Skip to content

Commit

Permalink
update some references
Browse files Browse the repository at this point in the history
  • Loading branch information
@darijgr
darijgr committed May 22, 2021
1 parent b168634 commit 1bb176d
Showing 1 changed file with 28 additions and 25 deletions.
53 changes: 28 additions & 25 deletions sols.tex
View file
Original file line number Diff line number Diff line change
Expand Up @@ -16,7 +16,7 @@
\usepackage[type={CC}, modifier={zero}, version={1.0},]{doclicense}
%TCIDATA{OutputFilter=latex2.dll}
%TCIDATA{Version=5.50.0.2960}
%TCIDATA{LastRevised=Saturday, May 22, 2021 23:07:22}
%TCIDATA{LastRevised=Saturday, May 22, 2021 23:40:04}
%TCIDATA{SuppressPackageManagement}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
Expand Down Expand Up @@ -11284,8 +11284,8 @@ \subsubsection{The setup and the problem}
(3)]{Swanso18} for the case when $\mathbb{A}=\mathbb{N}$; in \cite[Theorem
3.5.4 (3)]{Swanso18} for the case when $\mathbb{A}=\mathbb{Z}$; in
\cite[Theorem 3.6.4 (3)]{Swanso18} for the case when $\mathbb{A}=\mathbb{Q}$;
in \cite[Theorem 3.7.10]{Swanso18} for the case when $\mathbb{A}=\mathbb{R}$;
in \cite[Theorem 3.9.3]{Swanso18} for the case when $\mathbb{A}=\mathbb{C}$.}.
in \cite[Theorem 3.7.11]{Swanso18} for the case when $\mathbb{A}=\mathbb{R}$;
in \cite[Theorem 3.9.2]{Swanso18} for the case when $\mathbb{A}=\mathbb{C}$.}.

In Section \ref{sect.ind.gen-ass}, we have used Proposition
\ref{prop.ind.gen-ass-maps.fgh} to show that we can \textquotedblleft drop the
Expand Down Expand Up @@ -11322,8 +11322,8 @@ \subsubsection{The setup and the problem}
the case when $\mathbb{A}=\mathbb{N}$; in \cite[Theorem 3.5.4 (4)]{Swanso18}
for the case when $\mathbb{A}=\mathbb{Z}$; in \cite[Theorem 3.6.4
(4)]{Swanso18} for the case when $\mathbb{A}=\mathbb{Q}$; in \cite[Theorem
3.7.10]{Swanso18} for the case when $\mathbb{A}=\mathbb{R}$; in \cite[Theorem
3.9.3]{Swanso18} for the case when $\mathbb{A}=\mathbb{C}$.}.
3.7.11]{Swanso18} for the case when $\mathbb{A}=\mathbb{R}$; in \cite[Theorem
3.9.2]{Swanso18} for the case when $\mathbb{A}=\mathbb{C}$.}.

Furthermore, numbers can \textbf{always} be added, whereas maps can only be
composed if the domain of one is the codomain of the other. Thus, when we want
Expand Down Expand Up @@ -13035,8 +13035,8 @@ \subsubsection{Linearity of sums}
proven in \cite[Theorem 3.2.3 (6)]{Swanso18} for the case when $\mathbb{A}%
=\mathbb{N}$; in \cite[Theorem 3.5.4 (6)]{Swanso18} for the case when
$\mathbb{A}=\mathbb{Z}$; in \cite[Theorem 3.6.4 (6)]{Swanso18} for the case
when $\mathbb{A}=\mathbb{Q}$; in \cite[Theorem 3.7.13]{Swanso18} for the case
when $\mathbb{A}=\mathbb{R}$; in \cite[Theorem 3.9.3]{Swanso18} for the case
when $\mathbb{A}=\mathbb{Q}$; in \cite[Theorem 3.7.14]{Swanso18} for the case
when $\mathbb{A}=\mathbb{R}$; in \cite[Theorem 3.9.2]{Swanso18} for the case
when $\mathbb{A}=\mathbb{C}$.}.

