Applying UK visa along with my husband and my husband sponsoring my trip

We have already filled the UK visa application form and done with our biometrics. I wanted to know if we can send documents together? In the above thread i read we need to menton GWF numbers of dependents in the remarks where exactly is that to be done ? What if we havent added anything like that in the remarks is there any alternative option to that?

Projection of a polytope along 4 orthogonal axes

Consider the following problem:

Given an $ \mathcal{H}$ -polytope $ P$ in $ \mathbb{R}^d$ and $ 4$ orthogonal vectors $ v_1, …, v_4 \in \mathbb{R}^d$ , compute the projection of $ P$ to the subspace generated by $ v_1, …, v_4$ (and ouput it as an $ \mathcal{H}$ -polytope).

I know that the problem of computing projections along $ k$ orthogonal vectors in NP-hard (if $ k$ and $ d$ are part of the input), as shown in this paper. But does it help if $ k$ is a constant? Specifically, does it help if $ k \leq 4$ ? Do we have a polynomial algorithm in this case?

How to do permutation of characters and print it horizontally along that row in Google Sheets?

I am stuck here in printing permutation of two or more characters horizontally along that row in which the characters are. Is it possible? Because I searched it all over and only found vertical output along the column.

I have to make permutations of different people from different community and create their user names. The below is what I get after entering my formula

enter image description here

Here is the code:

function permuteEmails(a,b,c) {   var ssheet = SpreadsheetApp.getActive();   var sheet = ssheet.getSheetByName('Person Details');   var array = [];    var aVal = sheet.getRange(a).getDisplayValue();   var bVal = sheet.getRange(b).getDisplayValue();   var cVal = sheet.getRange(c).getDisplayValue();    var aVal_first = aVal.slice(0,1);   var bVal_first = bVal.slice(0,1);    array.push(aVal + "." + bVal + "@" + cVal);   array.push(bVal + "." + aVal + "@" + cVal);   array.push(aVal + "@" + cVal);   array.push(aVal + "." + bVal_first + "@" + cVal);   array.push(aVal + bVal_first + "@" + cVal);   array.push(aVal_first + bVal + "@" + cVal);    return array; } 

But I want it to look like, that after entering the details and applying the formula, all the combinations should print along the same row in which we entered the details.

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”?

How to have allocated pointer and size information along with memory allocation?

when I use heap space, along with the allocation I need the information of 1) allocated pointer, 2) size, 3) line number, 4) file name.

extendMalloc +—+—+—+—+———–+ | 1 | 2 | 3 | 4 | malloc | +—+—+—+—+———–+

When memory is freed, it has to be freed with extendMalloc information.

Multiple residencies along with multiple citizenships

Can a person have multiple residencies and each one in the country he/she is a citizen of? And can he/she have these residences printed each one in the respective country he/she is a citizen of? For example if one has a property/house in his/her name in every country he is citizen, is it internationally possible? Example to clarify: Let’s say I’m a citizen of Kuwait, Belgium and Morocco. Is it legally possible to put, as residence, Rabat in Moroccan passport, Kuwait city in Kuwaiti passport and Bruxelles in Belgian passport IF I have three houses and each one in one of those cities. Can it create me troubles when it comes to travel?