Would $\Sigma_i^P \neq \Pi_i^P$ imply that polynomial hierachy cannot collapse to the $i$-th level?

If $ \Sigma_i^P = \Pi_i^P$ , then it follows that the polynomial hierarchy collapses to the $ i$ -th level.

What about the case $ \Sigma_i^P \neq \Pi_i^P$ ? For example, consider the case of $ NP \neq coNP$ . As far as I understand, this would imply the polynomial hierarchy cannot collapse to the first level, since if $ PH =NP$ , then in particular, $ coNP \subseteq NP$ , which means $ NP = coNP$ . Can we expand this idea to proof the general case: $ \Sigma_i^P \neq \Pi_i^P$ implies $ PH$ cannot collapse to $ i$ -th level?

How do I collapse all subtasks in a SharePoint 2013 task list?

I’m using SharePoint 2013 to keep track of projects and their sub-tasks. But the view is getting pretty long and it would be nice to have a button to collapse all subtasks, leaving only the top-level parent tasks. Most to do lists have this functionality. I found a jQuery solution that didn’t work, but I haven’t found any other solutions. I’d rather not farm each project off to its own sub-site with it’s own list.

How exactly to adapt Brown’s collapse from monoids to algebras?

In The Geometry of Rewriting Systems, Brown describes a method to collapse the bar resolution of a monoid. Roughly:

  • Given a simplicial set $ X$ equipped with a collapsing scheme (a partition of the geometric realization $ \lvert X \rvert$ into essential, redundant and collapsible cells, which satisfy some properties) it is possible to collapse $ \lvert X \rvert$ into a smaller CW-complex with a cell for each essential cell of $ \lvert X \rvert$ .
  • A monoid $ M$ presented with a complete rewriting system (the set of relations $ R$ is terminating Church-Rosser) induces a collapsing scheme on the simplicial set $ BM$ .
  • The $ n$ -cells of $ \lvert BM \rvert$ are in correspondence with the generators of $ B_n$ in the normalized bar resolution of $ M$ . In particular, each $ B_n$ can be collapsed in a similar way that $ \lvert BM \rvert$ is.
  • If $ M$ has a good set of normal forms of finite type ($ M$ has a finite presentation $ (S,R)$ and $ R$ is terminating Church-Rosser), then the classifying space $ \lvert BM \rvert$ can be collapsed into a finite CW-complex
  • Under the assumptions on the last bullet, we can also collapse the bar resolution of $ M$ into one where all $ B_n$ are finitely generated. In particular, $ M$ is of type $ (FL)_{\infty}$ .

Brown states (emphasis mine):

The Method used in this section works, with no essential change, if the ring $ \mathbb{Z}[M]$ is replaced by an arbitrary augmented $ k$ -algebra $ A$ which comes equipped with a presentation satisfying the conditions of Bergman’s diamond lemma (The diamond lemma for ring theory, Theorem 1.2). Here $ k$ can be any commutative ring. One starts with the normalized bar resolution $ C$ of $ k$ over $ A$ , and one obtains a quotient resolution $ D$ , with one generator for each “essential” generator of the bar resolution. In particular, we recover Anick’s Theorem (On the homology of associative algebras, Theorem 1.4).

Let me state here the conditions of the diamond lemma:

Theorem: Let $ S$ be a reduction system for a free associative algebra $ k\langle X \rangle$ (a subset of $ \langle X \rangle$ \times k\langle X \rangle), and $ \leq$ a semigroup partial ordering on $ \langle X \rangle$ , compatible with $ S$ , and having descending chain condition. Then…

I can’t understand two aspects of this adaptation:

  • The conditions of the diamond lemma are stated for the free associative algebra $ k\langle X \rangle$ . What does it mean for the presentation of $ A$ to satisfy these conditions? If we assume $ A \cong k\langle X \rangle/I$ we can interpret Brown’s sentence as $ k\langle X \rangle$ satisfying the conditions, but what about the ideal $ I$ ? I assume it is related somehow to the reduction system, but how exactly?
  • What would be the essential generators of the normalized bar resolution? In the monoid setting, they originate from the essential cells of the classifying space of the monoid. In the algebra setting, we don’t have that tool anymore (unless we generalize classifying spaces for internal monoids over monoidal categories, which I don’t think is the case).

