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Drupal\Core\Cache\DatabaseBackend->getMultiple(Array, ) (Line: 126)
Drupal\Core\Cache\DatabaseBackend->get('preload-paths:/list-by-author/Helen Joyce ') (Line: 224)
Drupal\path_alias\AliasManager->getAliasByPath('/node/2200', 'und') (Line: 47)
Drupal\path_alias\PathProcessor\AliasPathProcessor->processOutbound('/node/2200', Array, Object, Object) (Line: 108)
Drupal\Core\PathProcessor\PathProcessorManager->processOutbound('/node/2200', Array, Object, Object) (Line: 403)
Drupal\Core\Routing\UrlGenerator->processPath('/node/2200', Array, Object) (Line: 305)
Drupal\Core\Routing\UrlGenerator->generateFromRoute('entity.node.canonical', Array, Array, 1) (Line: 108)
Drupal\Core\Render\MetadataBubblingUrlGenerator->generateFromRoute('entity.node.canonical', Array, Array, 1) (Line: 765)
Drupal\Core\Url->toString(1) (Line: 183)
Drupal\Core\Utility\LinkGenerator->generate(Object, Object) (Line: 102)
Drupal\Core\Render\Element\Link::preRenderLink(Array)
call_user_func_array(Array, Array) (Line: 113)
Drupal\Core\Render\Renderer->doTrustedCallback(Array, Array, 'Render #pre_render callbacks must be methods of a class that implements \Drupal\Core\Security\TrustedCallbackInterface or be an anonymous function. The callback was %s. See https://www.drupal.org/node/2966725', 'exception', 'Drupal\Core\Render\Element\RenderCallbackInterface') (Line: 870)
Drupal\Core\Render\Renderer->doCallback('#pre_render', Array, Array) (Line: 432)
Drupal\Core\Render\Renderer->doRender(Array, ) (Line: 248)
Drupal\Core\Render\Renderer->render(Array) (Line: 974)
Drupal\views\Plugin\views\field\EntityField->render_item(0, Array) (Line: 1214)
Drupal\views\Plugin\views\field\FieldPluginBase->advancedRender(Object) (Line: 238)
template_preprocess_views_view_field(Array, 'views_view_field', Array)
call_user_func_array('template_preprocess_views_view_field', Array) (Line: 261)
Drupal\Core\Theme\ThemeManager->render('views_view_field', Array) (Line: 491)
Drupal\Core\Render\Renderer->doRender(Array, ) (Line: 248)
Drupal\Core\Render\Renderer->render(Array) (Line: 1796)
Drupal\views\Plugin\views\field\FieldPluginBase->theme(Object) (Line: 767)
Drupal\views\Plugin\views\style\StylePluginBase->elementPreRenderRow(Array)
call_user_func_array(Array, Array) (Line: 113)
Drupal\Core\Render\Renderer->doTrustedCallback(Array, Array, 'Render #pre_render callbacks must be methods of a class that implements \Drupal\Core\Security\TrustedCallbackInterface or be an anonymous function. The callback was %s. See https://www.drupal.org/node/2966725', 'exception', 'Drupal\Core\Render\Element\RenderCallbackInterface') (Line: 870)
Drupal\Core\Render\Renderer->doCallback('#pre_render', Array, Array) (Line: 432)
Drupal\Core\Render\Renderer->doRender(Array, ) (Line: 248)
Drupal\Core\Render\Renderer->render(Array) (Line: 706)
Drupal\views\Plugin\views\style\StylePluginBase->renderFields(Array) (Line: 572)
Drupal\views\Plugin\views\style\StylePluginBase->renderGrouping(Array, Array, 1) (Line: 463)
Drupal\views\Plugin\views\style\StylePluginBase->render() (Line: 2177)
Drupal\views\Plugin\views\display\DisplayPluginBase->render() (Line: 1592)
Drupal\views\ViewExecutable->render() (Line: 201)
Drupal\views\Plugin\views\display\Page->execute() (Line: 1689)
Drupal\views\ViewExecutable->executeDisplay('page_1', Array) (Line: 81)
Drupal\views\Element\View::preRenderViewElement(Array)
call_user_func_array(Array, Array) (Line: 113)
