reference request – Choice of splitting in domain decomposition algorithms

When solving a PDE numerically by domain decomposition methods, what is the “optimal way” to split the domain? Are there any results stating that a particular partition of the domain yields “better” convergence than other partitions?

For concreteness, let us consider the classical Poisson equation $-Delta u = f(x)$ with homogeneous Dirichlet boundary conditions.

Default value for a multiple choice column

I have a custom list, in which there is one multiple choice column. I have 3 values possible (None, Accept,Reject). The default value is None, the column is hidden and it must contain value.
I did expect that would result in having “None” value when an item is created. However, it doesn’t. Instead it put “None” value when the item is edited.

Well, not exactly what I wanted because I have workflow based on this field. Workflow is triggered by change of an item. I can of course create one more workflow to set None value when item is created. But I hoped for some more elegant solution. Would appreciate your help

❓ASK – Will you choose to work in another country if given choice? | Proxies-free

There are people who strive to go for world tour as their retirement plan. People in our locality, dreams of doing job at foreign places as this will give them the opportunity to earn as well as travel in other countries.

As an Individual, would you prefer to join a job and go at another country for earning or still prefer to work at native places? I personally think that for the people who loves adventures, it is a perfect way to earn and travel around the world when you join a foreign firm.

time complexity – Most efficient algorithm to output the optimal choice of party invitees?

I am trying to design an efficient algorithm that takes as an input a list of n people and a list of pairs who know each other (as an adjacency list) and outputs the maximum number of invites given the following constraints:

  • Each person should have at least p other people they know at the party
  • Each person should have at least p other people they DONT know at the party

My attempt

I believe I have found a solution in $O(n(n + m))$ time. Here is the pseudocode:

1.) Loop through the adjacency list counting how many attendees each attendee knows that are invited. If any attendee doesn’t meet the criteria (num < p OR num > current_total - p), label them as a “0” in an array representing which invitees meet the criteria. This takes $O(n + m)$ time.

2.) Repeat the above loop until it finishes with no changes being made to the invitee array. This takes $O(n)$ time.

My question

Is there a way to beat this time complexity? If so, do I need to use a specialized data structure? Or perhaps delete nodes from the adjacency list if they dont fit the criteria as we go along?

vba – Transfering Choice of worksheets from Multile workbook into PDF

This query is part of a VBA automation project that our company uses internally,
But scenario wanted to convert worksheets from different workbooks like “Sheet3,4” from Workbook1 and “Sheet2” to “Sheet10” from different workbook and then “Sheet13” and “Sheet14” from Workbook1 again, then converted into oneingle PDF by the user choice of path with page number on it like page 1 of 10 or page 2 of 10 so on..

below is the code which is
Opening the file as per user input

Set xSFD = Application.FileDialog(msoFileDialogFolderPicker)
With xSFD
.Title = "Please select the folder contains the Excel files you want to convert:"
.InitialFileName = "S:"
End With

converting entire worksheet into destinantion folder like

If xBol Then
            xWbk.ExportAsFixedFormat Type:=xlTypePDF, Filename:=xRPath & xbwname & ".pdf"
            xWbk.Close SaveChanges:=False
       End If

into Seperate PDF as per excel file

Sub ExcelSaveAsPDF()
    Dim strPath As String
    Dim xStrFile1, xStrFile2 As String
    Dim xWbk As Workbook
    Dim xSFD, xRFD As FileDialog
    Dim xSPath As String
    Dim xRPath, xWBName As String
    Dim xBol As Boolean
    Set xSFD = Application.FileDialog(msoFileDialogFolderPicker)
    With xSFD
    .Title = "Please select the folder contains the Excel files you want to convert:"
    .InitialFileName = "S:"
    End With
    If xSFD.Show <> -1 Then Exit Sub
    xSPath = xSFD.SelectedItems.Item(1)
    Set xRFD = Application.FileDialog(msoFileDialogFolderPicker)
    With xRFD
    .Title = "Please select a destination folder to save the converted files:"
    .InitialFileName = "S:"
    End With
    If xRFD.Show <> -1 Then Exit Sub
    xRPath = xRFD.SelectedItems.Item(1) & ""
    strPath = xSPath & ""
    xStrFile1 = Dir(strPath & "*.*")
    Application.ScreenUpdating = False
    Application.DisplayAlerts = False
    Do While xStrFile1 <> ""
        xBol = False
        If Right(xStrFile1, 3) = "xls" Then
            Set xWbk = Workbooks.Open(Filename:=strPath & xStrFile1)
            xbwname = Replace(xStrFile1, ".xls", "_pdf")
            xBol = True
        ElseIf Right(xStrFile1, 4) = "xlsx" Then
            Set xWbk = Workbooks.Open(Filename:=strPath & xStrFile1)
            xbwname = Replace(xStrFile1, ".xlsx", "_pdf")
            xBol = True
        ElseIf Right(xStrFile1, 4) = "xlsm" Then
            Set xWbk = Workbooks.Open(Filename:=strPath & xStrFile1)
            xbwname = Replace(xStrFile1, ".xlsm", "_pdf")
            xBol = True
        End If
        If xBol Then
            xWbk.ExportAsFixedFormat Type:=xlTypePDF, Filename:=xRPath & xbwname & ".pdf"
            xWbk.Close SaveChanges:=False
       End If
        xStrFile1 = Dir
    Loop
    Application.DisplayAlerts = True
    Application.ScreenUpdating = True
End Sub

homotopy theory – Why does elliptic cohomology fail to be unique up to contractible choice?

