text:
\section{Representable constructible polygraphs} \label{sec:represent}

The theory developed in this work may serve as a basis for an approach to weak higher categories, modelled by constructible polygraphs that satisfy a \emph{representability} condition, similar to opetopic higher categories. In the light of \cite{henry2018regular}, this could be a good framework to study semi-strictification in the sense of C.\ Simpson's conjecture, beyond the ``groupoidal'' case (all cells are weakly invertible). In \cite{hadzihasanovic2018weak}, we developed this programme in the special case of dimension 2, proving an equivalence between bicategories and representable merge-bicategories --- a truncation of representable constructible polygraphs to dimension 2 --- which we then exploited in a semi-strictification argument.

For the interested reader, we give some basic definitions here, without much comment. In what follows, we identify a constructible directed complex with its embedding into $\cpol$.

\begin{dfn}
Let $X$ be a constructible polygraph and $U$ a constructible $n$\nbd molecule. A \emph{constructible $n$\nbd diagram} $x$ of shape $U$ in $X$ is a map $x: U \to X$. If $U$ is an atom, we call $x$ an \emph{$n$\nbd cell} of $X$.
\end{dfn}

\begin{dfn}
For $n > 0$, a \emph{ternary} $(n+1)$\nbd atom is a constructible $(n+1)$\nbd atom with three $n$\nbd dimensional elements. Necessarily, one boundary of $W$ splits into two atoms $W_+$ and $W_-$, with $W_+ \cap W_- \subseteq \bord{}{+}W_+ \cap \bord{}{-}W_-$, and the other boundary is a single atom $W_0$.

A \emph{horn} of $W$ is any of the following subsets:
\begin{equation*}
	\Lambda^W_0 := W_+ \cup W_-, \quad \quad \Lambda^W_- := W_+ \cup W_0, \quad \quad \Lambda^W_+ := W_- \cup W_0.
\end{equation*}
The horn $\Lambda^W_\zeta \incl W$ is called a \emph{composition horn} if $\zeta = 0$, a \emph{left division horn} if $\zeta = -$, and a \emph{right division horn} if $\zeta = +$.
\end{dfn}

\begin{dfn}
Let $X$ be a constructible polygraph. A \emph{horn of $W$ in $X$} is a pair of a horn $\Lambda \incl W$ and a morphism $\lambda: \Lambda \to X$. A \emph{filler} for the horn is an $(n+1)$\nbd cell $h: W \to X$ such that
\begin{equation*}
\begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
	\node[scale=1.25] (0) at (0,1.5) {$\Lambda$};
	\node[scale=1.25] (1) at (2,1.5) {$X$};
	\node[scale=1.25] (2) at (0,0) {$W$};
	\draw[1c] (0) to node[auto] {$\lambda$} (1);
	\draw[1cinc] (0) to (2);
	\draw[1c] (2) to node[auto,swap] {$h$} (1);
\end{tikzpicture}
\end{equation*}
commutes.

Let $x: U \to X$ be an $n$\nbd cell of $X$ and $V \sqsubseteq \bord{}{\alpha}U$ a constructible submolecule of its boundary. A \emph{horn for $x$ at $V$} is a pair of an inclusion $\imath: U \incl W$ such that $\imath(U) = W_\alpha$ and $\imath(V) = W_+ \cap W_-$, and a horn $\lambda: \Lambda^W_\zeta \to X$ with $\zeta \in \{-\alpha,0\}$ such that
\begin{equation*}
\begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
	\node[scale=1.25] (0) at (0,1.5) {$U$};
	\node[scale=1.25] (1) at (2,0) {$X$};
	\node[scale=1.25] (2) at (0,0) {$\Lambda^W_\zeta$};
	\draw[1c] (0) to node[auto] {$x$} (1);
	\draw[1cinc] (0) to (2);
	\draw[1c] (2) to node[auto,swap] {$\lambda$} (1);
\end{tikzpicture}
\end{equation*}
commutes.
\end{dfn}

\begin{remark} \label{rmk:dual_horn}
If $W$ is a ternary $(n+1)$\nbd atom, then $\oppn{n+1}{W}$ is still ternary, and every horn of $W$ is isomorphic to a horn of $\oppn{n+1}{W}$. Thus, for each horn for $x: U \to X$ at $V$, given by $U \incl \Lambda \incl W$ and $\lambda: \Lambda \to X$, there is a \emph{dual} horn, where $\Lambda$ is seen as a horn of $\oppn{n+1}{W}$.
\end{remark}

\begin{dfn} \label{dfn:universality}
Let $X$ be a constructible polygraph. Coinductively, we define families $\divis{n}{X}$ of pairs $(x \in X(U), V \sqsubseteq \bord{}{\alpha}U)$ of an $n$\nbd cell of shape $U$ and a constructible submolecule of $\bord{}{\alpha}U$, to be read ``$x$ is \emph{universal} at $V \sqsubseteq \bord{}{\alpha}U$''. If $\divis{n}{P}$ is defined, we also define a family $\equi{n}{X}$ of $n$\nbd cells, the \emph{$n$-equivalences} of $X$, by
\begin{equation*}
	\equi{n}{X} := \{x \in X(U) \,|\, \dmn{U} = n, (x, \bord{}{+}U) \text{ and } (x, \bord{}{-}U) \in \divis{n}{X}\}.
\end{equation*}

We let $(x, V \sqsubseteq \bord{}{\alpha}U) \in \divis{n}{X}$ if and only if all horns $\Lambda \incl W, \lambda: \Lambda \to X$ for $x$ at $V$ have a filler $h: W \to X$ which
\begin{enumerate}
	\item is an $(n+1)$\nbd equivalence, and
	\item is universal at $W_{-\alpha}$.
\end{enumerate}
\end{dfn}

\begin{remark}
Similarly to simplicial and opetopic approaches to higher categories, the idea is that an equivalence in $X$ of shape $W$, a ternary atom, exhibits the image of $W_0$ as a weak composite of the images of $W_+$ and $W_-$.

The notion of a cell of shape $U$ universal at $\bord{}{\alpha}U$ is analogous to a universal cell in the opetopic or multitopic approach \cite{baez1998higher}. One novelty of our approach is to consider, at the same time, universality at a part of the boundary, which captures universal properties such as those of Kan extensions and of Kan lifts.

We think that our second universality requirement on horn fillers may subsume, with a purely existential statement, what was achieved through a universal quantification on higher-dimensional universal cells in the opetopic approach. This allows us to obtain a proper coinductive definition, and tackle genuinely infinite-dimensional higher categories, whereas the opetopic approach had to resort to a truncation to dimension $n$.
\end{remark}

\begin{exm}
A 1-cell $e$ of a constructible polygraph $X$, which is of shape $\vec{I}$, can only be universal at $\bord{}{+}\vec{I}$ or at $\bord{}{-}\vec{I}$. If it is universal at $\bord{}{+}\vec{I}$, the relevant horns are those of the forms
\begin{equation*}
\input{img/a_1horn.tex}
\end{equation*}
which must have fillers
\begin{equation} \label{eq:1horn}
\input{img/a_1hornfiller.tex}
\end{equation}
all of which are both equivalences, and universal at the constructible submolecule indicated by the grey arrow.

In merge-bicategories, seen as ``2-truncated'' constructible polygraphs, $e$ is what we called (tensor and par) left universal in \cite{hadzihasanovic2018weak}. Intuitively, existence of fillers for the first horn mean that compatible 1-cells can be \emph{factorised} as $e$ followed by another 1-cell, while the existence of fillers for the second horn mean that compatible 1-cells can be \emph{composed} with $e$ on the left. The fact that the 2-cells in (\ref{eq:1horn}) have pairwise the same universal properties implies that by factorising then composing, or composing then factorising, we will obtain the same result, up to equivalence.

Dually, if $e$ is universal at $\bord{}{-}\vec{I}$, the relevant horns are
\begin{equation*}
\input{img/a_1hornR.tex}
\end{equation*}
with fillers
\begin{equation} \label{eq:1hornR}
\input{img/a_1hornfillerR.tex}
\end{equation}
in a merge-bicategory, $e$ is what we called (tensor and par) right universal. Finally, $e$ is an equivalence if both the fillers (\ref{eq:1horn}) and (\ref{eq:1hornR}) exist.

A 2-cell $t$ of $X$ has shape $U^{(n,m)}$ for some $n, m >0$, with $\bord{}{-}U^{(n,m)} = \#_0^n \vec{I}$ and $\bord{}{+}U^{(n,m)} = \#_0^m \vec{I}$. The submolecules $W_{[i_1,i_2]}$ of $\bord{}{-}U^{(n,m)}$ can be identified with sub-intervals $[i_1,i_2]$ of $[1,n]$, and the submolecules $W_{[j_1,j_2]}$ of $\bord{}{+}U^{(n,m)}$ with sub-intervals $[j_1,j_2]$ of $[1,m]$; in a merge-bicategory, universality of $t$ at $W_{[i_1,i_2]}$ or at $W_{[j_1,j_2]}$ is what we called universality at $\bord{[i_1,i_2]}{-}$ or at $\bord{[j_1,j_2]}{+}$ in \cite{hadzihasanovic2018weak}.

To see how the universality of horn fillers can subsume a condition of uniqueness of factorisations, consider the first 2-cell in $(\ref{eq:1horn})$, which we relabel
\begin{equation} \label{eq:uniqueness1}
\input{img/a_uniqueness1.tex}
\end{equation}
If there is another 2-cell
\begin{equation*}
\input{img/a_uniqueness2.tex}
\end{equation*}
the two together form a horn in $X$, which by the universal property of $t$ has a 3-dimensional filler
\begin{equation} \label{eq:uniqueness3}
\input{img/a_uniqueness3.tex}
\end{equation}
which is an equivalence and universal at the indicated submolecule. If $X$ is ``1-truncated'', for example if it is the nerve of a 1-category $C$ --- so all cells of dimension 2 or higher correspond to identities in $C$ --- then the existence of the filler (\ref{eq:uniqueness1}) implies that $x$ factors in $C$ as the composite $e \cp{0} e\backslash x$, while the existence of the filler (\ref{eq:uniqueness3}) implies that, if $x$ factors as $e \cp{0} z$ for some other $z$, then $e\backslash x = z$. Since $x$ is arbitrary, this is a ``unique factorisation'' property exhibiting $e$ as an isomorphism in $C$.

In a similar fashion, if $X$ is ``2-truncated'', for example if it is the nerve of a 2-category, the universality of (\ref{eq:uniqueness3}) where indicated by the grey arrow implies uniqueness of the factorisation of $s$ through $t$. The existence of the horn filler (\ref{eq:uniqueness3}) is equivalent to $t$ exhibiting $e \backslash x$ as a left Kan extension of $x$ along $e$; by considering more general horns, we can see in fact that $e \backslash x$ is an \emph{absolute} Kan extension.

In general, universality of 2-cells at different submolecules of their boundary captures the universal properties of natural isomorphisms, absolute Kan extensions and absolute Kan lifts in 2-categories.
\end{exm}

\begin{dfn}
A map $f: X \to Y$ of constructible polygraphs is \emph{strong} if it sends $n$\nbd equivalences of $X$ to $n$\nbd equivalences of $Y$.
\end{dfn}

\begin{dfn} \label{dfn:represent}
A constructible polygraph $X$ is \emph{representable} if, for all constructible $n$\nbd diagrams $x$ of $X$, there exist an $n$\nbd cell $\underline{x}$ and an $(n+1)$\nbd equivalence $c(x)$ with $\bord{}{\alpha}c(x) = x$ and $\bord{}{-\alpha}c(x) = \underline{x}$.

Representable constructible polygraphs and strong maps form a category $\rcpol$.
\end{dfn}

\begin{remark}
If $x$ is of shape $U$, then $c(x)$ is of shape $U \celto \tilde{U}$ or $\tilde{U} \celto U$, as in Construction \ref{cons:ou}. As in the 2-dimensional case, it should suffice to require that $\underline{x}$ and $c(x)$ exist when $x$ is a cell, or when it is a ``binary'' $n$\nbd diagram containing exactly two $n$\nbd cells.
\end{remark}

There is a functor $-_\cat{O}: \cpol \to \globset$, obtained by restricting presheaves on $\globe$ to its full subcategory $\cat{O}$. We expect the following to be true, for an adequate algebraic definition of weak higher category with an underlying $\omega$\nbd graph (possibly, a variant of Batanin's \cite{batanin1998monoidal} or Leinster's \cite{leinster2004higher}).
\begin{conj}
Let $f: X \to Y$ be a strong map of representable constructible polygraphs. Then $X_\cat{O}$ and $Y_\cat{O}$ admit the structure of an algebraic weak higher category, and $f_\cat{O}: X_\cat{O} \to Y_\cat{O}$ of a functor of weak higher categories.
\end{conj}


TITLE: Question about the Gram-Schmidt process on orthonormal vectors
QUESTION [0 upvotes]: My textbook derives the Gram-Schmidt process in many steps, and the final step being:

Suppose we have already constructed orthogonal vectors $W = \{v_1, ...., v_k\}$, and these vectors can be made orthonormal by dividing them by their norm to result in: $W = \left\{\frac{v_1}{\|v_1\|}, ...., \frac{v_k}{\|v_k\|}\right\}$, W is orthonormal.

