Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; JCommentsACL has a deprecated constructor in /var/www/astarmathsandphysics/components/com_jcomments/classes/acl.php on line 17 Call Stack: 0.0001 361048 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0514 1211216 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0514 1211216 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1303 4044352 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1318 4072048 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1328 4089760 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1328 4106768 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1337 4114504 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1337 4114504 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1848 4543024 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1877 4628576 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1883 4649504 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2928 9749144 13. JEventDispatcher->trigger() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:199 0.2930 9749544 14. plgContentJComments->update() /var/www/astarmathsandphysics/libraries/joomla/event/dispatcher.php:160 0.2930 9749544 15. plgContentJComments->onContentAfterDisplay() /var/www/astarmathsandphysics/libraries/joomla/event/event.php:70 0.2934 9757792 16. plgContentJComments->onAfterDisplayContent() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:339 0.2936 9759488 17. JComments::show() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:282 0.2943 9792576 18. JCommentsFactory::getACL() /var/www/astarmathsandphysics/components/com_jcomments/jcomments.php:188 0.2943 9792976 19. spl_autoload_call() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274 0.2944 9793056 20. JLoader::load() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274

Conjugate Iteration Sequences

The functionsandare conjugate to each other iffor some functioncalled the conjugating function. If the sequenceis defined by

for someandfor

then the sequencesatisfiesfor

andandare called conjugate iteration sequences.

In practice the functionis usually found by substitutingandintoand rearranging.

Since the sequenceis the image of the sequenceunder the functionboth sequences must have the same behaviour of convergence and continuity and ifis a fixed point ofthenis a fixed point of

If the conjugating function is to be one to one and entire then it must be of the form

Example: Show that the sequence

is conjugate to the sequencewith conjugating function

Note first thatis one to one onIfthen sobecomes

for

This simplifies to

so thatandare conjugate functions and the sequencesandare conjugate sequences.

Add comment

Security code
Refresh