions, which we shall
elaborate now. Also, the related question arises: given the many duality relations extant in string
 (and M) theories,
is there any connection between the different holonomy groups? We shall take up this question
 in the next section $4$.

\subsection{ Relation with octonions}
The necessary properties we need of the division algebra of the octonions $\mathbf {O}$ are
 described in the Appendix $2$. Here we recall that $G_2$ is the automorphism group of
  the octonions (as $SO(3)$ is $Aut(\mathbf{H})$ and $Z_2=Aut(\mathbf{C})$ ); the reals
   $ \mathbf {R}$ have not autos, hence the representation $8$ of $G_2$ in the octonions
    split naturally in $8=1+7$. Note that  $G_2$ acts transitively in the $S^6$ sphere of unit imaginary octonions.
     This implies the $6$-sphere acquires a quasi-complex structure (Borel-Serre). The
     sequence  reads as follows
\begin{equation}
\label{su3}
SU(3) \rightarrow  G_2 \rightarrow S^6 \quad (8+6=14)
\end{equation}

Now the octonionic product preserved by $G_2$, as any
 algebra $( xy=z)$, defines  an invariant  $T^1_2$  tensor  and  the conservation of the norm is like
 preserving a quadratic form.  The $T^1_2$ tensor can be seen  then as a $T^0_3$  tensor. Now the
  alternating property of the octonionic product is equivalent to this $T^0_3$ tensor to  become a 3-form in $R^7$, $ \wedge T^0_3$, which is generic, (i.e., they make up an open set). This implies
\begin{equation}
\mbox{dim}\; G_2= \mbox{dim}\; GL(7,R)-\mbox{dim} \;\wedge T^0_3=49-35=14.
\end{equation}
Besides, the dual form $\wedge T^0_4$ is also invariant, implying  $G_2$ is unimodular, i.e.
lies inside $SO(7)$. The dimension $14$ of this $G_2$ can of course be proved directly \cite{Rosenfeld}.

As with respect to $Spin(7)$, it has a real $8$-dimensional
representation as we said, and hence it acts in $S^7$, indeed
transitively. The little group acts in the $S^6$ equator, and it
 is certainly $G_2$. In fact, there is some sense, as explained in the Appendix $2$, to call $G_2$ and $Spin(7)$
  respectively $SOct(1)$ and $Oct(1)$.

\section{Relation between holonomy groups}
We know that strings,  M and F theories are related by different sorts of dualities and dimensional
 reductions. As a consequence we expect that also the different holonomy groups used in the different
  compactifications should be connected.
The aim of this section is to adress this question  using   exact sequences and commutative diagrams
for these groups.  To start,  we  note the following. If $H \subset G$  with (left)-coset space $X$,
 we write $H \to G  \to  X$ for  $ G/H =  X$; when $H$ is normal, $X$ becomes the quotient group.
   Some of these diagrams   have been already  given  in \cite{Boya3}.\\
The  first diagram   that we present here  comes form the inclusion  of the exceptional holonomy group
$ G_2 \subset Spin(7)$. The later acts transitively in all units in $\mathbf{O}$ (octonions of norm one
 and octonions with imaginary part of norm one), whereas $G_2= Aut(\mathbf{O})$  obviously leaves $1$
  invariant (the real part of the octonion).
 So  the main cross of the diagram  takes the following form
\bigskip

\begin{equation}
\begin{tabular}{lllll}
&  & Spin(6) &  &  \\
&  & $\quad \downarrow $ &  &  \\
$G_{2}$ & $\longrightarrow $ & Spin(7) & $\longrightarrow $ & $S^{7}$ \\
&  & $\quad \downarrow $ &  &  \\
&  & $\quad S^{6}$ &  &
\end{tabular}
\end{equation}
where the vertical line is elemental\footnote{The spin groups acting  on  the natural spheres via the $SO$
 (covered) groups.}. With the $A_3=D_3$ isomorphism $Spin(6)=SU(4)$  and the fact that
  $ SU(3)\subset G_2\bigcap (Spin(6) =SU(4))$, we can complete the previous diagram.  The result is given by
\begin{equation}
\label{seq}
\begin{tabular}{llllll}
$SU(3)$ & $\longrightarrow $ & $SU(4)$ & $=Spin(6)$ & $\longrightarrow $ & $
S^{7}$ \\
$\quad\downarrow $ &  &  & $\downarrow $ &  & $\parallel $ \\
$\quad G_{2}$ & $\longrightarrow $ &  & $Spin(7)$ & $\longrightarrow $ & $S^{7}$
\\
$\quad\downarrow $ &  &  & $\downarrow $ &  &  \\
$\quad S^{6}$ & === &  & $S^{6}$ &  &
\end{tabular}.
\end{equation}
From this picture   we  can see in particular the octonionic nature
of $SU(3)$. It is a group of
 automorphism of octonions, fixing  the product  $(ij)k$. There is a suspicion, still conjectural,
  that this is the reason why the gauge group of the strong forces  is  $SU(3)$ color.\\
To get  the second diagram,  we start by another  obvious cross, since   $Spin(7)$  is the (universal)
 double cover of  $SO(7)$.  In this way, we have the following diagram
\begin{equation}
\begin{tabular}{lllll}
&  & $Z_{2}$ &  &  \\
&  & $\downarrow $ &  &  \\
$G_{2}$ & $\longrightarrow $ & $Spin(7)$ & $\longrightarrow $ & $S^{7}$ \\
&  & $\downarrow $ &  &  \\
&  & $SO(7)$ &  &
\end{tabular}.
\end{equation}
It is  known  that $G_2$ does not  have a   centre, so $Z_2=Z_2$ must be  the upper row. The rest
 is easy  to complete  since  $S^7/Z_2$ is the real projective space $RP^7$.   We end up   with the
   following  picture
\begin{equation}
\begin{tabular}{lllll}
&  & $Z_{2}$ & $===$ & $Z_{2}$ \\
&  & $\downarrow $ &  & $\downarrow $ \\
$G_{2}$ & $\longrightarrow $ & $Spin(7)$ & $\longrightarrow $ & $S^{7}$ \\
$\parallel $ &  & $\downarrow $ &  & $\downarrow $ \\
$G_{2}$ & $\longrightarrow $ & $SO(7)$ & $\longrightarrow $ & $RP^{7}$
\end{tabular}.
\end{equation}
From this diagram one can  learn just the lower row, somewhat unexpected, until one sees the middle row.
The lower row is also a remainder that the orthogonal groups have torsion \cite{Borel}.

The  third, final,  diagram is obtained by  asking  the question that $Spin(7)$
 acts transitively and  isometrically in the seventh sphere $S^7$.  Indeed,  it must be a
 subgroup of $SO(8)$. What about the quotient (homogeneous space)?.  To answer this question,
   we   start  first   with the  the following incomplete cross
\begin{equation}
\begin{tabular}{lllll}
&  & $Spin(7)$ & $\longrightarrow $ & $S^{7}$ \\
&  & $\downarrow $ &  & $\parallel $ \\
$SO(7)$ & $\longrightarrow $ & $SO(8)$ & $\longrightarrow $ & $S^{7}$ \\
&  & $\downarrow $ &  &  \\
&  & $??$ &  &
\end{tabular}
\end{equation}
and then  try to  finish it. Indeed,  $G_2$ lies inside  both  $Spin(7)$  and  $SO(7)$,
 then it must be their intersection and must appear in the upper left corner. The rest
  of  the diagram   can be obtained easily, and the result is
\begin{equation}
\begin{tabular}{lllll}
$G_{2}$ & $\longrightarrow $ & $Spin(7)$ & $\longrightarrow $ & $S^{7}$ \\
$\downarrow $ &  & $\downarrow $ &  & $\parallel $ \\
$SO(7)$ & $\longrightarrow $ & $SO(8)$ & $\longrightarrow $ & $S^{7}$ \\
$\downarrow $ &  & $\downarrow $ &  &  \\
$RP^{7}$ & === & $RP^{7}$ &  &
\end{tabular}.
\end{equation}
The new result we learn is just the middle column.