So... shouldn't the projection mapping of a vector $v$ on subspace $W$ then be the orthonormal vector representation as in:
$$P_W(v) = \sum_{i=1}^k\frac{1}{\|v_i\|}\langle v_i, v \rangle v_i$$
Should it not? But my textbook says that the projection mapping of some vector $v$ on the orthonormal set we have should therefore be:
$$P_W(v) = \sum_{i=1}^k\frac{\langle v_i, v\rangle }{\langle v_i, v_i\rangle}v_i$$
But aren't we missing the required coefficient to make the set orthonormal?

REPLY [1 votes]: So actually $P_W(v) = \sum_{i=1}^k\langle\frac{v_i}{||v_i||}, v\rangle\frac{v_i}{||v_i||}$ right? Because the elements of $W$ are $\frac{v_i}{||v_i||}$.
So $P_W(v) = \sum_{i=1}^k\frac{1}{||v_i||^2}\langle v_i, v\rangle v_i=\sum_{i=1}^k\frac{1}{\langle v_i, v_i\rangle}\langle v_i, v\rangle v_i$ as your textbook says.


TITLE: Calculus of variations, Euler Lagrange problem
QUESTION [2 upvotes]: I have just started doing calculus of variations and have run into a wall trying to solve one of the first problems in solving the Euler-Lagrange equations.
$$\int_{0.1}^1 y'(1 + x^2y') \,dx,   y(0.1) = 19, y(1) = 1$$
I have to find the Euler-Lagrange equation which satisfies the boundary conditions but every time I am not able to reach the actual solution.

REPLY [1 votes]: You do not have a $y$ term so E-L reduces to $\frac{d}{dx}\frac{\partial L}{\partial y'}=0$.
Giving $\frac{d}{dx} (1+2 x^2 y')=0$. This easily integrates to give $1+2 x^2 y'=const$.
This rearranges to give $y'= const/x^2$. Can you take it from there?


text:
\section{Problem Definition}
\label{Sec:2}
In this section, we briefly describe the problem setting. Consider a set of new flight pairs (i.e., consecutive flights, specifically a flight from the hub to a new demand point and its return flight to hub) to be accommodated into the existing flight schedule in near real time. An airline needs to accommodate new flights into an existing flight schedule without excessively disrupting the existing schedule. Therefore, we only shift the departure times of existing flights within the intervals already determined by the airline. Moreover, any arrival tardiness of the existing flights due to inserting new flights is penalized in the objective. In hub-and-spoke networks, connecting passengers represent
a non-negligible percentage of the total number of passengers. Therefore, we also respect the connection of passengers at the hub airport while optimizing the
departure and cruise times of flights. There are two cases for a new flight: (1) If the new flight is assigned to a larger aircraft, then this assignment may  capture more passengers, providing a greater revenue, but  compressing the cruise times of flights to accommodate new flights may increase the fuel cost significantly; (2) if the new flight is assigned to a smaller fuel efficient aircraft to reduce the fuel burn, then there may be an additional cost of spilled passengers with a decrease in revenue. To increase the profit for an airline, we introduce a second model which additionally includes aircraft swapping decisions.   \ignore{While a swap may reduce the fuel burn, if a flight is assigned to a smaller aircraft after the swap, then some of the passengers of the subsequent flights may be spilled due to the low capacity of the aircraft. In this study, we consider the interplay among all decision variables with the goal of maximizing the airline's profit.}

In the following sections, we first define the fuel burn as a function of cruise time and the penalty function for arrival tardiness of flights, and then provide a numerical example to show how to utilize the cruise time controllability and aircraft swapping to \ignore{accommodate} open-up enough space for the block times of new flights.

\subsection{Fuel and CO$_2$ emission cost function} One of the main contributions of this study is to increase the aircraft cruise speed to compensate for the time required to operate new flights. However, we need to consider the adverse effect of increasing cruise speed on fuel and carbon emissions costs.  To estimate the fuel burn, we use the cruise stage fuel flow model developed by the Base of Aircraft Data (BADA) project of EUROCONTROL \cite{europaram}. This model has been widely used in the literature.  The fuel burn (kg) of a flight as a function of its cruise time  $f$ (minutes) and its aircraft type $t$ can be calculated as follows:

\begin{equation}
F^t\left(f\right)= \alpha^{t}  \frac{1}{f}  + \beta^{t}  \frac{1}{{f}^2}  + \gamma^t  {f}^3 + \nu^t  {f}^2. \label{eq:cost} \nonumber
\end{equation}
The coefficients \noindent $\alpha^{t}, \beta^{t},  \gamma^t,  \nu^t  > 0$

are expressed in terms of aircraft specific fuel consumption coefficients as
well as the mass of the aircraft, air density and gravitational
acceleration as provided in  \c{S}afak et al.\cite{Safak}. It is important to note that $F^t$ is a convex function whenever $f>0$. The minimizer of the fuel consumption function $F^t$ is represented by $u^t$, which is the ideal cruise time when an aircraft flies at the most fuel-efficient speed, referred to Maximum Range Cruise (MRC) speed. \ignore{In other words, MRC speed is the most fuel efficient speed of an aircraft.} Although the fuel burn is minimized at MRC speed, airlines may set higher cruise speed due to the scheduling constraints.

EUROCONTROL \cite{euro} states that each kg of fuel burn approximately produces 3.15 kg of CO$_2$ emission. Therefore, we can express fuel and CO$_2$ emission costs as a function of cruise time as follows:
\begin{equation}
c^t(f) = c_o F^t\left(f\right)

\end{equation}
\noindent where $c_o$ is the total cost of of fuel and CO$_2$ emitted by an aircraft per kg of fuel burned.

\subsection{Penalty function for arrival tardiness} In this paper, we aim to accommodate new flights with minimal
disruptions on the existing flight schedule. In this quest, we penalize the deviation from the original arrival times for the existing flights. Hoffman and Ball \cite{hoffman} suggest to use a nonlinear delay cost function to better reflect the reality since flight delay costs tend to grow with time at a greater rate than linear rate. Moreover, EUROCONTROL \cite{delay} performs a detailed investigation of airline cost functions and reports that the power curve provides a good fit to passenger costs as a function of delay duration. Therefore, we penalize the arrival tardiness with a convex increasing function of tardiness $b$ as

\begin{equation}
P(b) = \rho b^{\zeta} \label{eq:penaltyfunction}
\end{equation}

\noindent where $\rho \geq 0$. As in Akt{\"u}rk et al. (5), we let $\zeta = 1.5$ in our computational experiments.  \REV{{It is important to note that the US costs might differ from the ones provided in the EUROCONTROL \cite{delay}.}}

Arrival tardiness can be reduced by compressing the cruise time of flights with an additional fuel burn cost. Therefore, the assignment of fuel efficient aircraft to new flights becomes more critical in order to reduce both delay and fuel burn costs.

\subsection{Numerical example} In this section, first we will provide a numerical example to show how the cruise time controllability can be utilized to accommodate new flights into an existing airline  flight schedule. Then, we will extend the example to show how aircraft swapping together with the cruise speed control can be used to achieve a more profitable schedule.

We give a sample schedule for two aircraft in Table \ref{schedule1}. Tail numbers of the aircraft and the flight numbers along with the origin and destination airports, planned departure times in local ORD time, planned block times and demand for the flights are listed in the table. Each aircraft visits ORD at least once in a day.  Let us now introduce new round-trip flights from ORD to MSP and back.

\begin{table}[htbp]\scriptsize
  \centering
    \caption{Original schedule.}
  \label{schedule1}
    \begin{tabular}{l|cccccccc}
    \hline
    \hline
    & Tail \# & Flight \# & \multicolumn{1}{c}{From}  & \multicolumn{1}{c}{To}    & \multicolumn{1}{c}{Plan. Dep.} & \multicolumn{1}{c}{Plan. Dur.}  & \multicolumn{1}{c}{Plan. Arr.} &  \multicolumn{1}{c}{Demand} \\
    \hline
    &\multirow{4}*{N53442}		&1586	&ORD	&MCO	&08:00	&03:04 &11:04	&200 \\
Existing   & 			&633		&MCO	&ORD	&12:00	&03:07	&15:07	&180\\
flights   &			&451		&ORD	&IAH		&18:10	&03:08	&21:18	&190\\
  &			&584		&IAH		&ORD	&22:30	&02:50	&01:20	&186\\ \hline
  & \multirow{4}*{N45425}		&527		&ORD	&IAH		&08:45	&03:02	&11:47	&151\\
Existing   & 			&521		&IAH		&ORD	&12:32	&03:03	&15:35	&154\\
flights &			&623		&ORD	&MCO	&17:00	&03:02	&20:02	&160\\
&			&679		&MCO	&ORD	&21:10	&03:10	&00:20	&163\\
    \hline
  New &  		&1842	&ORD	&MSP	&13:15	&01:40	&14:55	&183\\
    flights &			&430		&MSP	&ORD	&16:10	&01:45	&17:55	&168\\
    \hline
    \hline
    \end{tabular}
\end{table}

Figure \ref{fig:origschedule} gives the time-space network representation of the original schedule. The red and blue arcs in Figure \ref{fig:origschedule} represent routes for aircraft N53442 and N45425 respectively. The flight arcs originate from the departure airport at the planned departure time and end at the destination airport after the planned block time. Ground arcs represent the aircraft turnaround times needed to prepare the aircraft for the next flight.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{OriginalScheduleModified}
\caption{Original time-space network.} \label{fig:origschedule}
\end{figure}

In this example, we assume that aircraft N53442 is a Boeing 767-300 and N45425 is an Airbus 320-212. The number of seats of the aircraft are 218 and 180, respectively. In the original schedule we assume that the aircraft fly at the most fuel efficient speed (MRC speed) and estimate the fuel burn rates as 87 kg/min and 40 kg/min for aircraft N53442 and N45425, respectively. The fuel burn rate is calculated using the fuel flow model of BADA as mentioned above. We assume that for each flight, non-cruise stages take 30 minutes. Then, cruise stages take 30 minutes less than flight block times given in Table \ref{schedule1}. Assuming $c_{fuel}$ = 1.2 \$/kg and $c_{CO_2}$ = 0.2 \$/kg, the total cruise stage fuel and carbon cost for the original schedule is \$100,593.

Let us assume that an airline wants to operate two new flights 1842 and 430. In order to open up sufficient time to accommodate these new flights, one approach is to compress the cruise times and adjust the departure times simultaneously. We refer to this approach with cruise time controllability as CTC. If the airline wants to meet the passenger demand of 183 for the new flight 1842, the only way to do so is to assign the new flights to aircraft N53442. These new flights can be placed between flights 633 and 451. The necessary block times for the new flights can be made available by left-shifting flights 1586 and 633 and right-shifting flights 451 and 584. However, we assume that the airline wishes to keep such a new schedule close to the original one. Thus, we only allow departure times to deviate at most 90 minutes from the planned departure times in the original schedule. In addition, if a flight arrives 15 minutes later than the original planned arrival times, we penalize the tardiness with a nonlinear function in \eqref{eq:penaltyfunction}. We let $\rho=5$ and $\zeta=1.5$ in the penalty function. Because of these scheduling limitations,
\ignore{operate business trips in the morning. Thus, we only allow departure times of morning flights to deviate 30 minutes from the planned departure times in the original schedule. In addition, we assume that airline wants to depart all flights before 11:00 pm in local times of departure airports. Therefore, }the new fights cannot be accommodated by only shifting the flight departure times. We also need to compress the cruise times of the existing flights 633, 451 and 584 by 17, 17 and 3 minutes, respectively. Besides, the cruise times of new flights 1842 and 430 are compressed by 8 minutes to satisfy the scheduling restrictions. However, flight 451 arrives 37 minutes later then the original planned arrival time, thus 22 minutes of this tardiness are penalized by \$552. In Figure \ref{fig:schedule1}, we give the time-space network representation of resulting schedule. In this figure, the dotted arcs represent the flights with original block times, whereas the line arcs represent the flights with compressed cruise times.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{NetworkCRS}
\caption{Time-space network with CTC.} \label{fig:schedule1}
\end{figure}

Compressing the cruise times of the flights incurs additional costs of fuel burn and CO$_2$ emission. The total fuel burn and CO$_2$ emission costs of the
existing flights increases to \$101,286. Moreover, for new  flights, the total
fuel burn and CO$_2$ emission cost is \$16,044. Therefore, the fuel and emission cost increment compared to the original schedule is \$16,737 calculated as \$16,737 = \$101,286 + \$16,044 - \$100,593. To reduce the fuel burn by reassigning the aircraft, we also propose an aircraft swapping mechanism together with the cruise time controllability, referred to as CTC-AS.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{NetworkCRSAS}
\caption{Time-space network with CTC-AS.} \label{fig:schedule2}
\end{figure}

In the CTC-AS approach, we swap the aircraft of flights 451 and 623 before departure. In the new schedule, aircraft N53442 operates flights 1586, 633, 623, and 679, whereas aircraft N45425 operates 527, 521, 451, and 584. We provide the time-space network representation of the resulting schedule in Figure \ref{fig:schedule2}. To reduce the fuel expenses, the new flights are assigned to the fuel efficient aircraft N45425. However 10, 6 and 3 passengers of flights 451, 584 and 1842, respectively are spilled due to the low capacity of the aircraft N45425, thus resulting in a cost of \$2,007. \ignore{However, 10 passengers of flight 451, 6 p  and 3 passengers of the new flight 1842 are spilled due to insufficient seat capacity of the aircraft N45425, thus resulting in \$3,514. }

Similarly, passengers' revenue obtained from ticket sales of new flights are reduced to \$49,923 from \$49,548, since 3 passengers of new flights are not served due to low seat capacity. An additional \$500 cost of swapping is incurred in the new schedule. On the other hand, the savings from the fuel burn and CO$_2$ emission costs may compensate for these additional cost of spilled passengers, revenue loss and cost of swapping. Indeed, the
fuel expenses and CO$_2$ cost of the new
flights are reduced to \$8,118, almost
half of the fuel expenses of the CTC approach. Even though the total fuel burn and CO$_2$
emission cost of existing flights slightly increases to \$101,590, the fuel and emission cost increment significantly decreases from \$16,737 to \$9,115. Penalization cost for arrival tardiness of flight 451 also decreases to \$115. Therefore, resulting schedule improves the airline's profit from \$30,234 to \$35,412.

