TITLE: Estimate for Operator Norm in Hilbert Space
QUESTION [3 upvotes]: Let $H$ a hilbert space with an orthonormal basis $(e_n)_{n\in \mathbb{N}}$ and $F$ a linear operator, such that $\langle e_k,F e_n\rangle =:\phi(n,k)$. Find a good estimate for $\lVert F\lVert$ in terms of $\phi(n,k)$. Apply your estimate to the special case of $\phi(n,k)=\frac{1}{n+k}$.
I tried applying parsevals identity and hölder's inequality:
\begin{align}
\lVert F x\lVert^2&=\lVert \sum_{n=1}^{\infty}\langle x,e_n\rangle F e_n\lVert^2=\lVert\sum_{n=1}^{\infty}\langle x,e_n\rangle  \sum_{k=1}^{\infty}\langle e_k,F e_n\rangle  e_k \lVert^2  \\&=\sum_{k=1}^{\infty} \left(\sum_{n=1}^{\infty} \langle x,e_n\rangle  \phi(n,k)\right)^2 \leq \sum_{k=1}^{\infty} \left [\left(\sum_{n=1}^{\infty} \langle x,e_n\rangle ^2\right)\left(\sum_{n=1}^{\infty}\phi(n,k)^2\right)\right]\\&= \lVert x\lVert^2 \sum_{k=1}^{\infty} \sum_{n=1}^{\infty} \phi(n,k)^2
\end{align}
Which implies $\lVert F\lVert \le \sum_{k=1}^{\infty} \sum_{n=1}^{\infty} \phi(n,k)^2$. But applied to $\phi(n,k)=\frac{1}{n+k}$ this sum doesn't converge at all.

REPLY [2 votes]: In fact the desired norm is the smallest constant $C$ such that for all $a,b\in\ell^2(\mathbb{N})$ we have
$$
\sum\limits_{m=1}^\infty\sum\limits_{n=1}^\infty \frac{a_n b_m}{m+n}\leq C\left(\sum\limits_{n=1}^\infty a_n^2\right)^{1/2}\left(\sum\limits_{m=1}^\infty b_m^2\right)^{1/2}.
$$
This is well known in narrow circles Hilbert's inequality. It is known that the smallest possible constant there is $\pi$, so $\Vert F\Vert=\pi$.
You can find several proofs of this fact in The Cauchy-Schwarz Master Class by J. Michael Steele (pages 155-165).
Also you should take a look at this discussion.


TITLE: Brownian Motion and Progressive Process
QUESTION [0 upvotes]: Let $B_t$ be a Brownian motion. Define sign function as follows. $sign(0) = 0$ and $sign(x) = \frac{x}{|x|}, \forall x \neq 0$. I do not know how to show the following two questions, especially on the progressive measurability one. Could anyone provide some detailed explanation on how to go about with them, please? Thank you!

Show that $sign(B_t)$ is a progressive process.
Show that $sign(B_t)\cdot B_t$ is a Brownian motion.

REPLY [2 votes]: A direct way to show that the process $X$ defined by $X_t=\mathrm{sign}(B_t)$ is progressively measurable is to consider, for every positive integer $n$, $$X^n_t=\frac{nB_t}{\sqrt{1+n^2B_t^2}}.$$ Each process $X^n$ is adapted with continuous paths hence progressively measurable. Furthermore, for every $t$, $X^n_t\to X_t$ almost surely, when $n\to\infty$, hence $X$ is progressively measurable.
(This answers item 1. Item 2. is absurd, for example because $X_tB_t=|B_t|$ and $t\mapsto E(|B_t|)$ is not constant hence $|B|$ is not a Brownian motion (much more is known but this simple argument suffices).)


TITLE: Derive the Weak form of the Conservation Law Problem
QUESTION [0 upvotes]: Derive the weak form of the conservation law problem
$u_t + u^2u_x = f(x,t)$
with the initial value $u(x,0)=\cos x$
I am trying to connect this problem with Burgers equation but failed. Anyone helps me to solve this problem? I am confused about where I need to stop. Do I need to find the formula for Rankine–Hugoniot conditions? Please help me to solve this problem. Thanks.

REPLY [0 votes]: The trick has already been mentioned in the comments: Note that by the chain rule $$\frac{\partial}{\partial x} \Bigg(\frac{1}{3} u(x,t)^3 \Bigg) = \frac{\partial u}{\partial x}\frac{\partial}{\partial u}\Bigg(\frac{1}{3} u(x,t)^3 \Bigg) = u_x u^2.$$
Then you are in usual business: Multiply the PDE
$$ u_t + \frac{1}{3} \big(u^3\big)_x = f(x,t)$$
with a suitable testfunction $\phi(x, t) \in C_c^1\big(\mathbb{R} \times \mathbb{R}_+ \big) $ i.e., a function that is (at least) once continuously differentiable in each variable and that vanishes outside of a compact set of $\mathbb{R} \times \mathbb{R}_+$ to obtain
$$ \Bigg(u_t + \frac{1}{3} \big(u^3\big)_x \Bigg) \phi(x, t) = f(x,t) \phi(x, t)$$
integrate over both space and time
$$ \int_\mathbb{R} \int_{\mathbb{R}_+} \Bigg(u_t + \frac{1}{3} \big(u^3\big)_x \Bigg) \phi(x, t) \:\mathrm dt \:\mathrm dx = \int_\mathbb{R} \int_{\mathbb{R}_+} f(x,t) \phi(x, t) \:\mathrm dt \:\mathrm dx$$
"Move" the derivatives (here first shown for time) onto the test function by means of integration by parts / product rule of differentiation:
$$ \int_\mathbb{R} \int_{\mathbb{R}_+} u_t \phi(x, t) + \Bigg(\frac{1}{3} \big(u^3\big)_x \Bigg) \phi(x, t) \:\mathrm dt \:\mathrm dx = \int_\mathbb{R} \int_{\mathbb{R}_+} -u \phi_t + \Bigg(\frac{1}{3} \big(u^3\big)_x \Bigg) \phi \:\mathrm dt  + u \phi \Big \vert^\infty_0 \:\mathrm dx$$
Using compact support of $\phi$ gives for bounded $u(x,t)$: $\lim_{t\to \infty} u(x, t) \phi(x, t) = 0$
and thus
$$\int_\mathbb{R} \int_{\mathbb{R}_+} \Bigg(u_t + \frac{1}{3} \big(u^3\big)_x \Bigg) \phi(x, t) \:\mathrm dt \:\mathrm dx = \int_\mathbb{R} \int_{\mathbb{R}_+} -u \phi_t + \Bigg(\frac{1}{3} \big(u^3\big)_x \Bigg) \phi \:\mathrm dt  - u(x, 0) \phi(x, 0) \:\mathrm dx$$
Analogous (after changing order of integration [Fubini]),
$$\int_\mathbb{R} \Bigg(\frac{1}{3} \big(u^3\big)_x \Bigg) \phi(x, t)  \:\mathrm dx = \int_\mathbb{R} -\Bigg(\frac{1}{3} \big(u^3\big)_x \Bigg) \phi_x  \:\mathrm dx + u\phi \Big \vert^\infty_{-\infty}$$
and again $u\phi \Big \vert^\infty_{-\infty}$ vanishes by the fact that $\phi$ has compact support.
Finally, the weak formulation is given by
$$\int_\mathbb{R} \int_{\mathbb{R}_+} -u \phi_t -\Bigg(\frac{1}{3} \big(u^3\big)_x \Bigg) \phi_x \:\mathrm dt  \:\mathrm dx - \int_\mathbb{R} \underbrace{u(x, 0)}_{u_0(x)} \phi(x, 0) \:\mathrm dx = \int_\mathbb{R} \int_{\mathbb{R}_+} f(x,t) \phi(x, t) \:\mathrm dt \:\mathrm dx$$


text:
\section{Multivalued Maps} \label{sec:multivalued-maps}

In this section we describe the concept of \emph{multivalued maps} and
obtain criteria that imply for such maps to be an \emph{enclosure} of a
map $f$. A \emph{cellular multivalued map} is defined to be constant
on each cell of a \emph{grid}, generalizing the cubical case of
\cite{KMM04}. A cellular map gives rise to a \emph{simplicial
  multivalued map} on the nerve of the grid. In the final part of this
section we show that one way to construct a grid is through an
\emph{$\alpha$-diagram}. In this case, the nerve is a geometrical
simplicial complex that is a deformation retract of the grid, and we
will show that the cellular map and the simplicial map induce the same
maps on homology.

Since we are interested in applications to data sets that
correspond to real-valued measurements of continuous dynamical
systems, we will assume that our time series $\Gamma$ is obtained from
a map $f$ on a submanifold $X \subset \bR^n$.  We will use the
Euclidean metric, $d(\cdot,\cdot)$, on $\bR^n$.  In the future, it might be
useful to consider more-general metrics on the submanifold $X$ itself.

