Show that,with the array representation for sorting an n-element heap, the leaves are the nodes indexed by n⌊n/2⌋+1,⌊n/2⌋+2,…,n

The Question of the CLRS $ 6.1-7$ exercise reads as:

Show that, with the array representation for sorting an n-element heap, the leaves are the nodes indexed by $ \lfloor n / 2 \rfloor + 1, \lfloor n / 2 \rfloor + 2, \ldots, n⌊n/2⌋+1,⌊n/2⌋+2,…,n$ .

I looked for the solution here: https://walkccc.github.io/CLRS/Chap06/6.1/

The solution was provided like this:

Let’s take the left child of the node indexed by $ \lfloor n / 2 \rfloor + 1.$

\begin{aligned} \text{LEFT}(\lfloor n / 2 \rfloor + 1) & = 2(\lfloor n / 2 \rfloor + 1) \ & > 2(n / 2 – 1) + 2 \ & = n – 2 + 2 \ & = n. \end{aligned}

I can’t understand this statement: $ LEFT(⌊𝑛/2⌋+1) > 2(𝑛/2−1)+2$

Please help me out. Thank you.

Moving Folders Between Libraries in Different Sites Leaves Copies in Both

When I moved a folder from Site 1 Doc Library to Site 2 Doc Library (modern experience in both, btw), I ended up with copies in both locations. Is this expected behavior, that it doesn’t actually MOVE the files? Does this mean that the files in the new location will not retain their Doc IDs and links, as well? Thanks, in advance!

Applying functions to leaves of nested list structure, when these leaves are more complex expression trees

Is there a way to apply a function h to the following nested list

{{a, b}, {c, d}, {{d, e }, {f, g}}}

where this should should become

{h[a], h[c], {h[d], h[f]}}

h is applied to the first element of each (deepest nested) sublist, replacing this sublist. The nesting is never deeper than the example displayed above. I.e. the expression tree for the list has at most depth 3.

The rest of the list structure pattern is preserved.

Here, a, c, d and f are not atoms. They are again expressions with heads. The heads are however not list-heads.

As an example, consider

{{u[a], u[b]}, {u[v[c]], d}, {{d, e}, {f, g}}}

applying h to this should yield:

{h[u[a]], h[u[v[c]]], {h[d], h[f]}}

In other words, h is applied to the leaves" of the list expression given above, where theseleaves” are more complicated expressions.

Does compiling a program in visual studio (windows form vb.net) leaves a mac address of the compiler?

I am not making an installer. It is just a form as a portable exe file. Asking because it’s generating links of a external Web API (not mine) that is a openload crawler. Although technically legal (streaming is also legal in my country) I still do not want to be linked to it in any way if it spreads around further from my friends circle. Its very useful if you’re lazy as me and hate ads/malware.

Programatically deleting items from cart leaves ‘ghost’ items;

I am using the observer checkout_cart_save_after to programatically delete some cart items.

public function execute(Observer $  observer) {     $  cart = $  observer->getCart();     $  quote = $  cart->getQuote();     $  cartItems = $  quote->getAllItems();      foreach ($  quote->getAllItems() as $  item) {         $  cart->removeItem($  item->getItemId())->save();     } } 

Whilst this does delete the cart items, it still shows the checkout page as an empty table like so:

Programatically deleted cart

But if I delete all the items manually on the frontend, I get a much nicer page after I delete the last item:

Manually deleted cart

What am I missing?

Dual Booting installation fails, leaves residual Ubuntu boot option, upon booting from it encounter GNU Grub menu

I have Windows 10 installed already, and am trying to dual-boot with Ubuntu desktop(19.04). However, after finishing the dual-boot installation (allocating memory), Ubuntu asked for a restart, and upon restarting, my screen was filled with “squashfs” errors. After a long period of waiting, I decided to force shutdown. After turning my computer on from the shutdown, there was no Ubuntu dual-boot menu, and Windows was immediately booted. I decided to go to Window’s disk management application (diskmgmt.msc) to delete the memory I allocated to my failed Ubuntu installation, and reattempt to install Ubuntu. However, when I am on the advanced startup page to attempt to boot from my USB to reinstall Windows, there is still an option to boot from Ubuntu, even though I already deleted the partition containing it. When I attempt to boot from it, (can also be accessed from default boot menu upon computer startup), I am greeted with this GNU Grub shell. Can anyone direct me on what I should do?