We give the operational cost components and revenues of two schedules achieved by CTC and CTC-AS approaches in Table \ref{eq:CostPublished}. Airline's profit is calculated as follows:
\begin{eqnarray}
\text{Profit } & = & \text{Revenue} - \left(\text{Fuel \& Emiss. Cost Increment}\right) - \text{Spilled Cost}  \nonumber \\
& & -  \text{Penalty Cost} - \text{Swap Cost} -  \left(\text{Crew \& Service Costs}\right). \nonumber
\end{eqnarray}

\begin{table}[ht]
		\setlength{\tabcolsep}{1pt}
  \centering
  \caption{Cost comparison.}
  \label{eq:CostPublished}
  \scalebox{0.9}{
\begin{tabular}{c r r r  r r   }
\hline
\hline
\multicolumn{1}{r}{} &\multicolumn{1}{r}{Fuel \& Emiss.}  &\multicolumn{1}{r}{Spilled}& \multicolumn{1}{r}{Deviation}& \multicolumn{1}{r}{Swap} \\
\multicolumn{1}{r}{} &\multicolumn{1}{r}{Cost Increment ($\$$)}&\multicolumn{1}{r}{ Cost ($\$$)} &\multicolumn{1}{r}{ Penalty ($\$$)} & \multicolumn{1}{r}{Cost ($\$$)} \\
\hline
 \multicolumn{1}{l}{CTC} &16,044           &0	&552.0  &0			\\

\multicolumn{1}{l}{CTC-AS} &8,118   &2,007	&115.0    &500		\\
 \hline  \hline
& \multicolumn{1}{r}{Crew \& Service}  &  \multicolumn{1}{r}{Passengers}&\multicolumn{1}{r}{} \\
& \multicolumn{1}{r}{Cost ($\$$)}   & \multicolumn{1}{r}{Revenue ($\$$)}&\multicolumn{1}{r}{Profit ($\$$)}\\
  \hline
\multicolumn{1}{l}{CTC}&4,400  &49,923  &30,234\\

\multicolumn{1}{l}{CTC-AS} &4,400	 &49,548   &35,412\\
  \hline  \hline
    \end{tabular}
}
\end{table}

config: arxiv


TITLE: How to exactly compute $dh_p(w)$?
QUESTION [0 upvotes]: Let $v\in R^3$ be a unit vector and let $h:S\to R$, $\text{ }h(p)=v\cdot p$, $\text{ }p\in S$, be the height function, where the dot denotes the usual inner product in $R^3$. $\text{ }h(p)$ is the height of $p\in S$ relative to a plane normal to $v$ and passing through the origin of $R^3$. To compute $dh_p(w),$ $\text{ }w\in T_p(S)$, choose a differentiable curve $\alpha:(-\epsilon,\epsilon)\to S$ with $\alpha(0)=p$, $\text{ }\alpha'(0)=w$.
This is the part that I don't quite understand:
Since $h(\alpha(t))=\alpha(t)\cdot v$, we obtain
$$dh_p(w)=\frac{d}{dt}h(\alpha(t))\lvert_{t=0}$$
How do we explicitly compute $dh_p(w)$ and end up getting $\frac{d}{dt}h(\alpha(t))\lvert_{t=0}$?
That is how can we go from $dh_p(w)$ to $\frac{d}{dt}h(\alpha(t))\lvert_{t=0}$?
I understand the basic ideas of the meaning of differentials but not exactly sure how to carry out the computation.
Some explanations are really appreciated. Thanks.

REPLY [1 votes]: Assuming $S$ is the unit sphere (and using the chain rule):
$\frac{d}{dt}h(\alpha(t))\lvert_{t=0}=dh_{\alpha(0)}(\dot \alpha(0))=dh_p(w)$
We can take $\alpha(t)=p+tw$, so $dh_p(w)=\frac{d}{dt}(v\cdot \alpha(t))=v \cdot\alpha'(0)=v \cdot w$


TITLE: Time-reversal symmetry for spin Hamiltonian
QUESTION [2 upvotes]: In the topology online course by TU Delft, the time-reversal operator acting on a system of spin-1/2 particles is introduced as
$$ \mathcal T = i\sigma_y\mathcal K. $$
I understand this acts on the matrix $H$ in
$$ \mathcal H = \vec c^\dagger H\vec c, $$
and $\sigma_y$ acts on the spin label.
My question is, does it have any significance that $i\sigma_y$ is chosen? Why not $\sigma_x$ or $\sigma_y$ only? Does it matter?

REPLY [4 votes]: In general the representation of the time-reversal operator depends on the system that you consider. Let us start with some basic remarks on time-reversal symmetry prior to the spinor question that you are asking.
Time-reversal means to go backwards in time, $t \rightarrow -t$.
When we want to know the time-evolution of a quantum-mechanical state, we need to look at the time-dependent Schrödinger equation.
\begin{equation}
- \frac{\hbar}{i} \frac{\partial \psi}{\partial t} = \mathcal{H} \psi
\end{equation}
The solutions of this equation are known and can be represented by
\begin{equation}
\psi(\vec{r}, t) = e^{-\frac{i}{\hbar} \mathcal{H}t} \psi(\vec{r}, 0) \quad.
\end{equation}
Using the time reversal operator $\mathcal{T}$ on the wave function above, results in
\begin{equation}
\mathcal{T}  \psi(\vec{r}, t) = \psi(\vec{r}, -t) = \psi^*(\vec{r}, t) \quad .
\end{equation}
From the action of the time reversal operator on the wave function we see, that it leads to a complex conjugation of the wave function. Thus, we can simply construct it in the case of spin-less wave functions as $\mathcal{T} = K$, where $K$ is the operator for complex conjugation. More generally we can write $$\mathcal{T} = UK \tag{1} \quad ,$$ where $U$ is a unitary operator. The determination of $U$ in the case of spin-$\frac{1}{2}$ particles, leaves us with the explicit representation for the time-reversal operator.
There are some basic properties, that every time-reversal operator needs to possess.

energy conservation: $[\mathcal{T}, \mathcal{H}] = 0$
anti-linearity: $\mathcal{T} i = -i$, where $i$ is the imaginary unit (in fact: one can show that $\mathcal{T}$ is antiunitary $\mathcal{T} \mathcal{T}^{\dagger} = - 1$)

So let us now consider the case of wave functions incorporation the spin degree of freedom.
For a spin-$\frac{1}{2}$ particle we need to fulfill the relation
$$\mathcal{T} \, \vec{\sigma} \, \mathcal{T}^{-1} = -\vec{\sigma} \tag{2} \quad ,$$
where $\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)^{T}$ is the Pauli-vector. This relation basically coincides with flipping the spin due to time-reversal ($\mathcal{T} \vec{S} = -\vec{S} \sim \mathcal{T} \vec{\sigma}$).
If we choose $\sigma_z$ to be diagonal, the matrices take the form:
$$\sigma_x = \begin{pmatrix} 0 & 1 \\
1 & 0 \end{pmatrix},\, \sigma_y = \begin{pmatrix} 0 & -i \\
i & 0 \end{pmatrix},\, \sigma_z = \begin{pmatrix} 1 & 0 \\
0 & -1 \end{pmatrix}\quad .$$
If we now use the form of the time-reversal that we found for the spin-less case we find
$$K \sigma_x K^{-1} = \sigma_x,\quad K \sigma_y K^{-1} = -\sigma_y,\quad K \sigma_z K^{-1} = \sigma_z \tag{3} \quad .$$
From $(1)$, $(2)$ and $(3)$ we can simply show that $U = c \sigma_y$. As we require unitarity for $U \Rightarrow UU^{\dagger} = 1$ $\Rightarrow$ $|c|^2 = 1$ and we can choose $c = 1$.
This leaves us with the desired result.
\begin{equation}
\boxed{\mathcal{T} = \sigma_y K}
\end{equation}
PS: that your time-reversal operator contains an additional $i$ could be because of the chosen representation of the Pauli matrices but either way your operator fulfills the criteria to serve as time-reversal operator, as $|i|^2 = 1$.
To develop a deeper understanding, I encourage you to read the relevant chapters in Dresselhaus' Group Theory - Application to the Physics of Condensed Matter.


TITLE: Which side of liquid gas interface is at high pressure?
QUESTION [1 upvotes]: With respect to what (whether liquid or gas) should I consider that the convex side has higher pressure and concave at lower pressure? As seen in the example of image, it is given that air bubble inside water has higher pressure which implies that concave side is at higher pressure but in text,  it is quite opposite. What I am thinking wrong?

REPLY [1 votes]: the pressure inside the bubble tends to increase surface area, the pressure outside tends to compress the bubble.
surface tension acts against expansion.
I.e. surface tension is compressive.
hence pressure inside has to be higher than outside.
the problem arises from defining "inside" and "outside".
for consistency "inside" a  surface is defined where the normals to the surface converge and "outside" is where they diverge.


text:
\begin{document}

\title{\textsc{{On the representation of multi-ideals by
tensor norms}}}
\author{Geraldo Botelho\thanks{Supported by CNPq Grant 306981/2008-4.}~, Erhan \c Caliskan and Daniel Pellegrino\thanks{Supported by INCT-Matem\'{a}tica, CNPq Grants 620108/2008-8 (Ed. Casadinho), 471686/2008-5 (Ed. Universal), 308084/2006-3 and PROCAD-NF Capes.\newline
2000 Mathematics Subject Classification: 46G25, 46B28.}}
\date{}
\maketitle
\begin{abstract} A tensor norm $\beta= (\beta_{n})_{n=1}^{\infty}$ is smooth if the natural correspondence
$(E_{1} \otimes\cdots\otimes E_{n} \otimes\mathbb{K},\beta_{n+1}) \longleftrightarrow
(E_{1} \otimes\cdots\otimes E_{n} ,\beta_{n})$ is always an isometric isomorphism. In this paper we study the representation of multi-ideals and of ideals of multilinear forms by smooth tensor norms.
\end{abstract}

\section{Introduction and notation}
The idea of describing the dual of a topological tensor product by means of a special class of bilinear mappings goes back to Grothendieck's celebrated R\'esum\'e \cite{resume}. For example, in his seminal work Grothendieck showed that to linear functionals on the injective tensor product correspond integral bilinear forms. With the emergence of the theory of ideals of multilinear mappings (multi-ideals) between Banach spaces, several instances of this kind of correspondence have appeared. For example, Matos in \cite{complutense} constructs a tensor norm such that to linear operators on the tensor product which are continuous with respect to this norm correspond exactly the class of nuclear multilinear mappings.\\
\indent In the theory of multi-ideals, the possibility of moving smoothly from spaces of $(n+1)$-linear mappings down to spaces of $n$-linear mappings turned out to be an important property (see, e.g., \cite{indagationes, studia, muro}). In this note we study
tensor norms in which this transition is smooth, as well as multi-ideals that correspond to such smooth tensor norms. To be more precise we need some definitions:

\begin{definition}\rm An {\it $n$-tensor norm} $\beta_{n}$ assigns to every $n$-tuple of normed spaces
$E_{1},\ldots,E_{n}$ a reasonable crossnorm $\beta_{n}(\cdot)$ on the full
$n$-fold tensor product $E_{1}\otimes\cdots\otimes E_{n}$ which satisfies the
metric mapping property. The resulting normed space is denoted by
$(E_{1}\otimes\cdots\otimes E_{n},\beta_{n})$. A {\it tensor norm} is a sequence
$\beta=(\beta_{n})_{n=1}^{\infty}$ where each $\beta_{n}$ is an $n$-tensor
norm.
\end{definition}