We begin by recalling some standard definitions for
multivalued maps that approximate a dynamical system.\footnote

{Following \cite{Day04, Day08}, we denote single-valued maps with lower-case letters (e.g., $f$), sets and set-valued maps with capital letters, e.g., $F$, and combinatorial objects and maps with calligraphic letters, e.g. \cF.}

\begin{definition}[Multivalued Map]
	A \emph{multivalued map}, $F:X \rightrightarrows X$, is a map from
    $X$ to its power set. That is, for each $x \in X$, $F(x)$ is a subset of $X$.\end{definition}

We use multivalued maps to approximate continuous maps $f:X \to X$,
and the approximation is taken to be ``good" if the action on homology
induced by $F$ is
equivalent to that induced by $f$. In order for this to be the case, the
action of $F$ must
\emph{enclose} the action of $f$ and not introduce any extra
homological structure. These requirements are spelled out in the
following definitions.

\begin{definition}[Outer Approximation]\label{def:outer:approximation}
	A multivalued map $F:X \rightrightarrows X$ is an \emph{outer approximation} of a continuous map $f:X \to X$ if $f(x) \in F(x)$ for each $x \in X$. In this case $f$ is said to be a \emph{continuous selector} for $F$.
\end{definition}

\noindent
The (weak) preimage of a multivalued map is itself a multivalued map defined as
\[
	F^{-1}(y) = \{ x \in X :  y \in F(x) \}.
\]

\begin{definition}[Semicontinuous]\label{def:semi:continuous}
	A multivalued map is (lower) \emph{semicontinuous} if the preimage of each open set is open.
\end{definition}

\noindent
As usual, the $n$-dimensional homology group of a set $X$ is denoted $H_{n}(X)$. We use, for simplicity, the homology over $\bZ_2$ so that the torsion subgroups are ignored. Given this, a multivalued map preserves homology if the image of each point is homologous to a point. This is captured in the following definition.

\begin{definition}[Acyclic] \label{def:acyclic}
	A multivalued map $F$ is \emph{acyclic} if for each $x \in X$,
\[
		H_{n}(F(x)) = \left\{\begin{array}{cc}
		                     \bZ_2\ ,\ &n = 0 \\
						    \mathbf{0}\ ,\ &n > 0
					   \end{array} \right. .
\]
\end{definition}

\noindent
The key point is that if $F$ is semicontinuous and acyclic, then every continuous selector for $F$ induces the same homomorphism in homology---a consequence of the acyclic carrier
theorem \cite[Thm.~13.3]{Munkres84}.

\begin{definition}[Enclosure] A semicontinuous, acyclic, multivalued map $F$ is an \emph{enclosure} of any continuous selector $f$.
\end{definition}

\noindent
Therefore, one can define the homology induced by an enclosure to be that of any of its continuous selectors.

In order that this homology be computable, however, it is necessary to obtain a finitely representable approximation of the map $f$, and it is this to which we turn next.

\subsection{Grids} \label{sec:Grids}

A grid allows one to construct finitely representable maps that can be
outer approximations of a map $f$ \cite{Kalies05, Mrozek99}. We will
consider generalizations of the cubical cells of \cite{KMM04} to a
grid constructed from a collection of cells $\cA = \{A_1,A_2, \dots\}$
in $X$.
Associated with any such collection is its geometrical realization---the union of these cells as subsets of $X$---denoted by
\beq{GeometricalRealization}
	|\cA|:= \bigcup_{A \in \cA} A.
\eeq
Since the shape and number of neighbors of each cell can
vary, such a grid may permit more-efficient computational algorithms than those
for cubes of fixed size and shape.\footnote

{See \App{Complexity} for more discussion of algorithms and associated computational complexity issues.}

Nevertheless, many of the results that appear here are easily adapted from the cubical case.

There are four basic properties for the cells of a grid.

\begin{definition}[Grid \cite{Arai09}] \label{def:abstract:grid}
	A family of nonempty compact sets \cA is a \emph{grid} on $X$ if
	\begin{enumerate}
		\item[a)] $X = |\cA|$;
		\item[b)] for all $A \in \cA $, $ A = \text{cl}(\Int(A))$;
		\item[c)] for all $A,B \in \cA$ if $A \neq B $ then $ A \cap \Int(B) = \emptyset$; and
		\item[d)] a finite subset of $\cA$ covers each compact $S \subset X$.
	\end{enumerate}
\end{definition}

\noindent
A prototypical grid is a lattice of closed cubes; indeed, this is the example studied extensively in \cite{KMM04}. An example of a more-general grid is shown in \Fig{CellularMap}. The cells in this grid are $\alpha$-cells, see \Sec{AlphaComplex}.

\InsertFig{CellularMapII}{Sketch of a a grid, $\cA_\alpha(L)$, of $\alpha$-cells (see \Sec{AlphaComplex}) based on a set of nine landmarks $\{l_i\}$ (red points) and the action of a cellular map $\FA$ \Eq{CellularMap} on two of these cells. Here the image of any $x \in \Int(A_1)$ is $A_5 \cup A_6 \cup A_8 \cup A_9$ and of any
  $x \in \Int(A_3)$ is $A_2 \cup A_3 \cup A_5 \cup A_6$.
  If $x \in A_1 \cap A_3$ then $\FA(x) = A_5 \cup
  A_6$.}{CellularMap}{6in}

\subsection{Cellular and Simplicial Maps}\label{sec:cellular:maps}
A cellular multivalued map is a map on the geometrical realization,
\Eq{GeometricalRealization}, of a grid $\cA$ that is constant on the interior of
each cell of $\cA$. It generalizes the cubical multivalued map of \cite{KMM04} to a
situation in which the cell boundaries need not be rectilinear.

\begin{definition}[Cellular Multivalued Map (CMM)]
A multivalued map $\FA: |\cA| \rightrightarrows |\cA|$ on the geometrical realization  of a grid \cA is a cellular multivalued map if it is the outer approximation of $f$ defined by
\beq{CellularMap}
	\FA(x) :=
	            \bigcap_{B\in \cA\,:\,x \in B }\{ A \in \cA: A \cap f(B) \neq \emptyset\} .
\eeq
\end{definition}

\noindent
The map \FA takes the interior of each cell to the
union of the cells that intersect its image and
the boundary shared by multiple cells to the cells that contain the intersection of their
images.  An illustration is shown in \Fig{CellularMap}.
An implication is that CMMs are
semicontinuous because they map the boundary of each cell to a subset
of the image of the cell itself---this is a straightforward generalization of
\cite[Prop. 6.17]{KMM04} for the related cubical case.

This construction is not easy to implement on a computer for two reasons:
it is a map on a continuum $|\cA|$, and to construct it we must
know $f(x)$ for each point $x \in |\cA|$.  The second problem is
addressed by the \emph{witness map} introduced in \Sec{WitnessMap}.
The first problem can be mitigated by defining a finite map, $\cF: \cK
\rightrightarrows \cK$, on a complex $\cK$ related to the grid
$\cA$. One such complex is the CW-complex formed from $\cA$: that is,
the collection of cells $\{A_i\}$, together with their faces $\{A_i \cap A_j\}$, their edges, and so forth.  An example is
the cubical CW-complex used in the approach of \cite{KMM04}. Since the
cellular multivalued map \FA is constant on each cell of the CW-complex, it naturally gives rise to an associated combinatorial map. For example
in \Fig{CellularMap} the two-cell $A_1$ would be mapped to
$\{A_5,A_6,A_8, A_9\}$ and the one-cell represented by $A_1\cap A_3$
would be mapped to $\{A_5,A_6\}$.

Our goal is to use a more easily described complex that is also
naturally suited to homology calculations: in particular, a simplicial
complex.
A natural simplicial complex to use is the \emph{nerve} $N(\cA)$, recall \App{Complexes}. Let $L$ denote a set of labels for the elements of $\cA$, and
$\sigma = \simplex{l_0,l_2,\ldots, l_k}$ be any finite subset of $L$. The intersection of the cells labeled by the elements of $\sigma$ is denoted
\beq{Atau}
	A_\sigma := \bigcap_{l \in \sigma} A_{l} .
\eeq
A set $\sigma$ is a simplex in the nerve when $A_\sigma \neq \emptyset$, thus
\beq{NerveComplex}
	\cK := N(\cA) = \{ \sigma = \simplex{l_0,l_1,\ldots, l_k}: A_\sigma \neq \emptyset\} .
\eeq
For example, the nerve of the grid in \Fig{CellularMap} is shown in \Fig{SimplicialMap}.