Leafwise de Rham cohomology(A true definition of Differential forms along leaves)

For a foliated space $ (M, \mathcal{F})$ , one associate a leafwise de Rham cohomology. This cohomology and trace class operators on this cohomology and trace interpretations for closed orbits of certain flow on $ M$ is the main object of this paper”Number theory and dynamical system of foliated manifolds.

But in the later paper, I did not find a very precise definition of “Differential forms along leaf”.

So I try to find other papers or talk to find a precise definition for this concept. Then I found a definition at page 8 of this talk “Lefschetz trace formula for flow on foliated manifolds” which give a local representation for such forms. But my problem is the following:

I think that such representation, which is quoted below, is NOT invariant under foliation charts:

$ $ \omega\sum_{\alpha_1<\alpha_2<\ldots<\alpha_k} a_{\alpha}(x,y) dx_{\alpha_1}\wedge dx_{\alpha_2}\wedge \ldots\wedge dx_{\alpha_k}$ $

Am I mistaken?

What is a precise definition and precise local representations of “Differential forms along leaves”?

Is an encrypted private key which never leaves my home directory more secure than an unencrypted one?

On a Linux system I’m running an utility like this:

$   /usr/bin/myapp myprivatekey Enter passphrase for the private key:... ...application runs and uses the private key 

My understanding is that if I have a private key encrypted with a passphrase it is more secure than an unencrypted one because the private key cannot be accessed even if the user account is compromised. So if the private key is loaded by a process running as a different user and the passphrase is typed manually by the user then one cannot intercept the above passphrase. Please note that the /usr/bin/myapp can only be written by root.

On the other hand a colleague argues that, if the user account is compromised then the private key is compromised too even if it’s protected by a passphrase, because if the account is compromised then the password typed by the user can be intercepted and one cannot be protect himself in such a situation.

Which one is correct? Is it possible to setup a system such that the private key is protected in the above situation?

Thanks!

Is there a $2 $ dimensional foliation $F$ of a 4 dimensional almost complex manifold such that $F$ and $JF$ have intersecting compact leaves?

Before we ask our question we present our motivation for this question;

Motivation: Obviously the following situation is impossible: A planar vector fields $ P\partial_x+Q\partial_y$ possess a closed orbit $ \gamma_1$ and its rotated vector field $ Q\partial_x-P\partial_y$ possess a closed orbits $ \gamma_2$ such that $ \gamma_1$ and $ \gamma_2$ have non empty intersection.

Question: What is an example of a $ 4$ dimensional almost complex manifold $ (M, J)$ , with 2 dimensional foliations $ F_1, F_2$ , whose tangent spaces are $ J$ -related and $ F_i$ possess a compact leaf $ L_i$ such that $ L_1$ has non trivial intersection with $ L_2$ ? Is there an example of such situation with extra condition that each $ L_i$ has non trivial holonomy?

didn’t get France Visa ( Schengen visa) yet but my plane leaves in two days

I’m Indian citizen with UK residence permit holder(my husband is British) we decide to visit France for this summer holidays, so i applied for my France visa as i’m not EU member but i have residence permit so i can applied for France visa over here. we already book hotel and flight tickets as per needed document to submit for France visa(with return ticket). but i didn’t get any appointment to the TLScontact centre. (till mid may they don’t have dates) they suggest us to do visa through agency. we selected agency and submitted my bio metric and application.(on 26th April early date provided by agency)

i didn’t get any confirmation yet about my visa status(still in progress), my travel date is 5th may sunday.

can anyone guide me what to do? i have one day to go (weekend is off) Is there any way to get temporary document to enter in France through embassy?