text:
\begin{document}

\title{Canonical toric Fano threefolds}
\author{Alexander M. Kasprzyk}
\address{Department of Mathematics and Statistics\\University of New Brunswick\\Fredericton NB\\E3B 5A3\\Canada.}
\email{kasprzyk@unb.ca}

\maketitle

\blankfootnotetext{2000 \textit{Mathematics Subject Classification}. Primary 14J45; Secondary 14J30, 14M25, 52B20.}
\blankfootnotetext{\textit{Key words and phrases}. Toric, Fano, threefold, canonical singularities, convex polytopes.}

\begin{abstract}
An inductive approach to classifying all toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are $674,\!688$ such varieties.
\end{abstract}

\section{Introduction}

Recall that a normal projective variety $X$ with log terminal singularities such that the anticanonical divisor $-K_X$ is an ample $\Q$-Cartier divisor is said to be \emph{Fano}. A nonsingular Fano surface is usually called a \emph{del Pezzo surface}. Their classification is well known: $\Proj^2$, $\Proj^1\times\Proj^1$, and $\Proj^2$ blown up in at most eight points (in general position). Of these, the first five are toric. Nonsingular Fano threefolds have also been classified. There are seventeen families with Picard number one, and eighty-nine other families~(\cite{Isk79a, Isk79b, MU82, Sok79, Cut89, Tak89, MM04}).

A great deal more can be said concerning nonsingular toric Fano varieties (\cite{Wis01,FS03}). There are eighteen smooth toric Fano threefolds (\cite{Bat81,Baty91,WW82}), and $124$ smooth toric Fano fourfolds (\cite{Bat99,Sat00}). An inductive algorithm for classifying the smooth toric Fano $n$-folds was recently described in~\cite{KN07}. This algorithm requires knowledge of the Gorenstein toric Fano $(n-1)$-folds and, using the data from~\cite{KS00}, allowed the classification of the five-folds. {\O}bro has presented a different algorithm based on the ingenious notion of \emph{special facets}; using this method dimensions six, seven, and eight have now been classified (see~\cite{Obr07}).

In~\cite{KMM92} it was shown that the degree $(-K_X)^n$ of any smooth Fano variety $X$ of dimension $n$ is bounded, as is the number of deformation types. Similar results are not known for Fano varieties in general; but the number of isomorphism classes of toric Fano varieties of fixed dimension and bounded discrepancy is known to be finite (see~\cite{BB92,Bor00}). It thus makes sense to look for complete classifications in the toric setting beyond the smooth cases.

Gorenstein toric Fano varieties have been classified up to dimension four. There are $16$, $4319$, and $473,\!800,\!776$  isomorphism classes in, respectively, dimension two, three, and four (see~\cite{KS97,KS98b,KS00}). These classifications are of particular interest: Gorenstein toric Fano varieties are used to construct mirror pairs of Calabi-Yau varieties (see~\cite{Bat94,BaBo96,KS02}).

One can also attempt to classify those toric Fano varieties with at worst terminal singularities. Every surface of this form is nonsingular, and so the classification reduces to the smooth case above. In three dimensions, the author showed in~\cite{Kas03} that there are (up to isomorphism) $634$ varieties, of which $233$ are $\Q$-factorial and $100$ are Gorenstein.

All the above classifications are subsets of a more general case: toric Fano varieties with at worst canonical singularities. Here the surface case reduces to the Gorenstein case. This paper describes an inductive approach to achieving a classification in higher dimensions. As an application, the classification for threefolds is calculated. There are $674,\!688$ isomorphism classes. As well as encapsulating the three--dimensional classifications mentioned above, it is worth observing that $12,\!190$ of the resulting varieties are $\Q$-factorial (of which the Picard number is bounded by $\rho\leq 7$). The classification is available online via the Graded Rings Database (\cite{Bro07}) at \href{http://malham.kent.ac.uk}{\texttt{http://malham.kent.ac.uk/}}.

The various classifications are summarised in Table~\ref{tab:classifications}.

\begin{table}[htdp]
\begin{center}
\begin{tabular}{|r||c||c|c|c||c|c|c|}
\hline
&&\multicolumn{3}{c||}{Terminal}&\multicolumn{3}{c|}{Canonical}\\
$n$&Smooth&Gorenstein&$\Q$-factorial&Total&Gorenstein&$\Q$-factorial&Total\\
\hline
2&5&5&5&5&16&16&16\\
3&18&100&233&634&4,319&12,190&674,688\\
4&124&&&&473,800,776&&\\
5&866&&&&&&\\
6&7,622&&&&&&\\
7&72,256&&&&&&\\
8&749,892&&&&&&\\
\hline
\end{tabular}
\end{center}
\caption{Known classifications of toric Fano $n$-folds.}
\label{tab:classifications}
\end{table}

\begin{acknowledge}
The author would like to express his gratitude to Dr.~G.~K.~Sankaran for his invaluable explanations and advice. A special acknowledgement is due to Professor Alexander Borisov for making~\cite{BB} available; the current paper was inspired by the ideas developed in that unpublished work. Thanks also to Dr.~Gavin Brown and the IMSAS at the University of Kent for hosting the final classification online in a searchable format, to Michael Kerber for assistance with the web interface, and to an anonymous referee for several useful observations.

A significant portion of this work was funded by an Engineering and Physical Sciences Research Council (EPSRC) studentship, and forms part of the author's PhD thesis~(\cite{KasPhD}). The author is currently funded by an ACEnet Postdoctoral Research Fellowship.
\end{acknowledge}

\section{Fano Polytopes}\label{sec:Fano_polytopes}

A \emph{toric variety} is a normal variety $X$ that contains an algebraic torus as a dense open subset, together with an action of the torus on $X$ which extends the natural action of the torus on itself. For further details see~\cite{Oda78,Dan78,Ful93}. We shall briefly review the properties we need, and in so doing fix our notation.

Let $M\cong\Z^n$ be the lattice of characters of the torus, with dual lattice $N:=\mathrm{Hom}(M,\Z)$. Every toric variety $X$ of dimension $n$ has an associated fan $\Delta$ in $N_\R:=N\otimes_\Z\R$. The converse also holds; to any fan $\Delta$ there is an associated toric variety $X(\Delta)$. Let $\set{\rho_i}_{i\in I}$ be the set of rays of $\Delta$. For each $i\in I$ there exists a unique primitive lattice element of $\rho_i$, which by a traditional abuse of notation we continue to denote $\rho_i$. $X$ is Fano if and only if $\set{\rho_i}_{i\in I}$ correspond to the vertices of a convex polytope in $N_\R$ (see, for example,~\cite{Dan78}).

A normal variety $X$ is \emph{$\Q$-factorial} if every prime divisor $\Gamma\subset X$ has a positive integer multiple
$c\Gamma$ which is a Cartier divisor. Once again, for the toric case there exists a well known description in terms of the fan. The toric variety $X$ is $\Q$-factorial if and only if the fan $\Delta$ is simplicial.

We say that a fan $\Delta$ is \emph{terminal} if each cone $\sigma\in\Delta$ satisfies the following:
\begin{enumerate}
\item The rays $\rho_1,\ldots,\rho_k$ of $\sigma$ are contained in an affine hyperplane $H:(u(v)=1)$ for some $u\in
 M_{\Q}$;
\item\label{cond:hyperplane_terminal} There are no other elements of the lattice $N$ in the part of $\sigma$ under or on $H$ (i.e. $N\cap\sigma\cap
 (u(v)\leq1)=\set{0,\rho_1,\ldots,\rho_k}$).
\end{enumerate}

A toric variety $X$ is \emph{terminal} (i.e. has at worst terminal singularities) if and only if the fan $\Delta$ is terminal. Relaxing condition~\eqref{cond:hyperplane_terminal} slightly to allow lattice points on $H$, one obtains the definition of a \emph{canonical} fan. $X$ has (at worst) canonical singularities if and only if the fan $\Delta$ is canonical~(\cite{Reid83M}).

\begin{defn}
Let $P\subset N_\R$ be a convex lattice polytope containing only the origin as a strictly interior lattice point (i.e.~$P^\circ\cap N=\set{0}$). We call such a polytope \emph{Fano}. If in addition the only boundary lattice points of $P$ are the vertices (i.e.~$\partial P\cap N=\mathrm{vert}\,{P}$) then we call $P$ a \emph{terminal Fano polytope}. Otherwise we call $P$ a \emph{canonical Fano polytope}.
\end{defn}

Clearly there is an equivalence between terminal (resp. canonical) Fano polytopes and toric Fano varieties with at worst terminal (resp. canonical) singularities. Two toric Fano $n$-folds are isomorphic if and only if the corresponding Fano polytopes are unimodular equivalent; i.e. equivalent up to a linear unimodular transformation from $GL(n,\Z)$.

In~\cite{Kas03} a classification of toric Fano threefolds with at worst terminal singularities was given. The method employed relied on an approach first outlined in~\cite{BB}. It depends on the polytopal description of a toric Fano variety, and can be summarised in two steps:
\begin{itemize}
\item[(i)] Classify all the ``minimal'' polytopes;
\item[(ii)] Inductively ``grow'' these minimal polytopes.
\end{itemize}

Let us explain this algorithm in more detail. First we shall define what we mean by minimal:

\begin{defn}\label{defn:canon_min}
Let $P$ be a canonical (resp. terminal) Fano $n$-tope. We say that $P$ is \emph{minimal} if, for all $\rho\in\mathrm{vert}\,P$, the polytope $\conv\!(P\cap N\setminus\set{\rho})$ obtained by subtracting $\rho$ from $P$ is not a canonical (resp. terminal) Fano $n$-tope.
\end{defn}

Notice that in the canonical case we are only required to check that the origin is not contained in the interior of any of the smaller polytopes obtained by subtracting a vertex. Our use of Fano and minimal will often be relative to some obvious subspace. Such occurrences should not cause any confusion. This is a common theme when considering lattice polytopes: for example, when talking about the volume of a face, one usually means the lattice volume of the face in the appropriate sublattice.

\begin{eg}\label{eg:minimal_examples}
Let $P:=\sconv{\pm e_1,\pm e_2}$, where $e_1$ and $e_2$ form a basis for $N$. $P$ is the terminal Fano polygon associated with $\Proj^1\times\Proj^1$. Let $P':=\sconv{\pm e_1}\subset P$. $P'$ is the one-dimensional terminal Fano polytope associated with $\Proj^1$. Both $P$ and $P'$ are examples of minimal Fano polytopes (in two and one dimension respectively).
\end{eg}

Given a Fano polytope $P$ one can enlarge (or ``grow'') it to $P'=\conv\!(P\cup\set{v})$ by the addition of a lattice point $v\in N$, and evaluate whether $P'$ is also a Fano polytope. Clearly, if one starts with the minimal Fano polytopes, one will achieve a complete classification using this technique.

The number of possible lattice points that can be added to $P$ to create a Fano polytope is finite. Assume that $P'$ is Fano, and consider the ray passing through the origin and $-v$. It will intersect $\partial P$ in a point $x$ on some face $F$ not containing $v$. Let $S\subset\mathrm{vert}\,P\cap F$ be of smallest size such that $x\in\conv S$; say $\abs{S}=d$, where $d\leq n$ is as small as possible. Then $\conv\!(S\cup\set{v})$ is a $d$-simplex containing the origin strictly in its (relative) interior. In other words, $\conv\!(S\cup\set{v})$ is a Fano $d$-simplex; there are finitely many of these by, for example,~\cite{BB92,Bor00}.

Thus we have an algorithm for finding all possible Fano polytopes $P'$ which can be obtained from $P$. What we require is a classification of the Fano $d$-simplices, for $d\leq n$ (actually it is sufficient to know the possible weights). Such a classification can be obtained from the techniques in~\cite{BB92} (see also~\cite{Con02,Kas08b}).

What remains to be described is a method for constructing the minimal Fano polytopes. We shall prove an inductive description of these minimal Fano polytopes in Proposition~\ref{prop:decomposition}. It shall be seen that an understanding of these minimal Fano polytopes reduces to an understanding of the Fano $d$-simplices for all $d\leq n$.

Finally, in Section~\ref{sec:min_canonical_threefolds}, we shall find all minimal canonical Fano $3$-topes. A computer can then be used to establish a complete classification of toric Fano threefolds with canonical singularities. The resulting classification is summarised in Section~\ref{sec:class_canonical_threefolds}.

\section{Decomposition of Minimal Fano Polytopes}

The results in this section should be compared with~\cite{KS97}. It should be stressed that the results ignore the lattice point structure of the Fano polytope; only the property that the Fano polytope contains the origin in its interior is relevant.

Let $x_0,\ldots,x_n\in N_\R\cong\R^n$ be such that $P:=\sconv{x_0,\ldots,x_n}$ is an $n$-simplex with $0\in P^\circ$. To this simplex we associate the complete fan $\Delta:=\Delta(P)$ given by the cones over the faces of $P$; i.e. generated by
$$\sigma_i:=\scone{x_0,\ldots,\hat{x}_i,\ldots,x_n},\qquad\text{ where }i=0,\ldots,n.$$
$\hat{x}_i$ indicates that the vertex $x_i$ is omitted.

