text:
\section{Proof of the main geometric statements} \label{s:main-geom}

In this section we prove the main geometric results of the
paper.

We will make use of the following variants of the notion of Gromov
width. Let $(M^{2n},\omega)$ be a symplectic manifold, $L\subset M$ a
Lagrangian submanifold and $Q\subset M$ a
subset. Following~\cite{Bar-Cor:Serre,Bar-Cor:NATO} we define the
Gromov width $\delta(L;Q)$ of $L$ relative to $Q$ as follows. Assume
first that $L \not \subset Q$. Define:
\begin{equation} \label{eq:delta1}
  \begin{aligned}
    \delta(L;Q)=\sup \bigl\{\pi r^{2} \in (0,\infty] \ \mid \ &
    \exists \; \text{a symplectic embedding} \; e:
    B(r) \longrightarrow M \\
    & \text{such that} \; e^{-1}(L) = B_{\mathbb{R}}(r) \; \text{and}
    \; e(B(r)) \cap Q = \emptyset \; \bigr\} ~.~
  \end{aligned}
\end{equation}
Here $B(r) \subset \mathbb{R}^{2n}$ is the standard $2n$-dimensional
closed ball of radius $r$, endowed with the standard symplectic
structure from $\mathbb{R}^{2n}$, and
$B_{\mathbb{R}}(r) := B(r) \cap (\mathbb{R}^n \times \{0\})$ is the
real part of $B(r)$. In case $L \subset Q$ we set $\delta(L;Q) := 0$.

Another variant of the Gromov width is associated to an immersed
Lagrangian. Let $\widehat{\mathbb{L}}$ be a smooth closed manifold
(possibly disconnected) and let
$\iota: \widehat{\mathbb{L}} \longrightarrow M$ be a Lagrangian
immersion with image $\mathbb{L} := \iota(\widehat{\mathbb{L}})$. We
would like to measure the "size" of a subset of the double points of
$\mathbb{L}$ relative to a given subset $Q\subset M$.  More precisely,
denote by $\Sigma(\iota) \subset \mathbb{L}$ the set of points that
have more than one preimage under the immersion $\iota$. Let
$\Sigma' \subset \Sigma(\iota)$ be a non-empty subset such that each
point in $\Sigma'$ is a transverse intersection of two branches of the
immersion. As before, let $Q \subset M$ be a subset. Assume first that
$\Sigma' \not \subset Q$. We define the Gromov width
$\delta^{\Sigma'}(\mathbb{L}; Q)$ of the self-intersection set
$\Sigma'$ relative to $Q$ by:
\begin{equation*} \label{eq:delta-sigma}
  \begin{aligned}
    \delta^{\Sigma'}(\mathbb{L}; Q)=\sup \bigl\{ \pi r^{2}\in
    (0,\infty] \; \mid \; & \forall \; x \in \Sigma', \; \exists \;
    \text{a symplectic embedding} \;
    e_{x}:B(r) \longrightarrow M \; \text{with},\\
    & e_{x}(0)=x, \; e_{x}^{-1}(\mathbb{L})= B_{\mathbb{R}}(r)\cup i
    B_{\mathbb{R}}(r), \;
    e_{x}(B(r)) \cap Q = \emptyset, \\
    & \mathrm{and} \; e_{x'}(B(r))\cap e_{x''}(B(r)) = \emptyset \;
    \text{whenever} \; x' \neq x'' \bigr\}.
  \end{aligned}
\end{equation*}
Here $iB_{\mathbb{R}}(r)$ stands for the imaginary part of the ball,
$iB_{\mathbb{R}}(r) := B(r) \cap (\{0\} \times \mathbb{R}^n)$.

In case $\emptyset \neq \Sigma' \subset Q$ we set
$\delta^{\Sigma'}(\mathbb{L}; Q) = 0$. In case $\Sigma'=\emptyset$ we
set $\delta^{\emptyset}(\mathbb{L};Q)=\infty$. In what follows, if
$Q=\emptyset$, then we omit the set $Q$ from the notation in both
$\delta(L;Q)$ and $\delta^{\Sigma'}(\mathbb{L};Q)$.

The next important geometric measurement is the {\em shadow} of a
cobordism, as defined in~\cite{Co-She:metric} and already mentioned in the introduction. Let
$V \subset \mathbb{R}^2 \times M$ be a Lagrangian cobordism. Denote by
$\pi: \mathbb{R}^2 \times M \longrightarrow \mathbb{R}^2$ the
projection. The shadow $\mathcal{S}(V)$ of $V$ is defined as:
\begin{equation} \label{eq:shadow} \mathcal{S}(V) = Area \bigl(
  \mathbb{R}^2 \setminus \mathcal{U} \bigr),
\end{equation}
where $\mathcal{U} \subset \mathbb{R}^2 \setminus \pi(V)$ is the union
of all the {\em unbounded} connected components of
$\mathbb{R}^2 \setminus \pi(V)$. Equivalently one can define
$\mathcal{S}(V)$ as follows. Fix a subset
$V_0 := V \cap \pi^{-1}([-R,R] \times \mathbb{R})$ outside of which
$V$ is cylindrical. Then $\mathcal{S}(V)$ is the infimum of $Area(D)$,
where $D$ runs over all subsets $D \subset \mathbb{R}^2$ which are
symplectomorphic to a $2$-dimensional disk and such that
$D \supset \pi(V_0)$.

We now turn to the main result which we restate here for the convenience of the reader. Recall that
$\mathcal{L}ag^{\ast}(M)$ denotes the collection of closed Lagrangian
submanifolds of $M$ of class $*$, where $*$ stands either for the
weakly exact Lagrangians ($* = \text{we}$ in short), or for the
monotone Lagrangians with given Maslov-$2$ disk count
$\mathbf{d} \in \Lambda_0$ ($*=(\text{mon}, \mathbf{d})$ in short) as
introduced in~\S\ref{sb:monotone}. Similarly, we have the collection
$\mathcal{L}ag^{\ast}(\mathbb{R}^2 \times M)$ of Lagrangian cobordisms
$V \subset \mathbb{R}^2 \times M$ of class $*$, where $*$ is as above.

\begin{thm} \label{thm:fission} Let
  $L,L_{1},\ldots, L_{k}\in \mathcal{L}ag^{\text{we}}(M)$ and
  $V:L \leadsto (L_{1},\ldots, L_{k})$ a weakly exact Lagrangian
  cobordism.  Denote $S := \cup_{i=1}^k L_i$ the union of the
  Lagrangians corresponding to the negative ends of $V$. Then
  \begin{equation} \label{eq:SV-delta} \mathcal{S}(V)\geq
    \tfrac{1}{2}\delta(L;S).
  \end{equation}

  For the next two points of the theorem we will use the following
  notation. Let $N \in \mathcal{L}ag^{\text{we}}(M)$ be another weakly
  exact Lagrangian submanifold and consider
  $S = \cup_{i=1}^k L_i \subset M$ and $N \cup S \subset M$ as
  immersed Lagrangians (parametrized by $\coprod_{i=1}^k L_i$ and
  $N \coprod (\coprod_{i=1}^k L_i)$ respectively).
  \begin{enumerate} [label=(\alph*),ref=\alph*]
  \item \label{i:ch-inters} Assume that $N$ intersects each of the
    Lagrangians $L_1, \ldots, L_k$ transversely and that
    $N \cap L_i \cap L_j = \emptyset$ for all $i \neq j$. Denote
    $\Sigma' := N \cap S$. If
    $\mathcal{S}(V) <\frac{1}{2}\ \delta^{\Sigma'}(N\cup S)$ then
    \begin{equation} \label{eq:N-cap-L}
      \#(N\cap L) \geq \sum_{i=1}^{k}\# (N\cap L_{i})~.~
    \end{equation}
  \item \label{i:hf-inters} Assume that the Lagrangians
    $L_{1},\ldots, L_{k}$ intersect pairwise transversely and that no
    three of them have a common intersection point (i.e.
    $L_i \cap L_j \cap L_r = \emptyset$ for all distinct indices
    $i,j,r$). Let $\Sigma''$ be the set of all double points of $S$,
    i.e.  $\Sigma'' := \cup_{1 \leq i < j \leq k} L_{i} \cap
    L_{j}$. If $\mathcal{S}(V) < \frac{1}{4}\ \delta^{\Sigma''}(S; N)$
    then
    \begin{equation} \label{eq:N-cap-L-HF}
      \# (N\cap L)\geq \displaystyle\sum_{i=1}^{k}
      \dim_{\Lambda} (HF(N,L_{i}))~.~
    \end{equation}
  \end{enumerate}
\end{thm}

The proof is given in~\S\ref{sb:prf-thm-fission} below.

Theorem~\ref{thm:fission} has an analog in the monotone case too.
Recall from~\S\ref{sb:monotone} the Maslov-$2$ disk count
$\mathbf{d} \in \Lambda_0$ associated to a monotone Lagrangian $L$ and
also its minimal disk area $A_L$ defined by~\eqref{eq:min-disk-area}
in~\S\ref{sb:monotone}.

\begin{thm} \label{thm:fission-mon} Let
  $L,L_{1},\ldots, L_{k} \subset M$ be monotone Lagrangians and
  $V:L \leadsto (L_{1},\ldots, L_{k})$ a connected monotone
  cobordism. Let $S$ be the same as in
  Theorem~\ref{thm:fission}. Denote by $\mathbf{d} \in \Lambda_0$ the
  Maslov-$2$ disk count of $L$ (hence by~\S\ref{sb:monotone} also of
  the $L_i$'s) and let $N \subset M$ be another monotone Lagrangian
  with $\mathbf{d}_N = \mathbf{d}$.  Then:
  \begin{equation} \label{eq:SV-delta-mon} \mathcal{S}(V)\geq \min \{
    \tfrac{1}{2}\delta(L;S), A_L\}.
  \end{equation}
  Moreover, under the above assumptions statements~\eqref{i:ch-inters}
  and~\eqref{i:hf-inters} continue to hold as stated in
  Theorem~\ref{thm:fission}.
\end{thm}
The proof is given in~\S\ref{sb:prf-thm-fission-mon}.
\subsection{Proof of
  Theorem~\ref{thm:fission}} \label{sb:prf-thm-fission}

We begin with the proof of~\eqref{eq:SV-delta}. We first assume that
the Lagrangians $L, L_0, \ldots, L_k$ intersect pairwise transversely,
and treat the general case afterwards.

\begin{figure}[htbp]
   \begin{center}
     \includegraphics[scale=0.5]{cob-v} \;
     \includegraphics[scale=0.5]{cob-w}
   \end{center}
   \caption{The cobordisms $W$ obtained from $V$ by bending the
     positive end.}
   \label{f:cob-v-w}
\end{figure}

We start by bending the positive end of $V$ by $180^{\circ}$ clockwise
in such a way as to get a cobordism $W$ without positive ends, and
whose negative ends are $(L_0,L_{1},\ldots, L_{k})$, where $L_0 :=
L$. See Figure~\ref{f:cob-v-w}. Clearly
$\mathcal{S}(W)=\mathcal{S}(V)$.

Fix $\epsilon>0$. Let $\gamma$, $\gamma'$ be two curves, as depicted
in Figure~\ref{f:gamma-gamma'}, and such that there exists a (not
compactly supported) Hamiltonian isotopy, horizontal at infinity,
$\varphi_t: \mathbb{R}^2 \longrightarrow \mathbb{R}^2$, $t \in [0,1]$,
with $\varphi_0 = \id$, $\varphi_1(\gamma) = \gamma'$ and with
\begin{equation} \label{eq:l-varphi-t} \text{length} \{\varphi_t\}
  \leq \mathcal{S}(W) + \epsilon/2,
\end{equation}
where $\text{length} \{\varphi_t\}$ stands for the Hofer length of the
isotopy $\{\varphi_t\}$.

\begin{figure}[htbp]
   \begin{center}
     \includegraphics[scale=0.5]{cob-w-gamma}
   \end{center}
   \caption{The curves $\gamma$ and $\gamma'$ and the cobordism $W$.}
   \label{f:gamma-gamma'}
\end{figure}

Put $S = \cup_{i=1}^k L_i$ and let
$e: B(r) \longrightarrow M \setminus S$ be a symplectic embedding as
in the definition of $\delta(L_0; S)$ in~\eqref{eq:delta1}, with
$$\delta(L_0, S) - \epsilon \leq \pi r^2 \leq \delta(L_0,S).$$
Next, let
\begin{equation} \label{eq:JB}
  B := \textnormal{image\,}(e), \quad q := e(0) \in L_0, \quad
  J^B := e_*(J_{\textnormal{std}}),
\end{equation}
where the latter is the complex structure on $B$ corresponding to the
standard complex structure $J_{\textnormal{std}}$ of $B^{2n}(r)$ via
the embedding $e$.

Next, we fix a symplectic identification between a small
\label{pp:function-f}open
neighborhood $U$ of $L_0$ in $M$ and a neighborhood $U'$ of the
zero-section in $T^*(L)$. Let $f: L \longrightarrow \mathbb{R}$ be a
$C^1$-small Morse function with exactly one local maximum at the point
$q \in L_0$.  We extend $f$ to a function
$\widetilde{f}: U' \longrightarrow \mathbb{R}$ by setting it to be
constant along the fibers of the cotangent bundle. Finally, let
$H_f^{L_0,L_0}: M \longrightarrow \mathbb{R}$ be a smooth function
such that $H_f^{L_0,L_0}|_U$ coincides with $\widetilde{f}$ via the
identification between $U$ and $U'$ that we have just fixed.

We now turn to the Fukaya categories relevant for this proof. Let
$\mathcal{C}$ be the collection of Lagrangians $L_0, \ldots, L_k$. We
will use the Fukaya categories $\fuk(\mathcal{C})$ and
$\fukcob(\widetilde{\mathcal{C}})$ associated to $\mathcal{C}$. More
specifically, we consider regular perturbation data
$p \in E'_{\textnormal{reg}}$ and $C^1$-small profile functions
$h \in \mathcal{H}'_{\textnormal{prof}}(\gamma)$ as
in~\S\ref{sb:inc-functors}. We impose two additional restrictions on
the admissible choices of perturbation data $p$ as follows. The first
one is that the datum $\mathscr{D}_{L_0,L_0}$ of the pair $(L_0,L_0)$
should have the function $H_f^{L_0,L_0}$ as its Hamiltonian function,
defined using any choice of a $C^1$-small Morse functions $f$ as
described above. Furthermore, we allow only for functions $f$ that are
sufficiently $C^1$-small such that
$\mathcal{O}(H_f^{L_0,L_0}) = \textnormal{Crit}(f)$. Note that for
every $y \in \mathcal{O}(H_f^{L_0,L_0})$ we have $A(y) = f(y)$.

The second restriction is that the Hamiltonian functions $H^{L_i,L_j}$
in the Floer data $\mathscr{D}_{L_i, L_j}$, $i \neq j$, are all
$0$. It is possible to impose these additional restriction and still
maintain regularity since we have assumed that the Lagrangians
$L_0, L_1, \ldots, L_k$ intersect pairwise transversely. With these
choices we have for every $i \neq j$:
$$\mathcal{O}(H^{L_i,L_j}) = L_i \cap L_j, \quad A(z) = 0,
\; \forall \, z \in \mathcal{O}(H^{L_i,L_j}).$$

We denote the space of all such regular choices of perturbation data
by $E''_{\textnormal{reg}} \subset E'_{\textnormal{reg}}$. We remark
that the Morse function $f$ is not fixed over $E''_{\textnormal{reg}}$
and each choice $p \in E''_{\textnormal{reg}}$ comes with its own
function $f$. Finally, note that $\mathcal{N}$ is still in the closure
of $E''_{\textnormal{reg}}$.