Let $E,E_1, \ldots, E_n,F$ be (real or complex) Banach spaces. By ${\cal L}(E;F)$ we denote the space of bounded linear operators from $E$ to $F$ endowed with the usual operator norm. When $F$ is the scalar field we simply write $E'$. By ${\cal L}(E_1, \ldots, E_n;F)$ we mean the space of continuous $n$-linear mappings from $E_1 \times \cdots \times E_n$ to $F$ endowed with the usual sup norm. When $E_1 = \cdots = E_n = E$ we write ${\cal L}(^nE;F)$.

\begin{definition}\rm \label{Definition 1.3} \rm An {\it ideal of multilinear mappings} (or {\it multi-ideal}) $\cal M$ is a
subclass of the class of all continuous
 multilinear mappings between Banach spaces such that for a positive integer $n$, Banach spaces $E_1, \ldots, E_n$ and $F$, the components ${\cal M}
 (E_1, \ldots, E_n;F) := {\cal L}(E_1, \ldots, E_n;F) \cap {\cal M}$ satisfy:
 \\
\noindent (i) ${\cal M}(E_1, \ldots, E_n;F)$ is a linear subspace of
${\cal L}(E_1, \ldots, E_n;F)$ which contains the $n$-linear mappings of finite type.\\
(ii) The ideal property: if $A \in {\cal M}(E_1, \ldots, E_n;F)$,
$u_j \in {\cal L}(G_j;E_j)$ for $j = 1, \ldots, n$ and $t \in
{\cal L}(F;H)$, then $t \circ A \circ (u_1, \ldots, u_n)$ is in
${\cal M}(G_1, \ldots, G_n;H)$.

\medskip

\noindent Moreover, there is a function $\|\cdot\|_{\cal M} \colon {\cal
M} \longrightarrow \mathbb{R}^+$ satisfying

\medskip

\noindent (i') $\|\cdot\|_{\cal M}$ restricted to ${\cal M}(E_1, \ldots,
E_n;F)$ is a norm for all
 Banach spaces $E_1, \ldots, E_n$ and $F$, which makes ${\cal M}(E_1, \ldots,
E_n;F)$ a Banach space. \\
\noindent (ii') $\|A \colon \mathbb{K}^n \longrightarrow \mathbb{K} : A(\lambda_1,\ldots,
 \lambda_n)
 = \lambda_1 \cdots \lambda_n \|_{\cal M} = 1$ for all $n$,\\
\noindent (iii') If $A \in {\cal M}(E_1, \ldots, E_n;F)$, $u_j \in
{\cal L}(G_j;E_j)$ for $j = 1, \ldots, n$ and $t \in {\cal
L}(F;H)$, then $\|t \circ A \circ (u_1, \ldots, u_n)\|_{\cal M}
\leq \|t\| \|A\|_{\cal M} \|u_1\|\cdots \|u_n\|$.

Of course the Banach spaces considered in this definition are all over the same fixed scalar field $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. We
define
\[
\mathcal{M}^{\mathbb{K}}:= \{\mathcal{M}(E_{1}, \ldots, E_{n}; \mathbb{K}): n
\in\mathbb{N} {\rm~and~} E_{1}, \ldots, E_{n} {\rm~are ~Banach~spaces}\},
\]
and say that $\mathcal{M}^{\mathbb{K}}$ is an {\it ideal of multilinear forms}.
\end{definition}

\begin{definition}\rm
\label{definition}We say that a tensor norm $\beta= (\beta_{n}
)_{n=1}^{\infty}$ {\it represents} the multi-ideal $\mathcal{M}$ - or
$\mathcal{M}$ is $\beta$-represented - if $\mathcal{M}(E_{1}, \ldots,
E_{n};F^{\prime})$ is {isometrically} isomorphic to $(E_{1}
\otimes\cdots\otimes E_{n}\otimes F, \beta_{n+1})^{\prime}$ through
the canonical mapping
\[
\varphi\colon\mathcal{M}(E_{1}, \ldots, E_{n};F^{\prime}) \longrightarrow
(E_{1} \otimes\cdots\otimes E_{n}\otimes F, \beta_{n+1})^{\prime}
\]
\[
T \mapsto\varphi(T)(x_{1} \otimes\cdots\otimes x_{n} \otimes y) = T(x_{1},
\ldots, x_{n})(y),
\]
for every $n$ and every Banach spaces $E_{1}, \ldots, E_{n}, F$. \\
\indent The ideal of multilinear forms $\mathcal{M}^{\mathbb{K}}$ is {\it represented} by $\beta$ - or $\mathcal{M}^{\mathbb{K}
}$ is $\beta$-represented - if the condition above holds for all components $\mathcal{M}(E_{1},
\ldots, E_{n}; \mathbb{K})$ of $\mathcal{M}^{\mathbb{K}}$.
\end{definition}

\indent In \cite[Theorem 4.5]{hf} it is proved that a multi-ideal
$\mathcal{M}$ is maximal if and only if $\mathcal{M}$ is represented by some
(finitely generated) tensor norm. It is well known (see, e.g., \cite[Exercise
12.1]{defantfloret}) that for every normed space $E$ and every 2-tensor norm
$\alpha$, $(E \otimes\mathbb{K}, \alpha)$ is isometrically isomorphic to $E$
via the correspondence $x \otimes\lambda\longleftrightarrow\lambda x$. Given a
tensor norm $\beta= (\beta_{n})_{n=1}^{\infty}$, this property can be
rewritten as $(E \otimes\mathbb{K}, \beta_{2}) = (E, \beta_{1})$ for every
$E$. As we will see along the paper, this property is no longer valid for larger $n$, that is, it is not always true that
$(E_1\otimes\cdots \otimes E_n, \beta_{n})$ is canonically isomorphic to
$(E_1\otimes\cdots \otimes E_n \otimes\mathbb{K}, \beta_{n+1}) $ for
every $n\geq2$. This phenomenon motivates the following definition:

\begin{definition}\rm
\label{def}\textrm{A tensor norm $\beta= (\beta_{n})_{n=1}^{\infty}$ is said
to be \textit{smooth} if, regardless of the natural $n$ and the normed spaces
$E_{1}, \ldots, E_{n}$, the natural map
\[
\psi\colon(E_{1} \otimes\cdots\otimes E_{n} \otimes\mathbb{K},\beta_{n+1})
\longrightarrow(E_{1} \otimes\cdots\otimes E_{n} ,\beta_{n}),
\]
\[
\psi(x_{1} \otimes\cdots\otimes x_{n} \otimes\lambda) =\lambda(x_{1}
\otimes\cdots\otimes x_{n} )
\]
is an isometric isomorphism. }
\end{definition}

In this paper we are concerned with the representation of multi-ideals by smooth tensor norms. The conclusion of our results/examples is that: (i) multi-ideals are rarely represented by smooth tensor norms, (ii) ideals of multilinear forms are more often represented by smooth tensor norms, (iii) the representation of an ideal of multilinear forms by a smooth tensor norm yields the representation of some of its vector-valued components by the same smooth tensor norm.

\section{Vector-valued case}
The aim of this section is to show that multi-ideals, including their vector-valued components, are rarely represented by smooth tensor norms. We start with the two first obvious examples.

\begin{examples}\rm
\label{example}(a) The projective tensor norm $\pi$ is smooth: given
$z = \sum_{j=1}^{m} x_{j}^{(1)}\otimes\cdots\otimes x_{j}^{(n)}\otimes
\lambda_{j} \in E_{1}\otimes\cdots\otimes E_{n}\otimes\mathbb{K}$, letting
$\bar z = \sum_{j=1}^{m} \lambda_{j}x_{j}^{(1)}\otimes\cdots\otimes
x_{j}^{(n)}\in E_{1}\otimes\cdots\otimes E_{n} $, it is easy to check that
$\pi_{n+1}(z) = \pi_{n}(\bar z)$. So $\psi\colon(E_{1} \otimes\cdots\otimes
E_{n} \otimes\mathbb{K},\pi_{n+1}) \longrightarrow(E_{1} \otimes\cdots\otimes
E_{n} ,\pi_{n})$ is an isometric isomorphism. So the multi-ideal $\mathcal{L}$
of all continuous multilinear mappings between Banach spaces, which is
obviously $\pi$-represented (see \cite[Proposition A.3.7]{dales}), is
represented by a smooth tensor norm. \\

\noindent (b) The injective tensor norm $\varepsilon$ is smooth.
Indeed, for $z$ and $\bar{z}$ as above,
\begin{align*}
\varepsilon_{n+1}(z)  & =\sup_{\varphi_{l}\in B_{E_{l}^{\prime}},\varphi\in
B_{\mathbb{K}^{\prime}}}\left\vert \sum_{j=1}^{m}\varphi_{1}(x_{j}
^{(1)})\cdots\varphi_{n}(x_{j}^{(n)})\varphi(\lambda_{j})\right\vert \\
& =\sup_{\varphi_{l}\in B_{E_{l}^{\prime}},\varphi\in B_{\mathbb{K}^{\prime}}
}\left\vert \sum_{j=1}^{m}\varphi_{1}(\lambda_{j}x_{j}^{(1)})\cdots\varphi
_{n}(x_{j}^{(n)})\varphi(1)\right\vert \\
& =\sup_{\varphi_{l}\in B_{E_{l}^{\prime}},\varphi\in B_{\mathbb{K}^{\prime}}
}|\varphi(1)|\left\vert \sum_{j=1}^{m}\varphi_{1}(\lambda_{j}x_{j}
^{(1)})\cdots\varphi_{n}(x_{j}^{(n)})\right\vert \\
& =\sup_{\varphi_{l}\in B_{E_{l}^{\prime}}}\left\vert \sum_{j=1}^{m}
\varphi_{1}(\lambda_{j}x_{j}^{(1)})\cdots\varphi_{n}(x_{j}^{(n)})\right\vert
=\varepsilon_{n}(\widetilde{z}),
\end{align*}
proving that $\psi\colon(E_{1}\otimes\cdots\otimes E_{n}\otimes\mathbb{K}
,\varepsilon_{n+1})\longrightarrow(E_{1}\otimes\cdots\otimes E_{n}
,\varepsilon_{n})$ is an isometric isomorphism. It is known since Grothendieck's R\'esum\'e that the
multi-ideal $\mathcal{L}_{\mathcal{I}}$ of integral multilinear mappings is $\varepsilon$-represented (the scalar-valued bilinear case can be found in \cite[Theorem 1.1.21]{diestelresume}). We give the details for the sake of completeness. Denoting by
$\mathcal{I}(E;F)$ the space of integral linear operators from $E$ to $F$ and
by $E_{1}\widehat{\otimes}_{\varepsilon}\cdots\widehat{\otimes}_{\varepsilon
}E_{n}$ the completion of $(E_{1}\otimes\cdots\otimes E_{n},\varepsilon_{n})$
we have that
\[
\mathcal{L}_{\mathcal{I}}(E_{1},\ldots,E_{n};F^{\prime})\overset{(1)}
{=}\mathcal{I}(E_{1}\widehat{\otimes}_{\varepsilon}\cdots\widehat{\otimes
}_{\varepsilon}E_{n};F^{\prime})\overset{(2)}{=}\mathcal{L}_{\mathcal{I}
}(E_{1}\widehat{\otimes}_{\varepsilon}\cdots\widehat{\otimes}_{\varepsilon
}E_{n},F;\mathbb{K})
\]
\[
\overset{(1)}{=}\mathcal{I}(E_{1}\widehat{\otimes}_{\varepsilon}\cdots
\widehat{\otimes}_{\varepsilon}E_{n}\widehat{\otimes}_{\varepsilon
}F;\mathbb{K})\overset{(3)}{=}(E_{1}\widehat{\otimes}_{\varepsilon}
\cdots\widehat{\otimes}_{\varepsilon}E_{n}\widehat{\otimes}_{\varepsilon
}F)^{\prime}\overset{(4)}{=}(E_{1}\otimes\cdots\otimes E_{n}\otimes
F,\varepsilon_{n+1})^{\prime}
\]
(1) See \cite[Proposition 2.2]{cdg}.\newline(2) See \cite[Proposition 3.22]{Ryan}.\newline(3) This is trivial
because $\mathcal{I}$ is an operator ideal.\newline(4) Since $E_{1}
\widehat{\otimes}_{\varepsilon}\cdots\widehat{\otimes}_{\varepsilon}
E_{n}\widehat{\otimes}_{\varepsilon}F$ is the completion of the normed space
$(E_{1}\otimes\cdots\otimes E_{n}\otimes F,\varepsilon_{n+1})$, we can
identify $(E_{1}\otimes\cdots\otimes E_{n}\otimes F,\varepsilon_{n+1}
)^{\prime}$ with $(E_{1}\widehat{\otimes}_{\varepsilon}\cdots\widehat{\otimes
}_{\varepsilon}E_{n}\widehat{\otimes}_{\varepsilon}F)^{\prime}$ through the
well known isometry $\varphi\longrightarrow\overline{\varphi}$, where
$\overline{\varphi}$ is the unique extension of $\varphi$.\newline
\indent Since all identifications above are done by the corresponding standard
mappings, their composition coincides with our mapping $\varphi$, which proves
that $\mathcal{L}_{\mathcal{I}}$ is $\varepsilon$-represented, hence
represented by a smooth tensor norm.
\end{examples}