\InsertFig{SimplicialMap}{Simplicial complex $\cK_\alpha(L)$ that is the nerve of the $\alpha$-cells of \Fig{CellularMap}, and the action of the simplicial map $\cFK$ \Eq{FKDefinition} on the simplices $\simplex{l_1}$ (green), $\simplex{l_3}$ (blue) and $\simplex{l_1,l_3}$ (red).}{SimplicialMap}{6in}

A cellular multivalued map $\FA$ induces a combinatorial map $\cFK$ on the nerve that is defined to commute with the correspondence between the grid and its nerve.

\begin{definition}[Simplicial Multivalued Map (SMM)] If $\FA$ is a CMM \Eq{CellularMap} on a grid $\cA$, the map
$\cFK: \cK \rightrightarrows \cK$, defined by
\beq{FKDefinition}
	\cFK(\sigma) := \left\{\tau:
		A_\tau \subset \FA(A_\sigma) \right\} ,
\eeq
is a simplicial multivalued map.
\end{definition}

\noindent
Note that \cFK is a combinatorial multivalued map. Its domain consists
of simplices and its range of \emph{sets} of simplices; moreover,
$\cFK(\sigma)$ is a sub-complex of $\cK$ (recall \App{Complexes}). As
an example, the simplicial map induced by the cellular map of
\Fig{CellularMap} is shown in \Fig{SimplicialMap}. In this case, since
$A_{\simplex{l_1}} = A_1$, then $
\{\simplex{l_5,l_6,l_8},\simplex{l_6,l_8,l_9}\} \subset
\cFK(\simplex{l_1})$ as are the nine faces of these two
$2$-simplices. Similarly, since $A_{\simplex{l_1,l_3}} = A_1 \cap
A_3$, then $\cFK(\simplex{l_1,l_3}) = \{\simplex{l_5,l_6},
\simplex{l_5},\simplex{l_6}\}$.

The definition \Eq{FKDefinition} satisfies the closed graph condition:

\begin{lemma}[Closed Graph Condition] \label{lem:closed:graph}
	If  $\cFK$ is an SMM and $\tau \le\sigma \in \cK$ (i.e., $\tau$ is a face of $\sigma$), then $\cFK(\tau) \supseteq \cFK(\sigma)$.
\end{lemma}
\begin{proof}
	Since $\tau \le \sigma$, then by \Eq{Atau} $A_\tau \supseteq A_\sigma$. Equation  \Eq{CellularMap} implies that $\FA(A_\tau) \supseteq \FA(A_\sigma)$, and so by \Eq{FKDefinition} $\cFK(\tau) \supseteq \cFK(\sigma)$.
\end{proof}

\noindent

When the nerve $\cK$ of a grid $\cA$ is a geometrical simplicial
complex (again, recall \App{Complexes}), it induces a natural cellular
multivalued map on the geometrical realization $|\cK|$, the union of the
convex hull $|\sigma| \in \bR^n$ of each of its geometrical simplices.  The
map $\FK: |\cK| \rightrightarrows |\cK|$ is the multivalued map
induced by \cFK; i.e., by analogy with \Eq{CellularMap}:
\beq{GeometricSimplicialMap} \FK(x) := \bigcap_{\sigma\in \cK \,:\, x \in
  |\sigma|} \{ |\cFK(\sigma)|\} \eeq

In certain cases, \FA and $\FK$ contain the same information about homology. We can show this when the geometric realization of the complex is a (strong) deformation retract of the geometric realization of the grid, i.e.,  when $|\cK| \subset |\cA|$ and there exists a continuous map $r:|\cA| \times [0,1] \to |\cA|$, such that
\bsplit{DeformationRetract}
		             r(x,0) &= x, \\
		             r(x,t) &= x \mbox{ if }  x \in |\cK|, \mbox{ and}\\
		             r(x,1) &:=\rho(x) \in |\cK| .
\esplit
As we will see in \Sec{AlphaComplex}, this assumption can be verified for an $\alpha$-grid and its nerve.
We begin with the following ``partial commutativity" lemma.

\begin{lemma} \label{lem:fk:subset:fc}
	Suppose that $\cK = N(\cA)$ is a geometrical simplicial complex. Let \FA and \FK be cellular multivalued maps as in \Eq{CellularMap} and \Eq{GeometricSimplicialMap}. If $|\cK|$ is a deformation retract of $|\cA|$ and $\rho(A_i) = r(A_i,1) \subset A_i$, then for any $x \in |\cA|$, $(\FK \circ \rho)(x) \subseteq (\rho \circ \FA)(x)$.
\end{lemma}

\begin{proof}
Note that since $\rho$ is onto $|\cK|$ and is the identity on $|\cK|$, $\rho(A_i) = |\cK| \cap A_i$ and it suffices to show that for any $x \in |\cA|$, $\FK(\rho(x)) \subseteq \FA(x)$. Furthermore, we note that it follows directly from \Eq{FKDefinition} that for any simplex $\sigma \in \cK$, $|\cFK \left(\sigma \right)| \subset \FA(A_\sigma)$.
For an $x \in A_\sigma$, $\rho(x)$ is a point in a geometric simplex $|\sigma|$.
Therefore, by \Eq{GeometricSimplicialMap},
\[
	\FK(\rho(x)) \subseteq |\cFK( \sigma) |
	\subseteq \FA(A_\sigma)
\]
as required.
\end{proof}

This result is exactly what is needed to show that
the maps on homology induced by \FK and \FA are isomorphic. In particular, we can prove the following:

\begin{theorem}\label{thm:acyclic} Under the hypotheses of \Lem{fk:subset:fc},
whenever $\FA$ is an acyclic multivalued map then $\FK$ induces the same map on homology as $\FA$.
\end{theorem}

\begin{proof}
Whenever $\FA$ is an acyclic multivalued map, then the
definition \Eq{FKDefinition} implies that $\FK$ is as well. Moreover,
since there exists a deformation retract \Eq{DeformationRetract}, $\rho_*$ is the identity; thus, the maps $\rho \circ \FA$ and $\FK \circ \rho$ are also acyclic. Now,
by \Lem{fk:subset:fc}, $\FK \circ \rho$ is a submap of $\rho \circ
\FA$; therefore, it follows that there is a continuous map
$u:|\cA| \to |\cK|$ that
is a continuous selector for both $\FK \circ \rho$ and $\rho \circ
\FA$. Thus, if $v$ and $w$ are continuous selectors for \FK
and \FA, respectively, then $(v \circ \rho)$ and $(\rho \circ w)$ are
continuous selectors carried by $(\FK \circ \rho)$ and $(\rho \circ \FA)$.
The acyclic carrier theorem \cite[Thm.~13.3]{Munkres84} then implies that
\[
	(v \circ \rho)_* = u_* = (\rho \circ w)_* \quad
	\Rightarrow \quad v_* \circ \rho_* = \rho_* \circ w_* \quad
	\Rightarrow \quad v_* = w_* .
\]
\end{proof}

By \Eq{CellularMap}, \FA is semicontinuous and an outer approximation
of $f$; therefore, if \FA is acyclic, it
induces a well-defined map on the
homology groups such that $(\FA)_* = f_*$.  That is, in order to determine
the Conley index (recall \App{Conley})
for an isolated invariant set of $f$, we need only to know the map on homology induced by \FA and check acyclicity. In \App{ComputingConley},
we recall the theoretical framework and algorithms from \cite{Day08} that
can be used to compute the discrete Conley index for a multivalued map.

Note that our construction of the cellular multivalued map, \FA, is still purely theoretical:
since we only know the points in a time series, we do not know
the image of every point in the state space and thus cannot
compute the outer approximation. Without this, there is no direct
method to compute the associated simplicial map $\cFK$ on the nerve.
In \Sec{witness:map} we define a new map, the witness map, that can be
computed algorithmically from time-series data. We then provide a set
of conditions under which the witness map and the CMM \FA contain
the same information about homology. These results allow us to
calculate the Conley index using the witness map rather than \FA.

\subsection{$\alpha$-Diagrams and Complexes}\label{sec:AlphaComplex}

The concepts in the previous sections can be specialized to the case
of a grid based on an $\alpha$-diagram\footnote

{See \App{Complexes} for a discussion of $\alpha$-diagrams and related simplicial complexes.}

for a finite set of \emph{landmarks}, $L = \{l_1,\ldots,l_{\ell}\}
\subset \bR^n$.  Recalling that $d(x,y)$ is the
Euclidean metric, we denote the closed ball of radius $\alpha$ centered at $l$ by
\beq{Ball}
	B_\alpha(l) = \{ x \in \bR^n: d(x,l) \le \alpha\} ,
\eeq
and the distance from $x$ to a set $S \subset \bR^n$ by
$d(x,S) = \inf_{y\in S} d(x,y)$.
For a given $\alpha >0$, the $\alpha$-cell
centered at a landmark point $l_i$ is the intersection of
$B_\alpha(l_i)$ with the Voronoi cell of $l_i$:
\beq{AlphaCell}
	A_i(\alpha) := B_\alpha(l_i) \cap \{x \in \bR^n: d(x,l_i) \le d(x,L) \}.
\eeq
We denote the collection of
$\alpha$-cells for a set of landmarks $L$ by
$\cA_\alpha(L)$. An example of a set of landmarks and their associated
$\alpha$-cells is shown in \Fig{CellularMap}.