The following lemma is immediate:

\begin{lem}\label{lem:minus_fan}
With notation as above, let $x\in N_\R$. Then $x\in(-\sigma_i)^\circ$ if and only if
$$P':=\sconv{x_0,\ldots,\hat{x}_i,\ldots,x_n,x}$$
is an $n$-simplex with $0\in{P'}^\circ$, and $\Delta(P')$ is a complete fan.
\end{lem}

We are now in a position to prove the main result of this section:

\begin{prop}\label{prop:decomposition}
Any minimal canonical (resp.~terminal) Fano $n$--tope $P$ is either a simplex, or can be written as $P=\conv\!(S\cup P')$ for some $S$ a minimal canonical (resp.~terminal) Fano $k$--simplex and $P'$ a minimal canonical (resp.~terminal) Fano $(n-k+r)$--tope, where $0\leq r<k<n$, moreover, $\dim(\mathrm\,S\cap\mathrm\,P')\leq r$, and $r$ equals the number of common vertices of $S$ and $P'$.
\end{prop}

\begin{proof}
We assume that $P$ is not a simplex. Let $x_0,\ldots,x_l$ be the vertices of $P$, where $l>n$. Without loss of generality we may assume that $x_0,\ldots,x_n$ do not lie in a hyperplane and that $0\in\sconv{x_0,\ldots,x_n}$.

Minimality of $P$ ensures that $0\notin\sconv{x_0,\ldots,x_n}^\circ$. Hence the origin must lie on some facet, and we may assume (with a possible reordering) that $0\in\sconv{x_0,\ldots,x_k}^\circ$ for some $k<n$. We obtain the $k$--simplex $S:=\sconv{x_0,\ldots,x_k}$. $S$ is minimal and Fano since $P$ is; if $P$ is terminal then $S$ must be terminal.

Let $P'':=\sconv{x_{k+1},\ldots,x_l}$, so $P=\conv\!(S\cup P'')$. Let $\Gamma$ be the $k$-dimensional subspace of $N_\R$ containing $S$. Since the $x_i$ are vertices we have that $\set{x_0,\ldots,x_k}\cap P''=\emptyset$, and since $P$ is minimal we have that $\set{x_{k+1},\ldots,x_l}\cap\Gamma=\emptyset$. It must also be that ${P''}^\circ\cap\Gamma\neq\emptyset$, otherwise $0$ would lie in a facet of $P$. Let $m:=\dim({P''}^\circ\cap\Gamma)$. A dimension count reveals that $\dim{P''}=n-k+m$.

By minimality of $P$ and Lemma~\ref{lem:minus_fan} we have that ${P''}^\circ\cap\Gamma\subset-\sigma$ for some $r$-dimensional simplicial cone $\sigma\in\Delta(S)$, where $k>r\geq m$. Since $\set{0}$ is the apex of $-\sigma$ we have that either $\set{0}={P''}^\circ\cap\Gamma$ or $0\notin{P''}^\circ\cap\Gamma$. The first case gives us that $P''$ is a minimal Fano $(n-k)$--tope (which is necessarily terminal if $P$ is terminal), so by setting $P'=P''$ we are done. For the second possibility we may assume that $\sigma=\scone{x_{k-r+1},\ldots,x_k}$ and construct the polytope $P':=\sconv{x_{k-r+1},\ldots,x_l}$. By construction $\dim P'=n-k+r$, and by Lemma~\ref{lem:minus_fan} we have that $0\in{P'}^\circ$. Hence $P'$ is our desired minimal Fano $(n-k+r)$--tope.
\end{proof}

From Proposition~\ref{prop:decomposition} we may conclude the following two corollaries, which are well-known results of Steinitz.

\begin{cor}\label{cor:max_minimal_vertices}
Any minimal Fano polytope $P$ has at most $2\dim{P}$ vertices.
\end{cor}

\begin{cor}\label{cor:max_vertices_centrally_sym}
Let $P$ be a minimal Fano polytope such that $\abs{\mathrm{vert}\,P}=2\dim{P}$. Then $P$ is centrally symmetric.
\end{cor}

For $k>1$, no $k$--simplex is centrally symmetric. Hence Corollary~\ref{cor:max_vertices_centrally_sym} is actually an ``if and only if''.

A characterisation of centrally symmetric simplicial \emph{reflexive} Fano polytopes is given in~\cite{Nill07}. These polytopes can always be embedded in the $n$-cube $\sconv{\pm e_1\pm\ldots\pm e_n}$.

\section{Minimal Canonical Fano Threefolds}\label{sec:min_canonical_threefolds}

For the convenience of the reader we begin by summarising the main results of this section in the following theorem (see also Tables~\ref{tab:minimal_tet} and~\ref{tab:minimal_canonical_dim3}):

\begin{thm}
There are $26$ minimum Fano polytopes in dimension three, up to the action of $GL(3,\Z)$. Of these sixteen are tetrahedra.
\end{thm}

First we shall describing which of the Fano tetrahedra are minimal. We do this by restricting the possible weights which may occur.

\begin{figure}[htbp]
\centering
\subfigure[]{\includegraphics[scale=.8]{P2}}
\hspace{.2in}
\subfigure[]{\includegraphics[scale=.8]{P112}}
\hspace{.2in}
\subfigure[]{\includegraphics[scale=.8]{P123}}
\hspace{.2in}
\subfigure[]{\includegraphics[scale=.8]{P112Z2}}
\hspace{.2in}
\subfigure[]{\includegraphics[scale=.8]{P2Z3}}
\caption{The five Fano triangles, with weights $(1,1,1)$, $(1,1,2)$, $(1,2,3)$, $(1,1,2)$, and $(1,1,1)$ respectively. Only (a) and (b) are minimal.}
\label{fig:Fano_triangles}
\end{figure}

\begin{defn}
Let $S$ be a Fano tetrahedron. We say that $S$ has \emph{weights} $(\lambda_0,\lambda_1,\lambda_2,\lambda_3)\in\Z_{>0}^4$ if:
$$\lambda_0x_0+\lambda_1x_2+\lambda_2x_2+\lambda_3x_3=0,$$
where the $x_i$ are the vertices of $S$, labelled in some order.
\end{defn}

Weights are unique up to reordering and scalar multiplication. It is useful to normalise them by insisting that $\lambda_0\leq\lambda_1\leq\lambda_2\leq\lambda_3$ and that $\gcd\!\set{\lambda_0,\lambda_1,\lambda_2,\lambda_3}=1$.

Before we continue, we need to be familiar with the Fano triangles. The Fano polytopes are well documented in the literature, more often than not appearing alongside an original method of proof. Consult, for example,~\cite{KS97, Sat00, PR00, Nill05}. The triangles are illustrated in Figure~\ref{fig:Fano_triangles}.

\begin{prop}\label{prop:min_canonical_tet_weights}
Let $P$ be a minimal Fano tetrahedron. The possible weights for $P$ are:
\begin{center}
\begin{tabular}{lllll}
$(1,1,1,1)$,&$(1,1,1,2)$,&$(1,1,1,3)$,&$(1,1,2,2)$,&$(1,1,2,3)$,\\
$(1,1,2,4)$,&$(1,1,3,4)$,&$(1,1,3,5)$,&$(1,1,4,6)$,&$(1,2,3,5)$,\\
$(1,3,4,5)$,&$(2,2,3,5)$,&$(2,3,5,7)$,&\multicolumn{2}{l}{or $(3,4,5,7)$.}
\end{tabular}
\end{center}
\end{prop}

\begin{proof}
If $P$ is terminal then the possible weights are listed in~\cite[Proposition 1.8]{Kas03}; they are $(1,1,1,1)$, $(1,1,1,2)$, $(1,1,2,3)$, $(1,2,3,5)$, $(1,3,4,5)$, $(2,3,5,7)$, and $(3,4,5,7)$.

Suppose that $P=\sconv{x_0,x_1,x_2,x_3}$ is not terminal. Minimality dictates that no edge of $P$ can contain more than one interior lattice point. Let $x\in\partial P\cap N\setminus\mathrm{vert}\,P$. Since the fan $\Delta$ of $P$ is complete, so $x\in-\sigma$ for some cone $\sigma\in\Delta$ of smallest possible dimension. In particular $\dim\sigma\leq 2$, otherwise $P$ is not minimal, hence $\sigma\preccurlyeq\scone{x_0,x_1}$ without loss of generality. Because of minimality we may suppose that any non-vertex lattice point in $\sconv{x_1,x_2,x_3}$ is contained in $-\scone{x_0,x_1}$.

\noindent\underline{$\dim\sigma=1$:}

Let $x=-x_0$, where $x$ is in the interior of the face $\sconv{x_1,x_2,x_3}$, and the line segment $x_1,x$ is lattice point free. There are two possibilities: either there is a second non-vertex lattice point in the face, or there isn't.
\begin{enumerate}
\item
If $x$ is the only non-vertex lattice point in the face then we may regard $\sconv{x_1,x_2,x_3}$ as the Fano triangle (a) in Figure~\ref{fig:Fano_triangles}, with $x$ playing the role of the origin. Hence:
$$\frac{1}{3}(x_1+x_2+x_3)=-x_0,$$
and we obtain weights $(1,1,1,3)$.
\item
Suppose that there exists a second non-vertex lattice point $x'\in\conv\set{x_1,x_2,x_3}$. Then $\sconv{x',x_0,x_1}$ is a Fano triangle with $x$ on the edge joining $x_1$ and $x'$. We may choose $x'$ to be as far from $x_1$ as possible.
\begin{enumerate}
\item
$x'$ lies on the edge joining $x_2$ and $x_3$. In which case, $x'=(1/2)(x_2+x_3)$. There are only two possible Fano triangles: (b) and (c) in Figure~\ref{fig:Fano_triangles}. The former gives:
$$2x_0+x_1+\frac{1}{2}(x_2+x_3)=0,$$
and hence $P$ has weights $(1,1,2,4)$. The latter gives:
$$3x_0+2x_1+\frac{1}{2}(x_2+x_3)=0,$$
yielding weights $(1,1,4,6)$.
\item
$x'$ does not lie on the edge joining $x_2$ and $x_3$. There are no lattice points on the line segment between $x_0$ and $x'$, hence the Fano triangle $\sconv{x',x_0,x_1}$ can only be (b) (observe that (c) is impossible since there are no lattice points between $x_1$ and $x=-x_0$), and so $x'=-2x_0-x_1$. In particular, $x'$ is the only lattice point in the triangle $\sconv{x,x_2,x_3}$, hence:
$$\frac{1}{3}(x_2+x_3-x_0)=-2x_0-x_1.$$
This gives weights $(1,1,3,5)$.
\end{enumerate}
\end{enumerate}

\noindent\underline{$\dim\sigma=2$:}

We have that $\sigma=\sconv{x_0,x_1}$ and may assume that $-x_0$ and $-x_1$ are not a lattice points in the polytope, otherwise we can reduce to the previous case. Let us choose $x$ to be as far from $x_1$ as is possible. Furthermore, minimality gives that any non-vertex lattice point in $\sconv{x_1,x_2,x_3}$ must be contained in $-\scone{x_0,x_1}$.
\begin{enumerate}
\item
Suppose that $x$ lies on that edge joining $x_2$ and $x_3$. Then $x=(1/2)(x_2+x_3)$.

In this case, since the edge joining $x_0$ and $x_1$ contains at most one interior lattice point, the Fano triangle $\sconv{x,x_0,x_1}$ must be equivalent to (a), (b), or (c) from Figure~\ref{fig:Fano_triangles} (note that (d) is impossible, since $-x_0$ or $-x_1$ would be lattice points in the polytope). (a) gives equation $x+x_0+x_1=0$, yielding weights $(1,1,2,2)$. For (b) we obtain $2x+x_0+x_1=0$, giving weights $(1,1,1,1)$.

Finally we consider (c). Notice that $-x_0$ is not in the face by assumption, hence either $x+2x_0+3x_1=0$ or $2x+x_0+3x_1=0$. The second possibility gives us the lattice points $-x_1$ and $x_0+x_1+x$ on the face $\sconv{x_0,x_2,x_3}$, where the second point is closer to $x_0$ than the first. This contradicts minimality. Hence the only possibility is $(1,1,4,6)$.
\item
If $x$ does not lie on the edge joining $x_2$ and $x_3$ then $x$ is, say, in the interior of $\sconv{x_1,x_2,x_3}$, and the only possible Fano triangles for $\sconv{x,x_0,x_1}$ are (a), (b), and (c) (since the edge joining $x_0$ and $x$ must be lattice point free). (a) tells us that $x$ is the only non-vertex lattice point in the face $\sconv{x_1,x_2,x_3}$, so we obtain:
$$\frac{1}{3}(x_1+x_2+x_3)=-x_0-x_1.$$
This gives weights $(1,1,3,4)$.

Since $-x_0$ is not in the face, (b) gives us that the face has only one non-vertex lattice point. Hence:
$$\frac{1}{3}(x_1+x_2+x_3)=-\frac{1}{2}(x_0+x_1),$$
yielding weights $(2,2,3,5)$.