\begin{definition}\rm Let a tensor norm
$\beta= (\beta_{n})_{n=1}^{\infty}$ be given. Define $\mathcal{L}_{\beta
}(E_{1}, \ldots, E_{n};F)$ as those multilinear mappings $A \in\mathcal{L}
(E_{1}, \ldots, E_{n};F)$ whose linearizations
\[
A_{L} \colon(E_{1} \otimes\cdots\otimes E_{n}, \beta_{n}) \longrightarrow F
\]
are continuous endowed with the norm $$\|A\|_{{\cal L}_{\beta}} :=
\|A_L \colon (E_1 \otimes \cdots \otimes E_n, \beta_n)
\longrightarrow F \|. $$
\end{definition}
It is easy to see that
$\mathcal{L}_{\beta}$ is a multi-ideal. The question
of whether or not ${\cal L}_{\beta}$ is $\beta$-represented is quite
natural. We shall treat it later.\newline\indent Next we define a
property which is closely related to property (B) of \cite{studia}:

\begin{definition}\rm
\textrm{Given $A \in\mathcal{L}(E_{1}, \ldots, E_{n}, \mathbb{K};F)$, define
$A1 \in\mathcal{L}(E_{1}, \ldots, E_{n};F)$ by $A1(x_{1}, \ldots, x_{n}) =
A(x_{1}, \ldots, x_{n},1)$. We say that a multi-ideal $\mathcal{M}$ has
property [B] if
\[
A \in\mathcal{M}(E_{1}, \ldots, E_{n}, \mathbb{K};F) \Longleftrightarrow A1
\in\mathcal{M}(E_{1}, \ldots, E_{n};F)
\]
{and in this case $\|A\|_{\cal M} = \|A1\|_{\cal M}$}, for every
$n$, $E_{1}, \ldots, E_{n},F$ and $A \in\mathcal{L}(E_{1}, \ldots,
E_{n};F)$. }
\end{definition}

\begin{proposition}
\label{prop} A tensor norm $\beta$ is smooth if and only if its corresponding
multi-ideal $\mathcal{L}_{\beta}$ has property [B].
\end{proposition}

\begin{proof} Assume that $\beta = (\beta_n)_{n=1}^\infty$ is a smooth tensor norm. Given
$A \in {\cal L}(E_1, \ldots, E_n,\mathbb{K};F)$, consider the chain
$$E_1 \otimes \cdots \otimes E_n \otimes \mathbb{K} \stackrel{\psi}{\longrightarrow} E_1 \otimes \cdots \otimes E_n
\stackrel{{\psi}^{-1}}{\longrightarrow} E_1 \otimes \cdots \otimes
E_n \otimes \mathbb{K} \stackrel{A_L}{\longrightarrow} F $$
It is
not difficult to see that $A_L = (A1)_L \circ \psi$ and $(A1)_L =
A_L \circ \psi^{-1}$. Since $\psi$ and $\psi^{-1}$ are continuous as
$\beta$ is smooth, it follows that $A_L$ is continuous if and only
if $(A1)_L$ is continuous, that is $A \in {\cal L}_\beta$ if and
only if $A_1 \in {\cal L}_\beta$. {In this case $\|A\|_{{\cal
L}_\beta} = \|A1\|_{{\cal L}_\beta}$ because $\psi$ and $\psi^{-1}$
are isometric
isomorphisms}, proving that ${\cal L}_\beta$ has property [B].\\
\indent Conversely, assume that ${\cal L}_\beta$ has property [B]. Given $E_1, \ldots, E_n$, consider $A
\colon E_1 \times \cdots \times E_n \times \mathbb{K} \longrightarrow (E_1 \otimes \cdots \otimes E_n,
\beta_n)$ defined by $A(x_1, \ldots, x_n,\lambda) = \lambda(x_1 \otimes \cdots \otimes x_n)$. Then
$$(A1)_L(x_1 \otimes \cdots \otimes x_n) = A1(x_1,
\ldots, x_n) = A(x_1, \ldots, x_n,1)= x_1 \otimes \cdots \otimes
x_n,$$ showing that $(A1)_L$ is the identity operator on $E_1
\otimes \cdots \otimes E_n$, hence {an isometric isomorphism} when
this space is endowed with $\beta_n$ on both sides. It follows that
$A1 \in {\cal L}_\beta(E_1, \ldots, E_n; (E_1 \otimes \cdots \otimes
E_n, \beta_n))$. So $A \in {\cal L}_\beta(E_1, \ldots, E_n,
\mathbb{K}; (E_1 \otimes \cdots \otimes E_n, \beta_n))$ {and
$$\|A\|_{{\cal L}_\beta} = \|A1\|_{{\cal L}_\beta} = \|(A1)_L \colon
(E_1 \otimes \cdots \otimes E_n, \beta_n) \longrightarrow (E_1
\otimes \cdots \otimes E_n, \beta_n) \| =1$$} as ${\cal L}_\beta$
has property [B]. Therefore
$$A_L \colon (E_1 \otimes \cdots \otimes E_n\otimes \mathbb{K},
\beta_{n+1}) \longrightarrow (E_1 \otimes \cdots \otimes E_n,
\beta_n) $$ is continuous. But $A_L = \psi$, so $\psi$ is
continuous and $$1 = \|A\|_{{\cal L}_\beta} = \|A_L \colon (E_1
\otimes \cdots \otimes E_n\otimes \mathbb{K}, \beta_{n+1})
\longrightarrow (E_1 \otimes \cdots \otimes E_n, \beta_n) \|. $$
Moreover, $\beta_n(\psi(z)) = \beta_n(A_L(z)) \leq \beta_{n+1}(z)$
for every $z \in E_1 \otimes \cdots \otimes E_n\otimes
\mathbb{K}$.\\
\indent Consider now $C \colon E_1 \times \cdots \times E_n \times
\mathbb{K} \longrightarrow (E_1 \otimes \cdots \otimes E_n\otimes
\mathbb{K}, \beta_{n+1})$ given by $C(x_1, \ldots, x_n, \lambda) =
x_1 \otimes \cdots \otimes x_n \otimes \lambda$. It is clear that
$C_L$ is the identity operator on $E_1 \otimes \cdots \otimes
E_n\otimes \mathbb{K}$, hence {an isometric isomorphism} when this
space is endowed with $\beta_{n+1}$ on both sides. Hence $C \in
{\cal L}_\beta(E_1, \ldots, E_n,\mathbb{K}; (E_1 \otimes \cdots
\otimes E_n\otimes \mathbb{K}, \beta_{n+1}))$. So $C1 \in {\cal
L}_\beta(E_1, \ldots, E_n; (E_1 \otimes \cdots \otimes E_n\otimes
\mathbb{K}, \beta_{n+1}))$ {and $$\|C1\|_{{\cal L}_\beta} =
\|C\|_{{\cal L}_\beta} = \|C_L \colon (E_1 \otimes \cdots \otimes
E_n\otimes \mathbb{K}, \beta_{n+1}) \longrightarrow (E_1 \otimes
\cdots \otimes E_n\otimes \mathbb{K}, \beta_{n+1}) \| =1$$} as
${\cal L}_\beta$ has property [B]. Therefore
$$(C1)_L \colon (E_1 \otimes \cdots \otimes E_n,
\beta_{n}) \longrightarrow (E_1 \otimes \cdots \otimes E_n\otimes
\mathbb{K}, \beta_{n+1}) $$ is continuous. But $(C1)_L = \psi^{-1}$,
so $\psi^{-1}$ is continuous  {and $$1 = \|C1\|_{{\cal L}_\beta} =
\|(C1)_L \colon (E_1 \otimes \cdots \otimes E_n, \beta_{n})
\longrightarrow (E_1 \otimes \cdots \otimes E_n\times \mathbb{K},
\beta_{n+1}) \|. $$ Moreover, $\beta_{n+1}(\psi^{-1}(w)) =
\beta_{n+1}((C1)_L(w)) \leq \beta_{n}(w)$ for every $w \in E_1
\otimes \cdots \otimes E_n$. Making $w = \psi(z)$ we obtain
$$\beta_n(\psi(z)) \leq \beta_{n+1}(z) \leq  \beta_n(\psi(z))$$
for every $z \in E_1 \otimes \cdots \otimes E_n\otimes \mathbb{K}$, proving that $\psi$ is an isometric
isomorphism, that is, $\beta$ is smooth.}
\end{proof}

Now we turn our attention to multi-ideals that can be represented by smooth
tensor norms.

\begin{theorem}
\label{theorem} {\rm(Uniqueness of the representation)}
{ The tensor norm that represents a given multi-ideal, if any, is unique.}
\end{theorem}

\begin{proof}{ Let $\cal M$ be a multi-ideal that is represented by the tensor norms
$\beta = (\beta_n)_{n=1}^\infty$ and $\gamma = (\gamma_n)_{n=1}^\infty$. Let $E_1, \ldots, E_{n-1}, E_n$ be
given. We have that the corresponding operators $\varphi_\beta \colon {\cal M}(E_1, \ldots,
E_{n-1};E_n^\prime) \longrightarrow (E_1 \otimes\cdots \otimes E_n,
\beta_n)^\prime$ and $\varphi_{\gamma} \colon {\cal M}(E_1, \ldots, E_{n-1};E_n^\prime) \longrightarrow
(E_1 \otimes\cdots \otimes
E_n,
\gamma_n)^\prime$ are isometric isomorphisms. So the composition $\varphi_\beta \circ (\varphi_{\gamma})^{-1}$,
which is clearly the formal identity, is an isometric isomorphism from $(E_1 \otimes\cdots \otimes
E_n,
\gamma_n)^\prime$ to $(E_1 \otimes\cdots \otimes
E_n,
\beta_n)^\prime$. By the Hahn-Banach theorem it follows that $ \gamma_n = \beta_n$ on $E_1
\otimes \cdots \otimes E_n$.}
\end{proof}

Given a maximal multi-ideal $\mathcal{M}$ (for the definition see
\cite{hf}), we have already mentioned that \cite[Theorem 4.5]{hf}
assures the existence of a tensor norm that represents
$\mathcal{M}$. Let us denote such tensor norm, {which is unique by
Theorem \ref{theorem}}, by $\beta ^{\mathcal{M}}$. Combining
Proposition \ref{prop} and {Theorem \ref{theorem}} we have:

\begin{proposition}
\label{max}
{The following are equivalent for a maximal multi-ideal $\cal M$:\\
{\rm (a)} $\cal M$ is represented by a smooth tensor norm.\\
{\rm (b)} $\beta^{\cal M}$ is smooth.\\
{\rm (c)} ${\cal L}_{\beta^{\cal M}}$ has property [B].\\
In particular, if $\cal M$ has property [B] and
${\cal M} = {\cal
L}_{\beta^{\cal M}}$ then $\cal M$ is represented by a smooth tensor norm.}
\end{proposition}

This result impels us to study the equality $\mathcal{M} = \mathcal{L}
_{\beta^{\mathcal{M}}}$. As to the projective norm, by Example
\ref{example}(a) and {Theorem \ref{theorem}} we know that
$\beta^{\mathcal{L}} = \pi$, so $\mathcal{L}_{\beta^{\mathcal{L}}} =
\mathcal{L}_{\pi}= \mathcal{L}$. We treat this question together with the question of whether or
not ${\cal L}_{\beta}$ is $\beta$-represented.

\begin{proposition}\label{injec}
{ The multi-ideal ${\cal L}_{\varepsilon}$ is not
$\varepsilon-$represented.}
\end{proposition}

\begin{proof} Assume for a while that ${\cal L}_{\varepsilon}$ is $\varepsilon-$represented. On the one hand,
by Example \ref{example}(b) we know that ${\cal L}_{\cal I}$ is $\varepsilon-$represented, so it follows
easily that ${\cal L}_{\varepsilon}(E_1, \ldots, E_n;F') = {\cal L}_{\cal I}(E_1, \ldots, E_n;F')$ for every
$n$ and $E_1, \ldots, E_n,F$.