We now appeal to the theory developed in~\S\ref{s:floer-theory}.
Consider the Fukaya category $\fuk(\mathcal{C}; p)$
(see~\S\ref{sb:fukaya-families}) as well as the Fukaya category of
cobordisms
$\fukcob(\widetilde{\mathcal{C}}_{1/2}; \iota_{\gamma}(p,h))$
(see~\S\ref{sb:ext-lcob} and~\eqref{eq:I-gamma}
in~\S\ref{sb:inc-functors}). Recall that we have an ``inclusion''
functor
$\mathcal{I}_{\gamma; p,h}: \fuk(\mathcal{C}; p) \longrightarrow
\fukcob(\widetilde{\mathcal{C}}_{1/2}; \iota_{\gamma}(p,h))$.

Denote by $\mathcal{W}$ the Yoneda module corresponding to the object
$W \in \textnormal{Ob}(\fukcob(\widetilde{\mathcal{C}}_{1/2};
\iota_{\gamma}(p,h)))$ and consider its pull-back by the functor
$\mathcal{I}_{\gamma; p,h}$:
$$\mathcal{M}_{W; \gamma, p, h} := \mathcal{I}_{\gamma; p,h}^*\mathcal{W}.$$

Recall from~\S\ref{sb:wf-fukaya},~\S\ref{sb:ext-lcob}
and~\S\ref{sb:inc-functors} that the $A_{\infty}$-categories
$\fuk(\mathcal{C}; p)$,
$\fukcob(\widetilde{\mathcal{C}}_{1/2}; \iota_{\gamma}(p,h))$ as well
as the $A_{\infty}$-functor $\mathcal{I}_{\gamma; p,h}$ are all weakly
filtered. Moreover, by~\S\ref{sb:wf-icones-cobs} the module
$\mathcal{M}_{W; \gamma, p, h}$ if weakly filtered too. By
Propositions~\ref{p:wf-M_V}, and
points~\eqref{pp:iota_gamma-lim},~\eqref{pp:discr-fukcob-12} on
page~\pageref{pp:iota_gamma-lim} the discrepancy of this module is
bounded from above by
$\bme(p,h) = (\epsilon_1(p,h), \epsilon_2(p,h), \ldots,
\epsilon_d(p,h), \ldots)$ which satisfies
$\lim \epsilon_d(p,h) \longrightarrow 0$ for every $d$, as
$p \longrightarrow p_0 \in \mathcal{N}$ and $h \longrightarrow 0$ (the
latter in the $C^1$-topology).

Throughout the proof we will repeatedly deal with quantities having
the same asymptotics as $\epsilon_d(p,h)$. In order to simplify the
text we introduce the following notation. Let
$\mathcal{N}_0 \subset \mathcal{N}$ and let $(p,h) \longmapsto C(p,h)$
be a real valued function defined for $p$ in a subset of
$E'_{\textnormal{reg}}$ whose closure contains $\mathcal{N}_0$, and
$h \in \mathcal{H}'_{\textnormal{prof}}$. We will write
$C(p,h) \in \onx{\mathcal{N}_0}$ to indicate that for every
$p_0 \in \mathcal{N}_0$ we have $\lim C(p,h) = 0$ as
$p \longrightarrow p_0$ and $h \longrightarrow 0$ (the latter in the
$C^1$-topology).

By Proposition~\ref{p:icones-M_j}
(and~\eqref{eq:filtered-cob-it-cone}) we have:
\begin{equation} \label{eq:filtered-cob-it-cone-W}
  \begin{aligned}
    S^{s_h} & \mathcal{M}_{W; \gamma, p, h} \\ = \; & \tcn
    (\mathcal{L}_k \xrightarrow{\; \overline{\phi}_k \;}
    \tcn(\mathcal{L}_{k-1} \xrightarrow{\; \overline{\phi}_{k-1} \;}
    \tcn( \cdots \tcn(\mathcal{L}_2 \xrightarrow{\; \overline{\phi}_2
      \;} \tcn(\mathcal{L}_1 \xrightarrow{\; \overline{\phi}_1 \;}
    \mathcal{L}_0 )) {\cdot}{\cdot}{\cdot}))),
  \end{aligned}
\end{equation}
where $s_h \longrightarrow 0$ as $h \to 0$. (Recall
from~\S\ref{sbsb:action-shifts} that
$S^{s_h}\mathcal{M}_{V; \gamma, p, h}$ stands for the module
$\mathcal{M}_{V; \gamma, p, h}$ with action-shift by $s_h$.) The
modules $\mathcal{L}_i$ in~\eqref{eq:filtered-cob-it-cone-W} are the
Yoneda modules of the $L_i$'s. The notation $\overline{\phi_i}$ stands
for $\overline{\phi}_i = (\phi_i, 0, \bm{\delta^{(i)}})$, with
$\phi_i$ being a homomorphism of modules that shifts action by
$\leq 0$ and has discrepancy $\leq \bm{\delta^{(i)}}(p,h)$ where for
every $d$ we have $\delta^{(j)}_d (p,h) \in \on$. For simplicity of
notation set
$\bm{\delta}(p,h) := \max \{\bm{\delta^{(1)}}(p,h), \ldots,
\bm{\delta^{(k)}}(p,h)\}$, so that the discrepancy of all the
$\phi_i$'s is $\leq \bm{\delta}(p,h)$ and we still have
$\delta_d(p,h) \in \on$ for all $d$.

Consider the filtered chain complex
$\mathscr{C}_{p,h} := S^{s_{h}}\mathcal{M}_{W; \gamma, p,
  h}(L_0)$ \label{pp:C-ph} endowed with the differential coming from
the $\mu_1$-operation of $\mathcal{M}_{W; \gamma, p, h}$. By
definition
$\mathscr{C}_{p,h} = S^{s_h}CF(\gamma \times L_0, W;
\mathscr{D}_{\gamma \times L_0, W})$, where
$\mathscr{D}_{\gamma \times L_0, W}$ is the Floer datum prescribed by
$\iota_{\gamma}(p,h)$. By~\eqref{eq:filtered-cob-it-cone-W} the Floer
complex of $(L_0,L_0)$ is a subcomplex of $\mathscr{C}_{p,h}$, or more
precisely, we have an {\em action preserving} inclusion of chain
complexes:
\begin{equation} \label{eq:CF-L_0L_0} CF(L_0,L_0; p) \subset
  \mathscr{C}_{p,h},
\end{equation}
where $\mathscr{D}_{L_0,L_0}$ is specified by $p$ and is subject to
the additional restrictions imposed earlier in the proof. To simplify
the notation, we will denote from now on for a pair of Lagrangians
$(L', L'')$ by $CF(L',L''; p)$ the Floer complex
$CF(L',L''; \mathscr{D}_{L',L''})$, where $\mathscr{D}_{L',L''}$ is
the Floer datum specified by $p$.

Recall that we also have the curve $\gamma' \subset \mathbb{R}^2$ with
$\gamma' \cap \pi(W) = \emptyset$. Choose a Floer datum $\mathscr{D}'$
for $(\gamma' \times L_0, W)$ with a sufficiently $C^2$-small
Hamiltonian function so that
$CF(\gamma' \times L_0, W; \mathscr{D}')=0$. Now $\gamma \times L_0$
can be Hamiltonianly isotoped to $\gamma' \times L_0$ via an isotopy
horizontal at infinity with Hofer length
$\leq \mathcal{S}(W) + \epsilon/2$. By standard Floer theory (see
e.g.~\cite[Section~5.3.2]{FO3:book-vol1}) the identity map on
$\mathscr{C}_{p,h}$ is null homotopic via a chain homotopy which shifts
action by $\leq \mathcal{S}(W) + \epsilon/2$. Translated to the
formalism of~\eqref{eq:Bh-1} in~\S\ref{s:filt-ch} this means that
$B_h(\id_{\mathscr{C}_{p,h}}) \leq \mathcal{S}(W) + \epsilon/2$, hence
by~\eqref{eq:beta-beta_h} we have:
\begin{equation} \label{eq:Bh-C} \beta(\mathscr{C}_{p,h}) \leq
  \mathcal{S}(W) + \epsilon/2,
\end{equation}
where $\beta(\mathscr{C}_{p,h})$ is the boundary depth of the
(acyclic) chain complex $\mathscr{C}_{p,h}$ as defined
in~\eqref{sec:bdry-depth}.

We now appeal to Theorem~\ref{t:itcones} applied to the weakly
filtered iterated cone~\eqref{eq:filtered-cob-it-cone-W}. We apply
this theorem with $X=L_0$ and $\rho_i=0$. We obtain a new weakly
filtered module $\mathcal{M}$ such that $\mathcal{M}(L_0)$ has a
differential $\mu_1^{\mathcal{M}}$ as described in that theorem
together with a filtered {\em chain isomorphism}
$\sigma_1: \mathscr{C}_{p,h} \longrightarrow \mathcal{M}(L_0)$. An
inspection of the sizes of shifts and discrepancies of the various
maps involved in Theorem~\ref{t:itcones} show that there exists a
constant $s^{\sigma}(p,h) \in \on$ such that $\sigma_1$ shifts
filtration by $\leq s^{\sigma}(p,h)$.  Additionally,
Theorem~\ref{t:itcones} implies that $CF(L_0,L_0; p)$ is also a
filtered subcomplex of $\mathcal{M}(L_0)$ and that
$\textnormal{pr}_0 \circ \sigma_1$ maps
$CF(L_0,L_0; p) \subset \mathcal{C}_{P,h}$ to
$CF(L_0,L_0; p) \subset \mathcal{M}(L_0)$ via the identity map:
$(\textnormal{pr}_0 \circ \sigma_1)|_{CF(L_0,L_0; p)} = \id$. Here
$\textnormal{pr}_0: \mathcal{M}(L_0) \longrightarrow CF(L_0,L_0;p)$ is
the projection onto the $0$'th factor of $\mathcal{M}(L_0)$.

Consider now the homology unit $e_{L_0} \in CF(L_0,L_0; p)$ as
constructed in~\eqref{eq:e_L}. By standard Floer theory $e_{L_0} = q$
(recall that $q$ is the unique maximum of
$f:L_0 \longrightarrow \mathbb{R}$).

Let $c \in CF(L_0,L_0; p)$ and $\gamma \in \mathcal{O}(H^{L_0,L_0})$ a
generator, where $H^{L_0.L_0}$ is the Hamiltonian function of the
Floer datum specified by $p$ for $(L_0,L_0)$. We denote by
$\langle c, \gamma \rangle \in \Lambda$ the coefficient of $\gamma$
when writing $c$ as a linear combination of elements of
$\mathcal{O}(H^{L_0,L_0})$ with coefficient in $\Lambda$.

We will need the following Lemma.
\begin{lem} \label{l:unit-we} For every chain $c \in CF(L_0,L_0; p)$
  we have $\langle \mu_1(c), q \rangle = 0$.
\end{lem}
We postpone the proof of the lemma and continue with the proof of
Theorem~\ref{thm:fission}.

Put $C_f := \max_{x \in L_0} |f(x)|$,
$C^{(1)}(p,h) := C_f + s^{\sigma}(p,h)$.  By~\eqref{eq:Bh-C} there
exists
\begin{equation}\label{eq:boundary-fund}
b' \in \mathscr{C}_{p,h}\ \textnormal{with}\
A(b'; \mathscr{C}_{p,h}) \leq A(e_{L_0};\mathscr{C}_{p,h}) +
\mathcal{S}(W) + \frac{\epsilon}{2} \leq C_f + \mathcal{S}(W) +
\frac{\epsilon}{2}\ ,
\end{equation} such that
$e_{L_0} = \mu^{\mathscr{C}_{p,h}}_1(b')$.

Recall from point~\eqref{pp:Delta_j-Delta_0} of
Theorem~\ref{t:itcones} that
$\textnormal{pr}_0 \circ \sigma_1|_{CF(L_0,L_0;p)} = \id$. Set
$b:=\sigma_1(b')$ and apply $\textnormal{pr}_0 \circ \sigma_1$ to the
equality $e_{L_0} = \mu^{\mathscr{C}_{p,h}}_1(b')$. We obtain:
\begin{equation} \label{eq:mu-1-b-A} e_{L_0} = \textnormal{pr}_0 \circ
  \mu_1^{\mathcal{M}}(b), \quad A(b; \mathcal{M}(L_0)) \leq
  C^{(1)}(p,h) + \mathcal{S}(W) + \epsilon/2,
\end{equation}
where $C^{(1)}(p,h) := C_f + s^{\sigma}(p,h)$. Obviously
$C^{(1)}(p,h) \in \on$. (Note that $f \longrightarrow 0$ as
$p \longrightarrow p_0 \in \mathcal{N}$.)

Using the splitting~\eqref{eq:M-direct-sum} write
$b = b_0 + \cdots + b_k$, with $b_i \in CF(L_0, L_i; p)$ and
$$A(b_i;CF(L_0, L_i; p)) \leq C^{(2)}(p,h) + \mathcal{S}(W) + \epsilon/2,$$
where $C^{(2)}(p,h)$ is a new constant such that
$\lim C^{(2)}(p,h) \in \on$.

Continuing to apply Theorem~\ref{t:itcones} we have:
\begin{equation} \label{eq:q-equals-sum} q = \textnormal{pr}_0 \circ
  \mu_1^{\mathcal{M}}(b) = \sum_{j=0}^k a_{0,j}(b_j) =
  \mu_1^{CF(L_0,L_0;p)}(b_0) + \sum_{j=1}^k a_{0,j}(b_j),
\end{equation}
where the operators $a_{i,j}$ are the entries of the matrix
representation of $\mu_1^{\mathcal{M}}$ with respect to the
splitting~\eqref{eq:M-direct-sum}. By Lemma~\ref{l:unit-we},
$\langle \mu_1(b_0), q \rangle = 0$, hence there exists
$1 \leq j_0 \leq k$ such that
\begin{equation} \label{eq:nu-b-j0} \langle a_{0,j_0}(b_{j_0}),
  q\rangle \neq 0, \quad \nu \bigl(\langle a_{0,j_0}(b_{j_0}),
  q\rangle \bigr) \leq \nu(1) = 0.
\end{equation}
Here $\nu$ is the standard valuation of $\Lambda$
(see~\eqref{eq:val-Nov}).

By Theorem~\ref{t:itcones} there exists chains
$c_{i',i''} \in CF(L_{i'}, L_{i''};p)$, for all $i' < i''$, with
$A(c_{i',i''}) \leq C^{(3)}(p,h)$, where $C^{(3)}(p,h) \in \on$ and
such that
$$a_{0,j_0}(b_{j_0}) = \sum_{2 \leq d, \, \underline{i}}
\mu_d^{\fuk(\mathcal{C};p)}(b_{j_0}, c_{i_d, i_{d-1}}, \ldots, c_{i_2,
  i_1}),$$ where $\underline{i} = (i_1, \ldots, i_d)$ runs over all
partitions $0=i_1 < i_2 \cdots < i_{d-1} < i_d = j_0$.