In our application it will be important to choose the landmarks to cover the time series $\Gamma$ with some accuracy; in particular, there should be a landmark within some prescribed distance from each element of $\Gamma$.  The landmarks should also be relatively sparse: the minimal distance between any two should not be too small. These two requirements are linked, as will be discussed in \Sec{witness:map}.  One selection method, advocated by \cite{deSilva04}, is to choose the landmarks from $\Gamma$ itself by a ``max-min" algorithm: choose $L$ recursively by selecting the farthest point in $\Gamma$ from the previous selection until the operative density and sparseness requirements are satisfied, if possible. For the simple example in \Sec{example}, we do not choose the landmarks from the time series, but uniformly in $\bR^n$.  The general problem of finding the most appropriate landmarks in an efficient way is an interesting question for future investigation.

It is straightforward to show that a collection of $\alpha$-cells is a grid:

\begin{lemma} \label{lem:alpha:grid}
	When $\alpha >0$, the set of $\alpha$-cells $\cA_\alpha(L)$ is a grid on $|\cA_\alpha(L)|$.
\end{lemma}

\noindent
This lemma allows us to define a cellular multivalued map \Eq{CellularMap} for any $\alpha$-grid with $\alpha >0$.

The nerve of an $\alpha$-diagram is the \emph{$\alpha$-complex} denoted
\beq{AlphaComplex}
	\cK_\alpha(L) = N(\cA_\alpha(L)).
\eeq
Since the $\alpha$-complex is a nerve, it is always an abstract simplicial complex, and, as for the general case of \Sec{cellular:maps}, there is an associated simplicial map  $\cFK$ on  \Eq{AlphaComplex}. Moreover, whenever the vertices $l_i$ are in ``general position" (recall \App{Complexes}), $\cK_\alpha(L)$ is a geometrical complex so that its simplices have dimension at most $n$ and their intersections are faces. We will always assume that the landmarks are selected to be in general position.

For this case, Edelsbrunner proved that the geometrical realization of the nerve $|\cK_\alpha(L)|$ is a deformation retract, \Eq{DeformationRetract}, of $|\cA_\alpha(L)|$ \cite{Edelsbrunner95}.\footnote

{See \Lem{Retract} and the discussion in \App{Complexes}.}

He also showed, since the $\alpha$-cells are convex, that $\rho(\cdot) = r(\cdot,1)$ can be chosen to preserve inclusion in each specific cell: $\rho(A_i) \subset A_i$.
Thus for an $\alpha$-grid, the hypotheses of \Lem{fk:subset:fc} hold. Consequently, \Thm{acyclic} implies that whenever the cellular map $F_\cA$ on an $\alpha$-grid is an enclosure of a dynamical system $f$, then the Conley index of an isolated invariant set can be computed from the map induced by $F_\cK$ on homology.

config: arxiv


TITLE: Is an oscillation the same as a period?
QUESTION [1 upvotes]: Is one oscillation from peak to trough to peak again or is it just peak to trough? Doing a homework question and want to be sure I have the right definition

REPLY [2 votes]: Is one oscillation from peak to trough to peak again or is it just peak to trough?

This is a definition issue.
An internet search through some of the free dictionaries out there gives (link):

Physics.
a.  an effect expressible as a quantity that repeatedly and regularly fluctuates above and below some mean value, as the pressure of a sound wave or the voltage of an alternating current.
b. a single fluctuation between maximum and minimum values in such an effect.

And (link):

1. (General Physics) physics statistics
a. regular fluctuation in value, position, or state about a mean value, such as the variation in an alternating current or the regular swinging of a pendulum.
b. a single cycle of such a fluctuation

It has two meanings in physics: "an oscillation" can mean the very act that something is "swinging" in general. But about the other meaning, it seems the two sources do not entirely agree if "an oscillation" is between maximum and minimum or if it is a single cycle (that is, from maximum to maximum).
We are close to the answer, but a proper place for such a question would rather be https://english.stackexchange.com/

Is an oscillation the same as a period?

Your title is another question. And the answer is no.
The word "period" means something very specific, and is not the same thing as an oscillation in either meaning. A "period" is not the act of swinging, but the time for a complete cycle of the oscillation (from maximum to maximum). Usually denoted $T$ with units of seconds (or seconds per cycle if you like).


TITLE: Pullbacks and pushouts in the category of digraphs?
QUESTION [7 upvotes]: By definition the category of digraphs is:

Objects are endomorphisms of the category $\mathbf{Rel}$ (that is sets equipped with a binary relation on that set).
Morphisms from an object $\mu$ to an object $\nu$ are discretely continuous functions, that is functions $f$ such that $f\circ\mu\subseteq\nu\circ f$. (This formula is equivalent to $\mu\subseteq f^{-1}\circ\nu\circ f$ and to $f\circ\mu\circ f^{-1}\subseteq\nu$.)
Composition is the composition of functions.

I have proved that in categories like this all small products and all small coproducts are defined.
To prove that this category is (co)complete it remains to show existence of pullbacks and pushouts.
Please help me to examine existence of pullbacks and pushouts in the category of digraphs.

REPLY [8 votes]: Since it might be of interest to porton or others, I'll expand on my earlier comment, that the category of digraphs is in fact a Grothendieck quasitopos. Hence it is not only complete and cocomplete, it is also locally cartesian closed, among other things.
First, recall that the category of 'multigraphs' $\mathbf{Gph}$ (as in Tom Hirschowitz's answer) is the topos of presheaves $G^{op} \to \mathbf{Set}$ on the category
$$G = (0 \stackrel{\overset{s}{\longrightarrow}}{\underset{t}{\longrightarrow}} 1).$$
Then, I claim that the inclusion $i: \mathbf{Dig} \to \mathbf{Gph}$ in Tom's answer is the same as the inclusion of separated presheaves for the $\neg\neg$-topology. It is well-known that separated presheaves for a Grothendieck topology form a Grothendieck quasitopos (see for example Johnstone's Elephant).
The claim is not very hard to check. The representable $\hom_G(-, 1)$ is the digraph $\bullet \to \bullet$, and the representable $\hom_G(-, 0)$ is the digraph $\bullet$; it may be calculated that the only $\neg\neg$-dense subpresheaf of $\hom(-, 0)$ is $\hom(-, 0)$ itself, and that the only $\neg\neg$-dense subpresheaf of $\hom(-, 1)$ besides itself is the digraph inclusion
$$(\bullet \;\;\; \bullet) \hookrightarrow (\bullet \to \bullet).$$
Now a presheaf $X$ on $G$ is separated if, for each object $c$ and each $\neg\neg$-dense inclusion $i: F \hookrightarrow \hom(-, c)$, the induced map
$$X(c) \cong \mathbf{Set}^{G^{op}}(\hom(-, c), X) \stackrel{\mathbf{Set}^{G^{op}}(i, X)}{\to} \mathbf{Set}^{G^{op}}(F, X)$$
is injective. Since we have only one non-trivial dense inclusion (the digraph inclusion displayed above), the separation condition on $X$ boils down to saying that the canonical map
$$X(1) \to \hom((\bullet \;\;\; \bullet), X) \cong X(0) \times X(0)$$
is injective, which is to say that the map taking each edge $e \in X(1)$ to the source-target pair $(s^\ast(e), t^\ast(e))$ is injective. But this just means $X$ is isomorphic to a digraph (in the sense of this post), so we are done. (Details can be made available on request.)


TITLE: BSplines: how many segments spanned by a basis function of order K
QUESTION [0 upvotes]: I'm trying to teach myself Basis splines, from what I have read a basis spline of order K is non-zero in K segments only. Is this correct? I ask because when I use the following code to plot basis splines in R, some of the basis functions only span 2 or 3 segments (first dotted line from left etc.). May be it is just a convention in R, I mainly just want to make sure I understand the concept correctly.
Edit: Based on this link, it seems not all the basis splines produced in this plot are of degree 3. I should add 3 dummy knots on either end to get cubic bsplines between my input data range. Is this correct.

library(splines)
x = seq(1,10,0.01)
b = bs(x, knots = c(2.5, 5, 7.5), degree = 3, intercept = TRUE)
matplot(x, b, type = 'l')
abline(v = c(2.5, 5, 7.5), lty = 2)

EDIT: Below I produce 2 examples of fitting cubic bsplines (line in both the charts) to hypothetical data(dots in both the charts) for x between [0,4]. With 2 imaginary knots (knots at -2,-1,0,1,2,3,4,5,6), the fit is not very good but with 3 imaginary knots (knots at -3,-2,-1,0,1,2,3,4,5,6,7) it is perfect.