On the other hand, by
Example \ref{example}(b) and {Theorem \ref{theorem}} we know that
$\beta^{\mathcal{L}_{\mathcal{I}}} = \varepsilon$ and from
\cite[Proposition 2.2]{cdg} we have $\mathcal{L}_{\mathcal{I}}
\subseteq\mathcal{L}_{\varepsilon}$, hence
$\mathcal{L}_{\mathcal{I}}
\subseteq\mathcal{L}_{\varepsilon}= \mathcal{L}_{\beta^{\mathcal{L}
_{\mathcal{I}}}}$. As to the converse inclusion, let $n \geq2$ and $F$ be an
infinite dimensional Banach space. Assume for a while that $\mathcal{L}
(\hat\otimes_{n,\varepsilon} c_{0};F) = \mathcal{I}(\hat\otimes_{n,\varepsilon
} c_{0};F)$. As integral linear opertors are absolutely summing
\cite[Proposition 5.5]{djt}, we have that every continuous linear operator
from $\hat\otimes_{n,\varepsilon} c_{0}$ to $F$ is absolutely summing. We know
that $\hat\otimes_{n,\varepsilon} c_{0}$ has unconditional basis because
$c_{0} = \hat\otimes_{n,\varepsilon} c_{0}$, so by a result due to
Lindenstrauss-Pe{\l }czy\'nski \cite[Theorem 4.2]{lp} it follows that
$\hat\otimes_{n,\varepsilon} c_{0} = c_{0}$ is isomorphic to some $\ell
_{1}(\Gamma)$, but this is absurd. Therefore there exists a non-integral
operator $u \in\mathcal{L}(\hat\otimes_{n,\varepsilon} c_{0};F)$. Define
\[
A \colon c_{0} \times\cdots\times c_{0} \longrightarrow F~,~A(x_{1}, \ldots,
x_{n}) = u(x_{1} \otimes\cdots\otimes x_{n}).
\]
So $A_{L} = u$ is $\varepsilon$-continuous, hence $A \in\mathcal{L}
_{\varepsilon}(^{n} c_{0};F)$, but $A$ fails to be integral because its
linearization $A_{L} = u$ fails to be integral in the injective norm
$\varepsilon$. Hence $\mathcal{L}_{\beta^{\mathcal{L}_{\mathcal{I}}}}(^{n}
c_{0};F) = \mathcal{L}_{\varepsilon}(^{n} c_{0};F) \neq
\mathcal{L}_{\mathcal{I}}(^{n} c_{0};F)$ - a contradiction.
\end{proof}

Later, in Proposition \ref{propos}, we shall go quite further. For the moment, the proof above shows, in particular, that $\mathcal{L}_{\beta^{\mathcal{L}_{\mathcal{I}}}}\neq
\mathcal{L}_{\mathcal{I}}$. Corollary \ref{coro} shall provide another example of the inequality $\mathcal{M}
\neq\mathcal{L}_{\beta^{\mathcal{M}}}$.

Now we proceed to present some multi-ideals that are represented by tensor norms but not by smooth tensor norms.
Before we give a general criteria:

\begin{proposition}
Let $\mathcal{M}$ be a multi-ideal such that $\mathcal{M}(E_{1},
\ldots, E_{n},F;\mathbb{K}) = \mathcal{L}(E_{1}, \ldots,
E_{n},F;\mathbb{K})$ and $\mathcal{M}(E_{1}, \ldots,
E_{n};F^{\prime}) \neq\mathcal{L}(E_{1}, \ldots, E_{n};F^{\prime}) $
for some Banach spaces $E_{1}, \ldots, E_{n}, F$ and some positive
integer $n$.

Then there is no smooth tensor norm that represents
$\mathcal{M}$.\label{p:111}
\end{proposition}
\begin{proof}Assume  that there is a smooth tensor norm $\beta =
(\beta_n)_{n=1}^\infty$ that represents  ${\cal M}$. Since $\beta$ is
smooth, the adjoint $\psi^*$ of $\psi$ is an isometric isomorphism from $(E_1
\otimes \cdots \otimes E_n\otimes F, \beta_{n+1})^\prime$ to $(E_1 \otimes \cdots
\otimes E_n \otimes F\otimes \mathbb{K}, \beta_{n+2})^\prime$.
We have the following chain of isomorphisms:
$$(E_1 \otimes \cdots \otimes
E_n\otimes F, \beta_{n+1})^\prime \stackrel{\psi^*}{\longleftrightarrow} (E_1
\otimes \cdots \otimes E_n \otimes F\otimes \mathbb{K}, \beta_{n+2})^\prime
$$
$$\stackrel{\varphi}{\longleftrightarrow}  {\cal M}(E_1, \ldots, E_n, F;\mathbb{K}^\prime) \longleftrightarrow
{\cal M}(E_1, \ldots, E_n, F;\mathbb{K}) $$
$$\stackrel{id}{\longleftrightarrow} {\cal L}(E_1, \ldots, E_n, F;\mathbb{K})\longleftrightarrow (E_1 \otimes \cdots
\otimes E_n\otimes F,\pi_{n+1})^\prime, $$ where the non-indicated mappings are the
canonical ones. It follows that the identity operator is an
algebraic isomorphism between $(E_1 \otimes \cdots \otimes E_n\otimes F,
\beta_{n+1})^\prime$ and $(E_1 \otimes \cdots \otimes E_n\otimes F,\pi_{n+1})^\prime$. Since the ideal $\mathcal{L}$ is $\pi$-represented, we have the  following chain of canonical isomorphisms:
$$
{\cal M}(E_1, \ldots, E_n;F^\prime) \longleftrightarrow (E_1 \otimes \cdots \otimes E_n\otimes F, \beta_{n+1})'$$
$$\longleftrightarrow (E_1 \otimes \cdots \otimes E_n\otimes F, \pi_{n+1})' \longleftrightarrow {\cal L}(E_1, \ldots,
E_n;F^\prime).
$$
It follows that the identity operator is an algebraic isomorphism between the spaces ${\cal M}(E_1, \ldots, E_n;F^\prime)$ and ${\cal L}(E_1, \ldots,
E_n;F^\prime)$, which is a contradiction.
\end{proof}

The next multilinear generalization of the ideal of absolutely
summing linear operators was introduced in \cite{port}:

\begin{definition}\rm
\textrm{Given $p\geq q\geq1$, a multilinear mapping $T\in\mathcal{L}
(E_{1},\ldots,E_{n};F)$ is said to be \textit{strongly multiple $(p,q)$
-summing} if there exists $C\geq0$ such that
\[
\left(  \sum\limits_{j_{1},\ldots,j_{n}=1}^{m}\| T(x_{j_{1}}^{(1)}
,\ldots,x_{j_{n}}^{(n)})\|^{p}\right)  ^{1/p}\leq C\left(  \underset
{}{\underset{\phi\in B_{\mathcal{L}(E_{1},\ldots,E_{n})}}{\sup}}
\sum\limits_{j_{1},\ldots,j_{n}=1}^{m}\mid\phi(x_{j_{1}}^{(1)},\ldots
,x_{j_{n}}^{(n)})\mid^{q}\right)  ^{1/q}
\]
for every $m\in\mathbb{N}$, $x_{j_{l}}^{(l)}\in E_{l}$ with
$l=1,\ldots,n$ and $j_{l}=1,\ldots,m.$ The space of all strongly
multiple $p$-summing $n$-linear mappings from
$E_{1}\times\cdots\times E_{n}$ to $F$ will be denoted by
$\mathcal{L}_{sm(p,q)}(E_{1},\ldots,E_{n};F)$. The infimum of the
constants $C$ for which the inequality always holds defines a
complete norm $\Vert \cdot\Vert_{sm(p,q)}$ on
$\mathcal{L}_{sm(p,q)}(E_{1},...,E_{n};F)$. When $p=q$ we shortly
write $\mathcal{L}_{sm,p}(E_{1},\ldots,E_{n};F)$ and
$\Vert\cdot\Vert_{s{f},p}$. }
\end{definition}

\begin{proposition}
\label{not} The multi-ideal $\mathcal{L}_{sm,p}$, $1 \leq p < +
\infty$, of strongly multiple
$p$-summing multilinear mappings is represented by a tensor norm but not by a smooth tensor
norm.
\end{proposition}

\begin{proof} Following the lines of \cite[Proposici\'on
4.37]{david} it is not difficult to prove that ${\cal L}_{sm,p}$ is maximal. So it follows from
\cite[Theorem 4.5]{hf} that ${\cal L}_{sm,p}$ is represented by a
tensor norm.

 Let us see that ${\cal L}_{sm,p}$ cannot be represented by a smooth
tensor norm. It is clear that $ {\cal L}_{sm,p}(E_1, \ldots,
E_n,F;\mathbb{K}) = {\cal L}(E_1, \ldots, E_n,F;\mathbb{K}) $ for
every integer $n$ and every Banach spaces $E_1, \ldots, E_n$ and
$F$. On the other hand, assuming that ${\cal L}_{sm,p}(^nE,;F^\prime) = {\cal L}(^nE;F^\prime)$ for every
integer $n$ and every Banach spaces $E$ and $F$,
by \cite[Proposition 5.2(iii)]{port} we would have ${\cal
L}(E;F^\prime) = \Pi_p(E;F^\prime)$ for every $E$ and $F$. This is
absurd because for every infinite-dimensional Banach space $E$ the
canonical injection $E \hookrightarrow E^{''} = (E')'$ fails to be
$p$-summing. Hence ${\cal L}(^nE;F^\prime) \neq {\cal L}_{sm,p}(^nE;F^\prime)$ for
some $n$ and some Banach spaces $E$ and $F$. By Proposition \ref{p:111} it
follows that ${\cal L}_{sm,p}$ cannot be represented by a smooth
tensor norm.
\end{proof}

\begin{corollary}\label{coro} For $\mathcal{M} = \mathcal{L}_{sm,p}$ we have
$\mathcal{M} \ne\mathcal{L}_{\beta^{\mathcal{M}}}$.
\end{corollary}

\begin{proof} It is easy to check that $\mathcal{L}_{sm,p}$ has property [B]. Indeed, it is enough to combine the definition of
${\cal L}_{sm,p}$ with the well-known fact that the space of
$(n+1)$-linear forms $\mathcal{L}(E_{1}, \ldots, E_{n},
\mathbb{K};\mathbb{K})$ is isometrically isometric to the space of
$n$-linear forms $\mathcal{L}(E_{1}, \ldots, E_{n};\mathbb{K})$ via
the obvious correspondence. Assuming that
$\mathcal{M} = \mathcal{L}_{\beta^{\mathcal{M}}}$, by Proposition \ref{max} $\mathcal{L}_{sm,p}$ would be represented by a smooth tensor
norm; but this is not true by
Proposition \ref{not}.
\end{proof}

\begin{remark}\rm For the sake of completeness, let us construct the tensor norm that represents the
multi-ideal ${\cal L}_{sm,p}$: Given normed spaces
$E_{1},\ldots,E_{n},F$ and $p\geq1$, define\\

$ \beta_{p}(u):=$
\[\inf\left[  \sum_{m=1}^{M}\|(b_{m,j^{1}_{m},\ldots
,j^{n}_{m}})_{j^{1}_{m},\ldots,j^{n}_{m}=1}^{I^{1}_{m},\ldots,I^{n}_{m}
}\|_{q} \left(  {\underset{\phi\in
B_{\mathcal{L}(E_{1},\ldots,E_{n})}}
{\sup}}\sum\limits_{j^{1}_{m},\ldots,j^{n}_{m}=1}^{I^{1}_{m},\ldots,I^{n}_{m}
}\mid\phi(x^{(1)}_{m,j^{1}_{m}},\ldots,x^{(n)}_{m,j^{n}_{m}})\mid^{p}\right)
^{1/p}\right]
\]
where the infimum is taken over the set of all representations of
the tensor $u\in E_{1}\otimes\cdots\otimes E_{n}\otimes F$ of the
form
$$u=\sum_{m=1}^{M}\sum\limits_{j^{1}_{m},\ldots,j^{n}_{m}=1}^{I^{1}_{m}
,\ldots,I^{n}_{m}} x^{(1)}_{m,j^{1}_{m}}\otimes\cdots\otimes x^{(n)}_{m,j^{n}
_{m}}\otimes b_{m,j^{1}_{m},\ldots,j^{n}_{m}} $$

with $M\in\mathbb{N}$, $x^{(l)}_{m,j^{l}_{m}}\in E_{l}$,
$l=1,\ldots,n$, $b_{m,j^{1}_{m},\ldots,j^{n}_{m}}\in F$,
$j^{l}_{m}=1,\ldots,I^{l}_{m}$, and $q\geq1$ with
$\frac{1}{p}+\frac{1}{q}=1$. Following the lines of \cite[Teorema 4.38]{david} one can prove that
$\beta_p$ is a tensor norm and that ${\cal L}_{sm,p}$ is
$\beta_p$-represented. From Proposition \ref{not} it follows that
$\beta_p$ is not smooth.\label{r:12}
\end{remark}