It follows that there exists a partition
$\underline{i}^0 = (i^0_1, \ldots, i^0_d)$ with $d \geq 2$, for which
$$\big \langle \mu_d^{\fuk(\mathcal{C};p)}(b_{j_0}, c_{i^0_d, i^0_{d-1}},
\ldots, c_{i^0_2, i^0_1}), q \big \rangle \neq 0, \quad \nu \Big( \big
\langle \mu_d^{\fuk(\mathcal{C};p)}(b_{j_0}, c_{i^0_d, i^0_{d-1}},
\ldots, c_{i^0_2, i^0_1}), q \big \rangle \Big) \leq 0.$$ Writing
$b_{j_0}$ as a linear combination (over $\Lambda$) of elements from
$L_0 \cap L_{j_0}$ and similarly for the $c_{i^0_r, i^0_{r-1}}$'s, we
deduce that there exist $x \in L_0 \cap L_{j_0}$, $P(T) \in \Lambda$,
and $z_{r} \in L_{i^0_r} \cap L_{i^0_{r-1}}$, $Q_r \in \Lambda$ for
$r=2, \ldots, d$, such that:
\begin{equation*} \label{eq:P-Q_r}
  \begin{aligned}
    & A(P(T)x) \leq C^{(2)}(p,h) + \mathcal{S}(W) + \epsilon/2, \\
    & A(Q_r(T)z_r) \leq C^{(3)}(p,h), \; \forall \, 2 \leq r \leq d, \\
    & \big \langle \mu_d^{\fuk(\mathcal{C};p)}(P(T)x, Q_d(T)z_d,
    \ldots, Q_2(T)z_2), q \big \rangle \neq 0, \\
    \nu \Big( & \big \langle \mu_d^{\fuk(\mathcal{C};p)}(P(T)x,
    Q_d(T)z_d, \ldots, Q_2(T)z_2), q \big \rangle \Big) \leq 0.
  \end{aligned}
\end{equation*}
Note that $d$, as well as the points $x$, $z_d, \ldots, z_2$, $q$, all
depend on $(p,h)$, but for the moment we suppress this from the
notation.

Since $A(P(T)x) = -\nu(P(T))$ and $A(Q_r(T)z_r)=-\nu(Q_r(T))$ we
obtain:
\begin{equation} \label{eq:nu-mu-d} \nu \Big( \big \langle
  \mu_d^{\fuk(\mathcal{C};p)}(x, z_d, \ldots, z_2), q \big \rangle
  \Big) \leq \mathcal{S}(W) + \epsilon/2 + C^{(4)}(p,h),
\end{equation}
where $C^{(4)}(p,h) \in \on$.

Denote by $\mathscr{D}(p) = (K(p), J(p))$ the perturbation datum
prescribed by $p \in E''_{\textnormal{reg}}$ for the tuple of
Lagrangians $(L_0, L_{j_0}, L_{i_{d-1}}, \ldots, L_{i_2}, L_0)$. It
follows from~\eqref{eq:nu-mu-d} that there exists a non-constant Floer
polygon $u \in \mathcal{M}(x, z_d, \ldots, z_2, q; \mathscr{D}(p))$
with
\begin{equation*} \label{eq:om-u-S} \omega(u) \leq \mathcal{S}(W) +
  \epsilon/2 + C^{(4)}(p,h).
\end{equation*}

Let $p_0 \in \mathcal{N}$ be any choice of perturbation data which
assigns to the tuple of Lagrangians
$(L_0, L_{j_0}, L_{i_{d-1}}, \ldots, L_{i_2}, L_0)$ the perturbation
data $\mathscr{D}(p_0) = (K=0, J(p_0))$, where $J(p_0)$ is a family of
almost complex structures that coincide with $J^B$ on $B$
(see~\eqref{eq:JB}).

Fix a generic $C^1$-small Morse function $f$ as on
page~\pageref{pp:function-f}. We now choose a sequence
$\{(p_n, h_n)\}$ in $E''_{\textnormal{reg}}$ with
$(p_n, h_n) \longrightarrow (p_0,0)$ as $n \longrightarrow \infty$,
and with the following additional property. The Hamiltonian function
$H^{L_0,L_0}(n)$ prescribed by $p_n$ for the Floer datum
$\mathscr{D}_{L_0,L_0}(p_n)$ of $(L_0,L_0)$ is
$H^{L_0,L_0}_{\frac{1}{n}f}$, i.e.~constructed as on
page~\pageref{pp:function-f} but with the function $\frac{1}{n}f$
instead of $f$. Consequently, the point $q$ (the maximum of
$\frac{1}{n}f$) does not depend on $n$.

Passing to a subsequence of $\{(p_n, h_n)\}$ if necessary we may
assume that both $d$ as well as the points $x, z_d, \ldots, z_2$ above
do not depend on $n$ either. (Note that by Theorem~\ref{t:itcones},
$d \leq k$, so there are only finitely many possible values for $d$.)

In summary, we obtain a sequence,
$u_n \in \mathcal{M}(x, z_d, \ldots, z_2, q; \mathscr{D}(p_n))$ with
$\omega(u_n) \leq \mathcal{S}(W) + \epsilon/2 + C^{(4)}(p_n,h_n)$. By
a compactness result~\cite{Oh-Zhu:thick-thin, Oh-Zhu:floer-traj} (see
also~\cite{Fuk-Oh:zero-loop, Oh:relative, Oh:spectral}) there exists a
subsequence of $\{u_n\}$ which converges to a union of Floer polygons
$v_0, v_1, \ldots, v_l$, $l \geq 0$, together with a (possibly broken)
negative gradient trajectory $\eta$ of $f$. (``Broken'' means that the
trajectory might pass through several critical points of $f$.)

The Floer polygons $v_i$ map the boundary components of their domains
of definition to some of the Lagrangians in the collection
$L_0, L_{j_0}, L_{i_{d-1}}, \ldots, L_{i_2}, L_0$. Moreover, $v_0$
maps one of its boundary components to $L_0$. The maps $v_i$ satisfy
the Floer equation corresponding to the perturbation data prescribed
by $p_0$. Consequently they are all genuine pseudo-holomorphic
(i.e.~without Hamiltonian perturbations) with respect to the
(domain-dependent) almost complex structures prescribed by $p_0$. In
particular, $\omega(v_i) \geq 0$ for every $i$.

Since
$\omega(u_n) \leq \mathcal{S}(W) + \epsilon/2 + C^{(4)}(p_n,h_n)$ for
every $n$, it follows that
$\sum_{i=0}^l \omega(v_i) \leq \mathcal{S}(W) + \epsilon/2$, hence
\begin{equation} \label{eq:om-v_0} \omega(v_0) \leq \mathcal{S}(W) +
  \epsilon/2.
\end{equation}

The other part of the limit of $\{u_n\}$, namely the negative gradient
trajectory $\eta$ of $f$, emanates from an $L_0$-boundary point of one
of the polygons, say $v_0$, and ends at the point $q$.

Consider now $v_0$ and $\eta$. Note that $\eta$ must be the constant
trajectory at the point $q$ since it goes into $q$ which is a maximum
of $f$. It follows that the polygon $v_0$ passes (along its boundary)
through the point $q$.

We now appeal to the special form of $J(p_0)$ over the ball
$B$. Recall that $v_0$ is $J(p_0)$-holomorphic. Thus restricting $v_0$
to the subdomain (of its definition) which is mapped to
$\textnormal{Int\,}(B)$ we obtain a {\em proper} $J^B$-holomorphic
curve $v_0'$ parametrized by a non-compact Riemann surface with one
boundary component. Moreover that boundary component is mapped to
$B \cap L_0$, and $q \in B$ which is the center of the ball is in the
image of that boundary component. Passing to the standard ball $B(r)$
via the symplectic embedding $e$ mentioned in~\eqref{eq:JB} we obtain
from $v'_0$ a proper $J_{\textnormal{std}}$-holomorphic curve $v''_0$
inside $B(r)$ which passes through $0$ and its boundary is mapped to
$B_{\mathbb{R}}(r) \subset B^{2n}(r)$. Applying a reflection along
$\mathbb{R}^n \times 0$ to $v''_0$, and gluing the result to $v''_0$
we obtain a proper $J_{\textnormal{std}}$-holomorphic curve (without
boundary) $\widetilde{v}''_0$ in $\textnormal{Int\,} B^{2n}(r)$ which
passes through $0$. By the Lelong inequality we have
$\pi r^2 \leq \omega_{\textnormal{std}}(\widetilde{v}''_0)$. Putting
everything together we obtain:
$$\delta(L_0,S) - \epsilon \leq
\pi r^2 \leq \omega(\widetilde{v}''_0)= 2\omega(v''_0) \leq
2\omega(v_0) \leq 2\mathcal{S}(W) + \epsilon.$$ Since this
inequality holds for all $\epsilon>0$ the desired
inequality~\eqref{eq:SV-delta} follows.

\medskip

We now turn to the proof of Lemma stated earlier in the proof.
\begin{proof}[Proof of Lemma~\ref{l:unit-we}]
  Recall that the Hamiltonian function in the Floer data
  $\mathscr{D}_{L_0,L_0}$ of $(L_0,L_0)$ is $H_f^{L_0,L_0}$ and we
  have $\mathcal{O}(H_f^{L_0,L_0}) = \textnormal{Crit}(f)$.

  Let $u: \mathbb{R} \times [0,1] \longrightarrow M$ be a Floer strip
  connecting $x_{-}$ to $x_{+}$, where
  $x_{\pm} \in \textnormal{Crit}(f)$. Identifying
  $(D \setminus \{-1,+1\}, \partial D \setminus \{-1,+1\})$ with
  $(\mathbb{R} \times [0,1], \mathbb{R} \times \{0\} \cup \mathbb{R}
  \times \{1\})$ we obtain from $u$ a map
  $u': (D \setminus \{-1,+1\}, \partial D \setminus \{-1,+1\})
  \longrightarrow (M,L_0)$ that extends continuously to a map
  $\overline{u}': (D, \partial D) \longrightarrow (M,L_0)$.  Since
  $L_0$ is weakly exact we have $\omega(\overline{u}') = 0$, hence
  $\omega(u) = 0$.

  By~\eqref{eq:E-omega-strips} it follows that
  $f(x_{-}) = f(x_{+}) + E(u)$, where $E(u)$ is the energy of $u$
  (see~\eqref{eq:floer-eq-1}). As $E(u)\geq 0$ we have
  $f(x_{-}) \geq f(x_{+})$ with equality iff $E(u)=0$.

  Suppose by contradiction that $\langle \mu_1(x), q \rangle \neq 0$
  for some $x \in \textnormal{Crit}(f)$. Let $u$ be a Floer strip that
  contributes to $\mu_1(x)$ and connects $x$ to $q$. By the above, we
  have $f(x) \geq f(q)$. Since $q$ is the unique maximum of $f$ it
  follows that $x = q$. Moreover, $E(u) = 0$. The latter implies that
  $\partial_su \equiv 0$. But this can happen only if $u$ is the
  constant strip at $q$ which contradicts the fact that $u$
  contributes to $\mu_1(x)$.
\end{proof}

To complete the proof of~\eqref{eq:SV-delta} it remains only to treat
the case when the Lagrangians $L_0, L_1, \ldots, L_k$ do not intersect
pairwise transversely.

Fix $\epsilon>0$. We apply $k$ Hamiltonian isotopies, one to each
Lagrangian $L_i$, $1\leq i \leq k$, such that the following holds:
\begin{enumerate}
\item The images $L'_1, \ldots, L'_k$ of $L_1, \ldots, L_k$ after
  these isotopies are such that $L_0, L'_1, \ldots, L'_k$ intersect
  pairwise transversely.
\item The Hofer length of each of these isotopies is
  $\leq \epsilon/k$.
\item $\delta(L_0; S) - \epsilon \leq \delta(L_0; S')$, where
  $S' = L'_1 \cup \ldots \cup L'_k$.
\end{enumerate}
Let $V: L_0 \leadsto (L_1, \ldots, L_k)$ be a weakly exact
cobordism. We now glue to each of the negative ends $L_i$ of $V$ the
Lagrangian suspension associated to the preceding Hamiltonian isotopy
used to move $L_i$ to $L'_i$. The result is a new cobordism
$V':L_0 \leadsto (L'_1, \ldots, L'_k)$ whose shadow satisfies:
$\mathcal{S}(V') \leq \mathcal{S}(V) + \epsilon$.

Since the ends of $V'$ intersect pairwise transversely it follows from
what we have already proved that
$\mathcal{S}(V') \geq \frac{1}{2}\delta(L_0; S')$. Therefore:
$$\tfrac{1}{2}\delta(L_0; S) - \tfrac{\epsilon}{2} \leq
\tfrac{1}{2}\delta(L_0; S') \leq \mathcal{S}(V') \leq \mathcal{S}(V) +
\epsilon.$$ As this holds for all $\epsilon>0$ the result readily
follows.

This completes the proof of~\eqref{eq:SV-delta}.

\medskip

We now turn to the proofs of the other two statements of the theorem.
\paragraph{{\sl Proof of statement~\eqref{i:ch-inters}}}
As in the previous part of the proof, we first assume that the
Lagrangians $L_1, \ldots, L_k$ intersect pairwise transversely.

Fix $\epsilon>0$ small enough such that
\begin{equation} \label{eq:SV-eps} \mathcal{S}(V) + \epsilon <
  \tfrac{1}{2}\delta^{\Sigma'}(N \cup S) - \tfrac{1}{2}\epsilon.
\end{equation}
Fix also $r>0$ with
\begin{equation} \label{eq:delta-sigma-p-N} \delta^{\Sigma'}(N\cup S)
  - \epsilon \leq \pi r^2 < \delta^{\Sigma'}(N \cup S).
\end{equation}
Write $\Sigma' = N \cap S = \{x_1, \ldots, x_m \}$ for the double
points of $N \cup S$, and let $e_{x_i}: B(r) \longrightarrow M$,
$i=1, \ldots, m$, be a collection of symplectic embeddings with the
properties as in the definition of $\delta^{\Sigma'}$ on
page~\pageref{eq:delta-sigma} (we take $\mathbb{L} = N \cup S$ and
$Q = \emptyset$ in that definition). Denote
$B := \cup_{i=1}^m \textnormal{image\,}(e_{x_i})$ and let $J^B$ be the
complex structure on $B$ whose value on
$\textnormal{image\,}(e_{x_i})$ is the push forward
$(e_{x_i})_* (J_{\textnormal{std}})$ of the standard complex structure
$J_{\textnormal{std}}$ of $B(r)$ via the map $e_{x_i}$.