REPLY [1 votes]: Suppose we're dealing with basis functions of order $k$ (i.e. degree = $k-1$) on the knot sequence $(t_i)$. Then the $i$-th b-spline basis function is non-zero on the interval $[t_i, t_{i+k}]$. If $t_i < \ldots < t_{i+k}$, then there are $k$ "segments" between $t_i$ and $t_{i+k}$. But if some of these knots are equal, then some of these $k$ segments will shrink to zero, so you may choose to not count them as segments at all.
In the example you plotted, it looks like your software automatically added some extra (and duplicated) knots at the ends of your curve. This is a common practice. So, though you asked for knots $(2.5, 5, 7.5)$, you actually got knots $(0,0,0,0,2.5,5,7.5,1,1,1,1)$. The repeated knots give you zero-length segments, and this throws off your segment-counting scheme.
More details in these notes.


TITLE: Consider $H=\Bbb{Z}_{30}$ and $G=\Bbb{Z}_{15}$ as additive abelian groups. Show that ${\rm Aut}(H) \cong {\rm Aut}(G)$.
QUESTION [2 upvotes]: Consider $H = \mathbb{Z}_{30}$ and $G = \mathbb{Z}_{15}$ as additive abelian groups. Then how do I show that ${\rm Aut}(H) \cong {\rm Aut}(G)$?
By the Chinese remainder theorem, I know that $\mathbb{Z}_{30} \cong \mathbb{Z}_{2} \times \mathbb{Z}_{15}$. Intuitively I know that the only automorphism of $\mathbb{Z}_{2}$ is the identity automorphism, so we can construct a bijective map between $\psi : {\rm Aut}(G) \rightarrow {\rm Aut}(H)$. By $\psi(\phi) = \phi^{*}$, where $\phi^{*}((a,b)) = (a, \phi(b))$, which is an isomorphism. (Here we consider  $\mathbb{Z}_{30} \cong \mathbb{Z}_{2} \times \mathbb{Z}_{15}$.)
Is the definition of this map sufficient to claim that ${\rm Aut}(G) \cong {\rm Aut}(H)$? Also, how does one in general find the automorphism group of $\bigoplus_{k=1}^{r} \mathbb{Z}_{n_{k}}$, where the $n_{k}$'s are not necessarily coprime.

REPLY [1 votes]: For your first question, we have $\Bbb Z_{30}^×\cong(\Bbb Z_2\times\Bbb Z_{15})^×\cong\Bbb Z_2^×\times\Bbb Z_{15}^×\cong\Bbb Z_{15}^×$.
And $\operatorname {Aut}(\Bbb Z_n)\cong\Bbb Z_n^×$.


TITLE: Differentiation help (lots of square roots)
QUESTION [1 upvotes]: help finding the first derivative of this question:
$y=\sqrt{ 1 + \sqrt{ 1+\sqrt{1+8x}}}$
I get confused, is it meant to be done implicitly or is it just a really long chain rule?

REPLY [1 votes]: That is definitely something that can be done with a really long chain rule.
$y=(((1+8x)^\frac{1}{2}+1)^\frac{1}{2}+1)^\frac{1}{2}$
$\frac{dy}{dx}=\frac{1}{2}(((1+8x)^\frac{1}{2}+1)^\frac{1}{2}+1)^\frac{-1}{2}*\frac{1}{2}((1+8x)^\frac{1}{2}+1)^\frac{-1}{2}*\frac{1}{2}(1+8x)^\frac{-1}{2}*8$


TITLE: Cooper pairing from repulsive potential
QUESTION [1 upvotes]: Suppose the Hamiltonian of a many-electron system consists of a potential which is repulsive : $\langle k_1, k_2 |\hat V |k_1',k_2' \rangle > 0$ where $k_1, k_2, \cdots$ are possible momenta that an electron can have. I want to know whether such potential can lead to cooper pairing. In particular I want to know whether the following statements make sense at all.
Let's consider a similar problem as the one cooper solved that is "consider a pair of electrons which interact above a quiescent Fermi sphere" with the interaction being repulsive here.
Let $K \equiv {\{k^i\}}$ denotes the space of all momentum.  Let's partition the $K$ in two parts say $K_A \equiv \{ k_A^i \}$ and $K_B \equiv \{ k_B^i \}$. Now consider the paired state $\Psi = 1/(\sqrt{2}) (\Psi_A + e^{i \phi} \Psi_B$) where $\Psi_A$ is a superposition of paired momentum states in $K_A$ i.e $\Psi_A \approx b_1| k^1_A, -k^1_A \rangle + b_2 | k^2_A, -k^2_A \rangle + \cdots$ and similarly $\Psi_B$ is a superposition of paired states of momenta in $K_B$. Now take the matrix element $\langle \Psi| \hat V|\Psi\rangle = 1/2\left(\langle \Psi_A| \hat V|\Psi_A\rangle + \langle \Psi_B| \hat V|\Psi_B + 2*\cos\phi \langle \Psi_A| \hat V|\Psi_B\rangle \right)$, assuming $\langle \Psi_A| \hat V|\Psi_B\rangle$ is real. Now if $\phi = \pi$ and $\hat V$ is such that $\langle k^i_A, -k^i_A| \hat V |k^j_A, -k^j_A \rangle \ll \langle k^i_A, -k^i_A| \hat V |k^j_B, -k^j_B \rangle$ then it looks like the paired state $\Psi$ might be energetically favorable.
Here I have suppressed the spins of the fermions.
Please let me know whether something is fundamentally wrong with this ?

REPLY [1 votes]: Yes, pairing is possible from a repulsive interaction. The reason behind this is that pairing has to occur in a certain angular momentum channel : $l=0$ for s-wave superconductivity, $l=1$ for p-wave, and so on. To see this, you can expand the repulsive $k$-dependent interaction on Legendre polynomials. Check this review that deals with the Kohn-Luttinger mechanism responsible for pairing by the screened Coulomb potential.


TITLE: What is the theory behind Fresnel's half period zones?
QUESTION [0 upvotes]: I was not convinced by exactly how does this work,why do we need to take concentric spheres around point P and how do those spheres cast circular areas on the wavefront?
Why is the radius OM1 and the first sphere originating from P not same?

REPLY [1 votes]: Consider the wavefronts (orange) of a plane wave moving in from the left.

Each point on a wavefront acts as a secondary source - Huygen's construction.
To get to point $F$ the wavelets from point $X$ on the incident wavefront have to travel a distance $XF$.
The wavelets from a point $Y$ on the incident wavefront travel a distance $YF$ and so do not arrive in phase with the wavelets from point $X$.
At point $F$ there is a superposition of these two sets of wavelets.
As long as the path difference between the two sets of wavelets is less than $\frac \lambda 2$, where $\lambda$ is the wavelength of the incident waves the wavelets will superpose constructively ie combine together to give a greater amplitude of the wave at $F$.
If $AF$ and $A'F$ differ in distance by less than $\lambda 2$ then the wavelets which originate from the wavefront $AA'$ will arrive with a phase difference of less than $\frac \lambda 2$ and add together constructively.
Now consider the incident wavefront between $AB$ and consider a point $Z$ producing wavelets.
The distance $ZF$ is larger that $XF$ by at least $\frac \lambda 2$ and so the wavelet originate from $Z$ will be out of phase by greater than $\frac \lambda 2$ and so when they arrive at $F$ will superpose in a destructive manner reducing the amplitude of the resultant wave at $F$.
So that this not happen all waves which would produce a phase difference between $\frac \lambda 2\,(A)$ and $\lambda \, (B)$, and between $\frac \lambda 2\, (A')$ and $\lambda \,(B')$ are removed by making $AB$ and $A'B'$ opaque.
The next zone defined by $BC$ and $B'C'$ is made transparent because all the waves which arrive from that zone are between $\lambda$ and $\frac{3\lambda}{2}$ out of phase with the waves which came directly from $X$ and so will superpose constructively.
The end result is the constructive superposition of wave at $F$.
My diagram is just a section through the zone plate which has rotational symmetry about the line $XF$.
The radii of the circles which define the transparent and opaque zones do not increase in equal increments, just think about the geometry of the situation.
And finally a zone plate can be constructed with the transparent and opaque sections transposed.