The ideal $\mathcal{L}_{m,p}$ of multiple $p$-summing multilinear mappings, introduced by Matos \cite{mapre} and, independently, by Bombal, P\'erez-Garc\'ia and Villanueva \cite{bpgv}, has played a central role in the theory of multi-ideals, providing even unexpected applications (see Remark \ref{remark}(a)).

\begin{proposition}
For $1 \leq p \leq2$, the ideal $\mathcal{L}_{m,p}$ of multiple
$p$-summing multilinear mappings is represented by a tensor norm
but not by a smooth tensor norm.
\end{proposition}

\begin{proof} That $\mathcal{L}_{m,p}$ is represented by a tensor norm is proved in \cite[Proposici\'on 4.39]{david}. From
$\mathcal{L}_{m,p}(^{2}\ell_{1};\mathbb{K}) = \mathcal{L}(^{2} \ell
_{1};\mathbb{K})$ \cite[Teorema 5.23]{david} and $\Pi_{p}(\ell_{1};
\ell_{\infty})= \mathcal{L}_{m,p}(\ell_{1}; \ell_{\infty}) \neq\mathcal{L}
(\ell_{1};\ell_{\infty})$ (obvious), the non-representability by a smooth tensor norm follows from Proposition \ref{p:111}.
\end{proof}

\begin{remark}\rm \label{remark}(a) As proved by Defant and P\'erez-Garc\'ia \cite{dpg}, the tensor norm that represents the ideal $\mathcal{L}_{m,p}$ of multiple $p$-summing multilinear mappings is the first example of a tensor norm that preserves unconditionality for ${\cal L}_p$-spaces.\\
\noindent (b) Everything we proved for the multi-ideal
$\mathcal{L}_{sm,p}$ of strongly multiple $p$-summing multilinear mappings can
be proved, \textit{mutatis mutandis}, for the multi-ideal $\mathcal{L}_{ss,p}$
of strongly $p$-summing multilinear mappings introduced in \cite{dimant}.
\end{remark}

We finish this section with another hint that the representation of a (vector-valued) multi-ideal by a smooth tensor norm is not commonplace.

\begin{proposition}\label{propos}
{ The multi-ideal ${\cal L}_{\varepsilon}$ is not represented by a
smooth tensor norm.}
\end{proposition}

\begin{proof}Assume that $\mathcal{L}_\varepsilon$ is represented by a smooth tensor norm $\beta=(\beta_n)_{n=1}^\infty$. Fix a positive integer $n$ and Banach spaces $E_1,\ldots,E_n$. Since $\mathcal{L}_\varepsilon$ is $\beta$-represented, the mapping

$$\varphi:(\mathcal{L}_\varepsilon(E_1,\ldots, E_n;F'),\parallel.\parallel_{\mathcal{L}_\varepsilon})\longrightarrow
(E_1\otimes\cdots\otimes E_n\otimes F,\beta_{n+1})'$$
$$T\longrightarrow\varphi(T)(x_1\otimes\cdots\otimes x_n\otimes y)=T(x_1,\ldots,x_n)(y)$$
is an isometric isomorphism for every $F$. In particular, taking $F=\mathbb{K}$, we obtain the following isometric isomorphisms:
 $$(\mathcal{L}_\varepsilon(E_1,\ldots, E_n;\mathbb{K}'),\parallel.\parallel_{\mathcal{L}_\varepsilon})\cong(E_1\otimes\cdots\otimes E_n\otimes \mathbb{K},\beta_{n+1})'\cong(E_1\otimes\cdots\otimes E_n,\beta_{n})'.$$
 In Corollary \ref{cor} we shall prove that following spaces are also isometric isomorphic, with the same canonical correspondences:
$$(\mathcal{L}_\varepsilon(E_1,\ldots, E_n;\mathbb{K}'),\parallel.\parallel_{\mathcal{L}_\varepsilon})\cong(E_1\otimes\cdots\otimes E_n\otimes \mathbb{K},\varepsilon_{n+1})'\cong(E_1\otimes\cdots\otimes E_n,\varepsilon_{n})'.$$
\noindent Hence the identity mapping $(E_1\otimes\cdots\otimes E_n,\varepsilon_n)'\longrightarrow(E_1\otimes\cdots\otimes E_n,\beta_n)'$ is an isometric isomorphism. Calling on Hahn-Banach once again we get that $\varepsilon_n$ and $\beta_n$ coincide on $E_1\otimes\cdots\otimes E_n$. It follows that the ideal $\mathcal{L}_\varepsilon$ is $\varepsilon$-represented, which contradicts Proposition \ref{injec}.
\end{proof}

\section{Scalar-valued case}

The aim of this section is show that smooth tensor norms are more suitable to represent
ideals of multilinear forms.

Given a tensor norm $\beta$, we write
$\mathcal{L}_{\beta}^{\mathbb{K}}:=(\mathcal{L}_{\beta})^{\mathbb{K}}
$.\newline\indent We shall write ${\cal M}^{\mathbb{K}}
\stackrel{1}{=} {\cal L}_{\beta}^{\mathbb{K}}$ if for every $n$ and
every $E_1, \ldots, E_n$, the linearization operator $\Phi = \Phi(n,
E_1, \ldots, E_n)$: $$A \in {\cal M}(E_1, \ldots, E_n; \mathbb{K})
\mapsto \Phi(A) := A_L,$$ is an isometric isomorphism from ${\cal
M}(E_1, \ldots, E_n; \mathbb{K})$ onto $(E_1 \otimes\cdots \otimes
E_n, \beta_{n})^\prime$.

\begin{theorem}
\label{th}Let $\mathcal{M}$ be a multi-ideal and $\beta= (\beta_{n}
)_{n=1}^{\infty}$ be a smooth tensor norm. Then the ideal of
multilinear forms $\mathcal{M}^{\mathbb{K}}$ is $\beta$-represented
if and only if ${\cal M}^{\mathbb{K}}
\stackrel{1}{=} {\cal L}_{\beta}^{\mathbb{K}}$.
\end{theorem}

\begin{proof} Let $n \in \mathbb{N}$ and $E_1, \ldots, E_n$ be Banach spaces. We shall say
that $\Phi$ is well defined, isometric and onto if $\Phi(A) = A_L
\in (E_1 \otimes\cdots \otimes E_n, \beta_{n})^\prime$ {for every $A
\in {\cal M}(E_1, \ldots, E_n; \mathbb{K})$} and $\Phi \colon {\cal
M}(E_1, \ldots, E_n; \mathbb{K}) \longrightarrow (E_1 \otimes\cdots
\otimes E_n, \beta_{n})^\prime$ is {isometric and} surjective.
Assume that $\Phi$ is well defined, {isometric} and onto. It is
clear that $\Phi$ is linear and injective, {so $\Phi$ is well
defined, isometric and onto if and only if $\Phi$ is an isometric
isomorphism from ${\cal M}(E_1, \ldots, E_n; \mathbb{K})$ onto $(E_1
\otimes\cdots \otimes E_n,
\beta_{n})^\prime$.}\\

\indent We continue assuming that $\Phi$ is well defined,{isometric}
and onto. Let $h \colon \mathbb{K}' \longrightarrow \mathbb{K}$ be
the isometric isomorphism given by $h(f) = f(1)$ for every $f \in
\mathbb{K}'$. It is clear that the linear mapping
$$\xi \colon {\cal M}(E_1, \ldots, E_n;\mathbb{K}^\prime) \longrightarrow {\cal M}(E_1, \ldots, E_n; \mathbb{K})~,~\xi(A) = h \circ A,  $$
is an isometric isomorphism as well. Considering the chain
$${\cal M}(E_1, \ldots, E_n;\mathbb{K}^\prime) \stackrel{\xi}{\longrightarrow} {\cal M}(E_1, \ldots, E_n; \mathbb{K})
\stackrel{\Phi}{\longrightarrow} (E_1 \otimes\cdots \otimes E_n, \beta_{n})^\prime$$
$$\stackrel{\psi^*}{\longrightarrow}
(E_1 \otimes\cdots \otimes E_n \otimes \mathbb{K},
\beta_{n+1})^\prime,$$ where $\psi^*$ is the adjoint of the linear
operator $\psi$ of the definition of smooth tensor norm, it is not
difficult to check that $\varphi = \psi^* \circ \Phi \circ \xi$
(hence $\Phi = (\psi^*)^{-1} \circ \varphi \circ \xi^{-1})$, where
$\varphi = \varphi(n, E_1, \ldots, E_n)$ is the operator of
Definition \ref{definition}. \\
\indent Hence ${\cal M}^{\mathbb{K}} \stackrel{1}{=} {\cal L}_{\beta}^{\mathbb{K}}$ if and only if $\Phi(n,
E_1, \ldots, E_n)$ is an isometric isomorphism for every $n, E_1, \ldots, E_n$ if and only if $\Phi(n, E_1,
\ldots, E_n)$ is well defined, isometric and onto for every $n, E_1, \ldots, E_n$ if and only
if $\varphi(n, E_1, \ldots, E_n)$ is an isometric isomorphism for every $n, E_1, \ldots, E_n$ if and only if
${\cal M}^{\mathbb{K}}$ is $\beta$-represented.
\end{proof}

As we saw before (see, e.g., Proposition \ref{injec} and Corollary \ref{coro}), the theorem above cannot be generalized to vector-valued multi-ideals.

\begin{corollary}
\label{cor} Let $\beta$ be a smooth tensor norm. Then the ideal of
multilinear forms $\mathcal{L}_{\beta}^{\mathbb{K}}$ is
{$\beta$-represented and thus is} represented by a smooth tensor
norm.
\end{corollary}

\begin{example}\label{scalar}\rm The ideal of multilinear forms ${\cal L}_{\varepsilon}^\mathbb{K}$ is $\varepsilon$-represented, hence represented by a smooth tensor norm.
\end{example}

\begin{corollary}
{Let ${\cal M}$ be a multi-ideal. If ${\cal M}^{\mathbb{K}}$ is
represented by a smooth tensor norm then ${\cal M}^{\mathbb{K}}$
contains the integral multilinear forms.}
\end{corollary}

\begin{proof}{By Theorem \ref{th} we have that ${\cal M}^{\mathbb{K}} = {\cal L}_{\beta}^{\mathbb{K}}$, where
$\beta$ is a smooth tensor norm. Since $\varepsilon \leq \beta$ because $\beta$ is a tensor norm, it follows
that $ {\cal L}_{\cal I}^{\mathbb{K}} = {\cal L}_{\varepsilon}^{\mathbb{K}} \subseteq {\cal
L}_{\beta}^{\mathbb{K}} = {\cal M}^{\mathbb{K}}$.}
\end{proof}

Next we see that sometimes we can construct explicitly the smooth tensor norm
that represents an ideal of multilinear forms.

\begin{definition}\rm
\textrm{Let $p \geq1$. An $n$-linear mapping $A \in\mathcal{L}(E_{1},\ldots,
E_{n};F)$ is $p-$\textit{semi-integral}, in symbols $A \in\mathcal{L}
_{si,p}(E_{1},\ldots,E_{n};F)$, if there exist $C\geq0$ and a regular
probability measure $\mu$ on the Borel $\sigma-$algebra of $B_{E_{1}
^{^{\prime}}}\times\cdots\times$ $B_{E_{n}^{^{\prime}}}$ endowed with the
product of the weak star topologies $\sigma(E_{l}^{\prime},E_{l}),$
$l=1,\ldots,n,$ such that
\[
\| A(x_{1},\ldots,x_{n})\|\leq C\left(  \int_{B_{E_{1}^{\prime}}\times
\cdots\times B_{E_{n}^{\prime}}}|\varphi_{1}(x_{1})\cdots\varphi_{n}
(x_{n})|^{p}d\mu(\varphi_{1},\ldots,\varphi_{n})\right)  ^{1/p}
\]
for every $x_{j}\in E_{j}$, $j=1,\ldots,n$. The infimum of the constants $C$
defines a norm $\|\cdot\| _{si,p}$ on $\mathcal{L}_{si,p}(E_{1},\ldots
,E_{n};F)$. It is well known that $\mathcal{L}_{si,p}$ is a multi-ideal (see
\cite{CD}). }
\end{definition}

\begin{proposition}
\label{semi} For $p \geq1$, the ideal of $p$-semi-integral multilinear forms
$\mathcal{L}_{si,p}^{\mathbb{K}}$ is represented by a smooth tensor norm.
\end{proposition}