We consider now two curves $\gamma, \gamma'$ of the the same shape as
in the earlier part of the proof (see Figure~\ref{f:gamma-gamma'-2})
and such that (similarly to~\eqref{eq:l-varphi-t}) there exists a
Hamiltonian isotopy, horizontal at infinity,
$\varphi_t: \mathbb{R}^2 \longrightarrow \mathbb{R}^2$, $t \in [0,1]$,
with $\varphi_0 = \id$, $\varphi_1(\gamma) = \gamma'$ and with
\begin{equation} \label{eq:l-varphi-t-2} \text{length} \{\varphi_t\}
  \leq \mathcal{S}(V) + \epsilon/2.
\end{equation}

\begin{figure}[htbp]
   \begin{center}
     \includegraphics[scale=0.5]{cob-v-2}
   \end{center}
   \caption{The curves $\gamma$ and $\gamma'$ and the cobordism $V$.}
   \label{f:gamma-gamma'-2}
\end{figure}

Next we set up the Fukaya categories involved in the proof. Let
$\mathcal{C}$ be the collection of Lagrangians $L_1, \ldots, L_k,
L$. We will work with the Fukaya $\fuk(\mathcal{C};p)$ defined with
choices of perturbation data $p$ with the following restrictions. The
Floer data of $(N, L_i)$, prescribed by $p$, are of the type
$\mathscr{D}_{N, L_i} = (H^{N,L_i}=0, J(p))$, where $J(p)=\{J_t(p)\}$
is a family of almost complex structures such that $J_t(p)|_{B} = J^B$
for all $t$. The Floer data $\mathscr{D}_{L_i,L_j}$ $i \neq j$ have
the $0$ Hamiltonian function. Finally, the perturbation data
$\mathscr{D}_{N, L_{i_d}, \ldots, L_{i_1}}$, $d \geq 2$, all have
vanishing Hamiltonian form, i.e.~they are of the type $(K=0, J)$.  Due
to the assumption that $N, L_1, \ldots, L_k$ intersect pairwise
transversely, regular choices of perturbation data with the above
properties do exist. We denote the space of such regular choices by
$E''_{\textnormal{reg}}$. (It is important to note that the
restriction that $J_t(p)|_{B} = J^B$ for every $t$ does not pose any
regularity problem since every Floer strip or polygon relevant for the
definition of $\fuk(\mathcal{C}; p)$ cannot have its image lying
entirely inside $B$, and outside of $B$ we have not posed any
restrictions on the choice of almost complex structures.)

We set up the Fukaya categories
$\fukcob(\widetilde{\mathcal{C}}_{1/2}, \iota_{\gamma}(p,h))$ and
$\fukcob(\widetilde{\mathcal{C}}_{1/2}, \iota_{\gamma'}(p,h))$ and the
associated inclusion functors in the same way as in the previous part
of the proof.

Let $\mathcal{V}$, $\mathcal{V}'$ be the Yoneda modules corresponding
to $V$, one time viewed as an object
$V \in \textnormal{Ob}(\fukcob(\widetilde{\mathcal{C}}_{1/2};
\iota_{\gamma}(p,h)))$ and one time as
$V \in \textnormal{Ob}(\fukcob(\widetilde{\mathcal{C}}_{1/2};
\iota_{\gamma'}(p,h)))$. Consider the pull-back modules
$$\mathcal{M}_{V; \gamma, p, h} := \mathcal{I}^*_{\gamma; p, h}
\mathcal{V}, \quad \mathcal{M}_{V ; \gamma', p, h} :=
\mathcal{I}^*_{\gamma; p, h} \mathcal{V}'.$$

By Proposition~\ref{p:icones-M_j}
(and~\eqref{eq:filtered-cob-it-cone}) we have:
\begin{equation} \label{eq:filtered-cob-it-cone-V} S^{s_h}
  \mathcal{M}_{V; \gamma, p, h} \\ = \; \tcn (\mathcal{L}_k
  \xrightarrow{\; \overline{\phi}_k \;} \tcn(\mathcal{L}_{k-1}
  \xrightarrow{\; \overline{\phi}_{k-1} \;} \tcn( \cdots
  \tcn(\mathcal{L}_2 \xrightarrow{\; \overline{\phi}_2 \;}
  \mathcal{L}_1 )) {\cdot}{\cdot}{\cdot})),
\end{equation}
where $s_h \longrightarrow 0$ as $h \to 0$, and similarly to what we
have had on page~\pageref{eq:filtered-cob-it-cone-W},
$\overline{\phi}_i = (\phi_i, 0, \bm{\delta^{(i)}})$, with $\phi_i$ a
homomorphism of modules that shifts action by $\leq 0$ and has
discrepancy $\leq \bm{\delta}(p,h)$, where for every $d$ we have
$\delta_d (p,h) \in \on$. By similar arguments,
$S^{s'_h} \mathcal{M}_{V ; \gamma', p, h} = \mathcal{L}$, where
$\mathcal{L}$ is the Yoneda module of $L$, and
$s'_h \longrightarrow 0$ as $h \longrightarrow 0$.

Consider now the chain complexes
$$\mathcal{C}_{p,h} := \mathcal{M}_{V; \gamma, p, h}(N), \quad
\mathcal{C}'_{p,h} := \mathcal{M}_{V; \gamma', p, h}(N)$$ endowed with
the differential coming from the $A_{\infty}$-modules
$\mathcal{M}_{V; \gamma, p, h}$, $\mathcal{M}_{V; \gamma', p, h}$.
(Note that $\mathcal{C}_{p,h}$ defined here is different than the
$\mathcal{C}_{p,h}$ from page~\pageref{pp:C-ph}.)

By definition
$\mathcal{C}_{p,h} = CF(\gamma \times N, V; \mathscr{D}_{\gamma \times
  N, V})$, where $\mathscr{D}_{\gamma \times N, V}$ is the Floer datum
prescribed by $\iota_{\gamma}(p,h)$. Similarly
$\mathcal{C}'_{p,h} = CF(\gamma' \times N, V; \mathscr{D}_{\gamma'
  \times N, V})$, where $\mathscr{D}_{\gamma' \times N, V}$ is the
Floer datum prescribed by $\iota_{\gamma'}(p,h)$. Consider now the
Hamiltonian isotopy
$\widetilde{\varphi}_t := \varphi_t \times \id: \mathbb{R}^2 \times M
\longrightarrow \mathbb{R}^2 \times M$, $t \in [0,1]$, where
$\varphi_t$ is the Hamiltonian isotopy from
page~\pageref{eq:l-varphi-t-2}. Note that $\{\widetilde{\varphi}_t\}$
is horizontal at infinity and by~\eqref{eq:l-varphi-t-2} has Hofer
length $\leq \mathcal{S}(V) + \tfrac{1}{2}\epsilon$. Since
$\widetilde{\varphi}_1(\gamma \times N) = \gamma' \times N$, by
standard Floer theory (see e.g.~\cite[Chapter~5]{FO3:book-vol1}) this
isotopy induces two chain maps:
$$\phi: \mathcal{C}_{p,h} \longrightarrow \mathcal{C}'_{p,h},
\quad \psi: \mathcal{C}'_{p,h} \longrightarrow \mathcal{C}_{p,h},$$
which are both filtered and such that $\psi \circ \phi$ is chain
homotopic to $\id$ by a chain homotopy that shifts action by
$\leq \mathcal{S}(V) + \epsilon$. More specifically
$$\psi \circ \phi = id + K d^{\mathcal{C}_{p,h}} +
d^{\mathcal{C}_{p,h}} K,$$ where $d^{\mathcal{C}_{p,h}}$ is the
differential of $\mathcal{C}_{p,h}$ and $K$ is a $\Lambda$-linear map
that shifts action by $\leq \mathcal{S}(V) + \epsilon$. Using the
formalism of~\eqref{eq:Bh-1} this means that
\begin{equation} \label{eq:Bh-psi-phi-id}
  B_h(\psi \circ \phi - \id) \leq \mathcal{S}(V) + \epsilon.
\end{equation}

We now appeal to Theorem~\ref{t:itcones}, by which we obtain the
following:
\begin{enumerate}
\item A chain complex $\mathcal{M}(N)$ whose underlying
  $\Lambda$-module coincides with $\mathcal{C}_{p,h}$ and whose
  differential $\mu_1^{\mathcal{M}(N)}$ is described by~\eqref{eq:aij}
  from Theorem~\ref{t:itcones}.
\item An isomorphism of chain complexes
  $\sigma_1: \mathcal{M}(N) \longrightarrow \mathcal{C}_{p,h}$ such
  that both $\sigma_1$ and its inverse
  $\sigma_1^{-1}:\mathcal{C}_{p,h} \longrightarrow \mathcal{M}(N)$
  shift action by $\leq C^{(1)}(p,h)$, where $C^{(1)}(p,h) \in \on$.
\end{enumerate}

We now estimate the action drop $\delta_{\mu_1^{\mathcal{M}(N)}}$ (as
defined in~\eqref{eq:delta-dC}) of the differential
$\mu_1^{\mathcal{M}(N)}$ of the chain complex $\mathcal{M}(N)$.
\label{pp:estim-delta_mu_1}
By Theorem~\ref{t:itcones} the differential $\mu_1^{\mathcal{M}(N)}$
comprises various $\mu_d$-operations, $1\leq d\leq k$, associated to
tuples of Lagrangians of the type
$(N, L_{i_d}, L_{i_{d-1}}, \ldots, L_{i_2}, L_{i_1})$, where
$i=i_1 < \cdots < i_d\leq j$, $1 \leq i\leq j \leq k$. Recall also
that the perturbation data $p \in E''_{\textnormal{reg}}$ were chosen
with vanishing Hamiltonian perturbation for tuple of Lagrangians as
above. Therefore, the above mentioned $\mu_d$-operations are defined
by counting (unperturbed) pseudo-holomorphic polygons $u$ with corners
mapped to intersection points between consecutive pairs of Lagrangians
in tuples as above. Each polygon $u$ contributing to these
$\mu_d$-operation has an intersection point in $N \cap L_j$ as one of
its inputs and an intersection point in $N \cap L_i$ as its
output. Moreover, these polygons are $J^B$-holomorphic over $B$.  We
thus obtain:
$$\omega(u) \geq \omega(\textnormal{image\,}(u) \cap B) \geq
\tfrac{1}{4} \pi r^2 + \tfrac{1}{4}\pi r^2 = \tfrac{1}{2}\pi r^2,$$
where the first inequality hold because $u$ is
unperturbed-pseudo-holomorphic over its entire domain, while the
second inequality follows from a Lelong-inequality type of argument
(see e.g.~\cite{Bar-Cor:Serre, Bar-Cor:NATO}). Combining the preceding
inequalities with~\eqref{eq:delta-sigma-p-N} we deduce that every
Floer polygon $u$ that participate in the calculation of the
differential $\mu_1^{\mathcal{M}(N)}$ must satisfy
$\omega(u) \geq \tfrac{1}{2}\delta^{\Sigma'}(N\cup S) -
\tfrac{\epsilon}{2}$. It follows that
\begin{equation} \label{eq:delta-mu-1-M}
  \delta_{\mu_1^{\mathcal{M}(N)}} \geq
  \tfrac{1}{2}\delta^{\Sigma'}(N\cup S) - \tfrac{\epsilon}{2}.
\end{equation}

In view of the map $\sigma_1$ and its inverse $\sigma_1^{-1}$,
mentioned earlier in the proof, we deduce the following estimate for
the action drop of the differential of $\mathcal{C}_{p,h}$:
$$\delta_{d^{\mathcal{C}_{p,h}}} \geq
\tfrac{1}{2}\delta^{\Sigma'}(N\cup S) - \tfrac{\epsilon}{2} -
2C^{(1)}(p,h).$$ As $C^{(1)}(p,h) \in \on$, by choosing
$p \in E''_{\textnormal{reg}}$ close enough to $\mathcal{N}$ and the
profile function $h$ small enough, we may assume in view
of~\eqref{eq:SV-eps} that
$\tfrac{1}{2}\delta^{\Sigma'}(N\cup S) - \tfrac{\epsilon}{2} -
2C^{(1)}(p,h) > \mathcal{S}(V) + \epsilon$. Combining the above
together with~\eqref{eq:Bh-psi-phi-id} we obtain:
$$\delta_{d^{\mathcal{C}_{p,h}}} >
\mathcal{S}(V)+ \epsilon \geq B_h(\psi \circ \phi - \id).$$ By
Lemma~\ref{l:rig-cplx-n} (applied with $C = \mathcal{C}_{p,h}$,
$f = \psi \circ \phi$, $g = \id$) we deduce that $\psi \circ \phi$ is
injective. It follows that
$\phi: \mathcal{C}_{p,h} \longrightarrow \mathcal{C}'_{p,h}$ is
injective too, hence
$\dim_{\Lambda} \mathcal{C}_{p,h} \leq \dim_{\Lambda}
\mathcal{C}'_{p,h}$. But
$$\mathcal{C}_{p,h} = \bigoplus_{i=1}^k \bigoplus_{x \in N \cap L_i}
\Lambda \cdot x, \quad \mathcal{C}'_{p,h} = \bigoplus_{x \in N \cap
  L} \Lambda \cdot x,$$ which implies the desired
inequality~\eqref{eq:N-cap-L}. This completes the proof of
statement~\eqref{i:ch-inters} under the additional assumption that
$L_1, \ldots, L_k$ intersect pairwise transversely.

It remains to treat the case when the Lagrangians $L_1, \ldots, L_k$
do not necessarily intersect pairwise transversely.

Let $V$ be a cobordism as in the statement of the theorem. Fix $r>0$
and $\epsilon>0$ with
\begin{equation} \label{eq:perturb-a-eps}
  \mathcal{S}(V) + \epsilon < \pi r^2 < \delta^{\Sigma'}(N \cup S).
\end{equation}
Let $e_x: B(r) \longrightarrow M$, $x \in \Sigma' = N \cap S$, be a
collection of symplectic embeddings as in the definition of
$\delta^{\Sigma'}(N \cup S)$ on page~\pageref{eq:delta-sigma}.  Since
$\Sigma' = \cup_{i=1}^k (N \cap L_i)$ and the latter union is disjoint
every $x \in \Sigma'$ belongs to {\em precisely} one of the
Lagrangians $L_1, \ldots, L_k$. Now let $y \in \Sigma'$ and assume
that $y \in N \cap L_i$. Let $j \neq i$. It is easy to see from the
assumptions imposed on the embeddings $e_x$ in the definition of
$\delta^{\Sigma'}(N \cup S)$ that $L_j \cap e_y(B(r)) = \emptyset$. In
particular $L_j \cap L_i$ lies outside of $e_y(B(r))$. It follows that
$\cup_{i'<i''} (L_{i'} \cap L_{i''})$ lies outside of
$B := \cup_{x \in \Sigma'} \textnormal{image\,} e_x(B(r))$.

Next, apply a small Hamiltonian perturbation to each of the
Lagrangians $L_1, \ldots, L_k$, keeping them fixed inside $B$, so as
to obtain new Lagrangians $L'_1, \ldots, L'_k$ that intersect pairwise
transversely. By taking these perturbations small enough we may also
assume that no new intersection points between $S$ and $N$ have been
created, i.e.~$L'_i \cap N = L_i \cap N$ for every $i$.  Moreover, we
take these Hamiltonian perturbations to be small in the Hofer metric
so that the Hofer length of each of the isotopies generating the above
perturbations is $\leq \tfrac{\epsilon}{2k}$.

We now glue to each of the negative ends $L_i$ of $V$ the Lagrangian
suspension associated to the preceding Hamiltonian isotopy used to
move $L_i$ to $L'_i$. The result is a new cobordism
$V':L \leadsto (L'_1, \ldots, L'_k)$ with
$\mathcal{S}(V') \leq \mathcal{S}(V) + \frac{1}{2}\epsilon$. Combining
with~\eqref{eq:perturb-a-eps} we get:
$$\mathcal{S}(V') \leq \delta^{\Sigma'}(N \cup S').$$
As the ends $L'_i$ of $V'$ intersect pairwise transversely, by what we
have proved earlier we have:
$$\#(N\cap L) \geq \sum_{i=1}^{k}\# (N\cap L'_{i}).$$ Since
$N \cap L'_i = N \cap L_i$ for all $i$, the results follows and
completes the proof of statement~\eqref{i:ch-inters}.

