Accueil /Mathématiques /Formes quadratiques dégénérées et vecteurs isotropes

Message d'erreur

  • Deprecated function: Return type of DatabaseStatementBase::execute($args = [], $options = []) should either be compatible with PDOStatement::execute(?array $params = null): bool, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in require_once() (line 2244 of /home/laurentp/www/includes/database/database.inc).
  • Deprecated function: Return type of DatabaseStatementEmpty::current() should either be compatible with Iterator::current(): mixed, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in require_once() (line 2346 of /home/laurentp/www/includes/database/database.inc).
  • Deprecated function: Return type of DatabaseStatementEmpty::next() should either be compatible with Iterator::next(): void, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in require_once() (line 2346 of /home/laurentp/www/includes/database/database.inc).
  • Deprecated function: Return type of DatabaseStatementEmpty::key() should either be compatible with Iterator::key(): mixed, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in require_once() (line 2346 of /home/laurentp/www/includes/database/database.inc).
  • Deprecated function: Return type of DatabaseStatementEmpty::valid() should either be compatible with Iterator::valid(): bool, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in require_once() (line 2346 of /home/laurentp/www/includes/database/database.inc).
  • Deprecated function: Return type of DatabaseStatementEmpty::rewind() should either be compatible with Iterator::rewind(): void, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in require_once() (line 2346 of /home/laurentp/www/includes/database/database.inc).
  • Deprecated function: strlen(): Passing null to parameter #1 ($string) of type string is deprecated in drupal_random_bytes() (line 2268 of /home/laurentp/www/includes/bootstrap.inc).
  • Deprecated function: Optional parameter $type declared before required parameter $key is implicitly treated as a required parameter in include_once() (line 1389 of /home/laurentp/www/includes/bootstrap.inc).
  • Deprecated function: Return type of DateObject::__wakeup() should either be compatible with DateTime::__wakeup(): void, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in include_once() (line 143 of /home/laurentp/www/sites/all/modules/date/date_api/date_api.module).
  • Deprecated function: Return type of DateObject::format($format, $force = false) should either be compatible with DateTime::format(string $format): string, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in include_once() (line 143 of /home/laurentp/www/sites/all/modules/date/date_api/date_api.module).
  • Deprecated function: Return type of DateObject::setTimezone($tz, $force = false) should either be compatible with DateTime::setTimezone(DateTimeZone $timezone): DateTime, or the #[\ReturnTypeWillChange] attribute should be used to temporarily suppress the notice in include_once() (line 143 of /home/laurentp/www/sites/all/modules/date/date_api/date_api.module).
  • Deprecated function: Optional parameter $input declared before required parameter $form_state is implicitly treated as a required parameter in include_once() (line 1389 of /home/laurentp/www/includes/bootstrap.inc).
  • Deprecated function : Creation of dynamic property SelectQuery::$alterTags is deprecated dans SelectQuery->addTag() (ligne 978 dans /home/laurentp/www/includes/database/select.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : rtrim(): Passing null to parameter #1 ($string) of type string is deprecated dans url() (ligne 2349 dans /home/laurentp/www/includes/common.inc).
  • Deprecated function : Creation of dynamic property SelectQuery::$alterTags is deprecated dans SelectQuery->addTag() (ligne 978 dans /home/laurentp/www/includes/database/select.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property PDOException::$query_string is deprecated dans DatabaseConnection->query() (ligne 770 dans /home/laurentp/www/includes/database/database.inc).
  • Deprecated function : Creation of dynamic property PDOException::$args is deprecated dans DatabaseConnection->query() (ligne 772 dans /home/laurentp/www/includes/database/database.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : strpos(): Passing null to parameter #1 ($haystack) of type string is deprecated dans url_is_external() (ligne 2393 dans /home/laurentp/www/includes/common.inc).
  • Deprecated function : str_replace(): Passing null to parameter #3 ($subject) of type array|string is deprecated dans url_is_external() (ligne 2395 dans /home/laurentp/www/includes/common.inc).
  • Deprecated function : ltrim(): Passing null to parameter #1 ($string) of type string is deprecated dans url() (ligne 2311 dans /home/laurentp/www/includes/common.inc).
  • Deprecated function : Creation of dynamic property SelectQuery::$alterTags is deprecated dans SelectQuery->addTag() (ligne 978 dans /home/laurentp/www/includes/database/select.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property SelectQuery::$alterTags is deprecated dans SelectQuery->addTag() (ligne 978 dans /home/laurentp/www/includes/database/select.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property SelectQuery::$alterTags is deprecated dans SelectQuery->addTag() (ligne 978 dans /home/laurentp/www/includes/database/select.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property SelectQuery::$alterTags is deprecated dans SelectQuery->addTag() (ligne 978 dans /home/laurentp/www/includes/database/select.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property SelectQuery::$alterTags is deprecated dans SelectQuery->addTag() (ligne 978 dans /home/laurentp/www/includes/database/select.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property SelectQuery::$alterTags is deprecated dans SelectQuery->addTag() (ligne 978 dans /home/laurentp/www/includes/database/select.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property SelectQuery::$alterTags is deprecated dans SelectQuery->addTag() (ligne 978 dans /home/laurentp/www/includes/database/select.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property SelectQuery::$alterTags is deprecated dans SelectQuery->addTag() (ligne 978 dans /home/laurentp/www/includes/database/select.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property SelectQuery::$alterTags is deprecated dans SelectQuery->addTag() (ligne 978 dans /home/laurentp/www/includes/database/select.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).
  • Deprecated function : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans DatabaseCondition->compile() (ligne 1865 dans /home/laurentp/www/includes/database/query.inc).