Possibility (c) contradicts the assumption that $-x_0$ and $-x_1$ are not in the polytope.
\end{enumerate}
\end{proof}

\begin{table}[htdp]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
$(1,1,1,1)$&$(1,1,1,1)$&$(1,1,1,1)$&$(1,1,1,2)$\\
\hline
$\begin{pmatrix}-1&1&0&0\\-1&0&1&0\\-1&0&0&1\\\end{pmatrix}$&
$\begin{pmatrix}-2&2&0&0\\-2&1&1&0\\-1&0&0&1\\\end{pmatrix}$&
$\begin{pmatrix}-5&5&0&0\\-3&2&1&0\\-2&1&0&1\\\end{pmatrix}$&
$\begin{pmatrix}-1&1&0&0\\-1&0&1&0\\-2&0&0&1\\\end{pmatrix}$\\
\hline\hline
$(1,1,1,3)$&$(1,1,2,2)$&$(1,1,2,3)$&$(1,1,2,4)$\\
\hline
$\begin{pmatrix}-1&1&0&0\\-1&0&1&0\\-3&0&0&1\\\end{pmatrix}$&
$\begin{pmatrix}-1&1&0&0\\-2&0&1&0\\-2&0&0&1\\\end{pmatrix}$&
$\begin{pmatrix}-1&1&0&0\\-2&0&1&0\\-3&0&0&1\\\end{pmatrix}$&
$\begin{pmatrix}-1&1&0&0\\-2&0&1&0\\-4&0&0&1\\\end{pmatrix}$\\
\hline\hline
$(1,1,3,4)$&$(1,1,3,5)$&$(1,1,4,6)$&$(1,2,3,5)$\\
\hline
$\begin{pmatrix}-1&1&0&0\\-3&0&1&0\\-4&0&0&1\\\end{pmatrix}$&
$\begin{pmatrix}-1&1&0&0\\-3&0&1&0\\-5&0&0&1\\\end{pmatrix}$&
$\begin{pmatrix}-1&1&0&0\\-4&0&1&0\\-6&0&0&1\\\end{pmatrix}$&
$\begin{pmatrix}-2&1&0&0\\-3&0&1&0\\-5&0&0&1\\\end{pmatrix}$\\
\hline\hline
$(1,3,4,5)$&$(2,2,3,5)$&$(2,3,5,7)$&$(3,4,5,7)$\\
\hline
$\begin{pmatrix}-3&1&0&0\\-4&0&1&0\\-5&0&0&1\\\end{pmatrix}$&
$\begin{pmatrix}-1&1&0&0\\-3&0&2&0\\-4&0&1&1\\\end{pmatrix}$&
$\begin{pmatrix}-3&2&0&0\\-4&1&1&0\\-5&1&0&1\\\end{pmatrix}$&
$\begin{pmatrix}-4&3&0&0\\-3&1&1&0\\-5&2&0&1\\\end{pmatrix}$\\
\hline
\end{tabular}
\caption{The sixteen minimal canonical Fano tetrahedra.}
\label{tab:minimal_tet}
\end{table}

Knowing the weights, we can find the associated tetrahedra. We shall require the following result:

\begin{prop}[\protect{\cite[Proposition~2]{BB92}}]\label{prop:generating_fan_unique}
For any weights $(\lambda_0,\lambda_1,\ldots,\lambda_n)$ such that $\gcd\!\set{\lambda_0,\lambda_1,\ldots,\lambda_n}=1$,  let $\rho_0,\rho_1,\ldots,\rho_n\in N$ be the primitive generators for the fan of $\Proj(\lambda_0,\lambda_1,\ldots,\lambda_n)$. Then:
\begin{itemize}
\item[(i)] $\lambda_0\rho_0+\lambda_1\rho_1+\ldots+\lambda_n\rho_n=0$;
\item[(ii)] The $\rho_i$ generate the lattice $N$.
\end{itemize}

Furthermore, if $\rho'_0,\rho'_1,\ldots,\rho'_n$ is any set of primitive lattice elements satisfying (i) and (ii) then there exists a transformation in $GL(n,\Z)$ sending $\rho_i$ to $\rho'_i$ for $i=0,1,\ldots,n$.
\end{prop}

\begin{thm}
There are sixteen minimal Fano tetrahedra, whose vertices are listed (up to the action of $GL(3,\Z)$) in Table~\ref{tab:minimal_tet}.
\end{thm}

\begin{proof}
The terminal Fano tetrahedra are listed in~\cite[Table~4]{Kas03}. We need only consider the canonical cases.

From the proof of Proposition~\ref{prop:min_canonical_tet_weights} we can see when the vertices of a minimal tetrahedron generate the lattice $N$. When this is the case, Proposition~\ref{prop:generating_fan_unique} tells us that the tetrahedron corresponds to weighted projective space. This is the only possibility for all weights except $(1,1,1,1)$ (in the notation of the proof, we are considering $\dim\sigma=2$, case~(1)(b)). This gives a tetrahedron whose vertices generate an index two sublattice. This corresponds to a fake weighted projective space of index two;~\cite{Con02} describes how to compute the vertices of the tetrahedron.

\end{proof}

It should be emphasised that not every Fano tetrahedron is minimal. As mentioned in~\cite[pg.~278]{BB92}, there are a total of $225$ Fano tetrahedra; see the appendix of~\cite{BB} for the complete list. This has been verified by the author using the bounds described in~\cite{Kas08b}. There are $104$ distinct weights, which are listed in Table~\ref{tab:all_tet_weights}.

\begin{table}[htdp]
\centering
\begin{tabular}{|c|c|}
\hline
Weights&Sum\\
\hline
$(1,1,1,1)$&$4$\\
$(1,1,1,2)$&$5$\\
$(1,1,1,3)$&$6$\\
$(1,1,2,2)$&$6$\\
$(1,1,2,3)$&$7$\\
$(1,1,2,4)$&$8$\\
$(1,2,2,3)$&$8$\\
$(1,1,3,4)$&$9$\\
$(1,2,3,3)$&$9$\\
$(1,1,3,5)$&$10$\\
$(1,2,2,5)$&$10$\\
$(1,2,3,4)$&$10$\\
$(1,2,3,5)$&$11$\\
$(1,1,4,6)$&$12$\\
$(1,2,3,6)$&$12$\\
$(1,2,4,5)$&$12$\\
$(1,3,4,4)$&$12$\\
$(2,2,3,5)$&$12$\\
$(2,3,3,4)$&$12$\\
$(1,3,4,5)$&$13$\\
$(1,2,4,7)$&$14$\\
$(2,2,3,7)$&$14$\\
$(2,3,4,5)$&$14$\\
$(1,2,5,7)$&$15$\\
$(1,3,4,7)$&$15$\\
$(1,3,5,6)$&$15$\\
\hline
\end{tabular}
\begin{tabular}{|c|c|}
\hline
Weights&Sum\\
\hline
$(2,3,5,5)$&$15$\\
$(3,3,4,5)$&$15$\\
$(1,2,5,8)$&$16$\\
$(1,3,4,8)$&$16$\\
$(1,4,5,6)$&$16$\\
$(2,3,4,7)$&$16$\\
$(2,3,5,7)$&$17$\\
$(1,2,6,9)$&$18$\\
$(1,3,5,9)$&$18$\\
$(1,4,6,7)$&$18$\\
$(2,3,4,9)$&$18$\\
$(2,3,5,8)$&$18$\\
$(3,4,5,6)$&$18$\\
$(3,4,5,7)$&$19$\\
$(1,4,5,10)$&$20$\\
$(1,5,6,8)$&$20$\\
$(2,3,5,10)$&$20$\\
$(2,4,5,9)$&$20$\\
$(2,5,6,7)$&$20$\\
$(3,4,5,8)$&$20$\\
$(1,3,7,10)$&$21$\\
$(1,4,7,9)$&$21$\\
$(1,5,7,8)$&$21$\\
$(2,3,7,9)$&$21$\\
$(3,5,6,7)$&$21$\\
$(1,3,7,11)$&$22$\\
\hline
\end{tabular}
\begin{tabular}{|c|c|}
\hline
Weights&Sum\\
\hline
$(1,4,6,11)$&$22$\\
$(2,4,5,11)$&$22$\\
$(1,3,8,12)$&$24$\\
$(1,6,8,9)$&$24$\\
$(2,3,7,12)$&$24$\\
$(2,3,8,11)$&$24$\\
$(2,5,8,9)$&$24$\\
$(3,4,5,12)$&$24$\\
$(3,4,7,10)$&$24$\\
$(3,6,7,8)$&$24$\\
$(4,5,6,9)$&$24$\\
$(4,5,7,9)$&$25$\\
$(1,5,7,13)$&$26$\\
$(2,3,8,13)$&$26$\\
$(2,5,6,13)$&$26$\\
$(2,5,9,11)$&$27$\\
$(5,6,7,9)$&$27$\\
$(1,4,9,14)$&$28$\\
$(1,5,8,14)$&$28$\\
$(3,4,7,14)$&$28$\\
$(3,7,8,10)$&$28$\\
$(4,6,7,11)$&$28$\\
$(1,4,10,15)$&$30$\\
$(1,6,8,15)$&$30$\\
$(2,3,10,15)$&$30$\\
$(2,6,7,15)$&$30$\\
\hline
\end{tabular}
\begin{tabular}{|c|c|}
\hline
Weights&Sum\\
\hline
$(3,4,10,13)$&$30$\\
$(4,5,6,15)$&$30$\\
$(4,7,9,10)$&$30$\\
$(5,6,8,11)$&$30$\\
$(2,5,9,16)$&$32$\\
$(4,5,7,16)$&$32$\\
$(3,5,11,14)$&$33$\\
$(5,8,9,11)$&$33$\\
$(3,4,10,17)$&$34$\\
$(4,6,7,17)$&$34$\\
$(1,5,12,18)$&$36$\\
$(3,4,11,18)$&$36$\\
$(3,7,8,18)$&$36$\\
$(7,8,9,12)$&$36$\\
$(3,5,11,19)$&$38$\\
$(5,6,8,19)$&$38$\\
$(5,7,8,20)$&$40$\\
$(1,6,14,21)$&$42$\\
$(2,5,14,21)$&$42$\\
$(3,4,14,21)$&$42$\\
$(4,5,13,22)$&$44$\\
$(5,8,9,22)$&$44$\\
$(3,5,16,24)$&$48$\\
$(7,8,10,25)$&$50$\\
$(4,5,18,27)$&$54$\\
$(5,6,22,33)$&$66$\\
\hline
\end{tabular}
\caption{The $104$ distinct weights occuring for the $225$ Fano tetrahedra.}
\label{tab:all_tet_weights}
\end{table}

Proposition~\ref{prop:decomposition} allows us to calculate the non-simplex minimal Fano $3$-topes. Assume we have chosen $S$ and $P'$ such that $k$ is as small as possible. If $k=1$ then $r=0$ and we have that $S$ is the polytope for $\Proj^1$, and $P'$ is a minimal Fano polygon (the minimal Fano polygons are the triangles (a) and (b) in Figure~\ref{fig:Fano_triangles} and the polygon associated with $\Proj^1\times\Proj^1$ mentioned in Example~\ref{eg:minimal_examples}). These possibilities are classified in Lemmas~\ref{lem:3and3}--\ref{lem:3and1}. The alternative is that $k=2$. Since the polygon for $\Proj^1\times \Proj^1$ contains the polytope for $\Proj^1$, it can be excluded; we need only consider the cases when $r=1$ and $P'$ is a minimal Fano triangle. Hence the Fano polytope has five vertices. These cases will be classified in Lemmas~\ref{lem:2and2}--\ref{lem:1and1}. We find that there are exactly ten non-simplex minimal Fano polytopes in dimension three. The results are collated in Table~\ref{tab:minimal_canonical_dim3}.

Once the minimal polytopes are known, the following result is immediate\footnote{My thanks to Professor Victor Batyrev for this observation.}:

\begin{thm}\label{thm:canonical_Fano_degree_bound}
Let $X$ be a toric Fano threefold with at worst canonical singularities. Then $(-K_X)^3\leq 72$. If $(-K_X)^3=72$ then $X$ is isomorphic to $\Proj(1,1,1,3)$ or $\Proj(1,1,4,6)$.
\end{thm}

\begin{proof}
Let $P_X$ be the polytope associated with $X$. There exists a minimal polytope $Q$ such that $Q\subset P_X$, hence $P_X^\vee\subset Q^\vee$. Inspection gives $\mathrm{vol}\,Q^\vee\leq 12$, hence $(-K_X)^3\leq 3!\cdot 12$.
\end{proof}

Theorem~\ref{thm:canonical_Fano_degree_bound} should be compared with the following result, conjectured by Fano and Iskovskikh and proved by Prokhorov:

\begin{thm}[\cite{Pro05}]
Let $X$ be a Gorenstein Fano threefold with at worst canonical singularities. Then $(-K_X)^3\leq 72$. If $(-K_X)^3=72$ then $X$ is isomorphic to $\Proj(1,1,1,3)$ or $\Proj(1,1,4,6)$.
\end{thm}

For the following two results minimality ensures that any such Fano polytope must be at worst terminal; these were classified in~\cite[Lemma~3.4 and~3.5]{Kas03}.

\begin{lem}\label{lem:3and3}
The minimal Fano polytopes obtained from adding the points $\pm x$ to a Fano square are equivalent to:
$$\begin{pmatrix}1&0&0&-1&0&0\\0&1&0&0&-1&0\\0&0&1&0&0&-1\end{pmatrix}\text{ or }
\begin{pmatrix}1&0&-1&0&1&-1\\0&1&0&-1&1&-1\\0&0&0&0&2&-2\end{pmatrix}.$$
\end{lem}

\begin{lem}\label{lem:3and2}
The minimal Fano polytopes containing a Fano triangle equivalent to Figure~\ref{fig:Fano_triangles}~(a), along with a pair of points $\pm x$ not lying in the plane containing the Fano triangle, are equivalent to:
$$\begin{pmatrix}1&0&0&0&-1\\0&1&0&0&-1\\0&0&1&-1&0\end{pmatrix}\text{ or }
\begin{pmatrix}1&0&-1&1&-1\\0&1&-1&2&-2\\0&0&0&3&-3\end{pmatrix}.$$
\end{lem}

\begin{lem}\label{lem:3and1}
Any minimal Fano polytope containing the minimal Fano triangle shown in Figure~\ref{fig:Fano_triangles}~(b), along with a pair of points $\pm x$ not lying in the same subspace as the triangle, is equivalent to one of:
$$\begin{pmatrix}1&0&0&0&-2\\0&1&0&0&-1\\0&0&1&-1&0\end{pmatrix}\text{ or }
\begin{pmatrix}1&0&-2&1&-1\\0&1&-1&1&-1\\0&0&0&2&-2\end{pmatrix}.$$
\end{lem}

\begin{proof}
Arrange matters such that $P:=\sconv{e_1,e_2,-2e_1-e_2,x,-x}$; $x:=\coord{a,b,c}$ is such that $0\leq a,b<c$. Clearly $a=0,b=0,c=1$ is a solution. Let us assume that $c>1$.