\medskip
\paragraph{{\sl Proof of statement~\eqref{i:hf-inters}}}
The proof is similar to the proof of statement~\eqref{i:ch-inters}
above, only that now we use Proposition~\ref{prop:rig-cplx2} instead
of Lemma~\ref{l:rig-cplx-n} to estimate $\# (N\cap L)$
in~\eqref{eq:N-cap-L-HF}. Below we will mainly go over the points in
the proof that differ from the proof of statement~\eqref{i:ch-inters}.

Fix $\epsilon>0$ small enough and $r>0$ so that:
\begin{equation} \label{eq:SV-pr2-delta} \mathcal{S}(V) + \epsilon <
  \tfrac{1}{4}\delta^{\Sigma''}(S;N) - \tfrac{1}{4}\epsilon, \quad
  \delta^{\Sigma''}(S; N)-\epsilon \leq \pi r^2 <
  \delta^{\Sigma''}(S;N).
\end{equation}
Next, fix symplectic embeddings $e_x: B(r) \longrightarrow M$,
$x \in \Sigma''$, as in the definition of $\delta^{\Sigma''}(S;N)$ on
page~\pageref{eq:delta-sigma}. Fix also curves $\gamma$, $\gamma'$ as
in the proof of statement~\eqref{i:ch-inters}. We set up the Fukaya
categories $\fuk(\mathcal{C};p)$,
$\fukcob(\widetilde{\mathcal{C}}_{1/2}, \iota_{\gamma}(p,h))$,
$\fukcob(\widetilde{\mathcal{C}}_{1/2}, \iota_{\gamma'}(p,h))$ and the
inclusion functors $\mathcal{I}_{\gamma; p, h}$,
$\mathcal{I}_{\gamma'; p, h}$, in the same way as in the proof of
statement~\eqref{i:ch-inters}. We then define the chain complexes
$\mathcal{C}_{p,h}$, $\mathcal{C}'_{p,h}$, and the two chain maps
$\phi: \mathcal{C}_{p,h} \longrightarrow \mathcal{C}'_{p,h}$,
$\psi: \mathcal{C}'_{p,h} \longrightarrow \mathcal{C}_{p,h}$, with
\begin{equation} \label{eq:psi-circ-phi-2}
  \psi \circ \phi = \id + K \circ d^{\mathcal{C}_{p,h}} +
  d^{\mathcal{C}_{p,h}} \circ K,
\end{equation}
where $K$ shifts action by $\leq \mathcal{S}(V) + \epsilon$.

As before, we now use Theorem~\ref{t:itcones} and obtain a chain
complex $\mathcal{M}(N)$ whose underlying $\Lambda$-module coincides
with $\mathcal{C}_{p,h}$ and equals
\begin{equation} \label{eq:MN-split}
  \mathcal{M}(N) = \oplus_{i=1}^k CF(N,L_i; \mathscr{D}_{N,L_i}).
\end{equation}
By Theorem~\ref{t:itcones} the differential $\mu_1^{\mathcal{M}(n)}$
can be written with respect to the splitting~\eqref{eq:MN-split} as an
upper triangular matrix of operators $(a_{i,j})$ with diagonal
elements $a_{i,i} = \mu_1^{CF(N,L_i; \mathscr{D}_{N,L_i})}$. Write
$\mu_1^{\mathcal{M}(N)} = d_0 + d_1$, where
$d_0 = \oplus_{i=1}^k \mu_1^{CF(N,L_i; \mathscr{D}_{N,L_i})}$ with
respect to the splitting~\eqref{eq:MN-split} and
$d_1: \mathcal{M}(N) \longrightarrow \mathcal{M}(N)$ is the operator
represented by the part of the matrix $(a_{i,j})$ that lies strictly
above the diagonal.

The operator $d_1$ consists of sums of $\mu_d$-operations, $d \geq 2$,
where among the inputs of each such operation there is at least one
point from $L_i \cap L_j$, $i<j$. A similar argument to the one used
on page~\pageref{pp:estim-delta_mu_1} in estimating
$\delta_{\mu_1^{\mathcal{M}(N)}}$ in the proof of
statement~\eqref{i:ch-inters} shows that
$\delta_{d_1} \geq \tfrac{1}{4}\pi r^2$. Here, $\delta_{d_1}$ is the
action drop of $d_1$ (see~\S\ref{s:filt-ch},
page~\pageref{pp:delta-f}). Combining with~\eqref{eq:SV-pr2-delta} we
get:
\begin{equation} \label{eq:delta-d_1-ineq}
  \delta_{d_1} \geq \tfrac{1}{4}\delta^{\Sigma''}(S;N) -
  \tfrac{1}{4}\epsilon.
\end{equation}

Put
$f':= \psi \circ \phi : \mathcal{C}_{p,h} \longrightarrow
\mathcal{C}_{p,h}$. By~\eqref{eq:psi-circ-phi-2} we have
$B_h(f'-\id) \leq \mathcal{S}(V) + \epsilon$. Recall from
Theorem~\ref{t:itcones} the isomorphism of chain complexes
$\sigma_1: \mathcal{C}_{p,h} \longrightarrow \mathcal{M}(N)$ such that
both $\sigma_1$ and its inverse $\sigma_1^{-1}$ shift action by
$\leq C^{(1)}(p,h)$, where $C^{(1)}(p,h) \in \on$. Consider
$$f:= \sigma_1 \circ f' \circ \sigma_1^{-1} : \mathcal{M}(N)
\longrightarrow \mathcal{M}(N).$$ We would like to apply
Proposition~\ref{prop:rig-cplx2} to $C = \mathcal{M}(N)$, $d_0$, $d_1$
as defined above and the map $f$. We have:
$$f - \id = (\sigma_1 \circ K \circ \sigma_1^{-1}) \circ
d^{\mathcal{C}_{p,h}} + d^{\mathcal{C}_{p,h}} \circ (\sigma_1 \circ K
\circ \sigma_1^{-1}),$$ hence
$B_h(f - \id) \leq \mathcal{S}(V) + \epsilon + 2 C^{(1)}(p,h)$. As
$C^{(1)}(p,h) \in \on$, by taking $p$ close enough to $\mathcal{N}$
and the profile function $h$ small enough, we may assume in view
of~\eqref{eq:SV-pr2-delta} that
$$\mathcal{S}(V) + \epsilon + 2 C^{(1)}(p,h) <
\tfrac{1}{4}\delta^{\Sigma''}(S;N) - \tfrac{1}{4}\epsilon.$$

Together with~\eqref{eq:delta-d_1-ineq} we now obtain:
\begin{equation} \label{eq:Bh-f-id} B_h(f - \id) < \delta_{d_1}.
\end{equation}

In order to apply Proposition~\ref{prop:rig-cplx2} it remains to check
that
\begin{equation} \label{eq:dim-H-ineq}
  \dim_{\Lambda} H_*(\mathcal{M}(N), d_0) \geq \dim_{\Lambda}
  H_*(\mathcal{M}(N), \mu_1^{\mathcal{M}(N)}).
\end{equation}
This follows from standard results in homological algebra since
$$H_*(\mathcal{M}(N), d_0) = \oplus_{i=1}^k HF(N, L_i), \quad
H_*(\mathcal{M}(N), \mu_1^{\mathcal{M}(N)}) \cong
H_*(\mathcal{C}_{p,h}, d^{\mathcal{C}_{P,h}})$$ and
$\mathcal{C}_{p,h}$ is an iterated cone of the type
\begin{equation*}
  \begin{aligned}
    \mathcal{C}_{p,h} = \tcn (CF(N,L_k)
    \to \tcn( & CF(N, L_{k-1}) \\
    & \to \tcn( \cdots \tcn(CF(N, L_2) \to CF(N,L_1)))
    {\cdot}{\cdot}{\cdot})).
  \end{aligned}
\end{equation*}

We are now in position to apply Proposition~\ref{prop:rig-cplx2}, by
which we obtain:
$$\dim_{\Lambda} (\textnormal{image\,}(f)) \geq \dim_{\Lambda}
H_*(\mathcal{M}(N), d_0) = \sum_{i=1}^k \dim_{\Lambda} HF(N,L_i).$$ On
the other hand:
$$\dim_{\Lambda} (\textnormal{image\,}(f)) \leq \dim_{\Lambda} \mathcal{M}(N)
= \sum_{i=1}^k \#(N \cap L_i).$$ Putting the last two inequalities
together yields~\eqref{eq:N-cap-L-HF} and concludes the proof of
statement~\eqref{i:hf-inters}.  \qed

\begin{rem}\label{rem:Misha-etc}
a. The intersection
 result at the point  (a) in Theorem \ref{thm:fission} implies variants of both inequalities (\ref{eq:SV-delta})  and
 (\ref{eq:N-cap-L-HF}) but with slightly different assumptions and for different constants $\delta$.
 This is obvious concerning (\ref{eq:N-cap-L-HF}) and for inequality (\ref{eq:SV-delta}) it is seen by applying (\ref{eq:N-cap-L}) to the  cobordism $W:\emptyset\cobto (L,L_{1},\ldots, L_{k})$ obtained by bending the positive end of $V$ half way clockwise - as in Figure \ref{f:cob-v-w} - and taking $N$ a Hamiltonian
 perturbation of $L$.

 b. The following argument, due to Misha Khanevsky, leads to a more direct proof of the first part of Theorem \ref{thm:fission} but gives a weaker inequality.  We reproduce the argument here with Misha's permission.
 A result of Usher (Theorem 4.9 in \cite{Usher3}) claims that, given two Lagrangians $V$ and $V'$ that intersect transversely and non-trivially, there exists $\delta>0$ depending on
 $V$ and $V'$, such that the energy required to disjoin $V$ from $V'$ is greater than $\delta$.
 This result was proven for compact or (tame at infinity) symplectic manifolds but can be adjusted without any difficulty to the case of  Lagrangians with cylindrical ends in $\C\times M$. Assume that
 $L\not\subset \cup_{i}L_{i}$ and let $T$ be a small Lagrangian torus, disjoint from all $L_{i}$'s, and
 such that $T$ intersects $L$ transversally and non-trivially. Let $\gamma'$ be a curve as in Figure \ref{f:gamma-gamma'-2} and let $V'=\gamma'\times T$.  Thus $V'$ and $V$ intersect non-trivially
 and transversely (see also Figure \ref{f:cob-v-w}). The isotopy taking the curve $\gamma$ to the curve $\gamma'$  in Figure \ref{f:gamma-gamma'-2} disjoins $V'$ from $V$ and thus its energy, that can be assumed to be as close as needed to $\mathcal{S}(V)$, has to exceed $\delta$.
By inspecting Usher's proof \cite{Usher3}, the constant $\delta$ is smaller than the smallest area of a non-trivial  $J$-disk, sphere or strip with boundaries on $V$ and on $V'$. By standard arguments, the latter areas can be shown to only depend on $L$ and $T$ and this finishes Khanevsky's argument. However, notice that the
dependence on $T$ of the constant $\delta$ here  means that it is generally smaller than $\delta(L;S)$ from the statement of Theorem \ref{thm:fission}.  Moreover, this argument does not imply the points (a) and (b) of the statement and it also can not be adjusted to estimate the algebraic measurements that we will see later in the paper in Corollary \ref{cor:alg-weights}.
   \end{rem}

\subsection{Proof of
  Theorem~\ref{thm:fission-mon}} \label{sb:prf-thm-fission-mon} The
proofs of statements~\eqref{i:ch-inters} and~\eqref{i:hf-inters} given
in~\S\ref{sb:prf-thm-fission} carry over to the monotone case without
any modifications.

We now explain how to adjust the proof of~\eqref{eq:SV-delta} given
in~\S\ref{sb:prf-thm-fission} in order to
prove~\eqref{eq:SV-delta-mon}.

We may assume throughout the proof that $\mathcal{S}(V) < A_L$, for
otherwise the inequality~\eqref{eq:SV-delta-mon} is trivially
satisfied. We need to prove that
$\mathcal{S}(V) \geq \tfrac{1}{2}\delta(L;S)$.

We fix $\epsilon>0$ as in the proof of~\eqref{eq:SV-delta} but we
require additionally that
\begin{equation} \label{eq:SV-AL-eps} \mathcal{S}(W) + \epsilon < A_L.
\end{equation}
The proof now goes along the same lines as the proof
of~\eqref{eq:SV-delta}, detailed in~\S\ref{sb:prf-thm-fission}, up to
the point where we had to use Lemma~\ref{l:unit-we} (see
page~\pageref{l:unit-we}). That lemma does not hold in the monotone
case, and we will now use the following lemma instead:
\begin{lem} \label{l:unit-mo} Let $c \in \textnormal{Crit}(f)$, viewed
  as an element of $\mathcal{O}(H^{L_0,L_0}_f)$. If
  $\langle \mu^{CF(L_0,L_0;p)}_1(c), q \rangle \neq 0$ then
  $\nu(\langle \mu_1^{CF(L_0,L_0;p)}(c), q\rangle) \geq A_{L_0}$.
\end{lem}
We postpone the proof for a while and continue with the proof of
Theorem~\ref{thm:fission-mon}. As in the proof of
Theorem~\ref{thm:fission-mon} we decompose the element $b$
from~\eqref{eq:mu-1-b-A} as $b=b_0 + \cdots + b_k$ with
$b_i \in CF(L_0,L_i; p)$. We cannot deduce, as earlier, that
$\langle \mu_1(b_0), q \rangle = 0$, however by Lemma~\ref{l:unit-mo}
and~\eqref{eq:SV-AL-eps} we still obtain that
$$\nu( \langle \mu_1(b_0), q\rangle) \geq A_{L_0} - \mathcal{S}(W) -
C^{(2)}(p,h) - \tfrac{1}{2}\epsilon > \tfrac{1}{2}\epsilon -
C^{(2)}(p,h),$$ where $C^{(2)}(p,h) \in \on$. By taking $p$ close
enough to $p_0 \in \mathcal{N}$ and $h$ small enough we may assume
that $\nu( \langle \mu_1(b_0), q\rangle) > 0$. In view
of~\eqref{eq:q-equals-sum} we can now deduce, as before, that there
exists $1 \leq j_0 \leq k$ such that~\eqref{eq:nu-b-j0} holds. From
this point on, the proof continues exactly as carried out in the
weakly exact case in~\S\ref{sb:prf-thm-fission}.

It remains to proof the preceding lemma.
\begin{proof}[Proof of Lemma~\ref{l:unit-mo}]
  Let $u \in \mathcal{M}(c,q; \mathscr{D}_{L_0,L_0})$ be a Floer strip
  that goes from $c$ to $q$ and contributes to
  $\mu^{CF(L_0,L_0;p)}_1(x)$. We need to show that
  $\omega(u) \geq A_{L_0}$.