\begin{proof}
Expand Down Expand Up @@ -14299,8 +14299,8 @@ \subsubsection{\label{subsect.ind.gen-com.prods}Finite products}
3.2.3 (7)]{Swanso18} for the case when $\mathbb{A}=\mathbb{N}$; in
\cite[Theorem 3.5.4 (7)]{Swanso18} for the case when $\mathbb{A}=\mathbb{Z}$;
in \cite[Theorem 3.6.4 (7)]{Swanso18} for the case when $\mathbb{A}%
=\mathbb{Q}$; in \cite[Theorem 3.7.13]{Swanso18} for the case when
$\mathbb{A}=\mathbb{R}$; in \cite[Theorem 3.9.3]{Swanso18} for the case when
=\mathbb{Q}$; in \cite[Theorem 3.7.14]{Swanso18} for the case when
$\mathbb{A}=\mathbb{R}$; in \cite[Theorem 3.9.2]{Swanso18} for the case when
$\mathbb{A}=\mathbb{C}$.}.

Proposition \ref{prop.ind.gen-com.fg} also has an analogue for multiplication:
Expand All @@ -14317,8 +14317,8 @@ \subsubsection{\label{subsect.ind.gen-com.prods}Finite products}
the case when $\mathbb{A}=\mathbb{N}$; in \cite[Theorem 3.5.4 (8)]{Swanso18}
for the case when $\mathbb{A}=\mathbb{Z}$; in \cite[Theorem 3.6.4
(8)]{Swanso18} for the case when $\mathbb{A}=\mathbb{Q}$; in \cite[Theorem
3.7.13]{Swanso18} for the case when $\mathbb{A}=\mathbb{R}$; in \cite[Theorem
3.9.3]{Swanso18} for the case when $\mathbb{A}=\mathbb{C}$.}.
3.7.14]{Swanso18} for the case when $\mathbb{A}=\mathbb{R}$; in \cite[Theorem
3.9.2]{Swanso18} for the case when $\mathbb{A}=\mathbb{C}$.}.

Proposition \ref{prop.ind.gen-com.distr} has an analogue for multiplication as
well (but note that $x$ now needs to be in $\mathbb{N}$, in order to guarantee
Expand Down Expand Up @@ -16946,7 +16946,7 @@ \subsection{Binomial coefficients and polynomials}
coefficients) having degree $n\geq0$ has at most $n$ roots. See, for example,
\cite[Corollary 1.8.24]{Goodman} or \cite[Theorem 1.58]{Joyce17} or
\cite[Corollary 4.5.10]{Walker87} or \cite[Corollary 33.7]{Elman18} or
\cite[Theorem 2.4.13]{Swanso18} or \cite[Corollary 1.14]{Knapp1} for a proof.
\cite[Theorem 2.4.15]{Swanso18} or \cite[Corollary 1.14]{Knapp1} for a proof.
Note that Swanson, in \cite{Swanso18}, works with polynomial functions instead
of polynomials; but as far as roots are concerned, the difference does not
matter (since the roots of a polynomial are precisely the roots of the
Expand Down Expand Up @@ -95345,7 +95345,7 @@ \subsection{\label{sect.sol.cauchy-det-lem}Solution to Exercise

\subsubsection{The solution}

Exercise \ref{exe.cauchy-det-lem} is the equality (12.57.3) in \cite[ancillary
Exercise \ref{exe.cauchy-det-lem} is the equality (13.70.3) in \cite[ancillary
PDF file]{Reiner}. The following solution is taken from \cite[ancillary PDF
file]{Reiner}.

Expand Down Expand Up @@ -128594,6 +128594,7 @@ \subsubsection{Second solution}
\det\left( \operatorname*{sub}\nolimits_{w\left( \widetilde{Q}\right)
}^{w\left( \widetilde{P}\right) }A\right) \prod_{i\in Q}d_{i}.
\]

\end{corollary}

\begin{proof}
Expand Down Expand Up @@ -132684,7 +132685,8 @@ \section{Appendix: Old citations}

\bibitem[BjoBre05]{BjoBre05}\href{http://doi.org/10.1007/3-540-27596-7}{Anders
Bj\"{o}rner, Francesco Brenti, \textit{Combinatorics of Coxeter Groups},
Graduate Texts in Mathematics \#231, Springer 2005.}
Graduate Texts in Mathematics \#231, Springer 2005.}\newline See
\url{https://www.mat.uniroma2.it/~brenti/correct.ps} for errata.