TITLE: Sylvester's sequence - series representation of Vardi's constant
QUESTION [4 upvotes]: First, a bit of background - Sylvester's sequence is defined recursively as
$$e_{n+1}=\prod_{j=0}^{n-1}e_j=e_n^2-e_n+1, \space\space\space e_0=2$$
and, miraculously, there exists a number $E\approx 1.26$ that satisfies
$$e_n=\lfloor E^{2^{n+1}}+1/2\rfloor$$
Wolfram gives a series representation for the constant $E$ (also called "Vardi's constant"), a proof of which I have been unable to find (or derive):
$$E=\frac{\sqrt 6}{2}\exp\Bigg[\sum_{j=1}^\infty \frac{1}{2^{j+1}}\ln\bigg(1+\frac{1}{(2e_j-1)^2}\bigg)\Bigg]$$
or, reduced to a product,
$$E=\frac{\sqrt 6}{2}\prod_{j=1}^\infty \bigg(1+\frac{1}{(2e_j-1)^2}\bigg)^{2^{-j-1}}$$
Reduced further, this is equal to
$$E=\frac{\sqrt 6}{2}\prod_{j=1}^\infty \bigg(\frac{4e_{j+1}-4}{4e_{j+1}-3}\bigg)^{2^{-j-1}}$$
How might one derive this property? I haven't been able to make much progress with its derivation.

REPLY [0 votes]: Caveat: this is a partial answer. I will try to explain the methodology for the construction of the generating function and the definition of the constant (it lacks detail but it should give an overview, if the reader finds errors please be free to fix them or let me know and I will try to fix them). I hope it will help as an idea of how Vardi's constant and its generating function can be developed. So below I will not include the construction of  $E\approx 1.26$ that satisfies $$e_n=\lfloor E^{2^{n+1}}+1/2\rfloor$$ but a similar construction based on another constant that I will call $L$ such that  $L\approx 1.597910225682488561085083110$ that satisfies
$$e_n=\lfloor L^{2^{n}}+1/2\rfloor$$

The difference is that the exponent of my generating function based on the constant $L$ is $2^{n}$ instead of $2^{n+1}$, as in the case of Vardi's $E$.

Basically we observe that the expression below is true:

$$(e_n-1)^2 \lt e_{n+1} \lt e_n^2$$
for Sylvester's sequence:
$2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, \dots$
(sequence A000058 in the OEIS).
e.g.
$$(e_0-1)^2 = (2-1)^2=1 \lt e_1=3 \lt e_0^2 = 2^2=4$$
$$(e_1-1)^2 = (3-1)^2=4 \lt e_2=7 \lt e_1^2 = 3^2=9$$
$$(e_2-1)^2 = (7-1)^2=36 \lt e_3=43 \lt e_2^2 = 7^2=49$$
$$\cdots$$

Following the classical Mills demonstration (1947) (please see details of the theorem and proof in the paper) we can conclude that we have a bounded monotone increasing sequence able to generate each $e_n$. The generation of the $L$ constant is as follows:

$$L_0={e_0^{\frac{1}{2^0}}}$$
$$L_1={e_1^{\frac{1}{2^1}}}$$
$$L_2={e_2^{\frac{1}{2^2}}}$$
$$\cdots$$
$$L_5={e_5^{\frac{1}{2^5}}}=1.597910225682488561085083110$$
Where $L$ will be: $$\lim_{n \to \infty} L_n \ .$$ As in the case of the classical results of Mills, we do no know if it is a rational number or whether a closed-form formula could be obtained.
So you will require more precision as $n$ grows higher. With $L_5$ we will be able to calculate accurately $\{e_0,e_1,e_2,e_3,e_4,e_5\}$
The generating function initially would be as follows (it will require a correction I will explain later):
$$e_n = \lfloor L^{2^n} \rfloor$$
e.g. (still without correction):
$$e_5 = \lfloor L^{2^5} \rfloor = 3263443$$
$$e_4 = \lfloor L^{2^4} \rfloor = \color{red}{1806}$$
$$e_3 = \lfloor L^{2^3} \rfloor = \color{red}{42}$$
$$e_2 = \lfloor L^{2^2} \rfloor = \color{red}{6}$$
$$e_1 = \lfloor L^{2^1} \rfloor = \color{red}{2}$$
$$e_0 = \lfloor L^{2^0} \rfloor = \color{red}{1}$$
As you can observe still $e_0 \dots e_4$ are not accurate, but if we add $1/2$ the correction is good enough to make the result of the floor function match the sequence (as you can see the same correction is done in the OP's generating function for Vardi's constant $E$).
$$e_n=\lfloor L^{2^{n}}+1/2\rfloor$$
And now:
$$e_5 = \lfloor L^{2^5} +1/2\rfloor = 3263443$$
$$e_4 = \lfloor L^{2^4} +1/2 \rfloor = 1807$$
$$e_3 = \lfloor L^{2^3} +1/2 \rfloor = 43$$
$$e_2 = \lfloor L^{2^2} +1/2 \rfloor = 7$$
$$e_1 = \lfloor L^{2^11/2} +1/2 \rfloor = 3$$
$$e_0 = \lfloor L^{2^0} +1/2 \rfloor = 2$$
My guess is that the correction is required because, due to the properties of the sequence, we are embedding the sequence into quadratics intervals $(e_n-1)^2 \lt e_{n+1} \lt e_n^2$ instead of intervals of the shape $e_n^2 \lt e_{n+1} \lt (e_n+1)^2$ (which is the same case as Mills' constant but for intervals based on powers of three instead of powers of two. In the case of Mills' constant the correction is not required). I know this point do require a formal explanation but I would dare to say that it is related with the definition of the quadratic intervals.

So we have been able to define the generating function:
$$e_n=\lfloor L^{2^{n}}+1/2\rfloor$$
where
$$L\approx 1.597910225682488561085083110$$

This is basically an overview and lacks a lot of details, but the point is that due to the characteristics of the sequence we can bound it into quadratic intervals. Due to this, there is a monotone increasing sequence that we can embed into a constant, such that a quadratic exponent applied to it (plus the floor function corrected by $+1/2$) is able to recover the embedded sequence. So I think it is safe to say that this has been done by Vardi as well in a similar way. About the calculations of Wolfram, I think they are done by Wolfram's engine backwards, once you have defined properly the generating function and the constant. It can be seen that the precision of $E$ depends on the iteration $e_n$ so as $n$ grows higher we need more precision of $E$ (and same for my example $L$) to recover accurately the element $e_n$ of the sequence.


TITLE: Uniform Convergence of Bounded Linear Operators is Bounded?
QUESTION [1 upvotes]: Let ${\cal{B}}(X,Y)$ be the set of all bounded linear operators from normed linear spaces $X$ to $Y$.
Hunter and Nachtegaele say that if bounded linear operators $(T_{n})$ satisfy  $\displaystyle\lim_{n \rightarrow \infty} ||T_{n}-T|| = 0$ for some $T \in {\cal{B}}(X,Y)$, then the $T_{n}$ converge uniformly to $T$.
My question is: Is it necessary to assume that $T \in {\cal{B}}(X,Y)$? If we just assume that $T$ is some linear operator from $X$ to $Y$, would we get automatically that it is bounded if we have that limit?
(I'm just thinking about how uniform convergence of continuous functions gives something still continuous and about how continuity of linear operators is equivalent to boundedness, so I'm not sure why the authors make a point to specify that $T \in {\cal{B}}(X,Y)$ when they define uniform convergence here.)
Thank you in advance for any insight!