  Indeed, as in the proof of Lemma~\ref{l:unit-we} on
  page~\pageref{l:unit-we}, after identifying
  $(\mathbb{R} \times [0,1], \mathbb{R} \times \{0\} \cup \mathbb{R}
  \times \{1\})$ with
  $(D\setminus \{-1,+1\}, \partial D \setminus \{-1,+1\})$ the map $u$
  extends continuously to a map
  $\overline{u}: (D, \partial D) \longrightarrow (M,L_0)$.  The
  dimension of the component of $u$ in the space
  $\mathcal{M}^*(c,q; \mathscr{D}_{L_0,L_0})$ of {\em
    non-parametrized} Floer trajectories connecting $c$ to $q$ is
  given by
  $$\dim \mathcal{M}^*_{u}(c,q; \mathscr{D}_{L_0,L_0}) =
  |c|-|q|-1+\mu([\overline{u}]) = |c|-n-1+\mu([\overline{u}]),$$ where
  $\mu$ is the Maslov index and $[\overline{u}] \in H_2^D(M,L_0)$ is
  the homology class induced by $\overline{u}$.  Since
  $\dim \mathcal{M}^*_{u}(c,q; \mathscr{D}_{L_0,L_0}) \geq 0$ we must
  have $\mu([\overline{u}]) \geq n+1-|c| > 0$. By monotonicity of
  $L_0$ we have $\omega([\overline{u}]) \geq A_{L_0}$, hence
  $\omega(u) \geq A_{L_0}$. This concludes the proof of the lemma.
\end{proof}

The proof of Theorem~\ref{thm:fission-mon} is now complete.
\qed

config: arxiv


TITLE: A very curious rational fraction that converges. What is the value?
QUESTION [19 upvotes]: Is there any closed form for the following limit?

Define the sequence
  $$ \begin{cases}
  a_{n+1} = b_n+2a_n + 14\\
  b_{n+1} = 9b_n+ 2a_n+70
\end{cases}$$
  with initial values $a_0 = b_0 = 1$. Then $\lim_{n\to\infty} \frac{a_n}{b_n} = ? $

The limit is approximately $0.1376$. My math teacher Carlos Ivorra says that this limit have a closed form involving the sine of an angle. What is the closed form for is limit?
NOTE: I have found this (and another series of converging sequences) by the use of an ancient method for calculating sines recently rediscovered. I'll give the details soon as a more general question.

REPLY [6 votes]: Here is a rigorous and systematic way to analyze the existence and value of the limit. (miracle173's answer didn't prove existence but uses essentially the same method for finding the value if it exists.)
Let $c_n = \frac{a_n}{b_n}$ (for each $n \in \mathbb{N}$).
Then $b_{n+1} c_{n+1} = b_n + 2 b_n c_n + 14$.
And $b_{n+1} = 9 b_n + 2 b_n c_n + 70$.
Thus $c_{n+1} = \dfrac{ b_n + 2 b_n c_n + 14 }{ 9 b_n + 2 b_n c_n + 70 } = \dfrac{ 1 + 2 c_n + \frac{14}{b_n} }{ 9 + 2 c_n + \frac{70}{b_n} }$.
Let $r = \dfrac{\sqrt{57}-7}{4}$ so that $r = \dfrac{1+2r}{9+2r}$, and let $d_n = c_n - r$.
Take any $ε > 0$, and let "$[x]$" denote "$\{ t : t \in \mathbb{R} \land |t| \le x \}$".
Then $c_{n+1} \in \dfrac{ 1 + 2 c_n }{ 9 + 2 c_n } + [ε]$ as $n \to \infty$, since $b_n \to \infty$ as $n \to \infty$.
Then $d_{n+1} \in \dfrac{ 1 + 2 (r+d_n) }{ 9 + 2 (r+d_n) } + [ε] - r = \dfrac{ 2 - 2 r }{ 9 + 2 (r+d_n) } d_n + [ε] \subseteq [\frac29 d_n] + [ε]$.
Thus $d_n \in [2ε]$ eventually as $n \to \infty$, since $\frac29 x + 1 < x - \frac12$ for any $x > 2$.
Therefore since $ε$ was arbitrary, $d_n \to 0$ as $n \to \infty$.


TITLE: bounded monotone sequences converge - why do we need irrational numbers?
QUESTION [1 upvotes]: we've proved that monotone sequences converge, there is a question afterwards which asks why do we need the completeness axiom for this? Why wouldn't it be true if we didn't have irrational numbers?
I can't seem to answer this, so I feel as though I'm missing something.

REPLY [1 votes]: To show completeness is necessary, you just need to construct an increasing sequence of rational numbers converging to an irrational number.  There are many ways of doing this, for example:

Use the decimal expansion of a well-known irrational number, as in David Mitra's comment.
Use the partial sums of $1 + \frac11 + \frac12 + \frac1{3!} + \cdots = e$.
Use rational approximations to $\sqrt{2}$.  For each $n$, let $q_n$ be the largest rational number with denominator at most $n$ such that $q_n^2 < 2$.


TITLE: Finite vs infinite dimensional vector spaces
QUESTION [15 upvotes]: What familiar and intuitive properties of finite dimensional vector spaces fails in infinite dimensions?
For example:

In infinite dimensions there are non-continuous linear maps.
In infinite dimensions always $\dim V <\dim V^*$ and, in particular, $V\not\simeq V^{**}$.

REPLY [0 votes]: Assuming the axiom of choice, any vector space admits a basis and so has a norm. Therefore, we may look at topological properties. With this viewpoint, there is the classical result:

Theorem: (Riesz) A normed vector space is locally compact iff it is finite dimensional.

But there are also less known results like:

Property: Let $X$ be a normed vector space and $K \subset X$ be a compact subset. Then $X \backslash K$ is path-connected.

It is clearly false in finite dimension (take a sphere for example).


TITLE: Connecting homomorphism is determined upto a sign $\pm 1$?
QUESTION [2 upvotes]: Let $0\to K'\to K\to K''\to 0$ be a short exact sequence of chain maps and suppose the boundary map is going from $K'_n\to K'_{n-1}$ and similarly for $K,K''$. Then this induces homology long exact sequence.
$\textbf{Q:}$ Suppose one knows naturality of homology functor and existence of long exact sequence. Then connecting homomorphism $\partial_\star:H_\star(K'')\to H_{\star-1}(K')$ is determined up to a sign. What is the meaning of the question? Or why do I see there is a sign? The connecting homomorphism is usually defined by a "switchback" map.(i.e. pull back along $K\to K''$ to a represenative and descend to the lower $K$ to see this defining a cycle in $K'$ which requires further descending.) What is the meaning of this $\pm 1$ here? If I have $\pm 1$ issue, then I would certainly have trouble for naturality.
Ref. Dold Algebraic Topology pg 23. 2.16 Exercise.2

REPLY [1 votes]: Let $0\to A\to B\to C \to 0$ be a SES of chain complexes.
Then we're given that there is a LES of homology, with $f_*: H_n(A)\to H_n(B)$, $g_*:H_n(B)\to H_n(C)$, and $\partial_*:H_n(C)\to H_{n-1}(A)$, all of which are natural (in the sense that they represent natural transformations of functors on SESs).
Now note that $-\partial_*$ is still natural, and the LES is still exact if we replace $\partial_*$ with $-\partial_*$.
That's all that the book is trying to say.
The goal of this exercise is to show that these two possibilities are the only possible options for natural maps $H_n(C)\to H_{n-1}(A)$ that still make the LES exact. However, you don't seem to be asking how to show the exercise, just where the $-1$ comes from, so I'll leave it here.


TITLE: How to decompose a matrix into the outer product of two vectors?
QUESTION [8 upvotes]: I have a matrix $M$ and I would like to find two vectors $u$ and $v$, that minimize
$$ \sum_{i,j} (M_{i,j}-u_iv_j)^2  $$
How can I do this (numerically)?
Actually this is very simplified compared to my actual problem. For instance, I dont really have a full matrix, but only values for some range of $(i,j)$. However, I think once I know how to solve it in the general case, it should be straightforward to skip certain entries.

REPLY [9 votes]: First, note that $\Sigma_{ij} (M_{ij}-u_iv_j)^2 = \|M-uv^T\|_F^2$, where $\|\cdot\|_F$ denotes the Frobenius norm of a matrix. If you're doing a Google search that may be helpful.
In the unconstrained case, there is an analytic solution given by the singular value decomposition. That is: let $M=U\Sigma V^T$ be the SVD of $M$, with the singular values arranged in nonincreasing order along the diagonal of $\Sigma$. Then the triple $(\sigma_1,u_1,v_1)$ formed from the first singular value, the first column of $U$, the first column of $V$, respectively, can be used to compute your $u$ and $v$. For instance, $u=\sigma_1 u_1$, $v=v_1$ would work, but really any $u=\alpha\sigma_1 u_1$, $v=\alpha^{-1} v_1$ for $\alpha\neq 0$ will work as well.
In the constrained case, unfortunately, you have a nonconvex optimization problem on your hands. Presumably you have a problem like this:
\begin{array}{ll}
\text{minimize} & \|M-uv^T\|_F^2 \\
\text{subject to} & L_{ij} \leq M_{ij} \leq U_{ij} \quad \forall i,j
\end{array}
where $L$ and $U$ are lower and upper bound matrices (possibly with infinite entries where you have missing information). This is an extreme (e.g., rank-1) version of a low-rank approximation problem.
There is a variety of literature about this, but again, it's nonconvex, which means that it's not easy to solve in practice. A common technique is to alternate between minimizing over $u$ (holding $v$ fixed) and minimizing over $v$ (holding $u$ fixed). There is no guarantee this gives you the best solution though.
EDIT: A simpler way to look at the problem where you know only a portion of the elements of $M$ is that you're minimizing only over the indices that you know: that is,
$$\text{minimize} ~ \sum_{(i,j)\in\mathcal{I}} (M_{ij}-u_iv_j)^2$$
for some index set $\mathcal{I}$. After all, for the elements you don't know, you might as well just set $M_{ij}=u_iv_j$ after you've determined the $u$ and $v$ you prefer, so they will contribute 0 to the final Frobenius objective. Alas, that doesn't make the problem any easier; it's still nonconvex.


TITLE: Derivative of step function is Dirac delta function
QUESTION [1 upvotes]: The derivative of a locally integrable function $f$ is by definition the linear functional
$$\phi \mapsto -\int_\mathbb{R} \phi'(x)f(x) dx$$
Using this definition, why is it that the derivative of the step function is the dirac delta function?
Since the derivative is a linear functional, I don't understand why we should get a function like the dirac delta. Linear functionals and functions are not the same...

REPLY [2 votes]: For simplicity let's assume the step function $f$ is $f = \chi_{[0,\infty)}$. Let $\phi$ be a test function, then
$$(Df)(\phi) = -\int \phi'(x) f(x) dx = -\int_0^\infty \phi'(x) dx  = - \phi(C) + \phi(0) = \phi(0)$$
where $C$ is any large number such that $\phi(x) = 0$ for all $x\geq C$. Thus $Df$ is the delta function at $0$.


TITLE: An easier method to calculate factorials?
QUESTION [3 upvotes]: To find the factorial of a number, $n$, you need to multiply $n$ by every number that comes before it.
For example, if $n = 4$, then $n! = 24$ since $4 \cdot 3 \cdot 2 \cdot 1 = 24$.
However, this method is very time consuming and, as $n$ gets larger, this method also become more difficult, so is there an easier method that I can use to find the factorial of any number?

REPLY [3 votes]: To answer your question, there are several methods that can help calculate factorials faster, though most are either approximations or shortcuts for calculation. I will be detailing some of these methods below:
Method 1
As mentioned by Joe in the comments, Stirling's approximation is a good method to approximate the value of a large factorial, and by rewriting the factorial as a Gamma function, the following formula is obtained:
$$n! \sim \sqrt{2\pi n} \cdot \biggl(\frac{n}{e} \biggr)^n$$
$$\\$$
Method 2
Again, as mentioned by Math101 in the comments, we can make use of the property $n! = n(n - 1)!$ to quickly calculate a factorial given the value of a previous factorial.
For example, given that $5! = 120$ (or by calculating $5!$ by using the same method):
$$6! = 6 \cdot 5! = 6 \cdot 120 = 720$$
This method does not work as efficiently as Method 1, especially at larger values of $n$, but it may be useful in some ways.
$$\\$$
Method 3
By pairing up the numbers needed to calculate a factorial, a shortcut can be used to quickly calculate the factorial by using the commutative property ($xy = yx$).
For example:
$$8! = 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = (8 \cdot 1) \cdot (7 \cdot 2) \cdot (6 \cdot 3) \cdot (5 \cdot 4) = 8 \cdot 14 \cdot 18 \cdot 20 = 40320$$
The largest value is always paired up with the smallest value, the second-largest with the second-smallest, and so on.
Please have a read through the source for this method for more detail on how to use this method more efficiently.
Please note that there is no clear method to easily calculate large factorials; approximations can only bring you fairly close to the true value.

I hope this helps! I'm still new to MSE, so I would really appreciate any and all feedback. Thank you!


TITLE: Then the value of the integral $ \iint_{A} f(x) g(y) \,d x\, d y $
QUESTION [0 upvotes]: Let $A=\{(x, y): x+y \leq 3\}$. Then the value of the integral
$$
\iint_A f(x) g(y) \,d x\, d y
$$
equals
where
$f(x)=\left\{\begin{array}{ll}x & \text { if } x \in[0,2] \\ 0 & \text { if } x \notin[0,2]\end{array}\right.$
$g(y)=\left\{\begin{array}{ll}1 & \text { if } x \in[0,2] \\ 0 & \text { if } x \notin[0,2]\end{array}\right.$
My approach:
by the form ($\frac{x}{a}+\frac{y}{b} = 1$) I am getting a triangle,
whose area should be $$\frac{1}{2}\cdot3 \cdot3$$
$$= \frac{9}{2}$$
Am I correct?

REPLY [0 votes]: Since $f(x) g(y)$ is non-zero only if $(x, y) \in [0,2]^2$, we can recast our domain of integration as
$$A' = [0,2]^2 \setminus T$$
where $T \subset [0,2]^2$ is the triangle with vertices at $(1, 2), (2, 2)$, and $(2, 1)$. It's convenient to evaluate the integral over the square and subtract this triangle, writing
$$\iint_A f(x) g(y) \; dx \, dy = \int_0^2 \int_0^2 f(x) g(y) \; dx \, dy - \int_1^2 \int_{3-x}^2 f(x) g(y) \; dy \, dx$$
The first integral on the RHS is straightforward,
$$\int_0^2 \int_0^2 x \; dx \, dy = 4$$
The second requires only slightly more work,
$$\int_1^2 x \int_{3-x}^2 \; dy \, dx = \int_1^2  x(x-1) \, dx = \int_0^1 (x^2 + x) \, dx = \frac{1}{3} + \frac{1}{2} = \frac{5}{6}$$
and hence we obtain the final answer
$$\iint_A f(x) g(y) \; dx \, dy = 4 - \frac{5}{6} = \frac{19}{6}$$


TITLE: Evaluate the integral $\int\limits_{-\infty}^{\infty} \frac{e^{ax}}{1 + e^x}dx\;$ where $\;0<a<1\;.$
QUESTION [7 upvotes]: I have to evaluate : $\displaystyle\int_{-\infty}^{\infty} \frac{e^{ax}}{1 +  e^x}dx\;$ where $\;0<a<1\;.$
I know that we have :
$\displaystyle\int_{-\infty}^{\infty} \frac{e^{ax}}{1 +  e^x}dx =\lim\limits_{R \to \infty}\int_{-R}^{R} \frac{e^{ax}}{1 +  e^x}dx =$
$\displaystyle\quad=\lim\limits_{R \to \infty}\int_{\gamma_R} \frac{e^{ax}}{1 +  e^x}dx - \int_{C_R} \frac{e^{ax}}{1 +  e^x}dx$
For C$_R$ half-circle runs $R$ from $-R$ and $\gamma_R$ a closed contour.
I have a hint: try other closures than C$_R\;.$

REPLY [1 votes]: $$I(a)=\int_{-\infty}^\infty\frac{e^{ax}}{1+e^x}\,dx$$
$u=1+e^x\Rightarrow dx=du/e^x=du/(u-1)$ so:
$$I(a)=\int_1^\infty\frac{(u-1)^a}{u(u-1)}du=\int_1^\infty(u-1)^{a-1}u^{-1}\,du$$
and this wouldnt be too hard to write in terms of the incomplete beta function so you could express it as the limit of that?