This completes our study of the holonomy groups which are  suitable for the compactification
 of  superstrings, M, F-theories  respectively.   In particular,   we have studied     their
 relations  with the divisor algebra of octonions. This  may  reinforce   the role of the octonions
  in physics of  strings and higher dimensional  objects  moving  on manifolds with non trivial holonomy groups.  We have also
  found three triple exact sequences   explaining  some  links between  these  groups. One of
  the nice results that one gets from   the  diagrams is that one can also   see   the possible
  connections between the corresponding geometries. Indeed, from the following sequence of inclusions
\begin{equation}
SU(3)\longrightarrow G_2\longrightarrow S^6,
\end{equation}
one can see that the manifold with $G_2$ holonomy  can  be constructed  in terms of Calabi-Yau
 three folds with the $SU(3)$  holonomy group. This has been already discussed in \cite{Joyce}.
  We can suppose the same thing  for the manifold  with  $Spin(7)$  holonomy. It  can be
    constructed either from  manifold with   $G_2$ holonomy  or  Calabi-Yau fourfolds. This
     can be easily seen from the subdiagram (\ref{seq}).

\section{Appendix I: Toroidal Compactification}
 String theory lives in ten dimensions, maximal supergravity and M-theory  in eleven,
  the original F-theory of   Vafa in $12$d.  As we have seen, if we want to get models
  with minimal supercharges, we need compact manifolds with special holonomy groups.

   To complete the study we give here some information about toroidal compactification.
    In particular we consider the case of M-theory.  We start by recalling the particle
     content from 11 dimensional supergravity down to 4 or 3 dimensions.  We suppose a
     step-by-step reduction, so the intermediate manifold  is always a circle. This was
     first found  by Cremmer (ca. 1980), who also showed that scalars make up a sigma model
      type of manifold, as  homogeneous quasi-euclidean spaces. In particular, M-theory compactified
      on $T^k$  has U-duality group $E_k(Z)$ and scalars taking
values in $\frac{E_k}{H_k}$, where $H_k$ is the maximal compact
subgroup of $E_k$.  In eleven dimensions, one has a graviton,
a 3-form gauge field and the gravitino.  We give  the moduli space of the
toridal compactification from this theory; starting in dim $9$, they are given in the
following Table
\begin{equation}
\begin{tabular}{ll}
d=9 & $\frac{SL(2,R)}{SO(2)}\times SO(1,1)$ ($3$ scalars) \\
d=8 & $\frac{SL(3,R)}{SO(3)}\times \frac{SL(2,R)}{SO(2)}$ ($7$ scalars) \\
d=7 & $\frac{SL(5,R)}{SO(5)}$  ($14$ scalars) \\
d=6 & $\frac{SO(5,5,R)}{SO(5)\times SO(5)}$ ($25$ scalars) \\
d=5 & $\frac{E_{6}}{USp(8)}$ ($42$ scalars) \\
d=4 & $\frac{E_{7}}{SU(8)}$ ($70$ scalars) \\
d=3 & $\frac{E_{8}}{SO(16)}$ ($128$ scalars)
\end{tabular}
\end{equation}

Notice for $d=3,4$ and $5$ the large group is "reconstructed" by a distinguished representation of the subgroup, as
\begin{eqnarray}
&E_8 \sim (adj+spin) \,  \mbox{of} \, O(16) : 120+128= 248 \\
&E_7 \sim (adj+[ 1^4]) \, \mbox{ of} \, SU(8) : 63+70= 133  \nonumber  \\
&E_6 \sim (adj+[ 1^4]') \, \mbox{of} \, Sp(4) : 36+(70-28)= 78 \nonumber
\end{eqnarray}

Beyond this, the descent from $E_6$ has two branches: the $E$-branch $E_6 \rightarrow D_5 \rightarrow A_4$ and
$Sp(4) \rightarrow Sp(2)^2 \rightarrow Sp(2)$ and the octonionic branch $E_8 \rightarrow E_7 \rightarrow E_6
\rightarrow F_4$ with subgroups $O(16) \rightarrow O(12) \times Sp(1) \rightarrow O(10) \times U(1) \rightarrow O(9)$.

\section{Appendix II: The octonions}
We recall here some properties of division algebras in relation with special holonomy
 groups and manifolds. Starting with the real numbers $\mathbf R$, the space $R^2$ becomes
  an algebra $i \equiv \{0,1 \}$ and $i^2=-1$: we get the complex number $\mathbf
  C$.
  It is a commutative and distributive division algebra. Adding a second unit
  $j$, $j^2=-1$ a third $ij$ is necessary, with $ij=-ji$, and we obtain the
   division algebra of quaternions $\mathbf H$ in $R^4$. It is anticommutative
    but still distributive. Adding another independent unit $k$ to $i$ and $j$,
     with $k^2=-1, ik=-ki, jk=-kj$, we have to complete with $e_7=(ij)k$ to the
      algebra of octonions $\mathbf O$ in $R^8$, with units $1,i,j,k;ij,jk,ki;(ij)k=-i(jk)$.
      It is neither commutative nor associative, but still a division algebra:
      if $o=u_0+ \Sigma_{i=1}^7 u_i e_i$ we have
\begin{equation}
\bar o:=u_0 -\Sigma_{i=1}^7 u_i e_i \qquad {\cal N}(o) = \mbox{norm}(o)\! := \bar o o;
\mbox{inverse}\, o^{-1}=\frac {\bar o}{{\cal N}(o)}.
\end{equation}

The associator $\{o_1,o_2,o_3 \}:=(o_1 o_2) o_3-o_1 (o_2 o_3)$ is
 completely antisymmetric. The four algebras
${\mathbf R}{\mathbf C}{\mathbf H}{\mathbf O}$ are composition algebras, that is, we have
 ${\cal N}(xy)={\cal N}(x) {\cal N}(y)$. The continuous
automorphism groups are easily seen to be
\be
\mbox{Aut}({\mathbf R})=1,\quad \mbox{Aut}({\mathbf C})=Z_2, \quad \mbox{Aut}({\mathbf H})=SO(3),
\quad \mbox{Aut}({\mathbf O}):= G_2.
\ee

The norm-one elements form, for ${\mathbf R}{\mathbf C}{\mathbf H}{\mathbf O}$, respectively
\be
O(1)=Z_2=S^0; \quad U(1)= SO(2)=S^1; \quad Sp(1)=SU(2)=Spin(3)=S^3; \quad \mbox {and}\, S^7.
\ee

Now $S^7$ has an invertible product structure, in particular is paralellizable,
 but is not a group, because nonassociativity. Let us name jokingly $'Oct(1)' =S^7$.
 One obtains a {\it bona fide} group by stabilizing $S^7$ by the octonion automorfism
 group $G_2$ \cite{Ramond}. The result is $Spin(7) \approx G_2 (\times S^7$; we shall name $Spin(7):=Oct(1)$.

We recall now the description of compact Lie groups as finitely twisted products
of odd dimensional spheres (Hopf 1941); for details see \cite{Boya4}. For example
in the quaternion case one gets the sequence
\begin{equation}
Sp(1)=Spin(3)=S^3, \quad Sp(1)^2=Spin(4)=S^3\times S^3,\quad Sp(2)=Spin(5)=S^3 (\times S^{7}.
\end{equation}
There are analogous results  for the  octonions, after $G_2$
stabilization.  The series goes up to dim $3$, but not beyond. This
is due to the  lack of associativity.
  We just write the results, adding the sphere exponents
\begin{eqnarray}
&G_2=SOct(1) \approx S^3 (\times S^{11}; \quad  Spin(8)=Oct(1)^2
\approx S^3 ( \times S^7 ( \times S^7 (\times S^{11} \\
&Spin(9):= Oct(2) \approx S^3 ( \times S^7 ( \times S^{11} (\times S^{15};\quad
F_4:=SOct(3) \approx S^3 ( \times S^{11} ( \times S^{15} (\times S^{23} \nonumber
\end{eqnarray}
where by the prefix "$S$" we mean the unimodular restriction (no $S^7$ factors).
This is similar to $SO$ and $SU$ for $\mathbf R$ and $\mathbf C$ respectively.
 The usefulness of the notation can be seen e.g. in the projective line and plane:
\begin{eqnarray}
&HP^1=S^4=Sp(2)/Sp(1)^2 \; \mbox {corresponds to} \; OP^1=S^8= Spin(9)/Spin(8), \\
&CP^2=S^5/S^1=SU(3)/SU(2) \; \mbox {corresponds to} \;
OP^2=SOct(3)/Oct(2)= F_4/Spin(9). \nonumber
\end{eqnarray}
The later is called the Moufang plane (Moufang 1933; to call it the Cayley plane is historically inaccurate).