Formes quadratiques dégénérées et vecteurs isotropes

Soumis par Laurent Petit le dim, 19/08/2012 - 13:51

Définitions:

1) une forme quadratique Q est dégénérée si et seulement si sa forme bilinéaire associée l'est.
2) un vecteur X est isotrope pour une forme quadratique Q si Q(X) = 0

exemple: sur ℝ² notons X=(x,y) X₁=(x₁, y₁) X₂= (x₂, y₂)

Q la forme quadratique et φ sa forme bilinéaire associée.

Elles sont liées par l'identité de polarisation:

Q(X₁+X₂) - Q(X₁) - Q(X₂) = 2 φ(X₁,X₂)

Si Q(X)=x² alors φ(X₁,X₂) = x₁x₂ qui est dégénérée, (0,1) étant orthogonal à tout vecteur

Si Q(X)= 2xy alors φ(X₁,X₂)= x₁y₂+y₁x₂ qui n'est pas dégénérée, le déterminant de sa matrice dans la base canonique de ℝ² vaut -1, pourtant cette forme quadratique possède des vecteurs isotropes comme (0,1) ou (3,0)

Conclusion: une forme quadratique peut posséder des vecteurs isotropes sans être dégénérée.

Uncaught exception thrown in session handler.

PDOException: SQLSTATE[42000]: Syntax error or access violation: 1142 INSERT command denied to user 'laurentpabcm'@'10.2.20.46' for table 'drupclonesessions': INSERT INTO {sessions} (sid, ssid, uid, cache, hostname, session, timestamp) VALUES (:db_insert_placeholder_0, :db_insert_placeholder_1, :db_insert_placeholder_2, :db_insert_placeholder_3, :db_insert_placeholder_4, :db_insert_placeholder_5, :db_insert_placeholder_6); Array ( [:db_insert_placeholder_0] => rK6ncVRyW_bkaBOyh7QHUbelZzej26PoQv7TxJYy3zw [:db_insert_placeholder_1] => rK6ncVRyW_bkaBOyh7QHUbelZzej26PoQv7TxJYy3zw [:db_insert_placeholder_2] => 0 [:db_insert_placeholder_3] => 0 [:db_insert_placeholder_4] => 3.139.80.42 [:db_insert_placeholder_5] => messages|a:1:{s:5:"error";a:6:{i:0;s:300:"<em class="placeholder">Deprecated function</em> : Creation of dynamic property SelectQuery::$alterTags is deprecated dans <em class="placeholder">SelectQuery->addTag()</em> (ligne <em class="placeholder">978</em> dans <em class="placeholder">/home/laurentp/www/includes/database/select.inc</em>).";i:1;s:317:"<em class="placeholder">Deprecated function</em> : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans <em class="placeholder">DatabaseCondition->compile()</em> (ligne <em class="placeholder">1865</em> dans <em class="placeholder">/home/laurentp/www/includes/database/query.inc</em>).";i:2;s:317:"<em class="placeholder">Deprecated function</em> : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans <em class="placeholder">DatabaseCondition->compile()</em> (ligne <em class="placeholder">1865</em> dans <em class="placeholder">/home/laurentp/www/includes/database/query.inc</em>).";i:3;s:300:"<em class="placeholder">Deprecated function</em> : Creation of dynamic property SelectQuery::$alterTags is deprecated dans <em class="placeholder">SelectQuery->addTag()</em> (ligne <em class="placeholder">978</em> dans <em class="placeholder">/home/laurentp/www/includes/database/select.inc</em>).";i:4;s:317:"<em class="placeholder">Deprecated function</em> : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans <em class="placeholder">DatabaseCondition->compile()</em> (ligne <em class="placeholder">1865</em> dans <em class="placeholder">/home/laurentp/www/includes/database/query.inc</em>).";i:5;s:317:"<em class="placeholder">Deprecated function</em> : Creation of dynamic property DatabaseCondition::$stringVersion is deprecated dans <em class="placeholder">DatabaseCondition->compile()</em> (ligne <em class="placeholder">1865</em> dans <em class="placeholder">/home/laurentp/www/includes/database/query.inc</em>).";}} [:db_insert_placeholder_6] => 1722064684 ) in _drupal_session_write() (line 209 of /home/laurentp/www/includes/session.inc).