Can you reduce yourself, crawl into the Tarrasque’s airway, and try to collapse its lung by enlarging inside of it?

A player tried to use the spell enlarge/reduce on his PC while in the mouth of the Tarrasque to shrink himself, then go deeper into its airway and to try to collapse its lung by enlarging himself inside of it. I imagine there’s no real rule for this but I felt silly just denying it, because it seemed like a cool idea that you could only probably do with a creature as large as the Tarrasque.

Can you crawl into a gargantuan creature’s mouth or into its airway via reduce/enlarge and attack its brain or internal organs?

How do I circumvent protected sheets to expand or collapse groups and rows in Google Sheets?

A budget I am working on has each month on the same sheet and I would like to control which months are visible when comparing expenditure and income. In order to do so, I grouped the rows and columns according to category and month. The only cells which should be edited are the planned expense/income values. It makes sense to protect the sheet except for those cells. However, by protecting the sheet, I am unable to collapse or expand categories and months. Is there a workaround to retain the design and experience I would like to implement?

Bootstrap Date Picker working fine, but an issue with Collapse (Time picker) and Bootstrap Collapse

I’m able to get the date picker and I’ve called the required files as per the documentation but unable to collapse between date and time.

I have read the documentation and also followed the steps mentioned here -Bootstrap Datetime Picker not working with Bootstrap 4 Alpha 6

Here is a video sample – https://youtu.be/poWXMm__ntQ

Any help is greatly appreciated.


Problemas com Navbar Collapse

Opa, tudo bem??? Estou com problemas ao fazer o navbar collapse, qdo clico no ícone não aparece o menu abaixo.

Segue abaixo o código:

<div class="container">      <nav class="navbar navbar-expand-lg navbar-light bg-light">          <a class="navbar-brand" href="">O Rei do Bojo</a>            <button class="navbar-toggler" type="button" data-toggle="collapse" data-target="navbarMenu">             <span class="navbar-toggler-icon"></span>           </button>             <div class="navbar-collapse collapse" id="navbarMenu">           <ul class="navbar-nav">             <li><a class="nav-item nav-link active" href="">Home</a></li>             <li><a class="nav-link nav-link" href="">Sobre</a></li>             <li><a class="nav-link nav-link" href="">Produtos</a></li>             <li><a class="nav-link nav-link" href="">Contato</a></li>            </ul>         </div>      </nav><!-- /navbar --> </div> 

E abaixo segue os links referências para boostrap no Body e os scripts do js.

<link rel="stylesheet" href="assets/css/bootstrap.min.css">  <script type="text/javascript" src="assets/js/jquery-3.3.1.min.js"></script> <script type="text/javascript" src="assets/js/jquery/bootstrap.bundle.min.js"></script> 

Bootstrap помощь с collapse и якорем

хотел бы поинтересоваться, как сделать так, что-бы при нажатии на кнопки меню, к примеру на “Отзывы”, открывался коллапс соответствующего названия. Якорь уже есть и коллапс рабочий введите сюда описание изображения

<ul>   <li class="nav-item">           <a class="nav-link" href="#heading3">Отзывы</a>   </li> </ul>  <div class="card"> 	<div class="card-header" role="tab" id="heading3"> 		<a data-toggle="collapse" data-parent="#accordionEx" href="#collapse3" aria-expanded="true" aria-controls="collapse3"> 		    <h5 class="mb-0"> 			Отзывы 		</h5> 		</a> 	</div>  	<div id="collapse3" class="collapse" role="tabpanel" aria-labelledby="heading3" data-parent="#accordionEx"> 		<div class="card-body feedback"> 			TextTextTextTextText 		</div> 	</div> </div>