In any case, this is just a notational convention, that we find useful, if carefully employed.
We finish by remarking that no use has been made so far of the fundamental {\it triality}
 property of the $O(8)$ group and the octonions, namely $Out[Spin(8)]\equiv Aut/Inner=S_3$,
 the order three symmetric group. Perhaps in a deeper analysis this triality will show up in particle physics.

{\bf Acknowledgments.}   AB  would like to thank  R. Ahl Laamara, B.
Belhorma, P.  Diaz,  L.B. Drissi, M. P. Garcia de  Moral,    J. Rasmussen, E.H. Saidi,A. Sebbar and M. B. Sedra for collaboration  on related subject.
He would also  thank  GNPHE and UFR-PHE-Rabat for hospitality (the end of 2007).
AS acknowledges discussions with P. Diaz, M.P. Garcia de Moral and E.H. Saidi.
This work has been supported by CICYT (grant FPA-2006-02315) and DGIID-DGA (grant 2007-E24/2), Spain.
We thank also the support by  Fisica de altas energias: Particulas, Cuerdas y Cosmologia, A9335/07.


TITLE: Rolling a dice until we get a 5 or a 6
QUESTION [0 upvotes]: I have some trouble with this problem:
I keep tossing a fair six-sided dice until I get a 5 or a 6. Let T be the number of times I roll the dice and X the result of the last toss that I do.

What is the value of $\Bbb{P}(T=3, X=5)$?
find the distribution of T
find the distribution of X
Are X and T independent?

For the first point I think that is $\Bbb{P}(T=3, X=5)$ = $(\frac46) (\frac46) (\frac26$) and I tried to do the second one in this way: $P(T=k)=(\frac23)^{k-1}(\frac13)$ , where $(\frac23)^{k-1}$ means that the first $k−1$ rolls of the dice gave a number other than 5 or 6, and the kth roll gave exacly 5 or 6.
From now on I really don't know what to do.

REPLY [1 votes]: $P(T=3, X = 5) = \left(\frac {4}{6}\right)\left(\frac {4}{6}\right)\left(\frac {1}{6}\right)$
Answer to the second question is correct.
The distriubtion of $x.$
$P(X=k)= \begin{cases} \frac 12 & k \in \{5,6\}\\ 0 & k\notin\{5,6\}\end{cases}$
Are they independent?  Should it matter how many times you rolled the dice whether you roll a $5$ or a $6$ on your next roll?


TITLE: What force causes the ball to move up?
QUESTION [12 upvotes]: Let's consider this super-simple example. The ball (mass $m$) is moving from position 1 to position 2 at constant horizontal velocity $v$ until the ball reaches the red arrow (in position 1, it has vertical velocity $0$). Gravity force is equal to $mg$.
When the ball reaches the point marked with the red arrow, it starts moving up, therefore its vertical speed is not $0$ anymore.
Its vertical speed has changed = there was vertical acceleration, so there had to be a non-zero net force in the vertical direction that acted on the ball, right? This is exactly what Newton's first law says.
My question is - what force caused the ball to accelerate vertically so it could reach point 2? What source caused that additional force that allowed it to move upward, up the hill?
In position 1, the gravity force acting on the ball is cancelled out by the ground reaction force, so the net force is zero. In position 2, the net force isn't zero anymore, directed 'down' (because gravity force is greater than ground reaction force acting on the ball). So we have two cases - in both the net vertical force is either zero or it is directed down. But it had to be directed up, because the ball moved up!
I know it can be explained with conservation of energy law, but I think there has to be an explanation in terms of forces and momentum only as well (the laws of dynamics are always true).

REPLY [0 votes]: I can start to answer your question with the following question:

If the vertical momentum does not change as the ball pass through this horizon, what happens?

The answer to this question is: the ball would cross the floor.
The key here is that exist a source of force that is the floor. This source is such that the ball would never cross the floor. The origin of this force is the geometric constraint of the particular system. Fundamentally, this force is only a idealization. Is actually a complicated system very close to electrostatic equilibria that resides in the structure of matter of the ball and the floor.
Ok. Let's answer your question now. When the ball are with some horizontal momentum in the flat floor the vertical momentum that this ball receives from the gravitational field is matched with the momentum of the floor necessary for obey the condition the ball can't cross the floor .As the floor start to be non-flat and get some curvature upwards, the setup of the floor's forces start to change in a way that the condition the ball would can't the floor be appreciated.
Let's understand this more closer. If we pick the starting point when the floor start to warp and take a small interval around this point we can approximate this warp to a tiny straight line with different angle. To the above condition be satisfied the momentum of the ball at this point needs to be aligned with the this straight line, and then, the floor need to supply an extra momentum in the vertical such that the net momentum, horizontal plus vertical, is aligned with the tiny warp. This angle is very tiny too in the initial of the warp, but hey increase as we pass to other points across the warp and the same analysis is the same for other points.
The floor do not delivers energy to the ball, so the kinetic energy can't be changed by this process. So, the magnitude of the momentum need to be preserved by this process. But, now, as this particle are getting vertical momentum upward, they start to getting upper. The gravitational field start to reduce his kinetic energy because of that and then, this ball can't getting upper indefinitely. If the gravitational field is not there, the above explanation about warp is yet valid.
Other way to see why the floor do not change the magnitude of the ball's momentum is that, by the warp discussion above, the instantaneous momentum is always perpendicular to the floor's force (the delivery of vertical momentum on the tiny straight wap), only changing the direction of the velocity. You can see that by thinking that at each point you can think the floor's force as a stretched rope tight to the ball fixed at some point.
Conclusion: is the floor that pushes the ball upward by conservation of your shape and not ghost property (the ball can't cross the floor) of this objects.


text:
\begin{document}

\title[Synchronization of Cellular Neural Networks]{Feedback Synchronization of FHN Cellular Neural Networks}

\author[L. Skrzypek]{Leslaw Skrzypek}
\address{Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA}
\email{skrzypek@usf.edu}
\thanks{}

\author[Y. You]{Yuncheng You}
\address{Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA}
\email{you@mail.usf.edu}
\thanks{}

\subjclass[2010]{34A33, 34D06, 37L60, 37N99, 92B20}

\date{June 18, 2020}

\keywords{Cellular neural network, discrete FitzHugh-Nagumo equations, feedback synchronization, dissipative dynamics}

\begin{abstract}
In this work we study the synchronization of ring-structured cellular neural networks modeled by the lattice FitzHugh-Nagumo equations with boundary feedback. Through the uniform estimates of solutions and the analysis of dissipative dynamics, the synchronization of this type neural networks is proved under the condition that the boundary gap signal exceeds the adjustable threshold.
\end{abstract}

\maketitle

\section{\textbf{Introduction}}

Cellular neural network (briefly CNN) was invented by Chua and Yang \cite{Chua1, Chua2}. CNN consists of network-like interacted processors of continuous-time analog or digital signals. In some sense the dynamics of CNN can be analyzed as a lattice dynamical system oftentimes generated by lattice differential equations in time \cite{Chua3, CR, S2}.

From the theoretical viewpoint, the theory of cellular neural networks and, more recently, of convolutional neural networks (also called CNN) and variants of complex neural networks is closely linked to discrete nonlinear partial differential equations and delay differential equations as well as the spatially discrete Fourier transform and constrained optimization \cite{Chow1, Chow2, Chua3}. In the dramatically broadened application fronts to machine leaning, deep learning, and general artificial intelligence, the most prominent area is the image processing especially in medical visualization techniques \cite{Chua2, CR, S1}.

Many complicated computational problems can be formulated as multi-layer and parallel tasks for processing signal values on a geometric grid with direct interaction and transmission in a local neighborhood. The cloning template of each cell model and the coupling design mimic biological pattern formation in the brain and nerves are the two features of CNN. The concepts and models of CNN are based on some aspects of neurobiology \cite{S2} in terms of sheet-like layers or arrays of massively interconnected excitable neurons and the implementation counterpart by electronic integrated circuit chips on the other hand.

Rapidly expanding applications of CNN and all kinds of complex neural networks \cite{CR, Hp, I, W} stimulate frontier and interdisciplinary researches from data analysis to mathematical and statistical modeling to deep learning algorithms, softwares, and robotics.

In this work, we consider a CNN model as the discrete replacement of the biological neural networks described by the partial diffusive FitzHugh-Nagumo equations with different coupling patterns \cite{FH, LY, S1, WLZ, Yong},
\begin{align*}
	\frac{\pdr u_i}{\pdr t} &= a \Gd u_i + f(u_i) - b w_i, \\
	\frac{\pdr w_i}{\pdr t} &= c u_i - \gd w_i.
\end{align*}
We shall use the discrete Laplacian templates to approximate the Laplacian operator on a 1D domai \cite{Chow1, S2}. For the prototype FitzHugh-Nagumo equations, the nonlinear term is
$$
	f(s) = s(s-\alpha)(1 - s)
$$
and the parameters $0 < \alpha < 1$ and $0 < c \ll 1$.

