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Main Authors:	Baek, Jineon, Koizumi, Junnosuke, Ueoro, Takahiro
Format:	Preprint
Published:	2024
Subjects:	Combinatorics Metric Geometry 52C15, 52C10
Online Access:	https://arxiv.org/abs/2411.07274
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_version_	1866913579685380096
author	Baek, Jineon Koizumi, Junnosuke Ueoro, Takahiro
author_facet	Baek, Jineon Koizumi, Junnosuke Ueoro, Takahiro
contents	Let $f(n)$ denote the maximum total length of the sides of $n$ squares packed inside a unit square. Erdős conjectured that $f(k^2+1)=k$. We show that the conjecture is true if we assume that the sides of the squares are parallel to the sides of the unit square.
format	Preprint
id	arxiv_https___arxiv_org_abs_2411_07274
institution	arXiv
publishDate	2024
record_format	arxiv
spellingShingle	A note on the Erdős conjecture about square packing Baek, Jineon Koizumi, Junnosuke Ueoro, Takahiro Combinatorics Metric Geometry 52C15, 52C10 Let $f(n)$ denote the maximum total length of the sides of $n$ squares packed inside a unit square. Erdős conjectured that $f(k^2+1)=k$. We show that the conjecture is true if we assume that the sides of the squares are parallel to the sides of the unit square.
title	A note on the Erdős conjecture about square packing
topic	Combinatorics Metric Geometry 52C15, 52C10
url	https://arxiv.org/abs/2411.07274

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// Anecdotal testing Today and Yesterday seems to indicate destroy() // is called by the garbage collector and everything is good. // Something to keep in mind though. return 0; } /** * Write function that is called when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ public function write($sessId, $data): bool { if ($this->writesDisabled) { return true; } return $this->saveSession($sessId, $data); } /** * A function that is called internally when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ abstract protected function saveSession($sessId, $data): bool; }

* * @return void */ public function writeClose() { // The assumption is that we're using PHP's ext/session. // session_write_close() will actually overwrite $_SESSION with an // empty array on completion -- which leads to a mismatch between what // is in the storage object and $_SESSION. To get around this, we // temporarily reset $_SESSION to an array, and then re-link it to // the storage object. // // Additionally, while you _can_ write to $_SESSION following a // session_write_close() operation, no changes made to it will be // flushed to the session handler. As such, we now mark the storage // object isImmutable. $storage = $this->getStorage(); if (! $storage->isImmutable()) { $_SESSION = $storage->toArray(true); session_write_close(); $storage->fromArray($_SESSION); $storage->markImmutable(); } } /** * Attempt to set the session name * * If the session has already been started, or if the name provided fails * validation, an exception will be raised. * * @param string $name * @return SessionManager * @throws Exception\InvalidArgumentException */ public function setName($name) { if ($this->sessionExists()) { throw new Exception\InvalidArgumentException( 'Cannot set session name after a session has already started'

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