\bibitem[Bona12]{Bona12}Mikl\'{o}s B\'{o}na, \textit{Combinatorics of
Permutations}, 2nd edition, CRC Press 2012.
Expand Down Expand Up @@ -132770,7 +132772,7 @@ \section{Appendix: Old citations}
\href{https://doi.org/10.1006/jcta.1999.3000}{J. Combin. Theory Ser. A 88
(1999), pp. 368--378}.

\bibitem[Elman18]{Elman18}Richard Elman, \textit{Lectures on Abstract
\bibitem[Elman19]{Elman18}Richard Elman, \textit{Lectures on Abstract
Algebra}, 20 September 2019.\newline\url{https://www.math.ucla.edu/~rse/algebra_book.pdf}

\bibitem[FadSom72]{FadSom72}D. Faddeev, I. Sominsky, \textit{Problems in
Expand Down Expand Up @@ -132861,9 +132863,9 @@ \section{Appendix: Old citations}
\bibitem[Grinbe16b]{Gri-lina}Darij Grinberg, \textit{Notes on linear algebra},
version of 26 September 2019.\newline\url{https://github.com/darijgr/lina}

\bibitem[GriRei18]{Reiner}Darij Grinberg, Victor Reiner, \textit{Hopf algebras
in Combinatorics}, version of 11 May 2018,
\href{http://www.arxiv.org/abs/1409.8356v5}{\texttt{arXiv:1409.8356v5}}.
\bibitem[GriRei20]{Reiner}Darij Grinberg, Victor Reiner, \textit{Hopf algebras
in Combinatorics}, version of 27 July 2020,
\href{http://www.arxiv.org/abs/1409.8356v7}{\texttt{arXiv:1409.8356v7}}.
\newline See also
\url{http://www.cip.ifi.lmu.de/~grinberg/algebra/HopfComb-sols.pdf} for a
version that gets updated.
Expand All @@ -132875,8 +132877,8 @@ \section{Appendix: Old citations}
\bibitem[Hammac15]{Hammac15}Richard Hammack, \textit{Book of Proof}, 3rd
edition 2018.\newline\url{https://www.people.vcu.edu/~rhammack/BookOfProof/}

\bibitem[Heffer17]{Hefferon}Jim Hefferon, \textit{Linear Algebra}, 3rd
edition, 1 January 2017,\newline\url{http://joshua.smcvt.edu/linearalgebra/} .
\bibitem[Heffer20]{Hefferon}Jim Hefferon, \textit{Linear Algebra}, 4th
edition, 8 May 2020,\newline\url{http://joshua.smcvt.edu/linearalgebra/} .

\bibitem[Heinig11]{Heinig}\href{http://arxiv.org/abs/1103.2717v3}{Peter
Christian Heinig, \textit{Chio Condensation and Random Sign Matrices},
Expand Down Expand Up @@ -133130,12 +133132,13 @@ \section{Appendix: Old citations}
\textit{Counting and Recounting: The Aftermath}, The Mathematical
Intelligencer \textbf{6} (1984), no. 2, pp. 44--45}.

\bibitem[Swanso18]{Swanso18}Irena Swanson, \textit{Introduction to Analysis},
with construction of the number systems, 18 November 2019.\newline\url{http://people.reed.edu/~iswanson/analysisconstructR.pdf}
\bibitem[Swanso20]{Swanso18}Irena Swanson, \textit{Introduction to Analysis},
with construction of the number systems, 25 July 2020.\newline\url{https://web.archive.org/web/20201012174324/http://people.reed.edu/~iswanson/analysisconstructR.pdf}

\bibitem[Tenner04]{Tenner-NMU}Bridget Eileen Tenner, \textit{A Non-Messing-Up
Phenomenon for Posets}, Annals of Combinatorics 11 (2007), pp.
101--114.\newline A preprint is available as
\bibitem[Tenner04]{Tenner-NMU}%
\href{https://doi.org/10.1007/s00026-007-0308-z}{Bridget Eileen Tenner,
\textit{A Non-Messing-Up Phenomenon for Posets}, Annals of Combinatorics 11
(2007), pp. 101--114}.\newline A preprint is available as
\href{http://arxiv.org/abs/math/0404396}{arXiv:math/0404396}.

\bibitem[Turnbu29]{Turnbu29}H. W. Turnbull, \textit{The Theory of
Expand Down

0 comments on commit 1bb176d

Please sign in to comment.