Consider a layer of cellular neural network, which consists of 1D cells at the grid points $\{ih: i = 1, 2, \cdots , n\}$ for $h > 0$, in a ring-structure. We shall study synchronization dynamics of the FitzHugh-Nagumo lattice equations:
\beq \bl{CN}
	\begin{split}
	\frac{dx_i}{dt} &= a (x_{i-1} - 2x_i + x_{i+1}) + f(x_i) - b y_i + p u_i, \quad 1 \leq i \leq n, \\
	\frac{dy_i}{dt} &= c x_i - \gd y_i, \quad 1 \leq i \leq n,
	\end{split}
\eeq
where $t > 0$, the integer $n \geq 4$, and the discrete Laplacian operator
$$
	D_i (x) = a (x_{i+1} - 2x_i + x_{i-1}),
$$
can be called the synaptic law of cell coupling. In this model of CNN, we impose the periodic boundary condition
\beq \bl{pbc}
	x_0(t) = x_n(t), \quad x_{n+1}(t) = x_1(t)
\eeq
and the boundary feedback control $\{u_i\}_{i=1}^n$,
\beq \bl{bfc}
	\begin{split}
	&u_1(t) = u_{n+1}(t) = x_n (t) - x_1(t),     \\
	&u_i (t) = 0, \quad 2 \leq i \leq n - 1, \\
	&u_{n}(t) = u_0(t) = x_1(t) - x_n(t).
	\end{split}
\eeq
for $t \geq 0$, where $p > 0$ is the controllable feedback constant and $x_n(t) - x_1(t)$ measures the boundary gap signal between the two endpoints of the cellular neural network. The initial conditions for the system \eqref{CN} are denoted by
\beq \bl{inc}
	x_i(0) = x_i^0 \in \mR \quad \text{and} \quad y_i (0) = y_i^0 \in \mR, \qquad 1 \leq i \leq n.
\eeq
All the parameters in this system \eqref{CN} are positive constants.

We make the following Assumption: The scalar function $f \in C^1 (\mR, \mR)$ satisfies
\beq \bl{Asp}
	\begin{split}
	&f(s) s \leq  - \gl s^4 + \beta, \quad s \in \mathbb{R}, \\
	& f^{\,\prime} (s) \leq \ga, \quad s \in \mathbb{R}, \\
	\end{split}
\eeq
where $\gl, \beta$ and $\ga$ are positive constants. The typical nonlinearity $f(s) = s(s - \alpha)(1 - s)$ shown above in the FitzHugh-Nagumo model satisfies the Assumption \eqref{Asp}:
\begin{align*}
	f(s)s &= - \alpha s^2 + (\alpha + 1)s^3- s^4 \leq -\alpha s^2 + \left(\frac{1}{2}s^4 + 2^3 (\alpha +1)^4\right) - s^4 \\
	&\leq - \left(\alpha s^2 + \frac{1}{2}s^4 \right) + 8(\alpha + 1)^4 \leq - \frac{1}{2} s^4 + 8(\alpha + 1)^4,  \\[3pt]
        f^{\, \prime} (s) &= - \alpha + 2(\alpha +1)s - 3s^2 \leq - \alpha + (\alpha +1)^2 - 2s^2 \leq 1 + \alpha + \alpha^2.
\end{align*}

Synchronization plays a significant role for biological neural networks and for the artificial neural networks as well. Fast synchronization may lead to enhanced functionality and performance of complex neural networks.

In recent years, the dynamical behavior and problems of complex and large-scale networks including convolutional neural networks in machine learning and deep learning, Internet networks, epidemic spreading networks, and social networks attract many interdisciplinary research interests, cf.\cite{A, I, W}. Synchronization for the CNN modeled by PDE, delay differential equations, or lattice differential equations is one of the essential topics in the theoretical analysis of artificial intelligence.

Synchronization for biological neural networks has been studied by several mathematical models and methods. This topic has been studied for the diffusive FitzHuigh-Nagumo networks of neurons coupled by clamped gap junctions \cite{AA, A, AC, IJ, Yong}, the mean field couplings of Hodgkin-Huxley and FitzHuigh-Nagumo neuron networks  \cite{Dick, QT}, and the chaotic neural networks and stochastic neural networks \cite{Dick, SG}.

Recently we proved results on the exponential synchronization of the boundary coupled Hindmarsh-Rose neural networks in \cite{PLY, PY} and the boundary coupled partly diffusive FitzHugh-Nagumo neural networks in \cite{LY}.

The feature of this work is to provide a sufficient condition for realization of the feedback synchronization of the proposed FitzHugh-Nagumo (FHN) cellular neural networks with boundary control. The quantitative threshold condition for synchronization is explicitly expressed in terms of the parameters and can be adjusted by the feedback strength coefficient $p$ in  applications.

\section{\textbf{Uniform Estimates and Dissipative Dynamics}}

Define the following Hilbert space:
$$
	H = \ell^2 (\mZ_n, \mR^{2n}) = \{ (x, y) = \{(x_i, y_i): 1 \leq i \leq n \}\}
$$
where $\mZ_n = \{1 ,2, \cdots, n\}$ and $n \geq 4$. The norm in $H$ is denoted and define by $\|(x, y)\|^2 = \|x\|^2 + \|y\|^2 =  \sum_{i=1}^n (| x_i  |^2 + |y_i |^2)$.
The inner-product of $H$ or $\mathbb{R}^n$ is denoted by $\inpt{\,\cdot , \cdot\,}$.

Since there exists a unique local solution in time of the initial value problem \eqref{CN}-\eqref{inc} under the Assumption \eqref{Asp} that the right-side functions in \eqref{CN} are locally Lipschitz continuous, in this section we shall first prove the global existence in time of the solutions in the space $H$. By the uniform estimates we show the dissipative dynamics of the solution semiflow.

\begin{theorem} \label{Tm}
Under the boundary feedback control \eqref{bfc}, for any given initial state $(x^0, y^0) = ((x_1^0, y_1^0), \cdots , (x_n^0, y_n^0)) \in H$, there exists a unique solution,
$(x(t), y(t)) = ((x_1 (t, x_1^0), y_1(t, y_1^0)), \cdots, (x_n (t, x_n^0)), y_n(t, y_n^0)), t \in [0, \infty)$,  of the initial value problem \eqref{CN}-\eqref{inc} for this cellular neural network.
\end{theorem}

\begin{proof}
Multiply the $x_i$-equation in \eqref{CN} by $C_1 x_i(t)$ for $1 \leq i \leq n$, where the constant $C_1 > 0$ is to be chosen, then sum them up and by the Assumption \eqref{Asp} to get
\begin{equation} \bl{u1}
	\begin{split}
	&\frac{C_1}{2} \frac{d}{dt} \sum_{i = 1}^n |x_i |^2 = C_1 \sum_{i=1}^n \left[ a(x_{i-1} -2x_i + x_{i+1}) x_i + f(x_i) x_i - b x_i y_i + p u_i x_i \right]   \\
	\leq &\, C_1 \sum_{i=1}^n a(x_{i-1} - 2x_i+ x_{i+1}) x_i  \\
	+ &\, C_1 \sum_{i=1}^n \left[- \gl |x_i |^4 + \beta + \frac{b}{2}\, |x_i|^2 + \frac{b}{2}\, |y_i |^2 \right] - C_1 p(x_1 - x_n)^2, \quad t \in I_{max},
	\end{split}
\end{equation}
where $I_{max} = [0, T_{max})$ is the maximal existence interval of the solution. According to the discrete "divergence" formula and $x_0(t) = x_n(t), \, x_{n+1}(t) = x_1(t)$ due to the periodic boundary condition \eqref{pbc}, we have
\beq \bl{key}
	\begin{split}
	&\sum_{i=1}^n (x_{i-1} -2x_i + x_{i+1})x_i = \sum_{i=1}^n (x_{i+1} - x_i)x_i - \sum_{i=1}^n (x_i - x_{i-1})x_i   \\
	= &\, \left[\sum_{i=1}^{n-1} (x_{i+1} - x_i)x_i - \sum_{i=2}^n (x_i - x_{i-1})x_i\right] + (x_{n+1} - x_n)x_n - (x_1 - x_0)x_1 \\
	= &\, - \sum_{i=2}^n (x_i - x_{i-1})^2 - (x_1 - x_0)^2 = - \sum_{i=1}^{n} (x_{i} - x_{i-1})^2 \leq 0.
	\end{split}
\eeq
Then \eqref{u1} with \eqref{key} yields the differential inequality
\beq \bl{u2}
	\begin{split}
	&C_1 \frac{d}{dt} \sum_{i = 1}^n |x_i (t)|^2 + 2C_1 \left[\sum_{i=2}^{n} a(x_{i} - x_{i-1})^2 + p (x_1 - x_n)^2\right]  \\[2pt]
	\leq &\,C_1 \sum_{i=1}^N \left[- 2\gl |x_i (t)|^4 + 2\beta + b |x_i(t)|^2 + b |y_i (t)|^2 \right],  \quad t \in I_{max}.
	\end{split}
\eeq