Since $x\neq e_3$ we cannot have $e_3\in P$, since then removing $x$ would yield a smaller canonical Fano polytope with vertex $e_3$, contradicting minimality.

Hence $e_3\notin P$ and consider the line connecting $e_3$ to the origin. If $a\geq 2b$ this line intersects $\sconv{-e_1,-2e_1-e_2,x}$ at the point $ke_3$, where $k=c/(a-b+1)$. This tells us that $k<1$, thus $a-b\geq c$, which contradicts our assumptions.

It must be that $a<2b$. The line joining $e_3$ and $0$ intersects $\sconv{e_1,-2e_1-e_2,x}$ at the point $ke_3$, where $k=c/(3b-a+1)$. Hence:
\begin{equation}\label{lem:3and1Eq1}
3b-a\geq c.
\end{equation}

As before $-e_3\notin P$.  The line joining the origin and $-e_3$ intersects $\sconv{e_1,e_2,-x}$ at the point $k(-e_3)$, where $k=c/(a+b+1)$. Thus we obtain:
\begin{equation}\label{lem:3and1Eq2}
a+b\geq c.
\end{equation}

As before $-e_1-e_3\notin P$. The line connecting the origin with $-e_1-e_3$ intersects $\sconv{-e_1,e_2,-x}$ at the point $k(-e_1-e_3)$, where $k=c/(c+b-a+1)$. Hence:
\begin{equation}\label{lem:3and1Eq3}
b\geq a.
\end{equation}

Finally, let us consider the point $-e_1-e_2-e_3$. This point must lie outside $P$, for otherwise $\sconv{e_1,e_2,-e_1-e_2-e_3,x}$ would be a Fano tetrahedron. We consider the line connecting $0$ and this point. If $2b-a>c$ then the line intersects $\sconv{-e_1,-2e_1-e_2,-x}$ at the point $k(-e_1-e_2-e_3)$, where $k=c/(b-a+1)$. But this yields $b-a\geq c$, a contradiction. Hence it must be that $2b-a\leq c$, and the line intersects $\sconv{e_1,-2e_1-e_2,-x}$. This occurs when $k=c/(a-3b+2c+1)$, and gives us:
\begin{equation}\label{lem:3and1Eq4}
c\geq 3b-a.
\end{equation}

Combining equations~\eqref{lem:3and1Eq1} and~\eqref{lem:3and1Eq4} tells us that $c=3b-a$, and by applying equation~\eqref{lem:3and1Eq2} we see that $a\geq b$. Of course equation~\eqref{lem:3and1Eq3} now tells us that $a=b$, and so $x=\coord{a,a,2a}$. This forces $a=1$.
\end{proof}

\begin{lem}\label{lem:2and2}
The minimal Fano polytopes containing two copies of the Fano triangle shown in Figure~\ref{fig:Fano_triangles}~(a) are equivalent to:
$$\begin{pmatrix}1&0&0&-1&1\\0&1&0&-1&1\\0&0&-1&0&1\end{pmatrix}.$$
\end{lem}

\begin{proof}
Let us fix the lattice such that $P:=\sconv{e_1,e_2,-e_1-e_2,x,y}$, where $x:=\sconv{a+1,b+1,c}, y:=\sconv{-a,-b,-c},$ and $0<a+1\leq b+1\leq c$. Clearly $a=0,b=0,c=1$ is a solution. Assume that $c>1$.

By minimality $-e_3$ lies outside $P$. The line connecting $-e_3$ with the origin intersects $\sconv{e_1,e_2,y}$ at the point $-ke_3$, where $k=c/(a+b+1)$. We see that:
\begin{equation}\label{lem:2and2Eq1}
c\leq a+b.
\end{equation}

Consider the point $e_1+e_2+e_3$. The line joining this point and the origin intersects $\sconv{e_1,e_2,x}$ at $k(e_1+e_2+e_3)$, where $k=c/(2c-(a+1)-(b+1)+1).$ If $e_1+e_2+e_3\notin P$ then $(a+1)+(b+1)\leq c$, contradicting equation~\eqref{lem:2and2Eq1}. Hence $e_1+e_2+e_3$ lies on the boundary of $P$, and $(a+1)+(b+1)-1=c$. But again we find that this contradicts equation~\eqref{lem:2and2Eq1}.
\end{proof}

\begin{lem}\label{lem:2and1}
Any minimal Fano polytope containing one copy of each of the two minimal Fano triangles (Figure~\ref{fig:Fano_triangles}~(a) and~(b)) is equivalent to:
$$\begin{pmatrix}1&0&0&-2&-1\\0&1&0&-1&0\\0&0&1&0&-1\end{pmatrix}.$$
\end{lem}

\begin{proof}
Arrange matters so that $P:=\sconv{e_1,e_2,-2e_1-e_2,x,y}$. There are two cases to consider:
\begin{enumerate}
\item[(i)] $x+y+e_2=0$;
\item[(ii)] $x+y+e_1=0$.
\end{enumerate}

Observe that in case (i), the line joining $e_1$ and $-2e_1-e_2$ intersects $\spn{e_2}$ at the point $-(1/3)e_2$, whereas the line joining $x$ and $y$ intersects $\spn{e_2}$ at $-(1/2)e_2$. Hence $P\setminus\set{-2e_1-e_2}$ is still Fano, which contradicts minimality of $P$. Indeed, this case reduces to those polytopes discussed in Lemma~\ref{lem:3and2}.

We now address case (ii).

We have that $x=\coord{a,b,c},y=\coord{-a-1,-b,-c},$ and can insist that $0\leq a,b<c$. Clearly $a=0,b=0,c=1$ is a solution, so suppose that $c>1$. By minimality $e_3\notin P$.

Note that the point $-e_1$ lies on the line joining $e_2$ and $-2e_1-e_2$, whilst the line joining $x$ to $y$ intersects the plane $\spn{e_1,e_2}$ at $-(1/2)e_1$. Hence this line (without the end points) is contained strictly in the interior of $P$.

The point $e_1+e_2+e_3$ lies outside $P$, otherwise $\sconv{e_1, -2e_1-e_2, e_1+e_2+e_3, y}$ is a Fano tetrahedron contained in $P$. The line connecting this point to $0$ must intersect $\sconv{e_1,e_2,x}$. This occurs at $k(e_1+e_2+e_3)$, where $k=c/(2c-a-b+1)$. We thus have:
\begin{equation}\label{lem:2and1Eq3}
a+b\leq c.
\end{equation}

The point $-e_1-e_3$ must lie outside $P$, otherwise $P$ contains the Fano tetrahedron $\sconv{e_1,-2e_1-e_2,-e_1-e_3,x}$, contradicting minimality of $P$. The line originating at $0$ and passing through $-e_1-e_3$ intersects $\partial P$ in either $\sconv{e_1,e_2,y}$ or $\sconv{-e_1,e_2,y}$. The first possibility gives the point of intersection to be $k(-e_1-e_3)$, where $k=c/(a+b-c+2)$, and we have that $a+b+1\geq 2c$. Combining this with equation~\eqref{lem:2and1Eq3} yields a contradiction.

Consider the second possibility; the line connecting $-e_1-e_3$ and the origin intersects $\sconv{-e_1,e_2,y}$ at the point $k(-e_1-e_3)$ where $k=c/(c+b-a)$. We have that:
\begin{equation}\label{lem:2and1Eq7}
b\geq a+1.
\end{equation}

Finally, consider the point $e_2+e_3$. This point must lie outside $P$; if $e_2+e_3$ were contained in $P$, then $\sconv{e_1,e_2+e_3,-2e_1-e_2,y}$ would be a Fano tetrahedron. The line joining the point with the origin intersects $\sconv{e_1,-2e_1-e_2,x}$ or $\sconv{-e_1,e_2,x}$. In the first case the point of intersection is given by $k(e_2+e_3)$, where $k=c/(c-a-b+1)$. Hence  $a+b\leq 0$, which is an impossibility (since $c\neq 1$).

The alternative is that the line intersects $\sconv{-e_1,e_2,x}$. This occurs at the point $k(e_2+e_3)$, where $k=c/(a-b+c+1)$, and we see that $a\geq b$. By considering equation~\eqref{lem:2and1Eq7} we obtain our final contradiction.
\end{proof}

\begin{table}[tdp]
\centering
\begin{tabular}[t]{|c|c|}
\hline
Comments&Vertices\\\hline
\begin{tabular}{c}$5$ Vertices\\Simplicial\\Terminal\end{tabular}&$\begin{pmatrix}1&0&0&0&-1\\0&1&0&0&-1\\0&0&1&-1&0\end{pmatrix}$\\\hline
\begin{tabular}{c}$5$ Vertices\\Simplicial\\Terminal\end{tabular}&$\begin{pmatrix}1&0&-1&1&-1\\0&1&-1&2&-2\\0&0&0&3&-3\end{pmatrix}$\\\hline
\begin{tabular}{c}$5$ Vertices\\Simplicial\end{tabular}&$\begin{pmatrix}1&0&0&0&-2\\0&1&0&0&-1\\0&0&1&-1&0\end{pmatrix}$\\\hline
\begin{tabular}{c}$5$ Vertices\\Simplicial\end{tabular}&$\begin{pmatrix}1&0&-2&1&-1\\0&1&-1&1&-1\\0&0&0&2&-2\end{pmatrix}$\\\hline
\begin{tabular}{c}$5$ Vertices\\Terminal\end{tabular}&$\begin{pmatrix}1&0&0&-1&1\\0&1&0&-1&1\\0&0&1&0&-1\end{pmatrix}$\\\hline
\end{tabular}
\begin{tabular}[t]{|c|c|}
\hline
Comments&Vertices\\\hline
\begin{tabular}{c}$5$ Vertices\\Simplicial\end{tabular}&$\begin{pmatrix}1&0&0&-2&-1\\0&1&0&-1&0\\0&0&1&0&-1\end{pmatrix}$\\\hline
$5$ Vertices&$\begin{pmatrix}1&0&0&-2&-2\\0&1&0&-1&0\\0&0&1&0&-1\end{pmatrix}$\\\hline
$5$ Vertices&$\begin{pmatrix}1&0&-2&1&-3\\0&1&-1&1&-1\\0&0&0&2&-2\end{pmatrix}$\\\hline
\begin{tabular}{c}$6$ Vertices\\Simplicial\\Terminal\end{tabular}&$\begin{pmatrix}1&0&0&-1&0&0\\0&1&0&0&-1&0\\0&0&1&0&0&-1\end{pmatrix}$\\\hline
\begin{tabular}{c}$6$ Vertices\\Simplicial\\Terminal\end{tabular}&$\begin{pmatrix}1&0&-1&0&1&-1\\0&1&0&-1&1&-1\\0&0&0&0&2&-2\end{pmatrix}$\\\hline
\end{tabular}
\caption{The non-simplex three-dimensional minimal Fano polytopes.}
\label{tab:minimal_canonical_dim3}
\end{table}

\begin{lem}\label{lem:1and1}
Any minimal Fano polytope containing two copies of the minimal Fano triangle of type $\Proj(1,1,2)$ is equivalent to:
$$\begin{pmatrix}1&0&0&-2&-2\\0&1&0&-1&0\\0&0&1&0&-1\end{pmatrix}\text{ or }
\begin{pmatrix}1&0&-2&1&-3\\0&1&-1&1&-1\\0&0&0&2&-2\end{pmatrix}.$$
\end{lem}

\begin{proof}
Fix the lattice such that $P:=\sconv{e_1,e_2,-2e_1-e_2,x,y}$. Again there are two cases to consider. If $x+y+2e_2=0$ then $-e_2$ is contained on the boundary of $P$. We already know that $-e_1$ lies on the boundary of $P$, and hence minimality reduced us to the case considered in Lemma~\ref{lem:3and1}. Thus $x+y+2e_1=0$ and $x=\coord{a,b,c}, y=\coord{-a-2,-b,-c}$, where $0\leq a,b<0$. Clearly $a=0,b=0,c=1$ is a solution. Let us assume that $c>1$.

By minimality $-e_3\notin P$. The line joining $-e_3$ to the origin intersects $\sconv{e_1,e_2,y}$ at the point $k(-e_3)$, where $k=c/(a+b+3)$. Hence we conclude that:
\begin{equation}\label{lem:1and1Eq1}
a+b+2\geq c.
\end{equation}

The point $e_1+e_2+e_3$ does not lie in $P$, otherwise either:
$$\sconv{e_1,-2e_1-e_2,e_1+e_1+e_3,y},$$
or:
$$\sconv{-e_1,-2e_1-e_2,e_1+e_2+e_3,y},$$
would be a Fano tetrahedron. Consider the line connecting $0$ and $e_1+e_2+e_3$. This line intersects $\sconv{e_1,e_2,x}$ at the point $k(e_1+e_2+e_3)$, where $k=c/(2c-a-b+1)$. In particular,
\begin{equation}\label{lem:1and1Eq2}
a+b\leq c.
\end{equation}

If $e_2+e_3\in P$ then $\sconv{e_1,-2e_1-e_2,e_2+e_3,y}$ would be a Fano tetrahedron. This is not permissible. The line connecting $e_2+e_3$ and the origin intersects $\sconv{-e_1,e_2,x}$ at the point $k(e_2+e_3)$, where $k=c/(a-b+c+1)$. We conclude that:
\begin{equation}\label{lem:1and1Eq3}
a\geq b.
\end{equation}
In particular $a\neq 0$, since the alternative would force $c=1$.