REPLY [0 votes]: Yes, if $T:X\to Y$ is a linear operator and $\exists S\in B(X,Y)$ such that
$$
\alpha := \sup_{\|x\| = 1} \|S(x) - T(x)\| < \infty
$$
Then for any $x\in X,\|x\| = 1$, you have
$$
\|T(x)\| \leq \|S(x) - T(x)\| + \|S(x)\| \leq \alpha + \|S\|
$$
and so $T \in B(X,Y)$.
This is applicable in your situation with $S=T_1$. However, if $T \notin B(X,Y)$ a priori, it does not make sense to write down an expression $\|T_n - T\|$, which is why the authors assume $T\in B(X,Y)$.


text:
(*  Title:       CCPO_Topology.thy
    Author:      Johannes Hölzl, TU Munich
*)

section \<open>CCPO topologies\<close>

theory CCPO_Topology
imports
  "HOL-Analysis.Extended_Real_Limits"
  "../Coinductive_Nat"
begin

lemma dropWhile_append:
  "dropWhile P (xs @ ys) = (if \<forall>x\<in>set xs. P x then dropWhile P ys else dropWhile P xs @ ys)"
  by auto

lemma dropWhile_False: "(\<And>x. x \<in> set xs \<Longrightarrow> P x) \<Longrightarrow> dropWhile P xs = []"
  by simp

abbreviation (in order) "chain \<equiv> Complete_Partial_Order.chain (\<le>)"

lemma (in linorder) chain_linorder: "chain C"
  by (simp add: chain_def linear)

lemma continuous_add_ereal:
  assumes "0 \<le> t"
  shows "continuous_on {-\<infinity>::ereal <..} (\<lambda>x. t + x)"
proof (subst continuous_on_open_vimage, (intro open_greaterThan allI impI)+)
  fix B :: "ereal set" assume "open B"
  show "open ((\<lambda>x. t + x) -` B \<inter> {- \<infinity><..})"
  proof (cases t)
    case (real t')
    then have *: "(\<lambda>x. t + x) -` B \<inter> {- \<infinity><..} = (\<lambda>x. 1 * x + (-t)) ` (B \<inter> {-\<infinity> <..})"
      apply (simp add: set_eq_iff image_iff Bex_def)
      apply (intro allI iffI)
      apply (rule_tac x= "x + ereal t'" in exI)
      apply (case_tac x)
      apply (auto simp: ac_simps)
      done
    show ?thesis
      unfolding *
      apply (rule ereal_open_affinity_pos)
      using \<open>open B\<close>
      apply (auto simp: real)
      done
  qed (insert \<open>0 \<le> t\<close>, auto)
qed

lemma tendsto_add_ereal:
  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> (f \<longlongrightarrow> y) F \<Longrightarrow> ((\<lambda>z. x + f z :: ereal) \<longlongrightarrow> x + y) F"
  apply (rule tendsto_compose[where f=f])
  using continuous_add_ereal[where t=x]
  unfolding continuous_on_def
  apply (auto simp add: at_within_open[where S="{- \<infinity> <..}"])
  done

lemma tendsto_LimI: "(f \<longlongrightarrow> y) F \<Longrightarrow> (f \<longlongrightarrow> Lim F f) F"
  by (metis tendsto_Lim tendsto_bot)

subsection \<open>The filter \<open>at'\<close>\<close>

abbreviation (in ccpo) "compact_element \<equiv> ccpo.compact Sup (\<le>)"

lemma tendsto_unique_eventually:
  fixes x x' :: "'a :: t2_space"
  shows "F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. f x = g x) F \<Longrightarrow> (f \<longlongrightarrow> x) F \<Longrightarrow> (g \<longlongrightarrow> x') F \<Longrightarrow> x = x'"
  by (metis tendsto_unique filterlim_cong)

lemma (in ccpo) ccpo_Sup_upper2: "chain C \<Longrightarrow> x \<in> C \<Longrightarrow> y \<le> x \<Longrightarrow> y \<le> Sup C"
  by (blast intro: ccpo_Sup_upper order_trans)

lemma tendsto_open_vimage: "(\<And>B. open B \<Longrightarrow> open (f -` B)) \<Longrightarrow> f \<midarrow>l\<rightarrow> f l"
  using continuous_on_open_vimage[of UNIV f] continuous_on_def[of UNIV f] by simp

lemma open_vimageI: "(\<And>x. f \<midarrow>x\<rightarrow> f x) \<Longrightarrow> open A \<Longrightarrow> open (f -` A)"
  using continuous_on_open_vimage[of UNIV f] continuous_on_def[of UNIV f] by simp

lemma principal_bot: "principal x = bot \<longleftrightarrow> x = {}"
  by (auto simp: filter_eq_iff eventually_principal)

definition "at' x = (if open {x} then principal {x} else at x)"

lemma at'_bot: "at' x \<noteq> bot"
  by (simp add: at'_def at_eq_bot_iff principal_bot)

lemma tendsto_id_at'[simp, intro]: "((\<lambda>x. x) \<longlongrightarrow> x) (at' x)"
  by (simp add: at'_def topological_tendstoI eventually_principal tendsto_ident_at)

lemma cont_at': "(f \<longlongrightarrow> f x) (at' x) \<longleftrightarrow> f \<midarrow>x\<rightarrow> f x"
  using at_eq_bot_iff[of x] by (auto split: if_split_asm intro!: topological_tendstoI simp: eventually_principal at'_def)

subsection \<open>The type class \<open>ccpo_topology\<close>\<close>

text \<open>Temporarily relax type constraints for @{term "open"}.\<close>
setup \<open>Sign.add_const_constraint
  (@{const_name "open"}, SOME @{typ "'a::open set \<Rightarrow> bool"})\<close>

class ccpo_topology = "open" + ccpo +
  assumes open_ccpo: "open A \<longleftrightarrow> (\<forall>C. chain C \<longrightarrow> C \<noteq> {} \<longrightarrow> Sup C \<in> A \<longrightarrow> C \<inter> A \<noteq> {})"
begin

lemma open_ccpoD:
  assumes "open A" "chain C" "C \<noteq> {}" "Sup C \<in> A"
  shows "\<exists>c\<in>C. \<forall>c'\<in>C. c \<le> c' \<longrightarrow> c' \<in> A"
proof (rule ccontr)
  assume "\<not> ?thesis"
  then have *: "\<And>c. c \<in> C \<Longrightarrow> \<exists>c'\<in>C. c \<le> c' \<and> c' \<notin> A"
    by auto
  with \<open>chain C\<close> \<open>C \<noteq> {}\<close> have "chain (C - A)" "C - A \<noteq> {}"
    by (auto intro: chain_Diff)
  moreover have "Sup C = Sup (C - A)"
  proof (safe intro!: order.antisym ccpo_Sup_least \<open>chain C\<close> chain_Diff)
    fix c assume "c \<in> C"
    with * obtain c' where "c' \<in> C" "c \<le> c'" "c' \<notin> A"
      by auto
    with \<open>c\<in>C\<close> show "c \<le> \<Squnion>(C - A)"
      by (intro ccpo_Sup_upper2 \<open>chain (C - A)\<close>) auto
  qed (auto intro: \<open>chain C\<close> ccpo_Sup_upper)
  ultimately show False
    using \<open>open A\<close> \<open>Sup C \<in> A\<close> by (auto simp: open_ccpo)
qed

lemma open_ccpo_Iic: "open {.. b}"
  by (auto simp: open_ccpo) (metis Int_iff atMost_iff ccpo_Sup_upper empty_iff order_trans)

subclass topological_space
proof
  show "open (UNIV::'a set)"
    unfolding open_ccpo by auto
next
  fix S T :: "'a set" assume "open S" "open T"
  show "open (S \<inter> T)"
    unfolding open_ccpo
  proof (intro allI impI)
    fix C assume C: "chain C" "C \<noteq> {}" and "\<Squnion>C \<in> S \<inter> T"
    with open_ccpoD[OF \<open>open S\<close> C] open_ccpoD[OF \<open>open T\<close> C]
    show "C \<inter> (S \<inter> T) \<noteq> {}"
      unfolding chain_def by blast
  qed
next
  fix K :: "'a set set" assume *: "\<forall>D\<in>K. open D"
  show "open (\<Union>K)"
    unfolding open_ccpo
  proof (intro allI impI)
    fix C assume "chain C" "C \<noteq> {}" "\<Squnion>C \<in> (\<Union>K)"
    with * obtain D where "D \<in> K" "\<Squnion>C \<in> D" "C \<inter> D \<noteq> {}"
      by (auto simp: open_ccpo)
    then show "C \<inter> (\<Union>K) \<noteq> {}"
      by auto
  qed
qed

lemma closed_ccpo: "closed A \<longleftrightarrow> (\<forall>C. chain C \<longrightarrow> C \<noteq> {} \<longrightarrow> C \<subseteq> A \<longrightarrow> Sup C \<in> A)"
  unfolding closed_def open_ccpo by auto

lemma closed_admissible: "closed {x. P x} \<longleftrightarrow> ccpo.admissible Sup (\<le>) P"
  unfolding closed_ccpo ccpo.admissible_def by auto

lemma open_singletonI_compact: "compact_element x \<Longrightarrow> open {x}"
  using admissible_compact_neq[of Sup "(\<le>)" x]
  by (simp add: closed_admissible[symmetric] open_closed Collect_neg_eq)

lemma closed_Ici: "closed {.. b}"
  by (auto simp: closed_ccpo intro: ccpo_Sup_least)

lemma closed_Iic: "closed {b ..}"
  by (auto simp: closed_ccpo intro: ccpo_Sup_upper2)

text \<open>
  @{class ccpo_topology}s are also @{class t2_space}s.
  This is necessary to have a unique continuous extension.
\<close>