Next multiply the $y_i$-equation in \eqref{CN} by $y_i(t)$ for $1 \leq i \leq n$ and then sum them up. By using Young's inequality, we obtain
\begin{equation} \bl{w1}
	\begin{split}
	\frac{1}{2} \frac{d}{dt} \sum_{i=1}^n |y_i(t) |^2 &= \sum_{i=1}^n ( cx_i y_i - \gd y_i^2) \leq \sum_{i=1}^n \left[\left(\frac{c^2}{\gd} x_i^2 + \frac{1}{4} \gd \,y_i^2\right) - \gd \,y_i^2\right]  \\[3pt]
	= &\, \sum_{i=1}^n \left[\frac{c^2}{\gd} \, |x_i (t)|^2 - \frac{3}{4} \gd \,|y_i(t)|^2\right], \quad \text{for} \;\, t \in I_{max}.
	\end{split}
\end{equation}
Now add the above two inequalities \eqref{u2} and doubled \eqref{w1}. We obtain
\beq \bl{uw}
	\begin{split}
        &\frac{d}{dt} \sum_{i = 1}^n \left(C_1 |x_i (t)|^2 + | y_i (t)|^2 \right)  + 2C_1 \left[\sum_{i=1}^{n} a(x_{i} - x_{i-1})^2 + p (x_1 - x_n)^2\right]  \\[2pt]
        \leq &\sum_{i=1}^n \left[ \left(C_1 b + \frac{2c^2}{\gd}\right) |x_i(t)|^2 - 2C_1 \gl |x_i (t)|^4 + 2C_1 \beta \right] \\[2pt]
        + &\sum_{i=1}^n \left[\left(C_1 b - \frac{3 \gd}{2}\right) |y_i (t)|^2 \right], \quad t \in I_{mac} = [0, T_{max}).
	\end{split}
\eeq
We choose constant
\beq \bl{C1}
	C_1 = \frac{\gd}{2b} \quad \text{so that} \quad C_1 b - \frac{3\gd}{2} = - \gd.
\eeq
Then from \eqref{uw} with the fact $2C_1 [ \cdots ] \geq 0$ on the left-hand side and from the choice \eqref{C1}, we have
$$
	 \frac{d}{dt} \sum_{i = 1}^n \left(C_1 |x_i |^2 + | y_i |^2 \right) \leq \sum_{i=1}^n \left[ \left(C_1 b + \frac{2c^2}{\gd}\right)|x_i(t)|^2- 2C_1 (\gl |x_i (t)|^4 + \beta) - \gd |y_i (t)|^2\right]
$$
and consequently,

\beq \bl{Cuw}
	\begin{split}
	&\frac{d}{dt} \sum_{i = 1}^n \left(C_1 |x_i(t) |^2 + | y_i (t)|^2 \right) + \gd  \sum_{i = 1}^n \left( C_1 |x_i (t) |^2 + | y_i (t)|^2 \right)   \\
	\leq &\, \sum_{i = 1}^n \left[ \left(C_1 b + C_1 \gd + \frac{2c^2}{\gd}\right)|x_i(t)|^2- 2C_1 (\gl |x_i (t)|^4 + \beta)\right]   \\
	= &\, \sum_{i = 1}^n  \left[ \left(\frac{\gd^2}{2b} + \frac{\gd}{2} + \frac{2c^2}{\gd}\right) |x_i(t)|^2- \frac{\gd \gl}{b} |x_i (t)|^4 + \frac{\gd \beta}{b} \right], \quad t \in I_{max}.
	\end{split}
\eeq
Completing square shows that
\begin{align*}
	 &\left(\frac{\gd^2}{2b} + \frac{\gd}{2} + \frac{2c^2}{\gd} \right) |x_i(t)|^2- \frac{\gd \gl}{b} |x_i (t)|^4  \\
	 = &\, - \frac{\gd \gl}{b} \left[ | x_i (t)|^2 - \frac{b}{2\gd \gl} \left(\frac{\gd^2}{2b} + \frac{\gd}{2} + \frac{2 c^2}{\gd}\right) \right]^2 + C_2
\end{align*}
and
\beq \bl{C2}
	C_2 = \frac{b}{4\gd \gl} \left(\frac{\gd^2}{2b} + \frac{\gd}{2} + \frac{2 c^2}{\gd}\right)^2.
\eeq
Therefore, \eqref{Cuw} yields
\beq \bl{Suw}
	\frac{d}{dt} \sum_{i = 1}^n \left(C_1 |x_i |^2 + | y_i |^2 \right) + \gd  \sum_{i = 1}^n \left( C_1 |x_i |^2 + | y_i |^2 \right) \leq n \left(C_2 + \frac{\gd \beta}{b}\right), \quad t \in I_{max}.
\eeq
Apply the Gronwall inequality to \eqref{Suw}. Then we get the following bounded estimate for all the solutions of the system of equations \eqref{CN}-\eqref{inc},
\beq \label{dse}
	\begin{split}
	& \sum_{i=1}^n \left(|x_i (t, x_i^0) |^2 + |y_i (t, y_i^0)|^2 \right) \\
	\leq &\, \frac{1}{\min \{C_1, 1\}} \left[e^{- \gd \, t} \sum_{i=1}^n \left(C_1 |x_i^0 |^2 + |y_i^0 |^2 \right) + \frac{n}{\gd} \left(C_2 + \frac{\gd \beta}{b}\right)\right], \;\; t \in [0, \infty). \\
	\end{split}
\eeq
Here it is shown that $I_{max} = [0, \infty)$ for all the solutions because they will never blow up at any finite time.  Thus it is proved that for any given initial state there exists a unique global solution $((x_1 (t, x_1^0), y_1(t, y_1^0)), \cdots, (x_n (t, x_n^0)), y_n(t, y_n^0))$ in $H$.
\end{proof}

The global existence and uniqueness of the solutions to the initial value problem \eqref{CN}-\eqref{inc} and their continuous dependence on the initial data enable us to define the solution semiflow $\{S(t): H \to H\}_{t \geq 0}$ of this system of the FitzHugh-Nagumo cellular neural network:
$$
	S(t): ((x_1^0, y_1^0) \cdots, (x_n^0, y_n^0)) \longmapsto ((x_1(t, x_1^0), y_1 (t, y_1^0)), \cdots, (x_n(t, x_n^0), y_n (t, y_n^0))).
$$
We call $\{S(t)\}_{t \geq 0}$ the semiflow of the FitzHugh-Nagumo CNN.

\begin{theorem} \label{Dsp}
	The semiflow $\{S(t)\}_{t \geq 0}$ of the FitzHugh-Nagumo CNN in the space $H$ is dissipative in the sense that there exists a bounded ball
\beq \label{abs}
	B^* = \{g \in H: \| g \|^2 \leq Q\}
\eeq
where the constant
\beq \bl{Q}
	Q = \frac{1}{\min \{C_1, 1\}} \left[1 +  \frac{n}{\gd} \left(C_2 + \frac{\gd \beta}{b}\right) \right]
\eeq
such that for any given bounded set $B \subset H$, there is a finite time $T_B > 0$ and all the solutions with the initial state inside the set $B$ will permanently enter the ball $B^*$ for $t \geq T_B$.
\end{theorem}

\begin{proof}
The uniform estimate \eqref{dse} implies that
	\beq \label{lsp}
	\limsup_{t \to \infty} \,\sum_{i=1}^n \left(|x_i (t, x_i^0) |^2 + |y_i (t, y_i^0)|^2 \right) < Q
	\eeq
for all solutions of \eqref{CN} with any initial data $((x_1^0, y_i^0), \cdots, (x_n^0, y_n^0)) \in H$. Indeed for any given bounded set $B = \{g \in H: \|g \|^2 \leq \rho \}$ in $H$, there is a finite time
$$
	T_B = \frac{1}{\gd} \log^+ (\rho \, \max \{C_1, 1\} )
$$
such that
$$
	e^{- \gd \, t} \sum_{i=1}^n \left(C_1 |x_i^0 |^2 + |y_i^0 |^2 \right) < 1, \quad \text{for} \;\, t \geq T_B,
$$
which means all the solution trajectories started from the set $B$ will permanently enter the bounded ball $B^*$ shown in \eqref{abs} for $t \geq T_B$. Therefore, this semiflow is dissipative.
\end{proof}