Finally we consider the point $-e_1-e_3$. The line connecting this point with the origin intersects $\sconv{-e_1,e_2,y}$ if $a+2\leq c$, or $\sconv{e_1,e_2,y}$ if $a+2>c$. The first possibility gives the point of intersection as $k(-e_1-e_3)$, where $k=c/(b+c-a-1)$. If $-e_1-e_3$ lies on the boundary of $P$, we see that $b=a+1$. This contradicts equation~\eqref{lem:1and1Eq3}. Hence it must be that $-e_1-e_3$ lies outside $P$. In this case, $b\geq a+2$, and once again this contradicts equation~\eqref{lem:1and1Eq3}. It must be that $a+2>c$, which implies that $a=c-1$. Equation~\eqref{lem:1and1Eq2} forces $b\leq 1$, and by applying equation~\eqref{lem:1and1Eq1} we see that the only possibility is $a=1,b=1,c=2$.
\end{proof}

\section{Canonical Toric Fano Threefolds}\label{sec:class_canonical_threefolds}

Using the results of Section~\ref{sec:min_canonical_threefolds} a computer classification of all canonical Fano polytopes of dimension three is possible. This is a significant undertaking; a month of computation on a parallel computing system was required. The code, written in C, is available from the author upon request. It should be emphasised that several known results exist as sub-classifications, and that the resulting list can be independently checked using packages such as PALP~\cite{KS04}. We summarise the algorithm below.

\algorithm\label{alg:classification}
For each of the $26$ minimal Fano polytopes given in Tables~\ref{tab:minimal_tet} and~\ref{tab:minimal_canonical_dim3}, perform the following recursive algorithm:
\begin{enumerate}
\item\label{algstep:build_list}\emph{Identifying unimodular equivalence:}
We have been given a canonical Fano polytope $P$, and inductively are constructing a set $\mathcal{P}$ which will ultimately contain all possible canonical Fano polytopes, up to unimodular equivalent. Thus for each $Q\in\mathcal{P}$, check whether there exists a transformation in $GL(3,\Z)$ sending the vertices of $P$ bijectively onto the vertices of $Q$. If $P$ is new then add it to $\mathcal{P}$ and proceed to step~\eqref{algstep:vertices}. Obviously invariants of the two polytopes such as their volume, degree, whether they are both simplicial, etc.\ can be used to greatly reduce the number of comparisons required.
\item\label{algstep:vertices}\emph{Successively choosing new vertices:}
We have been given a canonical Fano polytope $P$ and wish to extend $P$ via the addition of a new vertex.
\begin{enumerate}
\item For each vertex $v$ of $P$ such that $P':=\conv(P\cup\set{-v})$ is a canonical Fano polytope with $-v\in\mathrm{vert}\,P'$, recuse on step~\eqref{algstep:build_list} with $P'$.
\item For each pair of distinct vertices $v_1$ and $v_2$ check which of the following six sums give a lattice point $v\in N$ (cf.~Figure~\ref{fig:Fano_triangles}):
$$
\begin{array}{lll}
-v_1-v_2,\\
-2v_1-v_2,&-\frac{1}{2}v_1-\frac{1}{2}v_2,\\
-2v_1-3v_2,&-\frac{1}{2}v_1-\frac{3}{2}v_2,&-\frac{1}{3}v_1-\frac{2}{3}v_2.
\end{array}
$$
In each case, if $P':=\conv(P\cup\set{v})$ is a canonical Fano polytope with $v\in\mathrm{vert}\,P'$, then recuse on step~\eqref{algstep:build_list} with $P'$.
\item For each choice of pair-wise distinct vertices $v_1,v_2$, and $v_3$, and for each weight $(\lambda_0,\lambda_1,\lambda_2,\lambda_3)$ in Table~\ref{tab:all_tet_weights}, check whether any of the four sums:
$$
\begin{array}{ll}
-\frac{\lambda_1}{\lambda_0}v_1-\frac{\lambda_2}{\lambda_0}v_2-\frac{\lambda_3}{\lambda_0}v_3,&
-\frac{\lambda_0}{\lambda_1}v_1-\frac{\lambda_2}{\lambda_1}v_2-\frac{\lambda_3}{\lambda_1}v_3,\\
-\frac{\lambda_1}{\lambda_2}v_1-\frac{\lambda_0}{\lambda_2}v_2-\frac{\lambda_3}{\lambda_2}v_3,&
-\frac{\lambda_1}{\lambda_3}v_1-\frac{\lambda_2}{\lambda_3}v_2-\frac{\lambda_0}{\lambda_3}v_3,
\end{array}
$$
give a lattice point $v\in N$. In each case, if $P':=\conv(P\cup\set{v})$ is a canonical Fano polytope with $v\in\mathrm{vert}\,P'$, then recuse on step~\eqref{algstep:build_list} with $P'$.
\end{enumerate}
\end{enumerate}

The final classification is available online, in a searchable format, via the Graded Rings Database at \href{http://malham.kent.ac.uk}{\texttt{http://malham.kent.ac.uk/}}. The key results are summarised below; for further details consult the online database.

\begin{thm}
Up to isomorphism, there exist exactly $674,\!688$ toric Fano threefolds. Of these, $18$ are smooth, $634$ have at worst terminal singularities, $4,\!319$ are Gorenstein, and $12,\!190$ are $\Q$-factorial. Amongst the $\Q$-factorial varieties, the rank of the Picard group is bounded by $\rho\leq 7$; this bound is attained in exactly two cases -- once when the variety is terminal, once when the variety is canonical.
\end{thm}

\bibliographystyle{amsalpha}

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\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}}
\providecommand{\href}[2]{#2}


TITLE: A particle of mass $m$ moves under a retarding force that is proportional to the cube of the speed
QUESTION [2 upvotes]: A particle of mass $m$ moves under a retarding force that is proportional to the cube of the speed. Find how long it takes to travel a distance $d$ from the instant speed was $u$.

I've been stuck on this question for an hour now.  Can someone please point me in the right direction? The answer is
$$ \dfrac{d}{u} + \dfrac{1}{2}\dfrac{k}{m}d^2 $$
My initial equation is:
$ a = - \dfrac{kv^3}{m} $ then trying $\dfrac{dv}{dt}$ or $\dfrac{dv}{dx}v$ for $a$, without to much success.
Here is my thought process:
$$ v \dfrac{dv}{dx} = - \dfrac{kv^3}{m} $$
$$  \dfrac{dv}{dx} = - \dfrac{kv^2}{m}  $$
$$  \dfrac{dx}{dv} = - \dfrac{m}{kv^2}  $$
$$  x = - \dfrac{m}{k} \int v^{-2} dv $$
$$  x = \dfrac{m}{kv} + c $$
When $x = 0$ , $v = u$, therefor $c =  - \dfrac{m}{ku} $
So I have
$$  x = \dfrac{m}{kv} - \dfrac{m}{ku} $$
I've solved it......
$$  x + \dfrac{m}{ku} = \dfrac{m}{kv} $$
$$  \dfrac{m}{kv} = \dfrac{kux + m}{ku} $$
Cancel the $k$ on the numerator.
$$  \dfrac{m}{v} = \dfrac{kux + m}{u} $$
Take reciprocal of both sides.
$$  \dfrac{v}{m} = \dfrac{u}{kux + m} $$
$$  v= \dfrac{mu}{kux + m} $$
$$ \dfrac{dx}{dt} = \dfrac{mu}{kux + m} $$
Take reciprocal of both sides.
$$ \dfrac{dt}{dx} = \dfrac{kux + m}{mu} $$
$$ \dfrac{dt}{dx} = \dfrac{m + kux}{mu} $$
$$ \dfrac{dt}{dx} = \dfrac{1}{u} +  \dfrac{kx}{m}$$
$$ t = \int \Bigg[\dfrac{1}{u} +  \dfrac{kx}{m} \Bigg] dx $$
Using initial conditions $x = 0$ to $x = d$.
$$ t = \int_{x=0}^{x=d} \Bigg[\dfrac{1}{u} +  \dfrac{kx}{m} \Bigg] dx $$
$$ t = \Bigg[\dfrac{x}{u} +  \dfrac{kx^2}{2m} \Bigg]_{x=0}^{x=d}  dx $$
$$ t= \dfrac{d}{u} + \dfrac{1}{2}\dfrac{k}{m}d^2 $$

REPLY [1 votes]: Your solution is probably the more elegant one, but here is an alternative with tedious algebraic manipulations omitted for brevity.
From
$$
a = {\mathrm{d}v \over \mathrm{d}t} = -{kv^3 \over m}
$$
solve for $v(t)$
$$
\int{\mathrm{d}v \over v^3} = -\int{k \over m}\mathrm{d}t\\
{1 \over v^2} = {2kt \over m} + c.
$$
Applying the initial condition $v(0) = u$, we find $$c = {1 \over u^2}$$ and so $$v(t) = \sqrt{mu^2 \over 2ku^2t + m}.$$
Integrating $v(t)$ from $0$ to the unknown time $t_d$ should result in the desired distance $d$, so
$$
d = \int_0^{t_d}\sqrt{mu^2 \over 2ku^2t + m}\mathrm{d}t = {\sqrt{m} \over 2ku}\int_m^{2ku^2t_d + m}s^{-1/2}\mathrm{d}s = {\sqrt{2ku^2mt_d + m^2} - m\over ku}
$$
using the substitution $s = 2ku^2t+m$.
Solving for $t_d$ yields
$$
t_d = {k \over 2m}d^2 + {d \over u}.
$$


TITLE: Category of profunctors compact closed?
QUESTION [2 upvotes]: In the article "Coherence for compact closed categories", Kelly and Laplaza say

The “category” of small categories and profunctors fails to be a compact closed category only because it fails to be an honest category with associative composition.

What composition are they talking about? What do they mean by an honest category?

REPLY [0 votes]: There's no consistent way to define the composition, as it involves taking a coend. The traditional way around this is to identify functors between small categories with cocontinuous functors between their free cocompletions, where the composition becomes the standard composition of functors. Unfortunately, the category of locally small cocomplete categories and cocontinuous functors is not closed, as the categories of functors may not be locally small. This can be solved by viewing small categories abd profunctors as a compact closed bicategory.


TITLE: Does ZF prove the propositional compactness theorem for general Boolean algebras?
QUESTION [0 upvotes]: Propositional compactness theorem for $\{0,1\}$ algebra: suppose that $\Gamma$ is a set of sentences of a (not necessarily enumerable) propositional language $L$. If for every finite subset $\Gamma_0 \subseteq \Gamma$ there exists an evaluation function $V_0: \text{Prop}(L) \rightarrow \{0,1\}$ such that $V(A)=1$ for every $A \in \Gamma_0$ then there exists an evaluation function $V: \text{Prop}(L) \rightarrow \{0,1\}$ such that $V(A)=1$ for every $A \in \Gamma$.
The Propositional compactness theorem for $\{0,1\}$ algebra is not provable in ZF, in fact it implies that every set can be totally ordered and this is not true in ZF.
Propositional compactness theorem for general Boolean algebras: suppose that $\Gamma$ is a set of sentences of a (not necessarily enumerable) propositional language $L$. If for every finite subset $\Gamma_0 \subseteq \Gamma$ there exist a general Boolean algebra $B_0$ (not necessarily the $\{0,1\}$ Boolean algebra) and an evaluation function $V_0: \text{Prop}(L) \rightarrow B_0$ such that $V(A)=1$ for every $A \in \Gamma_0$ then there exist a general Boolean algebra $B$ (not necessarily the $\{0,1\}$ Boolean algebra) and an evaluation function $V: \text{Prop}(L) \rightarrow B$ such that $V(A)=1$ for every $A \in \Gamma$.
Is the Propositional compactness theorem for general Boolean algebras provable in ZF?
Otherwise, does it imply some contraddiction in ZF?