TITLE: Resolve a thin plate spline function
QUESTION [1 upvotes]: I am trying to keep the outline of an object in a video. So I have the coordinate of the outline of the object in the image $t$ and after computing the optical flow I have the coordinate in the image $t+1$. However, as the optical flow contains error, I would like to interpolate the outline using the thin plate spline function.
I would like to resolve the thin plate spline function in (3) p.5 from the paper http://www.tc.umn.edu/~baixx015/Xue_Bai_mirage_2011.pdf which is
$$
f(x,y) = c_0 + c_xx + c_yy + \sum_{i=1}^M{c_i \phi(||(x,y)-(x_i',y_i')||)}
$$
with M number of points and
$$
\phi(r)=r^2log(r)
$$
which gives me in matrices this equation
$$
 \left[
  \begin{array}{ c c }
     K &  T\\
     T' & 0
  \end{array} \right]
\left[
  \begin{array}{ c}
     c \\
     d
  \end{array} \right] =
\left[
  \begin{array}{ c}
     z \\
     0
  \end{array} \right]
$$
where $K$ is $r^2log(r)$, $T$ is $(1,x_i,y_i)$, $c$ is $(c_i)$, $d$ is $(c_0, c_x, c_y)$ and $z$ is what I am looking for and correspond to $f(x,y)$.
I want to try the Uzawa's algorithm described in this document p.33 http://alexandria.tue.nl/extra1/afstversl/wsk-i/ghosh2010.pdf which can resolve equations of type:
$$
 \left[
  \begin{array}{ c c }
     A &  B\\
     B' & 0
  \end{array} \right]
\left[
  \begin{array}{ c}
     x \\
     y
  \end{array} \right] =
\left[
  \begin{array}{ c}
     b \\
     c
  \end{array} \right]
$$
As Daryl said $b=z$ and $c=0$ but I still don't know $z$.
I don't understand how he knows the vectors $b$ (or $z$) If we choose $x_0$ and $y_0$, we can compute a $b_0$  but he does not explicitly say to do that and recompute $b$ at every iteration. Also he said p. 25 that Azawa's method allows to resolve directly the thin plate spline equation.
So in the algorythm, I compute vectors c and d and after that i can compute b.
EDIT:
I tried with the conjugate gradient to resolve $Ax=b$ where
$A=
\left[
  \begin{array}{ c c }
     K &  T\\
     T' & 0
  \end{array} \right] $, $x = \left[
  \begin{array}{ c }
     c\\
     d \end{array} \right] $ and $b = \left[
  \begin{array}{ c }
     z\\
     0 \end{array} \right] $ but it doesn't seem to converge. I think I don't understand what I have to put in $b$.

REPLY [2 votes]: $b$ and $c$ do not need to be computed. They are the block sections of the RHS vector of the equation. (See (4.9) on p. 33)
In your equation given above, $b=z$ and $c=0$.


TITLE: Let $\|\cdot\|_2·$ be the norm of $C^n$ defined by $\|x\|_2=(|x_1|^2+|x_2|^2+...+|x_n|^2)^{1/2}$.
QUESTION [1 upvotes]: Let $\|\cdot\|_2·$ be the norm of $C^n$ defined by
$\|x\|_2=\left(|x_1|^2+|x_2|^2+...+|x_n|^2\right)^{1/2}$.
Prove that for the subordinate matrix norm $∥⋅∥_2$ must be
$\|A\|_2=\max\{ \sqrt{|λ|}:λ \text{ is an  eigenvalue  of } AA^∗ \}$
where $A^*$ is the conjugate transpose of $A$.

REPLY [0 votes]: $ A^*A$ is a nonnegative matrix. Then there is an unitary matrix $P$ such that
$$A^*A=P^*\operatorname{diag}(\lambda_1,\ldots,\lambda_n)P$$
where the $\lambda_j$'s are the eigen values of $A^*A$ which happen to be all nonnegative. Then
$$\|Ax\|^2_2=x^*A^*Ax=x^*P^*\operatorname{diag}(\lambda_1,\ldots,\lambda_n) Px$$
Since $\|A\|=\sup_{\|x\|_2}\|Ax\|_2$
$$\begin{align}\sup_{\|x\|_2=1}\|Ax\|^2=\sup_{\|x\|_2=1}\Big((Px)^*\operatorname{diag}(\lambda_1,\ldots,\lambda_n) Px\Big)\tag{1}\label{1}\end{align}$$
Since and $P$ is unitary ($P^*P=I$), $\|Px\|_2=\|x\|_2$ for all $x\in\mathbb{C}^n$. Letting $u=Px=[u_1\ldots u_n]^\intercal$ in \eqref{1} yields
$$\begin{align}\sup_{\|x\|_2=1}\|Ax\|^2&=\sup_{\|u\|_2=1}\Big(u^*\operatorname{diag}(\lambda_1,\ldots,\lambda_n) u\Big)\\
&=\sup_{\|u\|_2=1}\lambda_1|u_1|^2+\ldots+\lambda_n|u_n|^2=\max_{1\leq k\leq n}\lambda_j \tag{2}\label{2}\end{align}$$
The conclusion follows by noticing that $AA^*$ and $A^*A$ have the same eigenvalues.


TITLE: Probability of creating"MISSISSIPPI"
QUESTION [1 upvotes]: The original problem is:
"You choose a letter at random from the word Mississippi eleven times without replacement. What is the probability that you can form the word Mississippi with the eleven chosen letters? Hint: it may be helpful to number the eleven letters as 1,2,...,11."
I've looked through other pages which tell me the answer should be: 2.88*10^-5
Yet my book tells me it's ${11 \choose 4}*{7 \choose 4}*{3 \choose 2}*4^4*4^4*2^2*(1/(11^{11}))$ = .0318...
Is my book wrong? How would you approach this question?

REPLY [1 votes]: I am assuming the problem is about letters taken from Mississippi with replacement, not without replacement as stated in the OP.  The following solution is similar to one previously posted, but with additional justification as to why the book solution is the correct choice.
In problems like this it is important to decide on the probability space, which often amounts to finding a collection of equally likely outcomes, then counting those that are favorable in one sense or another.  Here, we want to view Mississippi as if all the letters were distinct, perhaps by applying subscripts to the letters, so the starting word is
$$M_1 I_1 S_1 S_2 I_2 S_3 S_4 I_3 P_1 P_2 I_4$$
Then 11 letters taken from that word with replacement can be arranged in $11^{11}$ distinct ways, all of which we assume are equally likely.
The favorable outcomes we want to count are those sequences which contain 1 M, 4 Is, 4 Ss, and 2 Ps.  The number of sequences of 1 M, 4 Is, 4 Ss, and 2 Ps without the subscripts is
$$\frac{11!}{1! \;4! \;4! \;2!}$$
but we need to count the number of sequences with subscripts in order to be consistent with our definition of equally likely possibilities. Each $M$ can be subscripted in only $1$ way, each $I$ can be subscripted in $4$ ways, etc., so the number of favorable outcomes is
$$\frac{11!}{1! \;4! \;4! \;2!}\cdot 1 \cdot 4^4 \cdot 4^4 \cdot 2^2$$
Divide the number of favorable outcomes by $11^{11}$ to find the probability of a favorable outcome.
$$\frac{11!}{1! \;4! \;4! \;2!}\cdot \frac{1 \cdot 4^4 \cdot 4^4 \cdot 2^2}{11^{11}} = 0.0318364$$


TITLE: On the definition of highest weight
QUESTION [6 upvotes]: Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $M$ a finite-dimensional representation of $\mathfrak{g}$ (or more generally an element of the category $\mathcal{O}$).
Let $\mathfrak{h}$ be a Cartan subalgebra, $\Phi^+$ a system of positive root and consider the partial ordering on $\mathfrak{h}^*$ induced by $\Phi^+$. Let $\Psi(M)\subseteq\mathfrak{h}^*$ be the set of weights of $M$. Here are two standard definitions:

$(1)$ A highest weight is a maximal element of $\Psi(M)$ with respect to this partial ordering. That is, $\lambda$ is a highest weight if for all $\mu\in\Psi(M)$, either $\mu$ doesn't compare to $\lambda$, or $\mu\leq\lambda$.
$(2)$ $\lambda\in\Psi(M)$ is a highest weight if there exists a nonzero $v\in M_\lambda$ such that $e_\alpha\cdot v=0$ for all $\alpha\in\Phi^+$ where $e_\alpha$ is a root vector of $\alpha$.

I can easily see that $(1)\Rightarrow(2)$. Is the converse also true?

REPLY [4 votes]: Choose $\lambda$ such that both $\lambda$ and $\lambda-\alpha$ are dominant. Then $L(\lambda)\oplus L(\lambda-\alpha)$ is a counter example.


TITLE: $\mathbb{Z} \times \mathbb{Z}$ is cyclic.
QUESTION [4 upvotes]: Proof $\mathbb{Z} \times \mathbb{Z}$ is cyclic.

My intuition is that there are four generators in $\mathbb{Z} \times \mathbb{Z}: 1 \times 1, 1 \times -1, -1 \times 1, -1 \times -1$. And the group they generated are $(z,z)$ or $(z,-z)$, rather than two independent $(z_1, z_2)$, where $z_1, z_2 \in \mathbb{Z}$.
But is this the right way to think about it, and right way to write it? Thank you. :-)

REPLY [5 votes]: Any quotient of a cyclic group is cyclic but $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ is a non-cyclic quotient of $\mathbb{Z} \times \mathbb{Z}$.


TITLE: Positive matrices are open
QUESTION [1 upvotes]: An linear application $A:\mathbb R^n\to \mathbb R^n$ is positive when it is symmetric and besides that $\langle Ax,x\rangle\gt 0$ for every $x\neq 0$ in $\mathbb R^n$. I would like to prove the set of the positive linear applications is open in the set of symmetric applications.
I don't have any idea how to prove it.
How does this matrix look like? I think this question can be answered directly from that.
Any hint?
Thanks

REPLY [0 votes]: Is the following solution correct?
Let $\mathcal{A}$ be the set of positive linear applications. For each $x$ st $|x|=1$ define $ f_x: \mathcal{A} \to  \mathbb{R} $ by $\langle Ax,x\rangle$.
$f_x$ is continuous and $(0,\infty)$ is open. So
$f_x^{-1}((0,\infty))$ is Open. So,
$$
 \mathcal{A}=\bigcup_{|x|=1} f_x^{-1}((0,\infty))
 $$ is Open.


TITLE: Fixed Matrices over finite field by a map
QUESTION [1 upvotes]: Consider a set $M_n$ of all possible square matrices of dimension $n$ over a finite field $F_q$. Clearly the cardinality of the set $M_n$ is $q^{n^2}$. Let us consider a map $f:M_n$ $\longrightarrow$ $M_n$ defined as $f_C(X)=X^2+C$, $C$ $\in$ $M_n$. Here $X^2=X*X$, where  $'*'$ denotes the $q$-modulo matrix multiplications and $'+'$ is also standing for $q$-modulo addition.
I am interested to find out fixed points of the maps.
So I want to solve $f_C(X)=X$. It turns out a quadratic equation: $X^2+(-X)+C=0$.
Question is how many solutions are there of the equation in $M_n$? What are the solutions?

REPLY [1 votes]: The algebraic matrix Riccati equation is given by
$$
XBX+XA-DX-C=0
$$
with given matrices $A,B,C,D$ over a field $K$. For $B=I_n$, $A=-I_n$ and $D=0$ we obtain the equation $X^2-X-C=0$, which you want. All information about solutions and other things can be found in the book of P. Lancaster, L. Rodman: Algebraic Riccati equations. Clarendon Press, Oxford (1995). One method is based on so-called Jprdan-chains.


text:
\section{Maximizing information gain versus minimizing information loss} \label{subsec:max_info_gain_vs_min_info_loss}

When solving the design problem using sequential algorithms, is maximizing information gain necessarily  better than minimizing information loss? Are there circumstances when one ought to be preferred over the other?

These are natural questions that arise when contrasting the two approaches. Here we discuss nuances of the two formulations, using the Bayesian experimental design problem of \Cref{subsec:BIP_and_exp_design} as a template. Recall that the goal of this design problem is to select a subset of observations $Y_{\mathcal{P}}$ that best informs the parameters $X$.

The fundamental difference between the two approaches is most evident at the onset of each procedure.
When maximizing information gain, we first compare independent contributions given by $\mathcal{I}\lr{X;Y_i}$. When minimizing information loss, on the other hand, we first assess the relative magnitudes of
 $\mathcal{I}\lr{X;Y} - \mathcal{I}\lr{X;Y \setminus Y_i}$; here the amount of collective information provided by the observations $Y \setminus Y_i$ is the guiding factor. In the latter case, note how the relative importance of each observation $Y_i$ is dictated by its absence, in contrast to the former.
These differences suggest two ideas:
 \begin{enumerate}[\itshape i)\upshape]
 \item If the dominating contributor to the total mutual information  $\mathcal{I}\lr{X;Y}$ is the aggregate of  \emph{independent} contributions $\mathcal{I}\lr{X;Y_i}$, then maximizing information gain seems the better approach.
 \item Conversely, if interaction among the observations $Y_i$ is the dominating contributor to $\mathcal{I}\lr{X;Y}$, then minimizing information loss would appear as a better choice.
 \end{enumerate}

The above remarks are not entirely new in and of themselves. Several authors have discussed similar notions in the context of seeking symmetric multivariate correlation measures \cite{Watanabe_1960,Han_1978}, and analyzing the polymatroidal dependence structure of a set of random variables \cite{Han_1975,Fujishige_1978}.

If a subset of observations has already been selected/discarded, one would simply redefine the terms by appropriately conditioning on that set. The main caveat of course is the size of the conditioning set. If a substantial number of observations have already been chosen, then the strategy of maximizing information gain also implicitly accounts for any favorable interactions among the observations. Thus the desired cardinality constraint and any prior knowledge about the strength of the interaction can help in the choice of strategy.

From an optimization perspective, there are clear distinctions between the two approaches.
Maximizing information gain corresponds to maximizing a non-decreasing set function, and when the observations are conditionally independent the function is submodular.

Minimizing information loss corresponds to minimizing a non-decreasing set function, and when the observations are conditionally independent the function is supermodular.