\section{\textbf{Synchronization of the FitzHugh-Nagumo CNN}}

Define the differences of solutions for two adjacent indexed cells of the FitzHugh-Nagumo CNN \eqref{CN} to be
$$
	V_i (t) = x_i(t) - x_{i-1}(t), \quad W_i (t) = y_i(t) - y_{i-1}(t), \quad \text{for}\;\,  i = 1, \cdots, n.
$$
Consider the system of the \emph{differencing} equations for this CNN. For $i = 1, \cdots, n$,
\beq \bl{dHR}
	\begin{split}
		\frac{\pdr V_{i}}{\pdr t} & = a (V_{i-1} - 2V_{i} + V_{i+1})+ f(x_i) - f(x_{i-1}) - b W_{i} + p(u_i - u_{i-1}),  \\
		\frac{\pdr W_{i}}{\pdr t} & = c\, V_{i} - \gd W_{i}.
	\end{split}
\eeq
The periodic boundary condition $V_0 (t) = V_n (t), \, V_{n+1}(t) = V_1 (t)$ holds due to \eqref{pbc}.

Here is the main result on the feedback synchronization of the proposed FitzHugh-Nagumo cellular neural networks.

\begin{theorem} \bl{ThM}
	If the following threshold condition for the boundary gap signal of the FitzHugh-Nagumo cellular neural network \eqref{CN}-\eqref{bfc} is satisfied,
\beq \bl{SC}
	 \liminf_{t \to \infty} \, (x_n(t) - x_1(t))^2 > \left(1 + \frac{1}{p} \left(\gd + \ga + |c - b|\right)\right)Q,
\eeq
where the constant $Q > 0$ is given in \eqref{Q}, then this cellular neural network is asymptotically synchronized in the space $H$ at a uniform exponential rate.
\end{theorem}

\begin{proof}
	 Multiply the first equation in \eqref{dHR} by $V_{i} (t)$ and the second equation in \eqref{dHR} by $W_{i} (t)$. Then sum them up for all $1 \leq i \leq n$ and use the Assumption  \eqref{Asp} to get
\beq \bl{eG}
	\begin{split}
	&\frac{1}{2} \frac{d}{dt} \sum_{i=1}^n \left(|V_{i}|^2 + |W_{i}|^2\right) - \sum_{i=1}^n a (V_{i-1} - 2V_i + V_{i+1})V_i \\
	= &\, \sum_{i=1}^n \left[ (f(x_i) - f(x_{i-1})) V_{i} + (c - b)V_{i} W_{i} - \gd \,|W_{i}|^2 + p(u_i - u_{i-1})V_i \right] \\
	\leq &\,\sum_{i=1}^n \left[f^{\,\prime} (\xi x_i + (1-\xi) x_{i-1} ) V^2_{i} + (c - b)V_{i} W_{i} - \gd \,|W_{i}|^2 + p(u_i - u_{i-1})V_i \right]   \\
	\leq &\, \sum_{i=1}^n \left[\ga |V_{i}|^2 + |c - b |(|V_{i}|^2 + |W_{i}|^2) - \gd \,|W_{i}|^2 + p(u_i - u_{i-1})V_i \right],
	\end{split}
\eeq
where $ 0 \leq \xi \leq 1$. By the periodic boundary condition for the differencing equations \eqref{dHR} we have
\begin{align*}
	&- \sum_{i=1}^n a (V_{i-1} - 2V_i + V_{i+1})V_i = - \sum_{i=1}^n a (V_{i+1} - V_i)V_i + \sum_{i=1}^n a (V_i - V_{i-1})V_i   \\
	= &\,- \left[\sum_{i=1}^{n-1} a(V_{i+1} - V_i)V_i - \sum_{i=2}^n a(V_i - V_{i-1})V_i\right] - a(V_{n+1} - V_n)V_n + a(V_1 - V_0)V_1 \\
	= &\, \sum_{i=2}^{n} a(V_i - V_{i-1})^2 + a(V_1^2 + V_n^2) - a(V_{n+1}V_n + V_0 V_1)   \\
	= &\, \sum_{i=2}^{n} a(V_i - V_{i-1})^2 + a(V_1^2 + V_0^2) - 2a V_1 V_0 \\
	= &\, \sum_{i=2}^{n} a(V_i - V_{i-1})^2 + a(V_1^2 - V_0^2) = \sum_{i=1}^{n} a(V_i - V_{i-1})^2 \geq 0.
\end{align*}
From the above two inequalities, we obtain
\beq \bl{VW}
	\begin{split}
	&\frac{1}{2} \frac{d}{dt} \sum_{i=1}^n \left(|V_i (t)|^2 + |W_i (t)|^2\right)   \\
	\leq &\, \sum_{i=1}^n \left[\ga |V_{i}|^2 + |c - b |(|V_{i}|^2 + |W_{i}|^2) - \gd \,|W_{i}|^2 + p(u_i - u_{i-1})V_i \right].
	\end{split}
\eeq
The boundary feedback \eqref{bfc} and the perioc boundary condition \eqref{pbc} infer that
\beq \bl{pu}
	\begin{split}
	 & \sum_{i=1}^n \, p(u_i - u_{i-1})V_i = \sum_{i=1}^n \, p(u_i - u_{i-1})(x_i - x_{i-1})  \\
	 = &\, p \left[(u_1 - u_0)(x_1 - x_0) + (u_2 - u_1)(x_2 - x_1) + (u_n - u_{n-1})(x_n - x_{n-1})\right] \\[4pt]
	 = &\, p \left[(u_1 - u_n)(x_1 - x_n) - u_1(x_2 - x_1) + u_n(x_n - x_{n-1})\right]  \\[4pt]
	 = &\, p \left[2(x_n - x_1)(x_1 - x_n) - (x_n - x_1)(x_2 - x_1) + (x_1 - x_n)(x_n - x_{n-1})\right]  \\[2pt]
	 = &\, p \left[- 2(x_n - x_1)^2 + (x_n - x_1)(x_1 - x_2 + x_{n-1} - x_n)\right]  \\[2pt]
	 = &\, p \left[- 3(x_n - x_1)^2 + (x_n - x_1)(x_{n-1} - x_2)\right]  \\[2pt]
	 \leq &\, p \left[- 2(x_n - x_1)^2 + (x_{n-1} - x_2)^2 \right].
	 \end{split}
\eeq
Substitute \eqref{pu} into \eqref{VW}. Then we get the following differential inequality
\begin{equation*}
	\begin{split}
	&\frac{d}{dt} \sum_{i=1}^n \left(|V_i (t)|^2 + |W_i (t)|^2\right) + 4p (x_n(t) - x_1(t))^2 \\
	\leq &\,\sum_{i=1}^n 2\left[\ga |V_{i}|^2 + |c - b |(|V_{i}|^2 + |W_{i}|^2) - \gd \,|W_{i}|^2\right] + 2p(x_{n-1}(t) - x_2(t))^2.
	\end{split}
\end{equation*}
Hence it holds that
\begin{equation}  \bl{Kq}
	\begin{split}
	&\frac{d}{dt} \sum_{i=1}^n \left(|V_i (t)|^2 + |W_i (t)|^2\right) + 2\gd \sum_{i=1}^n (|V_i(t)|^2 + |W_i(t)|^2) + 4p (x_n(t) - x_1(t))^2 \\
	\leq &\,\sum_{i=1}^n 2\left[(\gd + \ga)|V_i(t)|^2 + |c - b |(|V_{i}|^2 + |W_{i}|^2) \right] + 2p(x_{n-1}(t) - x_2(t))^2,
	\end{split}
\end{equation}
for $t > 0$. Note that \eqref{lsp} in Theorem \ref{Dsp} confirms that for all solutions of \eqref{CN},
$$
	\limsup_{t \to \infty}\, \sum_{i=1}^n \left(|x_i(t, x_i^0)|^2 + |y_i(t, y_i^0|^2\right) < Q.
$$
Thus for any given bounded set $B \subset H$ and any initial data $((x_1^0, y_i^0), \cdots , (x_n^0, y_n^0)) \in B$, there is a finite time $T_B \geq 0$ such that
\beq \bl{bd}
	\begin{split}
	&\sum_{i=1}^n 2\left[(\gd + \ga)|V_i(t)|^2 + |c - b |(|V_{i}|^2 + |W_{i}|^2)\right] + 2p(x_{n-1}(t) - x_2(t))^2   \\
	< &\, 4 \left(\gd + \ga + |c - b| \right) Q + 4p\,Q = 4 \left(\gd + \ga + |c - b|  + p\right) Q, \quad \text{for} \;\,   t \geq T_B.
	\end{split}
\eeq
Combining \eqref{Kq} and \eqref{bd}, we have shown that
\beq \bl{Mq}
	\begin{split}
	\frac{d}{dt} \sum_{i=1}^n \left(|V_i (t)|^2 + |W_i (t)|^2\right) &+ 2\gd \sum_{i=1}^n (|V_i(t)|^2 + |W_i(t)|^2) + 4p (x_n(t) - x_1(t))^2   \\
	&< 4 \left(\gd + \ga + |c - b|  + p\right) Q, \quad \text{for}  \;\;  t \geq T_B.
	\end{split}
\eeq