REPLY [1 votes]: First, we need to clarify the theorem. We need all Boolean algebras mentioned to be nontrivial, where “trivial” means that $\top = \bot$. Otherwise, the theorem itself is trivial: just take $B$ trivial.
Let us briefly discuss $Prop(L)$. We can construct a canonical evaluation map $\pi : Prop(L) \to Prop(L) / \equiv$, where $\equiv$ is logical equivalence and $Prop(L) / \equiv$ has the obvious Boolean algebra structure. This is a universal map in the sense that all evaluation maps $Prop(L) \to B$ factor uniquely as $f \circ \pi$, where $f : Prop(L) / \equiv \to B$ is a Boolean algebra homomorphism; conversely, given any Boolean algebra homomorphism $f : Prop(L) / \equiv \to B$, $f \circ \pi$ is an evaluation map.
Now in particular, we have a nontrivial evaluation $Prop(L) \to B$ sending all elements of a set $\Phi$ to 1 if and only if we have a nontrivial Boolean algebra homomorphism $Prop(L) / \equiv$ sending all elements of $\pi(\Phi)$ to $1$.
So we can rephrase the problem as follows. Suppose we have a Boolean algebra $C$ and some subset $\Delta \subseteq C$ and for all finite $\Delta_0 \subseteq \Delta$, there is some homomorphism $C to B_0$ sending all elements of $\Delta_0$ to $\top$, where $B_0$ is nontrivial. Then there is some homomorphism $C \to B$ sending all elements of $\Delta$ to $\top$, where $B$ is nontrivial. To translate to this case, let $C = Prop(L) / \equiv$ and $\Delta = \pi(\Gamma)$.
Recall that a filter on $C$ is a set $F \subseteq C$ closed under finite meets (that is, containing $\top$ and closed under $\land$) and upwards closed (meaning $x \leq y$ and $x \in F$ implies $y \in F$). We can define an equivalence relation $\sim$ on $C$ by taking $x \sim y$ iff $\exists f \in F (x \land f = y \land f)$. Then $C / \sim$ can be made into a Boolean algebra in a unique way such that the quotient map $\pi : C \to C / \sim$ is a Boolean algebra homomorphism (in other words, the Boolean algebra operations respect $\sim$). We write the resulting Boolean algebra as $C / F$.
We say $F$ is trivial if and only if $0 \in F$. Note that $F$ is trivial if and only if $C / F$ is trivial.
Note that given any Boolean algebra homomorphism $f : C \to D$, $f^{-1}(\top)$ is a filter on C. Note that $f^{-1}(\top)$ is trivial iff $D$ is trivial. In particular, $\pi^{-1}(\top) = F$. Note that this implies that $F$ is nontrivial if and only if there exists some nontrivial $D$ and some $f : C \to D$ such that $F \subseteq f^{-1}(\top)$.
Finally, given a set $S \subseteq C$, let the filter generated by $S$ be the smallest filter containing $S$. We write this as $filter(S)$. It is easy to show that $filter(S) = \bigcup\limits_{S_0 \subseteq S, S_0 \text{ is finite }} filter(S_0)$.
Therefore, we see that $filter(\Delta)$ is trivial if and only if there exists finite $\Delta_0 \subseteq \Delta$ such that $filter(\Delta_0)$ is trivial. Conversely, $filter(\Delta)$ is nontrivial iff for all finite $\Delta_0 \subseteq \Delta$, $filter(\Delta_0)$ is nontrivial.
Now note that $filter(S)$ is nontrivial if and only if there is some nontrivial $D$ and homomorphism $f : C \to D$ such that $filter(S) \subseteq f^{-1}(\top)$. And by the definition of $filter$, we have $filter(S) \subseteq f^{-1}(\top)$ if and only if $S \subseteq f^{-1}(\top)$, that is, iff $f$ sends all elements of $S$ to $\top$.
Combining these facts, we see that for all finite $\Delta_0 \subseteq \Delta$, we have some nontrivial homomorphism $f : C \to B_0$ such that $\Delta_0 \subseteq f^{-1}(\top)$, and thus $filter(\Delta_0)$ is nontrivial. Therefore, $filter(\Delta)$ is nontrivial, and thus we have some nontrivial homomorphism $f : C \to B$ such that $\Delta \subseteq f^{-1}(\top)$.
This completes our proof.
Note: filters here act a lot like normal subgroups. You can prove analogues of the isomorphism theorems for Boolean algebras.


TITLE: What are "ordered functions"?
QUESTION [1 upvotes]: In the solution to Problem 156b at p.120 of Combinatorics Through Guided Discovery by Kenneth P. Bogart:
$\newcommand{\risefac}[2]{{#1}^{\overline{#2}}}$

$\risefac{n}{k}$ counts the number of ordered functions from a $k$-element set to an $n$-element set.

What are "ordered functions" ?

REPLY [1 votes]: Found the answer at (the obvious place):
https://bogart.openmathbooks.org/ctgd/s3-1-distrib-idea.html

More precisely, an ordered function from a set $S$ to a set $T$ is a function that assigns an (ordered) list of elements of $S$ to some, but not necessarily all, elements of $T$ in such a way that each element of $S$
appears on one and only one of the lists.

...

The phrase ordered function is not a standard one, because there is as yet no standard name for the result of an ordered distribution problem.


TITLE: Prove that the polyhedron has at least $2^n$ vertices
QUESTION [0 upvotes]: Let A be the n x n identity matrix and let b be the n x 1 vector with every entry equal to one. Prove that the polyhedron P(A, b) has at least 2$^n$ vertices.
What I have so far:
P(A, b)= set of x $\in$ $\mathbb{R}$ so that
$\begin{bmatrix} -I_n \\A\end{bmatrix}$ x $\leq$ $\begin{bmatrix} **0** \\**b**\end{bmatrix}$
is satisfied. The 0 is the zero vector.
Therefore we will have the constraints x$_1$, x$_2$, ..., x$_n$ $\geq$ 0 and x$_1$ $\leq$ 1, x$_n$ $\leq$ 1, ..., x$_n$ $\leq$ 1. We will have 2n constraints. To make the proof a little less wordy, I've called the first set of constraints "0 constraints" and the second set of constraints "1 constraints" - this is just for my draft proof.
The number of vertices will be: 1 + n - 1 + n$^2$
The 1 comes from the origin always being a vertex as it is where the 0 constraints all intersect
The n - 1 comes from there being an intersection between every 1 constraint with every other 1 constraint except when the constraints are the same
The n$^2$ comes from there being an intersection between every 0 constraint and every 1 constraint - since we have n of each constraints, we get n$^2$.
That's where I get stuck. I think I'm heading in the correct direction but my counting or something is off.

REPLY [1 votes]: For every variable, $x_i$, you can pick it to be $0$ or $1$ to make it a vertex, that is for each variable, you have two choices.
Since there are $n$ variables, you have $2^n$ choices.


TITLE: Proof that $\lim_{n\to\infty} a_n = L$ then $\lim_{n\to\infty} |a_n| = |L|$
QUESTION [3 upvotes]: In order to prove that
$$\lim_{n\to\infty} a_n = L \implies \lim_{n\to\infty} |a_n| = |L|$$
I started with my assumption:
$\lim_{n\to\infty} a_n = L$ we have:
$$n>n_0\implies |a_n - L| < \epsilon $$
but I can say that $$||a_n|-|L||<|a_n -L| < \epsilon$$ rigth?
Then my proof is done?

REPLY [3 votes]: Just a small technicality: you should write $||a_n| - |L|| \color{red}{\le} |a_n - L| < \epsilon$, for it may be the case that $a_n = L$ for some $n$.


TITLE: Composition of kinetic energy
QUESTION [0 upvotes]: As kinetic energy is the "energy of motion", it cannot exist without matter.  However, the energy in an E/M wave can.  I understand that E/M waves have been described in structure as electrical and magnetic waves in a sinusoidal configuration.  This means that kinetic energy cannot be made of EM waves or it would be mass-independent.
So my question is, what is kinetic energy made of?

REPLY [4 votes]: Energy is an extraordinarily elusive concept, so much so that experienced physicists will use the term only with care and in well defined circumstances.
For example it might seems obvious that if I fire a pistol then the bullet has kinetic energy. But Einstein's theory of special relativity tells us that all inertial frames are equally valid, and in the bullet's frame the bullet is stationary and it's me and the pistol that are moving. So in this frame where did the kinetic energy of the bullet go and where did my and the pistol's kinetic energy come from?
If you say the bullet has a kinetic energy of $\frac{1}{2} m v^2$ what this really means is that to make the bullet move at a velocity $v$ I had to supply an energy of $\frac{1}{2} m v^2$. If I have lost an energy of $\frac{1}{2} m v^2$ then conservation of energy means the bullet has gained an energy of $\frac{1}{2} m v^2$, and this is what I measure as it's kinetic energy.
This is the way to think of kinetic energy. If some observer, like you or me, measures a kinetic energy $E$ this means we would have to put an energy $E$ into making the object move. And this carries straight over into measuring the energy carried by light. If I create a photon then this costs me energy, and that energy goes into the photon. If/when the photon hits something the energy is then transferred to whatever it hits. So the photon carries energy just like a bullet does, even though the photon has zero mass.
Now you may say this all sounds a bit contrived, because what I've just described is the total energy of the photon and not its kinetic energy, and in fact you'd be absolutely correct. This is because kinetic energy is a low energy approximation and the concept doesn't really exist in relativity. In relativity we only talk about the total energy of an object, and the total energy is given by the equation:
$$ E^2 = p^2 c^2 + m^2 c^4 $$
where $m$ is the object's mass and $p$ is its momentum. The first term, $p^2c^2$ is sort of the kinetic energy and the second term, $m^2c^4$, is sort of the rest energy, though thi separation isn't clear for fast moving particles. Anyhow, for a photon the mass, $m$, is zero so the expression simplifies to:
$$ E^2 = p^2 c^2 $$
And this could be regarded as the kinetic energy because it's the same term that is considered the kinetic energy for a massive object.


text:
\section{Vertex cover polynomial}
\label{sec:vertex_cover_polynomial}

We now discuss a graph polynomial that is strongly related to the independence polynomial and even equivalent to it, the vertex cover polynomial. This graph polynomial is the generating function for vertex covers and was introduced by \textcite{dong2002}.

\begin{defi}
Let $G = (V, E)$ be a hypergraph. A vertex subset $W \subseteq V$ is a \emph{vertex cover} in $G$ if $\forall e \in E \colon e \cap W \neq \emptyset$.
\end{defi}
In other words, a vertex cover includes at least one vertex from each edge.

\begin{defi}
Let $G = (V, E)$ be a hypergraph. The \emph{vertex cover polynomial} $\Psi(G, x)$ is defined as
\begin{align}
\Psi(G, x) = \sumsr{\substack{W \subseteq V \\ W \text{ is a vertex cover in }G}}{x^{\abs{W}}}.
\end{align}
\end{defi}

Just as in the case of the independence polynomial, for edgeless graphs each vertex subset is a vertex cover and therefore $\Psi(E_n) = (1+x)^n$.

While an independent set misses at least one vertex of each edge, a vertex cover includes at least one vertex of each edge. Therefore, the vertices not in an independent set form a vertex cover and vice versa.

\begin{prop}
Let $G = (V, E)$ be a hypergraph with a vertex subset $W \subseteq V$. $W$ is an independent set if and only if $V \setminus W$ is a vertex cover.
\end{prop}

From this proposition the relations between both graph polynomials follows directly.

\begin{prop} \label{prop:rel_i_psi}
Let $G = (V, E)$ be a hypergraph. The independence polynomial $I(G, x)$ and the vertex cover polynomial $\Psi(G, x)$ are equivalent graph polynomials related by
\begin{alignat}{2}
& I(G, x) &&= x^{n(G)} \cdot \Psi(G, x^{-1}), \\
& \Psi(G, x) &&= x^{n(G)} \cdot I(G, x^{-1}).
\end{alignat}
\end{prop}

Furthermore, the recurrence relations from both are related. This is a special case of a more general theorem relating recurrence relations of graph polynomials \cite[Theorem 3.18]{trinks2012c}.

\begin{theo} \label{theo:rec_rel_i_psi}
Let $G = (V, E)$ be a hypergraph. The recurrence relations of the independence polynomial $I(G, x)$ and the vertex cover polynomial $\Psi(G, x)$ are related to each other by:
\begin{align}
\Psi(G, x) &= \sum_{i}{a_i(x^{-1}) \: x^{n(G)-n(G_i)} \cdot \Psi(G_i, x)} \\
\intertext{if and only if}
I(G, x) &= \sum_{i}{a_i(x) \cdot I(G_i, x)},
\end{align}
where $a_i(x)$ are polynomials in $x$ and $G_i$ are hypergraphs (not necessarily arising from graph operations).
\end{theo}

\begin{proof}
The statement follows direct from the relations of both graph polynomials given in Propostion \ref{prop:rel_i_psi}:
\begin{align*}
\Psi(G, x)
&= x^{n(G)} \cdot I(G, x^{-1}) \\
&= x^{n(G)} \cdot \sum_{i}{a_i(x^{-1}) \cdot I(G_i, x^{-1})} \\
&= \sum_{i}{a_i(x^{-1}) \: x^{n(G)} \: (x^{-1})^{n(G_i)} \cdot \Psi(G_i, x)} \\
&= \sum_{i}{a_i(x^{-1}) \: x^{n(G)-n(G_i)} \cdot \Psi(G_i, x)}. \qedhere
\end{align*}
\end{proof}

Thus, for each recurrence relation of the independence polynomial there is a corresponding one for the vertex cover polynomial.

\begin{coro}
Let $G = (V, E)$ be a hypergraph with a vertex $v \in V$ and an edge $e \in E$. The vertex cover polynomial $\Psi(G, x)$ satisfies
\begin{alignat}{2}
& \Psi(G, x)
&&= \begin{cases}
x \cdot \Psi(G_{\vdel v}, x) + \Psi(G_{\vhid v}, x) & \text{if } \{v\} \notin E, \\
x \cdot \Psi (G_{\vdel v}, x) & \text{else,}
\end{cases} \\
& &&= \begin{cases}
\Psi(G_{-e}, x) - \Psi(G_{-e \vshid e}, x) & \text{if } \nexists f \in E \colon f \subseteq e, \\
\Psi(G_{-e}, x) & \text{else.}
\end{cases} \\
& &&= \sumsr{\emptyset \subset B \subseteq e}{- (-x)^{\abs{B}} \cdot \Psi(G_{\vsdel B}, x)}, \\
& &&= \Psi(G_{-e}, x) - \Psi(G_{/e}, x) + x \cdot \Psi(G_{\vsdel e}, x). \label{eq:vc_rec_rel}
\end{alignat}
\end{coro}

\begin{proof}
The statements follow via Theorem \ref{theo:rec_rel_i_psi} from Theorem \ref{theo:rec_rel_hg_v}, Theorem \ref{theo:rec_rel_hg_e}, Corollary \ref{coro:rec_rel_hg_e_1} and Corollary \ref{coro:rec_rel_hg_e_2}, respectively.
\end{proof}

The recurrence relation corresponding to the standard recurrence relation of the independence polynomial is well-known \cite[Theorem 2.2]{dong2002}. (For a recurrence relation of $(-1)^{n(G)} \cdot \Psi(G, -x)$, see \cite[Equation (9)]{gutman1992}.) Equation \eqref{eq:vc_rec_rel} has already been derived in the case of graphs via other graph polynomials \cite[Corollary 30]{trinks2012}.

