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Commit 970fb732 authored by m-iancu's avatar m-iancu
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......@@ -7,6 +7,7 @@ as an import. However, in general, it can have a more complex behavior (e.g. see
\begin{example}[http://arxiv.org/pdf/1502.00573.pdf] \label{ex:quant}
\cite{quant-orig} discusses quantifier elimination in $C^*$-algebras and begins by defining the formulas for $C^*$-algebras after the
the introduction.
\begin{labeledquote}\label{def:quant}
\begin{definition}
The \emph{formulas} for $C^*$-algebras are recursively defined as follows.
In each case, $\overline{x}$ denotes a finite tuple of variables (which will later be interpreted as
......@@ -23,6 +24,7 @@ We think of $\mathrm{sup}_{\|y\| \leq R}$ and $\mathrm{inf}_{\|y\| \leq R}$ as r
$\forall$ and $\exists$, respectively. A formula constructed using only clauses (1) and (2) of the
definition is therefore said to be \emph{quantifier-free}.
\end{definition}
\end{labeledquote}
The authors refer to \cite{quant-fs14} for a more complete discussion although it uses a slightly different definition. In fact, the
authors explicitly acknowledge that later:
......@@ -36,19 +38,24 @@ definition is therefore said to be \emph{quantifier-free}.
\begin{example}{http://arxiv.org/pdf/1502.00978.pdf}\label{ex:calculi}
\cite{calculi-orig} proves some results about undecidable problems for propositional calculi.
In the preliminaries it introduces notations and definitions from scratch:
``Let us consider the language consisting of an infinite set of propositional variables
\begin{labeledquote}
Let us consider the language consisting of an infinite set of propositional variables
$V$ and the signature $\Sigma$, i.e., a finite set of connectives. Letters
$x, y, z, u, p$ , etc., are used to denote propositional variables.
Usually connectives are binary or unary such as $\neg, \vee, \wedge,$ or $\arr$.''
Usually connectives are binary or unary such as $\neg, \vee, \wedge,$ or $\arr$.
\end{labeledquote}
However, it still relies on the literature to fully define the local concepts:
``Propositional formulas or $\Sigma$-formulas are built up from the signature
$\Sigma$ and propositional variables $V$ \emph{in the usual way}. '' [emphasis ours]
The paper continues by introducing previous results, by mapping them (implicitly) to the definitions introduced
locally. For instance :
\begin{labeledquote}
\begin{theorem}[Theorem 2.1]
(Linial and Post, 1949). \textbf{Axm}, \textbf{Ext}, and \textbf{Cmpl} are undecidable for $\textbf{Cl}_{\{\neg, \wedge\}}$.
\end{theorem}
\end{labeledquote}
where \textbf{Axm}, \textbf{Axm}, \textbf{Ext}, \textbf{Cmpl} and \textbf{Cl} are all defined locally.
Having established local concepts and their properties (via the literature) the paper continues with its contribution.
......@@ -59,6 +66,7 @@ Having established local concepts and their properties (via the literature) the
\cite{covers-orig} discusses covers of the multiplicative group of an algebraically closed field which are formally defined in the beginning
of the paper as follows:
\begin{labeledquote}
\begin{definition}
Let $V$ be a vector space over $Q$ and let $F$ be an algebraically closed field
of characteristic $0$. A \emph{cover of the multiplicative group} of $F$ is a structure represented by
......@@ -67,6 +75,7 @@ $0 \arr K \arr V \arr F^∗ \arr 1$ ,
where the map $V \arr F^*$ is a surjective group homomorphism from $(V, +)$ onto
$(F^∗, \cdot)$ with kernel $K$. We will call this map \emph{exp}.
\end{definition}
\end{labeledquote}
When, establishing results, \cite{covers-orig} mentions ``Moreover, with an additional axiom (in $L_{\omega_1\omega}$) stating $K \cong Z$, the class is categorical
in uncountable cardinalities. This was originally proved in \cite{covers-13} but an error was later found
......@@ -82,11 +91,12 @@ extends the result in \cite{covers-13} and additionally fixes a hole in the proo
\begin{example}{http://arxiv.org/pdf/1502.02059.pdf}\label{ex:mnets}
\cite{mnets-orig} studies the properties of multinets. In the preliminaries section they are introduced with the following
definition:
\begin{labeledquote}
\begin{definition} The union of all completely reducible fibers (with a fixed partition
into fibers, also called blocks) of a Ceva pencil of degree $d$ is called a $(k, d)-\mathit{multinet}$
where $k$ is the number of the blocks. The base $X$ of the pencil is determined by the multinet structure and called the base of the multinet.
\end{definition}
\end{labeledquote}
Later in that section some properties of multinets are introduced with the phrase ``Several important properties of
multinets are listed below which have been collected from \cite{mnets-ref4, mnets-ref10, mnets-ref12}.''
The referenced papers all use slightly different definitions of multinets but they are assumed to be equivalent so that the properties
......
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