Commit d4be70b1 authored by Lawrence Paulson's avatar Lawrence Paulson

Formal_Puiseux_Series website

parent 74a42283f683
......@@ -9676,6 +9676,23 @@ abstract =
p<sub><em>i</em></sub> for
i&gt;k.</p>
[Formal_Puiseux_Series]
title = Formal Puiseux Series
author = Manuel Eberl <https://www21.in.tum.de/~eberlm>
topic = Mathematics/Algebra
date = 2021-02-17
notify = eberlm@in.tum.de
abstract =
<p>Formal Puiseux series are generalisations of formal power
series and formal Laurent series that also allow for fractional
exponents. They have the following general form: \[\sum_{i=N}^\infty
a_{i/d} X^{i/d}\] where <em>N</em> is an integer and
<em>d</em> is a positive integer.</p> <p>This
entry defines these series including their basic algebraic properties.
Furthermore, it proves the Newton–Puiseux Theorem, namely that the
Puiseux series over an algebraically closed field of characteristic 0
are also algebraically closed.</p>
[Gaussian_Integers]
title = Gaussian Integers
author = Manuel Eberl <https://www21.in.tum.de/~eberlm>
......@@ -10232,8 +10249,8 @@ author = Jakub Kądziołka <mailto:kuba@kadziolka.net>
topic = Logic/General logic/Logics of knowledge and belief
date = 2021-01-30
notify = kuba@kadziolka.net
abstract =
In a <a href="https://xkcd.com/blue_eyes.html">puzzle published by
abstract =
In a <a href="https://xkcd.com/blue_eyes.html">puzzle published by
Randall Munroe</a>, perfect logicians forbidden
from communicating are stranded on an island, and may only leave once
they have figured out their own eye color. We present a method of
......
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<h1> <font class="first">F</font>ormal
<font class="first">P</font>uiseux
<font class="first">S</font>eries
</h1>
<p>&nbsp;</p>
<table width="80%" class="data">
<tbody>
<tr>
<td class="datahead" width="20%">Title:</td>
<td class="data" width="80%">Formal Puiseux Series</td>
</tr>
<tr>
<td class="datahead">
Author:
</td>
<td class="data">
<a href="https://www21.in.tum.de/~eberlm">Manuel Eberl</a>
</td>
</tr>
<tr>
<td class="datahead">Submission date:</td>
<td class="data">2021-02-17</td>
</tr>
<tr>
<td class="datahead" valign="top">Abstract:</td>
<td class="abstract mathjax_process">
<p>Formal Puiseux series are generalisations of formal power
series and formal Laurent series that also allow for fractional
exponents. They have the following general form: \[\sum_{i=N}^\infty
a_{i/d} X^{i/d}\] where <em>N</em> is an integer and
<em>d</em> is a positive integer.</p> <p>This
entry defines these series including their basic algebraic properties.
Furthermore, it proves the Newton–Puiseux Theorem, namely that the
Puiseux series over an algebraically closed field of characteristic 0
are also algebraically closed.</p></td>
</tr>
<tr>
<td class="datahead" valign="top">BibTeX:</td>
<td class="formatted">
<pre>@article{Formal_Puiseux_Series-AFP,
author = {Manuel Eberl},
title = {Formal Puiseux Series},
journal = {Archive of Formal Proofs},
month = feb,
year = 2021,
note = {\url{https://isa-afp.org/entries/Formal_Puiseux_Series.html},
Formal proof development},
ISSN = {2150-914x},
}</pre>
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<td class="data"><a href="Polynomial_Interpolation.html">Polynomial_Interpolation</a> </td></tr>
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<p></p>
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<a href="../browser_info/current/AFP/Formal_Puiseux_Series/outline.pdf">Proof outline</a><br>
<a href="../browser_info/current/AFP/Formal_Puiseux_Series/document.pdf">Proof document</a>
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\ No newline at end of file
......@@ -149,7 +149,7 @@ division of integers in target languages.</td>
<td class="data"><a href="Sqrt_Babylonian.html">Sqrt_Babylonian</a> </td></tr>
<tr><td class="datahead">Used by:</td>
<td class="data"><a href="Deep_Learning.html">Deep_Learning</a>, <a href="Gauss_Sums.html">Gauss_Sums</a>, <a href="Polynomial_Factorization.html">Polynomial_Factorization</a> </td></tr>
<td class="data"><a href="Deep_Learning.html">Deep_Learning</a>, <a href="Formal_Puiseux_Series.html">Formal_Puiseux_Series</a>, <a href="Gauss_Sums.html">Gauss_Sums</a>, <a href="Polynomial_Factorization.html">Polynomial_Factorization</a> </td></tr>
......
......@@ -88,6 +88,14 @@ of a scientific journal, is indexed by <a href="http://dblp.uni-trier.de/db/jour
<tr>
<td class="head">2021</td>
</tr>
<tr>
<td class="entry">
2021-02-17: <a href="entries/Formal_Puiseux_Series.html">Formal Puiseux Series</a>
<br>
Author:
<a href="https://www21.in.tum.de/~eberlm">Manuel Eberl</a>
</td>
</tr>
<tr>
<td class="entry">
2021-02-10: <a href="entries/Laws_of_Large_Numbers.html">The Laws of Large Numbers</a>
......