Recurrence relations via vertex subsets can be derived using the same approach. Here is an example.

\begin{coro}
Let $G = (V, E)$ be a hypergraph with a vertex subset $U \subseteq V$. The vertex cover polynomial satisfies
\begin{align}
\Psi(G, x) &= \sumsr{\substack{W \subseteq U \\ W \text{ is independent in }G}}{x^{\abs{U \setminus W}} \cdot \Psi(G_{\vshid W \vsdel U}, x)} \\
&= \sumsr{\substack{W \subseteq U \\ V \setminus W \text{ is a vertex cover in }G}}{x^{\abs{U \setminus W}} \cdot \Psi(G_{\vshid W \vsdel U}, x)}.
\end{align}
\end{coro}

\begin{proof}
The statements follow from Theorems \ref{theo:rec_rel_i_psi} and \ref{theo:rec_rel_hg_vs}.
\end{proof}

config: arxiv


text:
\begin{document}

\title{Spatio-temporal evolution of global surface temperature distributions}

\author{Federico Amato}
\orcid{0000-0002-5886-9038}
\affiliation{
  \institution{University of Lausanne \\ Institute of Earth Surface Dynamics}
  \streetaddress{UNIL-Mouline}
  \city{Lausanne}
  \state{}
  \country{Switzerland}
  \postcode{CH-1015}
}
\email{federico.amato@unil.ch}

\author{Fabian Guignard}
\affiliation{
  \institution{University of Lausanne \\ Institute of Earth Surface Dynamics}
  \streetaddress{UNIL-Mouline}
  \city{Lausanne}
  \state{}
  \country{Switzerland}
  \postcode{CH-1015}
}
\email{fabian.guignard@unil.ch}

\author{Vincent Humphrey}
\affiliation{
 \institution{California Institute of Technology \\ Division of Geological and Planetary Sciences}
 \streetaddress{Rono-Hills}
 \city{Pasadena}
 \state{California}
 \country{USA}}
\email{vincent.humphrey@caltech.edu}

\author{Mikhail Kanevski}
\affiliation{
  \institution{University of Lausanne \\ Institute of Earth Surface Dynamics}
  \streetaddress{UNIL-Mouline}
  \city{Lausanne}
  \state{}
  \country{Switzerland}
  \postcode{CH-1015}
}
\email{mikhail.kanevski@unil.ch}

\renewcommand{\shortauthors}{Amato, et al.}

\begin{abstract}
Climate is known for being characterised by strong non-linearity and chaotic behaviour. Nevertheless, few studies in climate science adopt statistical methods specifically designed for non-stationary or non-linear systems. Here we show how the use of statistical methods from Information Theory can describe the non-stationary behaviour of climate fields, unveiling spatial and temporal patterns that may otherwise be difficult to recognize. We study the maximum temperature at two meters above ground using the NCEP CDAS1 daily reanalysis data, with a spatial resolution of $2.5^{\circ}$ by $2.5^{\circ}$ and covering the time period from 1 January 1948 to 30 November 2018. The spatial and temporal evolution of the temperature time series are retrieved using the Fisher Information Measure, which quantifies the information in a signal, and the Shannon Entropy Power, which is a measure of its uncertainty --- or unpredictability. The results describe the temporal behaviour of the analysed variable. Our findings suggest that tropical and temperate zones are now characterized by higher levels of entropy. Finally, Fisher-Shannon Complexity is introduced and applied to study the evolution of the daily maximum surface temperature distributions.
\end{abstract}

\keywords{Fisher Information Measure, Shannon Entropy Power, Statistical Complexity, Air Temperature Distributions, Spatio-Temporal Exploratory Data Analysis}

\maketitle

\section{Motivation}
In the context of anthropogenic greenhouse gas emissions, global climate has been the focus of extensive research over the last decades. Countries which have adhered to the 2016 Paris Agreement aim to limit the increase in global average temperature below $1.5 ^\circ C$ and mitigate the risks and impacts of climate change \cite{hulme20161}. Moreover, climate change is also a primary concern in the context of the Sustainable Development Goals (SDGs) \cite{assembly2015sustainable}. In the United Nations agenda, climate influence on sustainable development is closely related not only to environmental issues but also to the social and economic dimension of the SDGs. Similarly, the Intergovernmental Panel on Climate Change (IPCC) has investigated the interaction between sustainable development, poverty eradication and ethics and equity \cite{intergovernmental2018global}. Many of these interactions are related to the fact that warming has significant spatial and temporal patterns that could affect some regions more than others. As an example many locations, especially in the Northern Hemisphere at the mid-latitudes, are experiencing regional warming that is more than double the global average \cite{collins2013long}.

Because climate is characterised by strong nonlinearity and chaotic behaviour, many studies in climate science cannot rely on statistical methods valid only for stationary or linear systems \cite{franzke2014warming, ribes2020describing}. It has already been shown that warming trends are characterised by strong non-linearities, with an acceleration in the increase of temperatures since 1980 \cite{ji2014evolution}.

In the present research we attempt to further investigate the complex nature of surface temperature, showing how statistical methods from Information Theory can be used to describe the non-stationary behaviour of such phenomena. We study the maximum temperature at two meters above ground using the NCEP CDAS1 daily reanalysis data, with a spatial resolution of $2.5^{\circ}$ by $2.5^{\circ}$ and covering the time period from 1 January 1948 to 30 November 2018 \cite{kalnay1996ncep}. For each spatial location we track through a sliding window the evolution of the corresponding temperature time series using the Fisher Information Measure (FIM) \cite{Fisher1925}, which is a powerful tool to identify the behaviour of dynamical systems, and the Shannon Entropy Power (SEP) \cite{Shannon1948}, which can be used to quantify the uncertainty --- or the disorder --- of a signal. We show how these measures can be used to retrieve the spatial and temporal changes of the investigated time series distributions, highlighting the complex behaviour of the temperature spatio-temporal field.

\section{Methods}

This section provides details concerning the estimation of FIM and SEP. Briefly, FIM is used to measure the information in a given signal $x$, while SEP quantifies its degree of disorder and predictability. To unveil the spatial structures of the spatio-temporal values of FIM and SEP computed, an Empirical Orthogonal Function (EOF) decomposition will also be performed on the data of the two estimations. EOF analysis is the spatio-temporal version of principal component analysis. Through this approach, data are decomposed into orthogonal basis functions defined by spatial eigenvectors --- the Empirical Orthogonal Functions (EOFs) ---  and temporal coefficients described by the principal components (PC) time series.

\subsection*{Fisher Information Measure and Shannon Entropy Power}

\begin{figure*}[hbt!]
\begin{center}
\includegraphics[width=1\linewidth]{Figures/Hovmoller.pdf}
\caption{\textbf{Hovm\"{o}ller plots for the latitude for both FIM and SEP.} The figures represent the mean values of the two measures over land and oceans. Data have been standardized using a z-score tranformation.}
\label{fig:hovmoller}
\end{center}
\end{figure*}

SEP and FIM are summary quantities to partially describe a Probability Density Function (PDF). Let $X$ be an univariate continuous random variable following a PDF  $f(x)$. The SEP of $X$ is defined as $N_X$ through the relationship \cite{Dembo1991}
\begin{equation}\label{N}
N_X = \frac{1}{2\pi e} e^{2H_X}, \qquad H_X = \mathbb E \left[-\log f(x)\right].
\end{equation}
The SEP is a strictly increasing transformation of $H_X$, which is the differential entropy of $X$ \cite{Cover2006}.
The FIM of $X$, noted $I_X$, is defined as \cite{Dembo1991}.
\begin{equation}\label{I}
    I_X=\mathbb E \left[\left(\frac{\partial}{\partial x}\log f(X)\right)^2\right].
\end{equation}

SEP and FIM can be jointly visualized into the Fisher-Shannon Information Plane (FSIP) to study the PDF of $X$ \cite{Vignat2003, guignard2019advanced}. It can be shown that $N_X \cdot I_X \geq 1$, with the equality satisfied if and only if $X$ is a Gaussian random variable \cite{Dembo1991}. Hence, the only reachable points in the FSIP belong to the set
\[
\mathcal D = \left\{(N_X, I_X) \in \mathbb R^2 \;\middle|\;
\begin{aligned}
&N_X >0, \, I_X>0, \\ & N_X \cdot I_X \geq 1
\end{aligned} \right\}.
\]
The quantity $C_X = N_X \cdot I_X$, which is called Fisher-Shannon Complexity (FSC), is sometimes used as a statistical complexity measure \cite{Angulo2008, Esquivel2010}.

It has been shown that the FSC can be interpreted as a measure of non-Gausiannity of $X$ \cite{guignard2019advanced}.
Thus, the boundary of $\mathcal D$ is reached if and only if $X$ has a unitary FSC, in which case it is a Gaussian random variable.

In order to estimate SEP and FIM from data, $f(x)$ and its derivative $f'(x)$ were replaced by their kernel density estimators (KDE) in the integral forms of \eqref{N} and \eqref{I}, \cite{Bhattacharya1967, Dmitriev1973, PrakasaRao1983, Gyorfi1987, Joe1989}.
To this aim, given $n$ independent realizations $\{x_1, \dots, x_n\}$ of $X$, the PDF $f(x)$ is approximated as  \cite{Wand1994}

\begin{equation}\label{KDE}
    \hat f_h(x) = \frac{1}{nh}\sum_{i=1}^n K\left( \frac{x-x_i}{h} \right),
\end{equation}
where $h$ is a bandwidth parameter and $K(\cdot)$ is a kernel assumed to be a unimodal probability density function symmetric around zero and having integral over $\mathbb R$ equal to 1. By using a Gaussian kernel with zero mean and unit variance, the estimator \eqref{KDE} assumes the form
\begin{equation}\label{KDEGauss}
    \hat f_h(x) = \frac{1}{nh\sqrt{2\pi}}\sum_{i=1}^n e^{-\frac{1}{2}\left( \frac{x-x_i}{h} \right)^2}.
\end{equation}
By deriving $\hat f_h(x)$ one can estimate the PDF derivative $f'(x)$ as
\begin{equation}\label{KDEderivGauss}
    \hat f'_h(x) = \frac{1}{nh^3 \sqrt{2\pi}}\sum_{i=1}^n (x-x_i)e^{-\frac{1}{2}\left( \frac{x-x_i}{h} \right)^2}.
\end{equation}
By plugging estimates \eqref{KDEGauss} and \eqref{KDEderivGauss} into the equations \eqref{N} and \eqref{I}, the latter assumes the form
\[
\hat N_X = \frac{1}{2\pi e}e^{-2\int \hat f_h(x) \log \hat f_h(x) \, dx},
\]
and
\[
\hat I_X = \int \frac{\left(f'_{h}(x)\right)^2}{f_h(x)} \, dx.
\]
The FSC is estimated by multiplying $\hat N_X$ by $\hat I_X$.
The estimates are sensitive to the choice of a proper bandwidth. Here, this parameter is selected using the Sheather-Jones direct plug-in method \cite{Sheather1991}, which approximates the optimal bandwidth with respect to the Asymptotic Mean Integrated Squared Error of $\hat f_h$. Operationally, the non-parametric estimation of the SEP, FIM and FSC are obtained with the FiShPy package \cite{guignard2019advanced}.

\subsection*{Empirical Orthogonal Functions}

EOF analysis is an extremely popular approach to study the variability in a geophysical field
of interest \cite{hannachi2007empirical}. EOF analysis can be seen as an application of Principal Component Analysis in the case where data is a multivariate spatially indexed vector with multiple samples over time \cite{lorenz1956empirical, jolliffe2016principal}.

Let  $\textbf{X}_{t_j}=\big( X(\textbf{s}_1;t_j),\dots, X(\textbf{s}_m;t_j)\big)' \in \mathbb R^m$ be observations for the spatial locations $\{\textbf{s}_i: i = 1,\dots, m\}$ and times $\{t_j: j = 1,\dots,T\}$. The empirical spatial mean for all location is given by
\begin{equation*}
    \widehat{\boldsymbol{\mu}}=\frac{1}{T}\sum_{j=1}^T \mathbf{X}_{t_j}.
\end{equation*}
The empirical spatial covariance matrix \cite{wikle2019spatio} can then be computed as
\begin{equation*}
    \widehat{\mathbf{C}} = \frac{1}{T} \sum_{j=1}^T (\mathbf{X}_{t_j} - \widehat{\boldsymbol{\mu}}) (\mathbf{X}_{t_j} - \widehat{\boldsymbol{\mu}})'.
\end{equation*}

As this real matrix is symmetric and non-negative definite, it can be spectrally decomposed as
\begin{equation*}
    \widehat{\mathbf{C}} = \boldsymbol\Phi \boldsymbol\Lambda \boldsymbol\Phi'
\end{equation*}
where $\boldsymbol\Lambda = \textrm{diag}(\lambda_1,\dots, \lambda_m)$ is the diagonal matrix of the non-negative eigenvalues decreasing down the diagonal, and $\boldsymbol\Phi=(\boldsymbol\phi_1,\dots, \boldsymbol\phi_m)$ is the matrix of the corresponding spatially indexed eigenvectors $\boldsymbol\phi_k = (\phi_k(\mathbf{s}_1), \dots, \phi_k(\mathbf{s}_m))'$, for $k=1,\dots,m$, also called EOFs. The EOFs form a  discrete orthonormal basis

\cite{cressie2015statistics}. The $k$-th PC time series --- which is the time series of coefficient of the corresponding EOF, or equivalently the contribution of the $k$-th spatial basis at time $t_j$ --- is then given by $a_k(t_j)=\boldsymbol\phi_k'\mathbf{X}_{t_j}$ \cite{monahan2009empirical}.