It is often stated that the derived moduli stack of oriented elliptic curves $mathsf{M}^mathrm{or}_mathrm{ell}$ is the unique lift of the classical moduli stack of elliptic curves satisfying some conditions, meaning the moduli space $Z$ of all such lifts is connected. This is mentioned in Theorem 1.1 of Lurie’s “A Survey of Elliptic Cohomology” (Surv), for example.

In Remark 7.0.2 of Lurie’s “Elliptic Cohomology II: Orientations” ((ECII)), Lurie says “…beware, however, that $Z$ is not contractible“. In other words, $mathsf{M}^mathrm{or}_mathrm{ell}$ is not the unique lift up to contractible choice (the gold standard of uniqueness in homotopy theory).

(Side note: in (ECII) and (Surv), Lurie is talking about the moduli stack of smooth elliptic curves, but the uniqueness up to homotopy of a derived stack $overline{mathsf{M}}_mathrm{ell}^mathrm{or}$ lifting the compactification of the moduli stack of smooth elliptic curves is also stated in the literature; for example, in Theorem 1.2 of Goerss’ “Topological Modular Forms (after Hopkins, Miller, and Lurie)”. I am interested in the compactified situation mostly, but both are related.)

Although I do not hope that $mathsf{M}^mathrm{or}_mathrm{ell}$ does possess this much stronger form of uniqueness, I would like to understand the reason for this failure:

Why is the moduli space $Z$ not contractible? and Does a similar statement apply in the compactified case?

To be a little more precise, let $mathcal{O}^mathrm{top}$ be the Goerss–Hopkins–Miller–Lurie sheaf of $mathbf{E}_infty$-rings on the small affine site of the moduli stack of elliptic curves $mathsf{M}_mathrm{ell}$. Denote this site by $mathcal{U}$. The moduli space $Z$ can then be defined as the (homotopy) fibre product
$$Z=mathrm{Fun}(mathcal{U}^{op}, mathrm{CAlg})underset{mathrm{Fun}(mathcal{U}^{op}, mathrm{CAlg}(mathrm{hSp}))}{times}{mathrm{h}mathcal{O}^mathrm{top}},$$
where $mathrm{CAlg}$ is the $infty$-category of $mathbf{E}_infty$-rings, and $mathrm{CAlg(hSp)}$ is the 1-category of commutative monoid objects in the stable homotopy category. The presheaf $mathrm{h}mathcal{O}^mathrm{top}$ can be defined using the Landweber exact functor theorem (at least on elliptic curves whose formal group admits a coordinate), and hence $Z$ can be seen as the moduli space of presheaves of $mathbf{E}_infty$-rings recognising the classical Landweber exact elliptic cohomology theories.

To prove uniqueness up to homotopy, I am aware one should use some arithmetic and chromatic fracture squares to break down the problem into rational, $p$-complete, $K(1)$-local, and $K(2)$-local parts. The $K(2)$-local part of $mathcal{O}^mathrm{top}$ is unique up to contractible choice by the Goerss–Hopkins–Miller theorems surrounding Lubin–Tate spectra (see Chapter 5 of (ECII) for a reference which you might already have open). The $K(1)$-local part also seems to be unique up to contratible choice, as all of the groups occuring in the Goerss–Hopkins obstruction theory vanish (this is discussed at length in Behrens’ “The construction of $tmf$” chapter in the “TMF book” by Douglas et al). Similarly, the rational case also has vanishing obstruction groups; see ibid.

I’m then lead to believe that is something interesting (being a pseudonym for “I don’t know what’s”) going on in the chromatic/arithmetic fracture squares gluing all this stuff together. Are their calculable obstructions/invariants to see this? Or otherwise known examples that contradict the contractibility of $Z$?

Any thoughts or suggestions are appreciated!

drop down list – What is better: a disabled dropdown or a dropdown with only one choice?

Sometimes, a web application uses a dropdown to present the user with a number of choices. For example:

Choose your size:
    +---------------+
    | Small       ▼ |
    +---------------+

However, sometimes there is only one available choice.

The designer then has two choices:

  • To make the option the only option in the dropdown.
  • To make the dropdown disabled, with a notice that only the selected option is available.

Which is better?

busy beaver – Are the outcomes of the maximum shifts function fixed regardless of our choice of axiomatic system?

It is known that there is a $748$-state Turing machine that halts if and only if $mathsf{ZF}$ is inconsistent. So by Gödel’s second incompleteness, $mathsf{ZF}$ cannot find what $S(748)$ exactly is, where $S$ is the maximum shifts function (Also known as the “Frantic Frog”).

I’m rather confused by this fact. As $S$ is well-defined, that should mean that regardless of what axiomatic system we use, the exact value of $S(748)$ always stays the same. We just need a stronger axiomatic system to find $S(748)$, like $mathsf{ZFC}$, $mathsf{ZFC+CH}$, or $mathsf{ZFC+(V=L)}$.

If $mathsf{ZFC}$ and $mathsf{ZF¬C}$ entailed $S(748)$ to be different numbers, since Axiom of Choice is independent to $mathsf{ZF}$, it would be concluded that $mathsf{ZF}$ was inconsistent at the first place. So far, is my understanding correct?