The constrained maximization of non-decreasing set functions is well understood theoretically, especially when the function is submodular. Results for the minimization of non-decreasing set functions, however, are fewer and less satisfactory. Even in the restricted setting of the non-decreasing set function being supermodular, the analysis of the greedy algorithm is more involved \cite{Ilev_2001_backward}, and the approximation guarantee is given using \emph{steepness} of the function. For the more general case of minimizing a set function which is not necessarily sub/super-modular, there are no approximation guarantees of which we are aware.
Note that in \cite{Ilev_2001_backward} the result was given for any non-increasing supermodular function. We are dealing with a non-decreasing function since the function argument ($Y \setminus Y_{\mathcal{P}}$) involves the complement of the index set, $\mathscr{I}_{\mathcal{P}}$, that we seek. The result in \cite{Ilev_2001_backward} is then applicable with suitable modifications.

\iffalse

\subsection{Minimization of a non-decreasing supermodular set function} \label{subsec:minimizing_non_decreasing_function}

Suppose we are given a real-valued positive  set-function $F : 2^{\mathscr{V}} \rightarrow \mathbb{R}_{\geq 0}$ that is non-decreasing and submodular with the normalization  $F(\emptyset)=0 $.
The function $F^{\#}(\mathscr{X})   \coloneqq  F(\mathscr{V}) - F(\mathscr{V} \setminus \mathscr{X})$ is the supermodular dual of $F$ and is non-decreasing with respect to the cardinality of its argument $\mathscr{X}$.
Alternatively if we define $\mathscr{Y}  \coloneqq  \mathscr{V} \setminus \mathscr{X}$ and $\Psi(\mathscr{Y})  \coloneqq  F^{\#}(\mathscr{V} \setminus \mathscr{Y}) = F(\mathscr{V}) - F(\mathscr{Y})$, then  $\Psi$ is a real-valued positive supermodular function, non-increasing with respect to the cardinality of its argument, and with the normalization $\Psi(\mathscr{V}) = 0$.
It is quite easy to see that,  $\min_{\mathscr{X} \subseteq \mathscr{V}, \abs{\mathscr{X}} \geq k } F^{\#}(\mathscr{X}) = \min_{\mathscr{Y} \subseteq \mathscr{V}, \abs{\mathscr{Y}} \leq \abs{\mathscr{V}} -k  } \Psi(\mathscr{Y})$. It is a simple identity, and  easier to understand when written only in terms of $F$,
\begin{equation} \label{eqn:equivalence_non_increasing_non_decreasing_fun_opt}
	\min_{\mathscr{X} \subseteq \mathscr{V}, \abs{\mathscr{X}} \geq k }  F(\mathscr{V}) - F(\mathscr{V} \setminus \mathscr{X}) = \min_{\mathscr{Y} \subseteq \mathscr{V}, \abs{\mathscr{Y}} \leq \abs{\mathscr{V}} -k  } F(\mathscr{V}) - F(\mathscr{Y}).
\end{equation}

For non-increasing supermodular functions such as $\Psi$, the greedy descent algorithm was analyzed in \cite{Ilev_2001_backward}  and the approximation guarantee involving the steepness of the function. The parameter steepness in the context of supermodular functions is analogous to curvature (\Cref{def:total_curvature}) for submodular functions. While the curvature was defined as the largest ratio of certain incremental gains, the steepness is defined similarly using incremental losses.
\begin{definition} \label{def:incremental_loss} \textbf{(Incremental loss)}   We denote the incremental or marginal loss of a set $ \mathscr{A} \subset \mathscr{V}$ in context of a set $\mathscr{B} \subset \mathscr{V}$ as $\bar{\rho}_{\mathscr{A}}(\mathscr{B})  \coloneqq  \Psi(\mathscr{B})  - \Psi(\mathscr{A}\cup\mathscr{B})$. For $\nu  \in \mathscr{V}$, we use
the shorthand $\bar{\rho}_{\nu}(\mathscr{B})$ for $\bar{\rho}_{\{\nu\}}(\mathscr{B})$.
\end{definition}
\begin{definition} \label{def:steepness} \textbf{(Steepness \cite{Ilev_2001_backward})}  The steepness of a non-increasing supermodular set function $\Psi$ is the scalar $s$  s.t.,
\begin{displaymath}
s = \max_{\nu \in \mathscr{V}, \bar{\rho}_{\nu}(\emptyset) > 0} \frac{\bar{\rho}_{\nu}(\emptyset) - \bar{\rho}_{\nu}(\mathscr{V}\setminus \{ \nu \}) }{ \bar{\rho}_{\nu}(\emptyset) }.
\end{displaymath}
\end{definition}
\begin{theorem} \label{thm:greedy_descent}(Greedy descent approximation guarantee for minimizing non-increasing supermodular functions \cite[Theorem~2, Corollary~4]{Ilev_2001_backward})
Let $\Psi$ be a non-increasing supermodular function with steepness $s \geq 0$ and normalization $\Psi \lr{\mathscr{V}} = 0$, then the greedy descent algorithm  returns a solution $\mathscr{Y}_{\text{gd}}$   s.t.,
\begin{displaymath}
\Psi \lr{\mathscr{Y}_{\text{gd}}}  \leq \frac{e^{s} -1}{s} \Psi \lr{\mathscr{Y}_{\text{opt}}}
\end{displaymath}
\end{theorem}

Let $\mathscr{X}_{\text{gd}} = \mathscr{V} \setminus \mathscr{Y}_{\text{gd}}$, then by definition we have
\begin{equation} \label{eqn:minimizing_non_decreasing_function_helpful_identity}
\Psi \lr{\mathscr{Y}_{\text{gd}}} = F^{\#}(\mathscr{X}_{\text{gd}})   \coloneqq  F(\mathscr{V}) - F(\mathscr{V} \setminus \mathscr{X}_{\text{gd}}).
\end{equation}
Using \cref{eqn:equivalence_non_increasing_non_decreasing_fun_opt,eqn:minimizing_non_decreasing_function_helpful_identity} we can now co-opt the result in \Cref{thm:greedy_descent}  for minimizing the non-decreasing supermodular set function in question,
\begin{equation} \label{eqn:greedy_descent_non_decreasing_function}
F(\mathscr{V}) - F(\mathscr{V} \setminus \mathscr{X}_{\text{gd}}) \leq \frac{e^{s} -1}{s} 	\min_{\mathscr{X} \subseteq \mathscr{V}, \abs{\mathscr{X}} \geq k }  F(\mathscr{V}) - F(\mathscr{V} \setminus \mathscr{X}).
\end{equation}
It is worth emphasizing that the solution $\mathscr{X}_{\text{gd}}$ is obtained as the complement of the result $ \mathscr{Y}_{\text{gd}}$, the latter being the approximate solution obtained by using the greedy descent algorithm to solve the  minimization problem $\min_{\mathscr{Y} \subseteq \mathscr{V}, \abs{\mathscr{Y}} \leq \abs{\mathscr{V}} -k  } \Psi(\mathscr{Y})$. Recall that $\Psi(\mathscr{Y}) = F(\mathscr{V}) - F(\mathscr{Y})$, and the steepness parameter $s$ in \cref{eqn:greedy_descent_non_decreasing_function} is associated with the function $\Psi$.

\subsection{The linear objective as difference of two submodular functions} \label{appendix:diff_two_submod_functions}

The expression for mutual information $\mathcal{I}(X;Y_{\mathcal{P}}) $ is quite revealing upon closer scrutiny; it can be written as the difference of log principal determinant of the data marginal and observation error covariance operators, $\frac{1}{2} \lr{ \log \det \lr{\mathcal{P}^{\top}\Gamma_{Y} \mathcal{P}} - \log \det \lr{\mathcal{P}^{\top} \Gamma_{Y|X} \mathcal{P}}}$, \cref{eqn:mutualInfoGaussians_reduced_data,eqn:ReducedDataMarginal}.  It is thus the difference between two set functions, each one defined as log determinant of principal submatrix of a definite matrix.  Here the set is constituted by the indices of rows/columns of the principal submatrix, with each index corresponding to a unique observation. Each of those set functions can be inferred to be submodular \cite{Gantmacher_Krein_1960,Kotelyanskii_1950,Fan_1967,Fan_1968,Kelmans_1983,Johnson_1985}. That is $\forall A \in \mathbb{R}^{m \times m} \succ 0, \mathscr{I} \subseteq \mathscr{V}$, if $A[\mathscr{I}]$ represents the principal submatrix obtained from $A$ by deleting rows and columns in the set $\mathscr{V} \setminus \mathscr{I}$, then $\forall \mathscr{I}_1, \mathscr{I}_2 \subset \mathscr{V}$,
\begin{equation} \label{eqn:log_det_principalsubmat_submodular}
		\log \det A[\mathscr{I}_1] + \log \det A[\mathscr{I}_2] \geq  \log \det A[\mathscr{I}_1 \cup \mathscr{I}_2] + \log \det A[\mathscr{I}_1 \cap \mathscr{I}_2].
\end{equation}
Mutual information can thus be interpreted as the difference between two submodular functions (which we refer to as DS functions following \cite{IyerBilmes2012_diff2SM} ). This is not entirely surprising since it has been shown how every set function can be decomposed as the difference between two submodular functions  \cite[Lemma~4]{NarBil2005}, \cite[Lemma~3.1]{IyerBilmes2012_diff2SM}.
This is akin to how any continuous function, subject to weak conditions, can be expressed as the sum of a convex and concave part \cite[Theorem~1]{Yuille_Rangarajan_2003}.
Such a decomposition is not unique and can be exponentially hard to compute \cite{IyerBilmes2012_diff2SM}, but in our case it is readily apparent.

Interpreting mutual information as a DS function can appear at odds with the result in \Cref{prop:MI_SM_uncorrelatedCase}; however this is easily reconciled. When observation errors are uncorrelated the corresponding covariance operator $\Gamma_{Y|X}$ is diagonal.

For such a diagonal matrix, \cref{eqn:log_det_principalsubmat_submodular} is always an equality and log determinant of principal submatrix of a diagonal definite matrix is a modular function (\Cref{def:modular_function}). Mutual information in this case is the difference between a submodular function and a modular function -- a simple translation of a submodular system -- and hence submodular  \cite{fujishige2005,Bach2013}.

Viewing mutual information as a DS function, as opposed to monotone set function is not necessarily helpful algorithmically. Optimizing DS functions is in general a more difficult problem, and concrete hardness results on multiplicative inapproximability of such optimization problems have been shown \cite{IyerBilmes2012_diff2SM}. The main strategy in attacking such problems is to seek tight modular bounds for one (or both) of the terms in the difference, and subsequently optimize the resulting submodular (or modular) objective using existing methods \cite{NarBil2005,IyerBilmes2012_diff2SM}. The core ideas can be understood in the framework of the MM (minorize-maximize or majorize-minimize)  principle \cite{Lange_2016_MMOpt}, and such connections have been made precise in related works \cite{IyerJegelkaBilmes2012_mirror,IyerJegBil2013}.

\begin{remark}
\Cref{prop:MI_SM_uncorrelatedCase} was proven using information theoretic arguments to achieve more generality, but submodularity for the case of Gaussian random variables can be easily demonstrated through strictly linear algebra arguments which we do not provide. Interestingly, starting from \Cref{prop:MI_SM_uncorrelatedCase} with the random variables specified to be multivariate normal, and with appropriate choices of covariance operators we can obtain the determinant inequality \cref{eqn:log_det_principalsubmat_submodular} as a corollary. Such an approach of realizing a determinant inequality as a special case of information theory inequality

is a handy tool which has been used to reproduce other famous results including Minkowski's determinant inequality, Hadamard's inequality (\Cref{thm:hadamard_inequality}) and its generalizations \cite{cover_thomas_1988}, \cite[Chapter~17]{cover_thomas_2005}.
\end{remark}

\fi

\iffalse

\subsection{Perspectives on the generalized eigenvalues defining mutual information} \label{subsec:perspectives_gen_eig}

If we consider two selection operators, $\mathcal{P}_1, \mathcal{P}_2$ such that $\mathscr{I}\lr{\mathcal{P}_1} \subseteq \mathscr{I}\lr{\mathcal{P}_2} \subset \mathscr{V}$, then we can claim the inequality,
\begin{equation} \label{eqn:mutual_info_non_decreasing}
\mathcal{I}(X;Y_{\mathcal{P}_1}) \leq \mathcal{I}(X;Y_{\mathcal{P}_2}) \leq \mathcal{I}(X;Y).
\end{equation}
A  proof of \cref{eqn:mutual_info_non_decreasing} is straightforward using the chain rule for mutual information \cite[Theorem~2.5.2]{cover_thomas_2005} and expanding the terms appropriately. The inequality however is very intuitive since by incorporating more observations we can only add information.  We could also infer \cref{eqn:mutual_info_non_decreasing} as a consequence of the inequality concerning the generalized eigenvalues corresponding to the case when using selection operators $\mathcal{P}_1$ and $\mathcal{P}_2$,
\begin{equation} \label{eqn:gev_inequality}
\widehat{\sigma}_{j,\mathcal{P}_1}^{\downarrow} \ \leq \  \widehat{\sigma}_{j,\mathcal{P}_2}^{\downarrow} \ \leq \ \sigma_j^{\downarrow} \qquad \forall j.
\end{equation}
We have appended subscripts $\mathcal{P}_1,\mathcal{P}_2$ to the eigenvalues for necessary clarity, and the superscript $\downarrow$ implies  the individual eigenvalues are ordered descendingly. The proof for \cref{eqn:gev_inequality} follows from Cauchy's interlacing theorem \cite{horn_johnson_2012,bhatia1997},\cite[Theorem~2.3]{Kressner2014},\cite[Theorem~2.1]{KovacStriko_Veselic_1995},

while \cref{eqn:mutual_info_non_decreasing} follows readily from \cref{eqn:gev_inequality} since mutual information is a monotone function of the eigenvalues.

\fi


TITLE: Mathematical journals that accept long papers (up to 100 pages)
QUESTION [25 upvotes]: I have a problem with finding a suitable mathematical journal (maybe not of the highest level) that accepts longer paper (say 100 pages). I know only two such journals: Memoirs of AMS and Dissert.Math.
But both journals now are in Q1 quartile, so are very selective, without any chances to be accepted there.
In fact, "Dissertationes Mathematicae" was created in 1951 (as "Rozprawy Matematyczne") with the purpose of publishing good Ph.D. Theses and indeed some time ago it was quite real to be accepted there. But since 2019 when they entered to the Q1 quartile this journal accepts only papers of exceptional quality (so far higher than an average Ph.D. Thesis).
So, what would be your suggestion concerning journals for longer papers (at the moment I would not like to consider splitting the long paper into smaller pieces). Thanks for the advise (I suspect that many mathematicians had such problems in mind, so the collective experience had to elaborate some helpful answer here).

REPLY [2 votes]: I recall here my answer to a similar question: a journal which accepts long papers and has also an illustrious history is the Journal published by Castelnuovo Institute of Mathematics of the Sapienza University of Rome, i. e. the "Rendiconti di Matematica e delle sue Applicazioni". I recall the statements one can read on the homepage of the Journal:

The Journal "Rendiconti di Matematica e delle sue Applicazioni" is regularly issued since 1914.
The Journal traditionally publishes high-quality research articles in Pure and Applied Mathematics and adheres to the EMS Code of Practice.

Articles of any length are considered for publication. Submission of surveys, of articles of foundational nature, of doctoral dissertations etc. is also encouraged.
Every article submitted is subjected to a first screening by the Editorial board: if the manuscript meets the journal’s basic requirements, it will be sent to a referee for a single blind peer review process. Once a paper is accepted it goes immediately into production and published online shortly after the approval of galley proofs.