\section{Results}

\subsection*{Temporal changes of temperature distributions}

\begin{figure*}[ht!]
\begin{center}
\includegraphics[width=1\linewidth]{Figures/FS_plane.pdf}
\caption{\textbf{Fisher-Shannon Information Plane.} The FSIP constructed with the estimated values of FIM and SEP for the first (left) and last (right) sliding windows. Each point corresponds to a spatial location, while the dashed line represents the theoretical Gaussian limit. Color indicates the absolute value of the latitude of the points.}
\label{fig:fsplane}
\end{center}
\end{figure*}

To study the NCEP CDAS1 daily data, we first consider separately the temperature time series of the single spatial locations. For each of them, we computed time-dependent measures of SEP and FIM using a sliding temporal window of width equal to five years and a sliding factor equal to one month.

To make spatial and temporal variabilities comparable, the obtained measures for each location have been standardized with a z-score transformation. The spatial distribution of FIM and SEP shown in Figure \ref{fig:hovmoller} highlights two different behaviour. In the period from 1948 to 1979 the latitudes higher than $60^{\circ}$ and lower than $-60^{\circ}$ are characterized by high levels of SEP and low levels of FIM. The situation has completely reversed starting from 1979, and in a faster way for the latitudes higher than $60^{\circ}$, so that tropical and temperate zones are now characterized by higher levels of entropy. The northern mid-latitude seems to have slightly stronger growth of SEP.

The FSIP is used for a simultaneous analysis of FIM and SEP to track the non-stationary behaviour of the temperature time series \cite{Vignat2003}. Figure \ref{fig:fsplane} shows the FSIP of the non z-scored data for the first sliding window (from 1948/01/01 to 1952/12/31) and for the last one (from 2013/12/01 to 2018/11/30). The Figure highlights how the characterization of points in the FSIP is strongly linked to their latitude. The latitudes higher than $60^{\circ}$ and lower than $-60^{\circ}$ are easily recognizable for having a behaviour more distant from the one expected in the Gaussian case. Moreover, it is possible to observe a significant movement of these points from the first to the last window. The displacements of the points in the plane can be interpreted as a change of the distribution of temperature.

FSC is analyzed in Figure \ref{fig:fsc}. The Hovm\"{o}ller plot highlights how the equatorial areas exhibit a FSC fluctuating around values not far from 1, implying temperature distributions almost Gaussian. Latitudes higher than $60^{\circ}$ and lower than $-60^{\circ}$ show a reduction of the FSC over time.
Figure \ref{fig:fsc} also investigates the FSC of the point having Longitude = 232.0 and Latitude = 56.2, located in the western Canada.
This is one of the point experiencing the greater overall change in the measured complexity over time among all the investigated spatial locations. The distribution changes of temperature at this location is clearly observed in the FSIP. The trajectory defined in the plane highlights different stages in its behaviour, which is also recognizable in the timeseries of the FSC. The latter show decreasing complexity values since September 1989. FSC can be used as a clear indicator of a changing pattern in the distribution of data. As an example, three distributions are shown, computed on the temporal windows from 1951-12-01 to 1956-11-0, from 1989-09-01 to 1994-08-31 and from 2009-02-01 to 2014-01-31, respectively.
All the distributions exhibit bimodality, where the modes are determined by the seasonals variabilities. The first distribution, associated with a FSC of 3.32, is actually not far from being a mixture of two Gaussian distributions.
Indeed, it can be empirically observed that for a mixture of two Gaussian distributions having the same variance, the FSC goes to 4 when the modes move away from each other. Differently, the second distribution plot, corresponding to an FSC of 29.37, is dominated by the mode at 273K. This is still true in the last plot, corresponding to FSC of 6.67, although since September 1989 a reduction of the complexity has been registered, indicating a behaviour closer to Gaussian.

\begin{figure*}[ht]
\begin{center}
\vspace{10mm}
\includegraphics[width=1\linewidth]{Figures/C_plots.pdf}
\caption{\textbf{Fisher-Shannon Complexity.} Top left: Hovm\"{o}ller latitudinal plot for FSC. Top right: Trajectory in the FSIP of the point represented by a red triangle in Figure \ref{fig:eofs} and having Longitude =232.0 and Latitude = 56.2. The dashed line represents the theoretical Gaussian limit. Middle: FSC of the same point over time. The grey region covers the temporal period from 1948-01-01 to 1979-01-01. Bottom row: Distributions of the temperature measurements in the temporal windows resulting into the FSC points highlighted with a vertical dashed line in the previous plot.}
\label{fig:fsc}

\end{center}
\end{figure*}

\subsection*{Regional behaviour of temperature distributions}

\begin{figure*}[ht!]
\begin{center}
\includegraphics[width=1\linewidth]{Figures/EOFs.pdf}
\caption{\textbf{Empirical Orthogonal Function decomposition.} First two EOFs and corresponding PC time series for the FIM (top) and the SEP (bottom). The red triangle in the maps corresponds to the point explored in Figure \ref{fig:fsc}. The grey regions in the PC time series plots cover the temporal period from 1948-01-01 to 1979-01-01. For the PC time series corresponding to the first components, a regression line is fitted to the data from January 1979 and reported with a dashed line.}
\label{fig:eofs}
\end{center}
\end{figure*}

To further improve the understanding of the spatial patterns of the variability in the spatio-temporal values of the estimated SEP and FIM, EOF decompositions were performed. Figure \ref{fig:eofs} shows the first two EOFs and the corresponding PC time series for both SEP and FIM estimates.

The first EOFs reproduce the pattern of reduction of SEP and growth of FIM in the extreme northern and southern latitudes. The first EOFs also show a drastic difference between oceans and lands surfaces, with the former generally characterized by significant increases of SEP in the period after 1979. This may be related to the fact that the land surface experiences a stronger warming than the ocean surface, in relation to the higher evaporation and heat storage capacity of the oceans \cite{guemas2013retrospective}. It might also be related to changes in the atmospheric circulation (e.g. the  North Atlantic oscillation and the related arctic oscillation). This pattern might also reflect some spatial properties of the observing system. Because weather stations  and atmospheric soundings are typically land-based, the atmospheric reanalysis assimilates much more observations over land than over the ocean, particularly until the first satellite observations became available in 1979. The PC time series corresponding to the first EOFs show how the data are increasingly projecting on these spatial patterns starting from 1979.

The second EOFs for SEP and FIM highlight interesting spatial patterns, once again related to significant changes starting from the late eighties as it shown by the corresponding PC time series. The most evident spatial trends concern the continental interiors of Asia, the western North America and central Brazil. These regions are described in the second EOFs by growing values of FIM up to the late eighties, with a drastic change from that period onward.

\section{Conclusions}

This paper discussed how statistical methods derived from Information Theory could be used to investigate the properties of a climate field, specifically the surface air temperature. We found that the three measures applied, FIM, SEP and FSC, could provide meaningful insight about the global and local properties of the mentioned time series. Specifically, we were able to recover spatio-temporal structures in the data, depicting behaviours that, otherwise, would have been difficult to highlight.
Indeed, the results presented in the previous section underpin the capability of the applied measures to detect the degree to which observational products, such as the reanalyses ones, are affected by step changes in the underlying observing system. The detection of such changes is relevant, as a lack of their documentation may lead scientist to misleading conclusions regarding, for instance, climate trends. An example is the detection of two phases in the reanalysis dataset studied in this paper. The two are clearly separated by a behavioural change in 1979 recognized by both FIM and SEP. Indeed, starting from 1980 with the introduction of the Operational Vertical Sounder (TVOS), satellite data became part of the reanalysis model. The PC time series in Figure \ref{fig:eofs} are particularly effective to visualize this changing point.
Nonetheless, even just considering data from 1979 it is possible to clearly identify spatial and temporal patterns in the two measures computed - as shown in Figure \ref{fig:eofs} by the regression lines on the PC time series of the first components of SEP ($R^2$ = 0.889) and FIM ($R^2$ = 0.902).

Warming is not a smooth monotonous process, and the chaotic nature of climate systems reduces the possibility of performing reliable forecasts of future temperature scenarios \cite{sevellec2018novel}. Previous studies shown that entropy can be used to spatially identify unpredictability patterns in surface temperature data \cite{arizmendi2017identifying}. The joint use of SEP and FIM highlighted how the spatial and temporal evolution of this predictability level may not be constant over the entire globe and along time, stressing how the increase of observational data could not always lead to a decrease of such unpredictability \cite{zidek2003uncertainty}. However, the effect of the distribution changes along time on the medium to long term forecasts will have to be further investigated.

Future work will also have to focus on the study of the trend recognized in SEP and FIM since 1979. It is not possible to infer the causal factors inducing this trend using only the NCEP CDAS1 data, as it is not possible to distinguish the effects induced by the growing number of available observations used by the model and those caused by a changing climate. Hence, future analysis will be based on factorial climate simulations from the Coupled Model Intercomparison Project, to compare the information/entropy behaviour of pre-industrial control simulations versus historical simulations including the effects of land use change and greenhouse gas emissions. This could recognize specific patterns due to e.g. GHG emissions.

\begin{acks}
The research presented in this paper was partly supported by the National Research Program 75 "Big Data" (PNR75, project No. 167285 "HyEnergy") of the Swiss National Science Foundation (SNSF). V.H. is supported by the SNSF grant no. P400P2\_180784.
\end{acks}

\bibliographystyle{ACM-Reference-Format}
\bibliography{sample-base}

\end{document}


TITLE: Quadratic Diophantine equation $x^2+6y^2-xy=47$ has no solutions.
QUESTION [1 upvotes]: I am trying to show that $x^2 + 6y^2 - xy = 47$ has no integer solutions. I know that the an efficient way is to look at this equation modulo $n$; other equations can be easily be solved this way. I tried this for $n = 2,3,4,5,6$ so far and I still cannot conclude that no solutions exist. Is there an efficient way of knowing what $n$ to try? Can you give some ideas for $n$ not large? Thanks.

REPLY [0 votes]: The equation $x^2-xy+y^2-47=0$ represents an ellipse and it is easy to verify that for $y\ge3$ and $y\le-3$ there are not real roots for the quadratic resultant.
Consequently it is enough to verify the corresponding values for $x$ when $y$ take the values $y=\pm2,\pm1,0$. We have
$$y=-2\Rightarrow x^2+2x-23=0\\y=-1\Rightarrow x^2+x-41=0\\y=0\Rightarrow x^2-47=0\\y=1\Rightarrow x^2-x-41=0\\y=2\Rightarrow x^2-2x-23=0$$
None of these five equations have integer roots. Thus the given diophantine equation has no solution.


TITLE: Is the function $f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ defined by $f(x, y) = 7x + 3y$ injective? surjective?
QUESTION [0 upvotes]: Define the mapping $\ \ f:\mathbb{N} \times \mathbb{N} \to \mathbb{N}\ $ by $\ f(x,y) = 7x + 3y$.
Show by example that $\,f\,$ is not one-to-one.
Is $\ f$ a surjection? $\ $Compute $\ f^{\tiny{-1}\ \ \ \ }\ (\{16\})\ $ and $\ f^{\tiny{-1}\ \ \ \ }\ (\{11\})$
I have been trying to do the function for single $\,f(x)\,$ only. I really have no idea on double function like $\,f(x,y)\,$ – can someone show me how to do it? Thanks in advance.

REPLY [1 votes]: The function $f$ is not one-to-one. For, the equation $7x+3y = 38$ admits at least these two positive-integer solutions $(x,y) = (5,1), (2,8)$.
The function $f$ is not surjective. For, the equation $7x+3y=1$ is not solvable in positive integers.
The last two questions require you to solve the equations $7x+3y = 16$ and $7x+3y = 11$ in positive integers, respectively. You can try these ones yourself. Well, for example, the pair $(x,y) = (1,3) \in \mathbb{N}$ and such that $7x + 3y = 16$; and no other pair of positive integers delivers the equation. So $f^{-1}\{ 16 \} = \{ (1, 3) \}$.

REPLY [1 votes]: The function is not surjective since $\ f(m,n)\geq 10 \ \  \forall\  (m,n)\in \mathbb{N}\times\mathbb{N}$,  and hence $ f(\mathbb{N}\times \mathbb{N})\neq \mathbb{N}$. Also $3\cdot8+7\cdot1=3\cdot1+7\cdot4=31$ so the function is not one one.


TITLE: How does QFT help with entanglement?
QUESTION [22 upvotes]: I'm a bit confused. QFT is claimed to incorporate both Quantum Mechanics and Special Relativity. Therefore it should address the problem of non-locality caused by entanglement. However when I search for an answer on the Internet, I found nothing.
I'm not complaining. But it seems that most people only use QFT to do some fancy particle stuff and forgot we should care more about the more fundamental stuff.

REPLY [3 votes]: Quantum mechanics is a local physical theory. The idea of non-locality is a misunderstanding that is common even among physicists.
In classical physics, a system can be described by a set of numbers whose values can all be measured using a single instance of that system. There is a mathematical result called Bell's theorem saying that no local theory can reproduce the predictions of quantum mechanics using classical physics. Quantum mechanics is not classical physics and so it is not surprising that they give rise to different predictions.
In quantum mechanics, a system is characterised by the values of observables where those values are represented by mathematical objects called Hermitian matrices. To describe how information is transferred between quantum systems you have to describe the ways in which the observables of one system depend on those of another. In general, an observable does not represent just a single valued measurable quantity changing over time. Rather, it represents a more complex structure that involves multiple different versions of that quantity interfering with one another. And if there are going to be multiple versions of each system, then any given system has to carry information about how a particular version of that system will interact with a particular version of another system. In general, you can't get that sort of information by measuring just one system and for that reason it is called locally inaccessible information. An explanation of how locally inaccessible information gives rise to EPR correlations, teleportation etc by entirely local interactions is given here:
http://arxiv.org/abs/quant-ph/9906007.
See also
http://arxiv.org/abs/1109.6223.
An account of how this mechanism works in quantum field theory is given here:
https://arxiv.org/abs/quant-ph/0204024.
For popular treatments see "The Fabric of Reality" and "The Beginning of Infinity" by David Deutsch.