	Under the threshold condition \eqref{SC} of this theorem, for any given initial state $(x^0, y^0) = ((x_1^0, y_1^0), \cdots, (x_n^0, y_n^0)) \in H$ as a set $B$ of single point, there exists a finite time $T_{(x^0, \, y^0)}  > 0$ such that the differential inequality \eqref{Mq} holds for $t > T_{(x^0, \, y^0)}$ and
$$
	\sum_{i=1}^n \left(|x_i (t, x_i^0) |^2 + |y_i (t, y_i^0)|^2 \right) < Q, \quad \text{for} \;\; t \geq T_{(x^0, \, y^0)}.
$$
Moreover,
$$
	(x_n(t) - x_1(t))^2 > \left(1 + \frac{1}{p} \left(\gd + \ga + |c - b|\right)\right)Q,  \quad \text{for} \;\; t \geq T_{(x^0, \, y^0)},
$$
so that
\beq \bl{pl}
	p (x_n(t) - x_1(t))^2 > \left(\gd + \ga + |c - b| + p \right)Q,  \quad \text{for} \;\; t \geq T_{(x^0, \, y^0)}.
\eeq
It follows from \eqref{Mq} and \eqref{pl} that
\beq \bl{Gwq}
	\frac{d}{dt} \sum_{i=1}^n \left(|V_i (t)|^2 + |W_i (t)|^2\right) + 2\gd \sum_{i=1}^n (|V_i(t)|^2 + |W_i(t)|^2) < 0, \;\; \text{for} \;\, t \geq T_{(x^0, \, y^0)}.
\eeq
Finally, the Gronwall inequality applied to \eqref{Gwq} shows that
\beq \bl{Syn}
	\begin{split}
	\sum_{i=1}^n (|V_i(t)|^2 + |W_i(t)|^2) &\leq e^{- \gd (t - T_{(x^0, \, y^0)})} \sum_{i=1}^n (|V_i (T_{(x^0, \, y^0)})|^2 + |W_i (T_{(x^0, \, y^0)})|^2)  \\[2pt]
	&\leq 2 e^{- \gd (t - T_{(x^0, \, y^0)})}\,Q \to 0,  \quad \text{as} \;\; t \to \infty.
	\end{split}
\eeq
Then it is proved that for all solutions of the problem \eqref{CN}-\eqref{bfc} for this FitzHugh-Nagumo CNN with the boundary feedback,
\beq \bl{gij}
	\lim_{t \to \infty} \sum_{i = 1}^n \left(|(x_i (t, x_i^0) - x_{i-1}(t, x_{i-1}^0))|^2 + |(y_i (t, y_i^0) - y_{i-1}(t, y_{i-1}^0))|^2 \right) = 0.
\eeq
This FHN cellular neural network with boundary feedback is asymptotically synchronized in the space $H$ at a uniform exponential rate.
\end{proof}

This result provides a sufficient condition for feedback synchronization of the FitzHugh-Nagumo complex neural networks with boundary control. The threshold condition \eqref{SC} needs to be satisfied by the boundary gap signal $\liminf_{t \to \infty}\, (x_n(t) - x_1(t))^2$ between the two boundary cells. And the threshold in \eqref{SC} is adjustable by the designed feedback coefficient $p$ in applications.

\bibliographystyle{amsplain}

config: arxiv


TITLE: Limit of solution of linear system of ODEs as $t\to \infty$
QUESTION [4 upvotes]: I am completely stuck on the following problem:
Consider the linear system: $x'(t)=A(t)x(t)$ where $A(t)$ is an $n$ by $n$ matrix. Assume that $\lim_{t\to \infty}A(t)=B$. Suppose that each eigenvalue of $B$ has strictly negative real part. Let $y(t)$ be a solution to the linear system. Does anyone how to show: $\lim_{t\to \infty}|y(t)|=0$
I know that if $y(t)$ is a solution to the linear system and all eigenvalues of $A$ have  strictly negative real parts, then $y(t)\to 0$ as $t\to \infty$. However, this is true when $A$ is independent of $t$. The issue for this problem is that $A$ is dependent on $t$, and it's the limiting matrix $B$ that has eigenvalues with strictly negative real parts. I don't know how to approach this problem. Thanks to anyone who help me out!

REPLY [1 votes]: The answer is positive. You can consider your system as
$$
\dot x=Bx+C(t)x,\tag{1}
$$
where $\lim_{t\to\infty}\|C(t)\|=0$, use the integral representation of the solution and the fact that the fundamental matrix solution $\Phi(t)$ to $\dot x=Bx$ satisfies the estimate $\|\Phi(t)\|\leq Ce^{-\mu t}$ for some $\mu>0$.
A word of caution: If matrix $B$ is stable (i.e., $\mbox{Re}\,\lambda\leq 0$) then the condition $\lim_{t\to\infty}\|C(t)\|=0$ is not enough to guarantee the stability of the trivial solution.
Edit: Outline of a proof. Let
$$
\dot \Phi(t)=B\Phi(t).
$$
In $(1)$ I make the substitution $x=\Phi(t)z$ and find that
$$
\dot z=\Phi^{-1}(t)C(t)\Phi(t)z.
$$
I integrate, return to the original variable and get that instead of $(1)$ I have
$$
x(t)=\Phi(t)x_0+\int_{t_0}^t\Phi(t-\tau+t_0)C(\tau)x(\tau)d\tau.
$$
Now you estimate the norm $\|x\|$ in the usual way, use a Grownwall's inequality and get the conclusion (it does require some work to fill in the gaps).


TITLE: Why can water wicks act like siphons?
QUESTION [2 upvotes]: Take two empty cups (cup A and cup B) and fill cup A with water.  Take a length of wet cloth and run it from the bottom of cup A to the bottom of cup B, while the cups are standing next to each other on a table.  Wait a long time.  The cloth will wick water from cup A to cup B until the water levels are equal in the two cups, just like a siphon would.  My question is how does the information of the water level in cup A get transmitted through the cloth to the water in cup B?
In the case of a siphon, this information is transmitted as a pressure gradient through the tube that is acting as the siphon.  This requires that the tube walls sustain compressive (and sometimes tensile) forces, so that water pressure varies with height throughout the tube length.  But in the case of a cloth wick it is capillary forces that pull the water along.  The water all along the cloth has free surfaces exposed directly to air, and so presumably at or close to atmospheric pressure.  So if there are no semi-rigid external walls to contain the water flowing along the cloth, how doe sit transmit pressure information?

REPLY [2 votes]: Water flows by capillarity both ways. But the flow from the glass with higher level will be greater because it needs a smaller hill to climb. When levels are equal, flow from both directions equalize.