subclass t2_space
proof
  fix x y :: 'a assume "x \<noteq> y"
  show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
  proof cases
    { fix x y assume "x \<noteq> y" "x \<le> y"
      then have "open {..x} \<and> open (- {..x}) \<and> x \<in> {..x} \<and> y \<in> - {..x} \<and> {..x} \<inter> - {..x} = {}"
        by (auto intro: open_ccpo_Iic closed_Ici) }
    moreover assume "x \<le> y \<or> y \<le> x"
    ultimately show ?thesis
      using \<open>x \<noteq> y\<close> by (metis Int_commute)
  next
    assume "\<not> (x \<le> y \<or> y \<le> x)"
    then have "open ({..x} \<inter> - {..y}) \<and> open ({..y} \<inter> - {..x}) \<and>
        x \<in> {..x} \<inter> - {..y} \<and> y \<in> {..y} \<inter> - {..x} \<and> ({..x} \<inter> - {..y}) \<inter> ({..y} \<inter> - {..x}) = {}"
      by (auto intro: open_ccpo_Iic closed_Ici)
    then show ?thesis by auto
  qed
qed

end

lemma tendsto_le_ccpo:
  fixes f g :: "'a \<Rightarrow> 'b::ccpo_topology"
  assumes F: "\<not> trivial_limit F"
  assumes x: "(f \<longlongrightarrow> x) F" and y: "(g \<longlongrightarrow> y) F"
  assumes ev: "eventually (\<lambda>x. g x \<le> f x) F"
  shows "y \<le> x"
proof (rule ccontr)
  assume "\<not> y \<le> x"

  show False
  proof cases
    assume "x \<le> y"
    with \<open>\<not> y \<le> x\<close>
    have "open {..x}" "open (- {..x})" "x \<in> {..x}" "y \<in> - {..x}" "{..x} \<inter> - {..x} = {}"
      by (auto intro: open_ccpo_Iic closed_Ici)
    with topological_tendstoD[OF x, of "{..x}"] topological_tendstoD[OF y, of "- {..x}"]
    have "eventually (\<lambda>z. f z \<le> x) F" "eventually (\<lambda>z. \<not> g z \<le> x) F"
      by auto
    with ev have "eventually (\<lambda>x. False) F" by eventually_elim (auto intro: order_trans)
    with F show False by (auto simp: eventually_False)
  next
    assume "\<not> x \<le> y"
    with \<open>\<not> y \<le> x\<close> have "open ({..x} \<inter> - {..y})" "open ({..y} \<inter> - {..x})"
        "x \<in> {..x} \<inter> - {..y}" "y \<in> {..y} \<inter> - {..x}" "({..x} \<inter> - {..y}) \<inter> ({..y} \<inter> - {..x}) = {}"
      by (auto intro: open_ccpo_Iic closed_Ici)
    with topological_tendstoD[OF x, of "{..x} \<inter> - {..y}"]
         topological_tendstoD[OF y, of "{..y} \<inter> - {..x}"]
    have "eventually (\<lambda>z. f z \<le> x \<and> \<not> f z \<le> y) F" "eventually (\<lambda>z. g z \<le> y \<and> \<not> g z \<le> x) F"
      by auto
    with ev have "eventually (\<lambda>x. False) F" by eventually_elim (auto intro: order_trans)
    with F show False by (auto simp: eventually_False)
  qed
qed

lemma tendsto_ccpoI:
  fixes f :: "'a::ccpo_topology \<Rightarrow> 'b::ccpo_topology"
  shows "(\<And>C. chain C \<Longrightarrow> C \<noteq> {} \<Longrightarrow> chain (f ` C) \<and> f (Sup C) = Sup (f`C)) \<Longrightarrow> f \<midarrow>x\<rightarrow> f x"
  by (intro tendsto_open_vimage) (auto simp: open_ccpo)

lemma tendsto_mcont:
  assumes mcont: "mcont Sup (\<le>) Sup (\<le>) (f :: 'a :: ccpo_topology \<Rightarrow> 'b :: ccpo_topology)"
  shows "f \<midarrow>l\<rightarrow> f l"
proof (intro tendsto_ccpoI conjI)
  fix C :: "'a set" assume C: "chain C" "C \<noteq> {}"
  show "chain (f`C)"
    using mcont
    by (intro chain_imageI[where le_a="(\<le>)"] C) (simp add: mcont_def monotone_def)
  show "f (\<Squnion>C) = \<Squnion>(f ` C)"
    using mcont C by (simp add: mcont_def cont_def)
qed

subsection \<open>Instances for @{class ccpo_topology}s and continuity theorems\<close>

instantiation set :: (type) ccpo_topology
begin

definition open_set :: "'a set set \<Rightarrow> bool" where
  "open_set A \<longleftrightarrow> (\<forall>C. chain C \<longrightarrow> C \<noteq> {} \<longrightarrow> Sup C \<in> A \<longrightarrow> C \<inter> A \<noteq> {})"

instance
  by intro_classes (simp add: open_set_def)

end

instantiation enat :: ccpo_topology
begin

instance
proof
  fix A :: "enat set"
  show "open A = (\<forall>C. chain C \<longrightarrow> C \<noteq> {} \<longrightarrow> \<Squnion>C \<in> A \<longrightarrow> C \<inter> A \<noteq> {})"
  proof (intro iffI allI impI)
    fix C x assume "open A" "chain C" "C \<noteq> {}" "\<Squnion>C \<in> A"

    show "C \<inter> A \<noteq> {}"
    proof cases
      assume "\<Squnion>C = \<infinity>"
      with \<open>\<Squnion>C \<in> A\<close> \<open>open A\<close> obtain n where "{enat n <..} \<subseteq> A"
        unfolding open_enat_iff by auto
      with \<open>\<Squnion>C = \<infinity>\<close> Sup_eq_top_iff[of C] show ?thesis
        by (auto simp: top_enat_def)
    next
      assume "\<Squnion>C \<noteq> \<infinity>"
      then obtain n where "C \<subseteq> {.. enat n}"
        unfolding Sup_eq_top_iff top_enat_def[symmetric] by (auto simp: not_less top_enat_def)
      moreover have "finite {.. enat n}"
        by (auto intro: finite_enat_bounded)
      ultimately have "finite C"
        by (auto intro: finite_subset)
      from in_chain_finite[OF \<open>chain C\<close> \<open>finite C\<close> \<open>C \<noteq> {}\<close>] \<open>\<Squnion>C \<in> A\<close> show ?thesis
        by auto
    qed
  next
    assume C: "\<forall>C. chain C \<longrightarrow> C \<noteq> {} \<longrightarrow> \<Squnion>C \<in> A \<longrightarrow> C \<inter> A \<noteq> {}"
    show "open A"
      unfolding open_enat_iff
    proof safe
      assume "\<infinity> \<in> A"
      { fix C :: "enat set" assume "infinite C"
        then have "\<Squnion>C = \<infinity>"
          by (auto simp: Sup_enat_def)
        with \<open>infinite C\<close> C[THEN spec, of C] \<open>\<infinity> \<in> A\<close> have "C \<inter> A \<noteq> {}"
          by auto }
      note inf_C = this

      show "\<exists>x. {enat x<..} \<subseteq> A"
      proof (rule ccontr)
        assume "\<not> (\<exists>x. {enat x<..} \<subseteq> A)"
        with \<open>\<infinity> \<in> A\<close> have "\<And>x. \<exists>y>x. enat y \<notin> A"
          by (simp add: subset_eq Bex_def) (metis enat.exhaust enat_ord_simps(2))
        then have "infinite {n. enat n \<notin> A}"
          unfolding infinite_nat_iff_unbounded by auto
        then have "infinite (enat ` {n. enat n \<notin> A})"
          by (auto dest!: finite_imageD)
        from inf_C[OF this] show False
          by auto
      qed
    qed
  qed
qed

end

lemmas tendsto_inf2[THEN tendsto_compose, tendsto_intros] =
  tendsto_mcont[OF mcont_inf2]

lemma isCont_inf2[THEN isCont_o2[rotated]]:
  "isCont (\<lambda>x. x \<sqinter> y) (z :: _ :: {ccpo_topology, complete_distrib_lattice})"
  by(simp add: isCont_def tendsto_inf2 tendsto_ident_at)

lemmas tendsto_sup1[THEN tendsto_compose, tendsto_intros] =
  tendsto_mcont[OF mcont_sup1]

lemma isCont_If: "isCont f x \<Longrightarrow> isCont g x \<Longrightarrow> isCont (\<lambda>x. if Q then f x else g x) x"
  by (cases Q) auto

lemma isCont_enat_case: "isCont (f (epred n)) x \<Longrightarrow> isCont g x \<Longrightarrow> isCont (\<lambda>x. co.case_enat (g x) (\<lambda>n. f n x) n) x"
  by (cases n rule: enat_coexhaust) auto

end

subset_name: curated