Drupal\Core\Render\Renderer->doTrustedCallback(Array, Array, 'Render #pre_render callbacks must be methods of a class that implements \Drupal\Core\Security\TrustedCallbackInterface or be an anonymous function. The callback was %s. See https://www.drupal.org/node/2966725', 'exception', 'Drupal\Core\Render\Element\RenderCallbackInterface') (Line: 870)
Drupal\Core\Render\Renderer->doCallback('#pre_render', Array, Array) (Line: 432)
Drupal\Core\Render\Renderer->doRender(Array, ) (Line: 248)
Drupal\Core\Render\Renderer->render(Array, ) (Line: 238)
Drupal\Core\Render\MainContent\HtmlRenderer->Drupal\Core\Render\MainContent\{closure}() (Line: 638)
Drupal\Core\Render\Renderer->executeInRenderContext(Object, Object) (Line: 239)
Drupal\Core\Render\MainContent\HtmlRenderer->prepare(Array, Object, Object) (Line: 128)
Drupal\Core\Render\MainContent\HtmlRenderer->renderResponse(Array, Object, Object) (Line: 90)
Drupal\Core\EventSubscriber\MainContentViewSubscriber->onViewRenderArray(Object, 'kernel.view', Object)
call_user_func(Array, Object, 'kernel.view', Object) (Line: 111)
Drupal\Component\EventDispatcher\ContainerAwareEventDispatcher->dispatch(Object, 'kernel.view') (Line: 186)
Symfony\Component\HttpKernel\HttpKernel->handleRaw(Object, 1) (Line: 76)
Symfony\Component\HttpKernel\HttpKernel->handle(Object, 1, 1) (Line: 53)
Drupal\Core\StackMiddleware\Session->handle(Object, 1, 1) (Line: 48)
Drupal\Core\StackMiddleware\KernelPreHandle->handle(Object, 1, 1) (Line: 28)
Drupal\Core\StackMiddleware\ContentLength->handle(Object, 1, 1) (Line: 32)
Drupal\big_pipe\StackMiddleware\ContentLength->handle(Object, 1, 1) (Line: 191)
Drupal\page_cache\StackMiddleware\PageCache->fetch(Object, 1, 1) (Line: 128)
Drupal\page_cache\StackMiddleware\PageCache->lookup(Object, 1, 1) (Line: 82)
Drupal\page_cache\StackMiddleware\PageCache->handle(Object, 1, 1) (Line: 48)
Drupal\Core\StackMiddleware\ReverseProxyMiddleware->handle(Object, 1, 1) (Line: 51)
Drupal\Core\StackMiddleware\NegotiationMiddleware->handle(Object, 1, 1) (Line: 36)
Drupal\Core\StackMiddleware\AjaxPageState->handle(Object, 1, 1) (Line: 51)
Drupal\Core\StackMiddleware\StackedHttpKernel->handle(Object, 1, 1) (Line: 741)
Drupal\Core\DrupalKernel->handle(Object) (Line: 19)

List by Author: Helen Joyce

Natural born mathematicians

Neuropsychologist Brian Butterworth tells us about research showing that even newborn babies have a basic understanding of number. It seems we are all mathematicians!

Mathematical mysteries: Painting the Plane

Suppose you have an infinitely large sheet of paper (mathematicians refer to this hypothetical object as the plane). You also have a number of different colours - pots of paint, perhaps. Your aim is to colour every point on the plane using the colours available. That is, each point must be assigned one colour.

Mathematical mysteries: Chomp

Chomp is a simple two-dimensional game, played as follows. Cookies are set out on a rectangular grid. The bottom left cookie is poisoned. Two players take it in turn to "chomp" - that is, to eat one of the remaining cookies, plus all the cookies above and to the right of that cookie.

Adam Smith and the invisible hand

Adam Smith is often thought of as the father of modern economics. In his book "An Inquiry into the Nature and Causes of the Wealth of Nations" Smith decribed the "invisible hand" mechanism by which he felt economic society operated. Modern game theory has much to add to Smith's description.

Faster than light

Scientists at the NEC Research Institute in Princeton have carried out an experiment in which a pulse of light appeared to emerge from a cloud of gas before it even entered.

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List by Author: Helen Joyce