TITLE: How is the volume of a tetrahedron one sixth that of a parallelepiped ? (Please read the whole post before closing)
QUESTION [1 upvotes]: Now some of the readers in a rush might think that this a duplicate of Tetrahedron volume relation to parallelepiped and pyramid
But it is not. In the froe-mentioned question it has been asked to derive a mathematical relation between the 2, which I already have derived(using the scalar triple product). But, I couldn't visually wrap my mind around the fact that you can fit 6 tetrahedrons in a parallelepiped ?  Am I interpreting the result right ? that you can fit 6 tetrahedrons whose volume are equal into a parallelepiped whose volume is the sum of all of those tetrahedrons ? I have looked all over the internet, but couldn't find a figure that would help me understand this visually, and develop a visual intuition for it. A figure would be much appreciated.

REPLY [8 votes]: You can see below how three equal-volume tetrahedra can be combined to form a prism with triangular base $ABCA'B'C'$. If you want a prism with a parallelogram as base, just add points $D$ and $D'$ and repeat the same scheme: you'll end with six tetrahedra packed into the prism.


TITLE: parallel=time, perpendicular=space? on multiple time dimensions
QUESTION [0 upvotes]: There are some discussions on more than one time dimensions, e.g., Intuition for multiple temporal dimensions and More than one time dimension.
If we define that the parallel direction is time, of course, we can have only one time dimension but multiple space dimensions.
Then, the questions are:
Q1. Is the above definition reasonable?
Q2. If not, then why time is so different from space? And, how we define time?
Q3. Is multiple time dimensions possible? For example, are there any GUT support it?
Q4. If not, why only one time dimension? Or, which principle rules multiple time dimensions out?

REPLY [2 votes]: Re q1 and q2: in special relativity the key invarient is the line element $ds$ defined by:
$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 $$
In this expression the timelike dimensions have a negative sign while the spacelike dimensions have a positive sign, so in the above expression there is only one timelike dimension, $dt$. This is how we define which dimensions are timelike. I'm not sure what you mean by saying "the parallel direction is time", but this is unlikely to be a helpful definition. The negative sign of the time dimension is responsible for all the "weird" effects in SR e.g. time dilation and length contraction, and it also implies a constant maximum speed (i.e. the speed of light).
Re q3: adding a second (macroscopic) timelike dimension is mathematically possible, though as far as I know no GUT uses this because it leads to physically unreasonable consequences, which takes us to ...
Re q4: a second macroscopic timelike dimension allows closed timelike curves and hence violations of causality, so any theory of this type wouldn't be a good description of the universe we observe. Various ways around this have been suggested. Itzhak Bars has suggested a theory with two time dimensions - I don't know the work well enough to know how he gets round causality problems. F Theory can be interpreted as having two time dimensions, but they are not macroscopic so there are no causality problems.


TITLE: If $(u_n)\subset C^\infty _c(\mathbb R^n)$ converge to $u\in C^1_c(\mathbb R^n)$ in $L^p$ doesn $u_n\to u$ in $L^q$?
QUESTION [0 upvotes]: Let $(u_n)_n$ a sequence of $\mathcal C_c^\infty (\mathbb R^n)$ that converge in $L^p$ to $u\in \mathcal C^1_c(\mathbb R)$. If $q>1$, is there a subsequence that converge to $u$ in $L^q$ ?
Attempts
I know that $(u_n)$ is in $L^q$, that $u\in L^q$ also. Now if $q\leq p$, then by Jensen $$\|u_n-u\|_{L^p}\geq \left(\int |u_n-u|^q\right)^{p/q}=\|u_n-u\|_{L^q}^p,$$
and thus $u_n\to u$ in $L^q$. But what happen if $p<q$ ?

REPLY [1 votes]: No, you don't get that $u_k\to u$ in $L^q$. Consider the case  $p<q$ first. Let $u\equiv 0$, let $u_1$ be any bump function. and define
$$
u_k(x) = k^{n/q} u_1(kx)
$$
so that $\|u_k\|_q = \|u_1\|_q$ for all $k$ (use the change of variable to see this). Clearly, $u_k$ does not converge to $0$ in $L^q$ norm. But it does converge in $L^p$ for any $p<q$, because $$\|u_k\|_p = k^{n/q-n/p}\|u_1\|_p\to 0$$
The case $p>q$ is similar: let $u$ and $u_1$ be as above, but now
$$
u_k (x) = k^{-n/q}u_1(k^{-1}x)
$$
Still, $\|u_k\|_q = \|u_1\|_q$ for all $k$. But $$\|u_k\|_p = k^{n/p-n/q}\|u_1\|_p\to 0$$


TITLE: The Equivalence of Non-Singularity and Invertibility for Finite-Dimensional Vector Spaces
QUESTION [0 upvotes]: In Lipschutz's Linear Algebra, he provides the attached discussion (see below) on invertible operators in $A(V$). Now, I understand the sketch of the proof that non-singularity and invertibility are equivalent, but I suppose I'm confused about the counterexample that he provides to show that non-singularity does not imply invertibility in the general case of potentially infinite-dimensional vector spaces.
My thinking was: $F$ is clearly non-singular because the zero vector maps to the zero vector. However, it is not onto because the constant $a_0$ is not present in the image of $F$. Is that it? It seems to me then that I am missing something, because then I think about the case of a finite polynomial vector space, $P_n(t)$. This would also not have $a_0$ in its image. But maybe this isn't a valid counter example because $F$ maps $P_n(t)$ out of $P_n(t)$ and into $P_{n+1}(t)$ and therefore this is not a valid counterexample to the assertion made about finite-dimensional vector spaces.
Lipschutz

REPLY [0 votes]: You are right. Since $a_0$ is not in the image of $F$, the map $F$ is not onto.
The situation with $P_n(t)$ is no counterexample since $F$ maps $P_n(t)$ into $P_{n + 1}(t)$ and these two spaces have not the same dimension. The equivalence of surjectivity and injectivity of a linear map $F : V \longrightarrow W$ is only given, if $\dim(V) = \dim(W) < \infty$.


TITLE: Find vector rotated on 3D plane
QUESTION [0 upvotes]: I don't have access to a computer so I can't give pictures but I will try to make this easy to visualize. Suppose I have a vector $n$. I will now draw a plane normal to this vector that and have that plane intersect the origin. I can find a vector $m$ on that plane with no z component that is denoted by $(n_2/n_1,1,0)$ (by making $z=0,y=1$, but for the purpose of this question all that is important is that $m$ is a vector on the $n$ vector defined plane with no z component. How would I rotate $m$ by $\theta$ degrees with respect to the 3d plane? I don't need magnitude, only direction.

REPLY [1 votes]: If $n = (n_1, n_2, n_3)$ is a non-zero vector, then $m = (n_2, -n_1, 0)$ is orthogonal to $n$ (and non-zero so long as $n$ does not lie on the third coordinate axis). The cross product $p = n \times m$ is orthogonal to both $n$ and $m$, and the vectors $\mathbf{e}_1 = m/\|m\|$ and $\mathbf{e}_2 = p/\|p\|$ are an orthonormal basis of $\Pi = n^\perp$.
If "rotate $m$ by $\theta$ degrees with respect to the $3$d plane" means "rotate $\Pi$ counterclockwise about $n$ by $\theta$ degrees, the desired vector is
$$
m' = \|m\|(\cos\theta \mathbf{e}_1 + \sin\theta \mathbf{e}_2),
$$
with the trig functions "in degrees mode".


TITLE: What is the sufficient statistic for this function?
QUESTION [1 upvotes]: $f(x)=\theta/x^2$ where $\theta<x<\infty$
Given that the likelihood is $lik(\theta)=\theta^n \prod x_i^{-2}$
It seems to me that $T(X)=\prod x_i^{2}$ is the sufficient statistic, but intuitively, I think it should be $T(X)=\min(x_1...x_n)$

REPLY [2 votes]: well your distribution function is
$f(x)=\dfrac{\theta}{x^2} \mathbb{1}_{\{x>\theta\}}$
so your likelihood is
$\prod\limits_{i=1}^n \dfrac{\theta}{x_i^2} \mathbb{1}_{\{x_i>\theta\}}$
Fisher's factorisation criterion says $l(\theta|x) = h(x) g_\theta(T(x))$ then T(x) is a sufficient statistic.
instead of what you wrote down, $\prod\limits_{i=1}^n \dfrac{1}{x_i^2}$ looks like $h(x)$
what you have left is the product of the indicator function is $\prod\limits_{i=1}^n\theta\mathbb{1}_{\{x_i>\theta\}}$, well how can you write that using min?
EDIT: It is a common error to forget the indicator function. This is very very important, especially when the range of the distribution is dependent on the parameter.