......@@ -9,7 +9,24 @@
and larger scientific developments, mechanically checked
in the theorem prover Isabelle.
</description>
<pubDate>10 Feb 2021 00:00:00 +0000</pubDate>
<pubDate>17 Feb 2021 00:00:00 +0000</pubDate>
<item>
<title>Formal Puiseux Series</title>
<link>https://www.isa-afp.org/entries/Formal_Puiseux_Series.html</link>
<guid>https://www.isa-afp.org/entries/Formal_Puiseux_Series.html</guid>
<dc:creator> Manuel Eberl </dc:creator>
<pubDate>17 Feb 2021 00:00:00 +0000</pubDate>
<description>
&lt;p&gt;Formal Puiseux series are generalisations of formal power
series and formal Laurent series that also allow for fractional
exponents. They have the following general form: \[\sum_{i=N}^\infty
a_{i/d} X^{i/d}\] where &lt;em&gt;N&lt;/em&gt; is an integer and
&lt;em&gt;d&lt;/em&gt; is a positive integer.&lt;/p&gt; &lt;p&gt;This
entry defines these series including their basic algebraic properties.
Furthermore, it proves the Newton–Puiseux Theorem, namely that the
Puiseux series over an algebraically closed field of characteristic 0
are also algebraically closed.&lt;/p&gt;</description>
</item>
<item>
<title>The Laws of Large Numbers</title>
<link>https://www.isa-afp.org/entries/Laws_of_Large_Numbers.html</link>
......@@ -584,49 +601,5 @@ development, we present the verified checker Pastèque that is obtained
by synthesis via the Refinement Framework. This is the formalization
going with our FMCAD&#39;20 tool presentation.</description>
</item>
<item>
<title>Some classical results in inductive inference of recursive functions</title>
<link>https://www.isa-afp.org/entries/Inductive_Inference.html</link>
<guid>https://www.isa-afp.org/entries/Inductive_Inference.html</guid>
<dc:creator> Frank J. Balbach </dc:creator>
<pubDate>31 Aug 2020 00:00:00 +0000</pubDate>
<description>
&lt;p&gt; This entry formalizes some classical concepts and results
from inductive inference of recursive functions. In the basic setting
a partial recursive function (&#34;strategy&#34;) must identify
(&#34;learn&#34;) all functions from a set (&#34;class&#34;) of
recursive functions. To that end the strategy receives more and more
values $f(0), f(1), f(2), \ldots$ of some function $f$ from the given
class and in turn outputs descriptions of partial recursive functions,
for example, Gödel numbers. The strategy is considered successful if
the sequence of outputs (&#34;hypotheses&#34;) converges to a
description of $f$. A class of functions learnable in this sense is
called &#34;learnable in the limit&#34;. The set of all these
classes is denoted by LIM. &lt;/p&gt; &lt;p&gt; Other types of
inference considered are finite learning (FIN), behaviorally correct
learning in the limit (BC), and some variants of LIM with restrictions
on the hypotheses: total learning (TOTAL), consistent learning (CONS),
and class-preserving learning (CP). The main results formalized are
the proper inclusions $\mathrm{FIN} \subset \mathrm{CP} \subset
\mathrm{TOTAL} \subset \mathrm{CONS} \subset \mathrm{LIM} \subset
\mathrm{BC} \subset 2^{\mathcal{R}}$, where $\mathcal{R}$ is the set
of all total recursive functions. Further results show that for all
these inference types except CONS, strategies can be assumed to be
total recursive functions; that all inference types but CP are closed
under the subset relation between classes; and that no inference type
is closed under the union of classes. &lt;/p&gt; &lt;p&gt; The above
is based on a formalization of recursive functions heavily inspired by
the &lt;a
href=&#34;https://www.isa-afp.org/entries/Universal_Turing_Machine.html&#34;&gt;Universal
Turing Machine&lt;/a&gt; entry by Xu et al., but different in that it
models partial functions with codomain &lt;em&gt;nat
option&lt;/em&gt;. The formalization contains a construction of a
universal partial recursive function, without resorting to Turing
machines, introduces decidability and recursive enumerability, and
proves some standard results: existence of a Kleene normal form, the
&lt;em&gt;s-m-n&lt;/em&gt; theorem, Rice&#39;s theorem, and assorted
fixed-point theorems (recursion theorems) by Kleene, Rogers, and
Smullyan. &lt;/p&gt;</description>
</item>
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......@@ -697,6 +697,7 @@
<a href="entries/Hybrid_Systems_VCs.html">Hybrid_Systems_VCs</a> &nbsp;
<a href="entries/Subset_Boolean_Algebras.html">Subset_Boolean_Algebras</a> &nbsp;
<a href="entries/Power_Sum_Polynomials.html">Power_Sum_Polynomials</a> &nbsp;
<a href="entries/Formal_Puiseux_Series.html">Formal_Puiseux_Series</a> &nbsp;
<a href="entries/Matrices_for_ODEs.html">Matrices_for_ODEs</a> &nbsp;
<a href="entries/Smith_Normal_Form.html">Smith_Normal_Form</a> &nbsp;
</div>
......
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