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Trigger Element Click

JSFiddle Live Demo
Script Explanation
My script is to sort and/or filter a mailbox whereas you can select one of each options and the sort option upon clicking, if it's active it'll switch the icon indicating going from A-Z to Z-A and visa-versa.
The filter option changes the button text and upon clicking on the button it'll animate the button width to the drop down width and then slide down the drop down. Upon clicking again, it'll reverse the process, checking to see if the text has been changed to adjust for the width.
What I Wish To Implement
Upon clicking one of the options apart from the .Active filter/view options, I wish to trigger the click event to reverse the show progress. I have tried to trigger this click hover with using e.target, I'm not sure how to trigger this correctly as my attempts have been doing nothing.

Your trigger event seems to be working fine, you're only triggering it on the wrong element. This seems to be working:
$('.dropdown-toggle').trigger("click");
I've updated the plunker

You can trigger the click this way.
if ($(e.target).closest('.Sort').length) {
if ($(e.target).hasClass('Active')) {
$(e.target).parent().find('i').toggleClass("fa-arrow-down fa-arrow-up");
} else {
$('.dropdown-menu .Sort').find('.Active').removeClass('Active');
$('.dropdown-menu .Sort').find('i').remove();
$(e.target).addClass('Active');
$(e.target).prepend('<i class="fa fa-arrow-down" aria-hidden="true"></i> ');
}
//Here.
//Ideally add another unique class to the main dropdown button and use that instead, otherwise you will have problems when there are multiple elements with dropdown-toggle class.
$('.dropdown-toggle').click();
}

Call javascript after choosing from Bootstrap Dropdown

All Bootstrap Dropdown menu examples show the items having an link like:
<li>a</li>
How can I make a Bootstrap dropdown menu where choosing an item triggers a javascript method?
(If it makes a difference, the menu items will also be added to the menu using javascript/jquery initially)
Is this just a matter of adding a click function to each <li>? Or does Bootstrap provide special handling for its dropdown items?
If I add a click function, do I still need each item to also have the <a href="#"> tag around the text?

Looks like the answer is:
Bootstrap does not supply any special handling for clicks on its
dropdown menu items; and
I could add a click function to each item in the list; or
I could use event delegation to add the click function to the list
itself, and it will be triggered on each child's click
Thanks to #Pevara and #spaceman for getting me rolling.

Do something with the particular element even though click event is fired on class name

I have a scenario where a dropdown launcher menu should appear on every row in a list page. I have tweaked the code here.
I want the popover (open) behavior be restricted for the click on the particular launcher-icon, though close action is perfect here.
the problem is when any icon is clicked, it shows all menus. My page has rows inflated from a database and every row has three such launcher icons.
I guess this block of code needs some tweaks:
// Click event handler to toggle dropdown
$(".button").click(function(event){
event.stopPropagation();
// How can I call toggle for the specific div element?
$(".app-launcher").toggle();
});
How can it be done?

You need to traverse the DOM to find the .app-launcher instance which is related to the clicked .button element, to do that use closest() and find(), like this:
$(".button").click(function(event){
event.stopPropagation();
$(this).closest('.launcher').find(".app-launcher").toggle();
});
Updated CodePen
I would also suggest looking in to using a single instance of .app-launcher and moving it around the DOM as required to DRY up your HTML code.

jCarousel with keyboard : wrong focus using the tab key

I'm using the 0.3 version of the jQuery jCarousel script to display three carousels on the webpage I'm working on. Those three carousels work just fine.
The trouble is : you can't use them properly using the keyboard.
If you tab through the page, the focus goes to the first link of the first item in the carousel, even if this item is not visible ! (For example : click on a "next" link in one of the carousels, then use the tab key to browse : the focus will go to a link which is not visible inside of the carousel).
If you keep using the "tab" key, the focus will successively go to all the links of all the items in the carousel.
Instead of that : the focus should go to the first link of the first visible item ; then, if the last link of the last visible item is reached, then the focus should go out of the carousel (the next link outside of it, in fact).
A solution could be to use tabindex... but some parts of the page are shared with other pages of the website, so I just can't use tabindex in all the links of all my pages...
Instead of that, I had tried things like this :
$("#carousel-editos li a").focusin(function () {
$("#carousel-editos li.jcarousel-item-first .post-title a").focus();
return false;
});
but then it prevents any further use of the "tab" key...
I hope this is clear... Thanx for any help !

I think you need a combination of the answers that you've already provided. It seems like you should be able to use Javascript to dynamically set tabindex attributes on the HTML that you need to be tabbable (heh, new word). I'm thinking of something like this:
On page load, find all visible items in the carousel. Use jQuery to set the tabindex property for each item that you want in the tab cycle.
Assign tabindex properties to all other links on the page that you want to cycle through.
Add some jQuery to modify the tabindex attributes when the user changes the items in the carousel (click the next/prev buttons).
It would be much easier to help you if you made a simplified example in jsFiddle.

On the carousel createEnd and scrollEnd functions you can reset the contents of .jCarousel so that only the visible carousel items are "tabbable". I have done that in my code as follows:
var bannerSlider_scrollEnd = function(event, carousel) {
var $carousel = carousel.element(),
$items = carousel.items(),
$bannerContent,
$visibleItemsContent = carousel.visible().find('.bannerContent');
$items.each(function (index) {
$bannerContent = $(this).find('.bannerContent');
disableTabbing($bannerContent);
});
reenableTabbing($visibleItemsContent);
$visibleItemsContent.find(':focusable').eq(0).focus();
};
The disableTabbing($container) and reenableTabbing($container) lines refer to helper functions I coded into my site which basically find all :focusable elements in a given container and set the tabindex to "-1", then "0" respectively.
After this processes, users will be left tabbing only through visible carousel items instead of all carousel items.

How to handle Android's Back Button click in a single HTML page webapp using PhoneGap?

I read from the documentation that we can handle the back button click using the following code:
document.addEventListener("backbutton", backKeyDown, true);
function backKeyDown() {
// Call my back key code here.
alert('go back!');
}
My concern is that I have a single HTML5 web page in which I have multiple div tags which I animate using jQuery as per the navigation option selected by the user from the menu options.
How can I, in this single page webapp, handle the back button click using PhoneGap and show the user the previously animated div. Clicking on the back button again would again take him to the previous div of the current previous div :-)
Thanks.

I solved the problem by creating a global array variable as
var myStack = new Array();
Then whenever I clicked on the div tag, I inserted the function prototype along with the arguments inside the myStack variable. For eg:
myStack.push(\"myfunction(args1, args2);\");
Then, using the code which I posted in my question, inside the BackButton handler, I wrote the following code:
var divToShow = myStack.pop();
eval(divToShow);
Hope this helps others.

I did an implementation in a similarly structured phonegap app. My situation was a bit more complex because I was loading in html as well as external data via ajax (rather than just unhiding divs). I created a global array called history which I used to keep track of current position as well as previous positions (position here being the most recent ajax function called, so the array was actually storing function names as text). The right sequence and combination of .pop and .push array methods got me a fully functioning js back button that scaled nicely and handled any kind of back and forth navigation I could think of.

I will just post my overall idea of handling this situation. Hope you can improvise and change it to suit your needs.
Have a global variable to remember the current div id that is
visible. For example, when a menu item x is clicked, set this global
variable to the div id that is currently visible (before showing the next div corresponding to menu item x).
When the back button is pressed, use the global variable's value to identify the previous div. Hide the current div and show the previous one.