\begin{proof} Given $u\in
E_{1}\otimes\cdots\otimes E_{n}$, define
\[
\sigma_{p}^{n}(u):=\inf\Vert(\lambda_{j})_{j=1}^{m}\Vert_{q}\left(
\sup_{\varphi_{l}\in B_{E_{l}^{^{\prime}}}}\sum_{j=1}^{m}|\varphi_{1}
(x_{1,j})\cdots\varphi_{n}(x_{n,j})|^{p}\right)  ^{1/p}
\]
where the infimum is taken over all representations of the form $u=\sum
\limits_{j=1}^{m}\lambda_{j}x_{1,j}\otimes\cdots\otimes x_{n,j}$, where
$m\in\mathbb{N}$, $x_{l,j}\in E_{l}$, $l=1,\ldots,n$, $\lambda_{j}
\in\mathbb{K}$, $j=1,\ldots,m$, and $\frac{1}{p}+\frac{1}{q}=1$. Standard
techniques show that $\sigma_{p}=(\sigma_{p}^{n})_{n=1}^{\infty}$ is a tensor
norm (hardwork!). Let us show that $\sigma_{p}$ is smooth. Given $u=\sum
_{j=1}^{m}\lambda_{j}x_{1,j}\otimes\cdots\otimes x_{n,j}\otimes b_{j}$ in
$E_{1}\otimes\cdots\otimes E_{n}\otimes\mathbb{K} $, we have
\begin{align*}
({\sigma}_{p}^{n+1}(u))^{p}  & \leq\Vert(\lambda_{j})_{j=1}^{m}\Vert_{q}
^{p}\sup_{\varphi_{l}\in B_{E_{l}^{^{\prime}}},\varphi\in B_{\mathbb{K}
^{^{\prime}}}}\displaystyle\sum_{j=1}^{m}\mid\varphi_{1}(x_{1,j})\cdots
\varphi_{n}(x_{n,j})\varphi(b_{j})\mid^{p}\\
& =\Vert(\lambda_{j})_{j=1}^{m}\Vert_{q}^{p}\sup_{\varphi_{l}\in
B_{E_{l}^{^{\prime}}},\varphi\in B_{\mathbb{K}^{^{\prime}}}}\displaystyle\sum
\limits_{j=1}^{m}\mid\varphi_{1}(b_{j}x_{1,j})\cdots\varphi_{n}(x_{n,j}
)\varphi(1)\mid^{p}\\
& \leq\Vert(\lambda_{j})_{j=1}^{m}\Vert_{q}^{p}\sup_{\varphi_{l}\in
B_{E_{l}^{^{\prime}}},\varphi\in B_{\mathbb{K}^{^{\prime}}}}\displaystyle\sum
\limits_{j=1}^{m}\mid\varphi_{1}(b_{j}x_{1,j})\cdots\varphi_{n}(x_{n,j}
)\mid^{p}\Vert\varphi\Vert^{p}\mid1\mid^{p}\\
& \leq\Vert(\lambda_{j})_{j=1}^{m}\Vert_{q}^{p}\sup_{\varphi_{l}\in
B_{E_{l}^{^{\prime}}}}\displaystyle\sum\limits_{j=1}^{m}\mid\varphi_{1}
(b_{j}x_{1,j})\cdots\varphi_{n}(x_{n,j})\mid^{p}.
\end{align*}
Since $\sum_{j=1}^{m}\lambda_{j}b_{j}x_{1,j}\otimes\cdots\otimes x_{n,j}$ is a
representation of $\psi(u)\in E_{1}\otimes\cdots\otimes E_{n}$, it follows
that ${\sigma}_{p}^{n+1}(u)\leq{\sigma}_{p}^{n}(\psi(u))$. A similar
computation shows that ${\sigma}_{p}^{n}(\psi(u))\leq{\sigma}_{p}^{n+1}(u)$,
completing the proof of the smoothness of $\sigma_{p}$.\\
\indent The proof that $\mathcal{L}_{si,p}^{\mathbb{K}}$ is $\sigma_{p}$-represented is a combination of the arguments of the proofs of \cite[Theorem 4.8]{AlencarMatos} and \cite[Theorem
1]{CD}. Let $E_1, \ldots, E_n$ and $f \in(E_{1}\otimes\cdots\otimes E_{n},\sigma_{p}^n)^{\prime}$ be given.
Consider the $n$-linear form
$A$ on $E_{1}\times\cdots\times E_{n}$ given by $A(x_{1}
,\ldots,x_{n})=f(x_{1}\otimes\cdots\otimes x_{n})$. For appropriate $\lambda_{j}\in\mathbb{K}$ with
$|\lambda_{j}|=1$,$j=1,\ldots,m$, we have that
$$ \sum_{j=1}^{m}|
A(x_{1,j},\ldots,x_{n,j})|^{p}= \sum_{j=1}^{m}\left| | f(x_{1,j}\otimes\cdots\otimes
x_{n,j})|^{p-1}f(x_{1,j}\otimes \cdots\otimes x_{n,j})\right| \hspace*{30em}$$
$$ =\left|
\sum_{j=1}^{m}|f(x_{1,j}\otimes\cdots\otimes x_{n,j})|^{p-1}\lambda_{j}f(x_{1,j}\otimes\cdots\otimes
x_{n,j})\right| \hspace*{30em} $$
$$=\left|f\left(  \sum_{j=1}
^{m}\lambda_{j}|f(x_{1,j}\otimes\cdots\otimes x_{n,j})|^{p-1}
x_{1,j}\otimes\cdots\otimes x_{n,j}\right)  \right|\hspace*{30em}$$
$$ \leq\|f\|_{(E_1\otimes\cdots\otimes
E_n,\sigma_p^n)'}\sigma_{p}^n \left(  \sum _{j=1}^{m}\lambda_{j}|f(x_{1,j}\otimes\cdots\otimes x_{n,j})|
^{p-1}x_{1,j}\otimes\cdots\otimes x_{n,j}\right)\hspace*{30em}$$
$$ \leq\|f\|_{(E_1\otimes\cdots\otimes
E_n,\sigma_p^n)'}\left\|  \left(  \lambda_{j}|
f(x_{1,j}\otimes\cdots\otimes x_{n,j})|^{p-1}\right)  _{j=1}
^{m}\right\|  _{q}\left( \sup_{\varphi_{l}\in B_{E_{l}^{^{\prime}}}}\sum\limits_{j=1}^{m}|\varphi_{1}
(x_{1,j})\cdots\varphi_{n} (x_{n,j})|^{p}\right)  ^{1/p}\hspace*{30em}$$
$$
=\displaystyle\|f\|_{(E_1\otimes\cdots\otimes E_n,\sigma_p^n)'}\left(  \sum_{j=1}
^{m}|f(x_{1,j}\otimes\cdots\otimes x_{n,j})|^{p}\right) ^{1/q}\left(  \sup_{\varphi_{l}\in
B_{E_{l}^{^{\prime}}}}\sum\limits_{j=1}^{m}|\varphi_{1} (x_{1,j}
)\cdots\varphi_{n} (x_{n,j})|^{p}\right)  ^{1/p}.\hspace*{30em}$$
Hence we have that
$$\left(  \sum_{j=1}^{m}| A(x_{1,j},\ldots,x_{n,j})|^{p}\right) ^{1/p}\leq\|f \|_{(E_1\otimes\cdots\otimes E_n,
\sigma_p^n)'}\left( \sup_{\varphi_{l}\in
B_{E_{l}^{^{\prime}}}}\sum\limits_{j=1}^{m}|\varphi_{1}
(x_{1,j})\cdots\varphi_{n} (x_{n,j})|^{p}\right) ^{1/p},$$ which
shows that $A\in \mathcal{L}_{si,p}(E_{1},\ldots,E_{n})$ and $$\|
A\|_{si,p}\leq\|f\|_{(E_1\otimes\cdots\otimes E_n,\sigma_p^n)'} {=
\|\Phi(A)\|_{(E_1\otimes\cdots\otimes E_n,\sigma_p^n)'}}.$${ In
particular $\Phi$ is surjective. To show the reverse inequality, take
$A\in {\mathcal L}_{si,p}(E_{1},\ldots,E_{n};\mathbb{K})$ and
let $u = \sum_{j=1}^{m}\lambda_{j}x_{1,j}\otimes\cdots\otimes
x_{n,j} \in E_{1}\otimes\cdots\otimes E_{n}$, where
$\lambda_{j}\in\mathbb{K}$, $x_{l,j}\in E_{l}$, $l=1,\ldots,n$,
$j=1,\ldots,m$. From \cite[Theorem 1]{CD} it follows that
\begin{eqnarray*}|\Phi(A)(u)|^{p}&=&\left|
\sum_{j=1}^{m}\lambda_{j}A(x_{1,j},\ldots,x_{n,j})\right|  ^{p}\leq \|(\lambda_{j})_{j=1}^{m}\|^{p}_{q}
\sum\limits_{j=1}^{m}| A(x_{1,j},\ldots,x_{n,j})|^{p}\\
&\leq&\| (\lambda_{j})_{j=1}^{m}\|^{p}_{q}\| A\|^{p}_{si,p} \sup_{\varphi_{l}\in
B_{E_{l}^{^{\prime}}}}\sum\limits_{j=1}^{m}|\varphi_{1} (x_{1,j})\cdots\varphi_{n} (x_{n,j}
)|^{p},
\end{eqnarray*}
so $|\Phi(A)(u)| \leq\| A\|_{si,p}\cdot\sigma_{p}(u)$. Since $u$ is arbitrary it follows that $\|\Phi(A)
\|_{(E_1\otimes\cdots\otimes E_n,\sigma_p)'}\leq\| A\|_{si,p}$.}
\end{proof}

Summing up the information:

\medskip

\noindent (i) The multi-ideal ${\cal L}_{\varepsilon}$ is not represented by a
smooth tensor norm (Proposition \ref{propos}), whereas ${\cal L}_{\varepsilon}^\mathbb{K}$ is represented by the smooth tensor norm $\varepsilon$ (Example \ref{scalar}),\\
(ii) The ideal $\mathcal{L}_{sm,p}$ of strongly multiple $p$-summing multilinear mappings is represented by a tensor norm but not by a smooth
tensor norm (Proposition \ref{not}) and, similarly to what we did in Proposition \ref{semi}, it can be proved that $\mathcal{L}_{sm,p}
^{\mathbb{K}}$ is represented by a smooth tensor norm;\\
(iii) The ideal $\mathcal{L}_{si,p}^{\mathbb{K}}$ of $p$-semi-integral multilinear forms is represented by a smooth tensor norm (Proposition \ref{semi}), the multi-ideal $\mathcal{L}_{si,p}$ of $p$-semi-integral multilinear mappings is represented by a tensor norm (\cite[Proposition 2]{note}) but not by a smooth tensor norm (hardwork!),

\medskip

\noindent it seems that the typical behavior of a maximal multi-ideal ${\cal M} \neq {\cal L}$ is that ${\cal M}$ is not represented by a smooth tensor norm whereas ${\cal M}^\mathbb{K}$ is represented by a smooth tensor norm.

We finish the paper showing how the representation of the ideal of multilinear functionals ${\cal M}^{\mathbb{K}}$ by
a smooth tensor norm provides information about the representation of certain vector-valued components of ${\cal M}$. Given Banach spaces
$E_1, \ldots, E_n,F$ and a tensor norm $\beta=(\beta_n)_{n=1}^\infty$, we shall say that ${\cal M}(E_1, \ldots, E_n;F')$ is $\beta$-represented if
$\mathcal{M}(E_{1}, \ldots, E_{n};F^{\prime})$ and $(E_{1} \otimes\cdots\otimes E_{n}\otimes F, \beta_{n+1})^{\prime}$ are isometrically isomorphic via the canonical mapping of Definition \ref{definition}.

\begin{proposition}\label{propo} Let $\mathcal{M}$ be a multi-ideal such that $\mathcal{M}^{\mathbb{K}}$ is represented by a smooth tensor norm $\beta$. Then ${\cal M}(E_1, \ldots, E_n;F')$ is $\beta$-represented whenever the spaces $\mathcal{M}(E_1,\ldots,E_n;F')$ and $\mathcal{M}(E_1,\ldots,E_n,F;{\mathbb{K}})$ are canonically isometric isomophic.

\end{proposition}

\begin{proof} Consider the following chain of canonical mappings:
\begin{eqnarray*}\mathcal{M}(E_1,\ldots,E_n;F')& \stackrel{\rm I}= &\mathcal{M}(E_1,\ldots,E_n,F;\mathbb{K})\\
& \stackrel{\rm II}= & \mathcal{M}(E_1,\ldots,E_n,F;\mathbb{K}')\\
& \stackrel{\rm III}= & (E_1\otimes\cdots \otimes E_n\otimes F\otimes\mathbb{K}, \beta_{n+2})' \\
& \stackrel{\rm IV}= &(E_1\otimes\cdots \otimes E_n\otimes F, \beta_{n+1})'.
\end{eqnarray*}
I is an isometric isomorphism by assumption, II because the canonical mapping
$$h \colon \mathbb{K} \longrightarrow \mathbb{K}'~,~h(\lambda)(\alpha) = \lambda \cdot \alpha $$
is an isometric isomorphism, III because $\mathcal{M}^{\mathbb{K}}$ is $\beta$-represented and IV because the tensor norm $\beta$ is smooth. Routine computations show that the composition of all these mappings ends up in the canonical mapping between $\mathcal{M}(E_1,\ldots,E_n;F')$ and $(E_1\otimes\cdots \otimes E_n\otimes F, \beta_{n+1})'$.
\end{proof}