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## The Witt group, or the cohomology of the periodic table of n-categories March 30, 2009

Posted by Noah Snyder in Category Theory, conferences, homological algebra, quantum groups, talks.

A very popular topic at the Modular Categories conference was the a generalization of the Witt group which is being developed by Davydov, Mueger, Nikshych, and Ostrik. What is this Witt group? Well it’s the simplest case of the cohomology of the periodic table of n-categories!

In this post I want to explain the definition of this cohomology theory and explain why it generalizes the classical Witt group.

First recall the Baez-Dolan periodic table.

Periodic Table

Here all the functors are the generalized “center” construction defined in HDA-1. In the first column, the center means exactly what you think it means. In the second column the first arrow, which we’ll call Z_1, is the Drinfel’d center, while the second arrow is the Mueger center.

A result of Mueger’s says that under some niceness conditions Z_2(Z_1(C)) is automatically just vector spaces! Hence the second column is a complex. The natural thing to do is to take homology. If you restrict your attention to fusion categories (which have duals, are semisimple, and have finitely many simple objects) of non-zero dimension, then the kernel of Z_2 is exactly modular categories. So what you’re doing is taking all modular categories (which form a monoid under external tensor product) modulo those modular tensor categories which are Drinfel’d centers. A priori this only gives you a monoid, but Davydov, Mueger, Nikshych, and Ostrik prove that it is a group.

What about the other columns? I think Kitaev mentioned in passing in his talk that the other columns also form complexes. But I’m a little confused about why. In particular there’s something fishy going on somewhere. You can continue the periodic table downward past the diagonal, but it stablizes. In particular, if you look at Z^2 in that region you get the identity instead of the functor sending everything to vector spaces. Anyone know what’s going on? Mueger’s paper mentions an analogue (due to Baez) in the 1st column where you extend the table upwards by one row. In the first column this gives you sets where the center becomes “the monoid of all endomorphisms.” What’s the homology there? We could also extend the other columns upwards. Then the second column would start with categories, and Z_0 would be the monoidal category of all endofunctors. Is Z_1(Z_0(C)) always vector spaces? If so, why? Can you extend this picture leftwards by looking at (-1)-categories?

Enough questions that I can’t answer, let’s go to one I can answer: “Why is this called the Witt group?” Well, one nice class of tractable examples of tensor categories are the pointed tensor categories (this means that $V \otimes V^* \cong \mathbf{1}$ for every V). Any braided pointed fusion category comes from an abelian group with a bilinear form. The abelian group is the Grothendieck group, and the bilinear form tells you by which scalar the braiding acts on the object $V \otimes W = W \otimes V$. (You should think of this as a kind of strictification.) Modularity becomes the condition that the bilinear form is nondegenerate, while Drinfel’d doubles consist of precisely the hyperbolic spaces. Hence the “Witt group” restricted to the pointed case is nothing other than the classical Witt group.

1. Dylan Thurston - March 31, 2009

The definition of pointed tensor category has some weird symbol which presumably should be fixed.

2. Scott Carnahan - March 31, 2009

Z certainly passes between equivalent (n-)categories in the stable range, but I don’t see why Z is necessarily the identity functor. You should be able to embed everything as infinity-categories with certain truncations and get the same answer. As far as I can tell, you’re taking E_n Hochschild cohomology for successive n.

By the way, \mathbb{1} is famously badly behaved. I usually use \mathbf{1}.

3. Scott Carnahan - March 31, 2009

Maybe I should clarify. The notion of center changes depending on your source category, and once you hit the stable range, it maps everything to the trivial object. In particular, if you take the “abelian monoid center” you will typically get a very different result than if you take the “associative monoid center” of a monoid that happens to be abelian. This is a bit easier to see in higher categorical settings, where commutativity is an extra structure rather than a property.

Standard references (all available online): A preprint from last year by Ben-Zvi, Francis, Nadler. Also, material from Lurie’s DAG and HTT.

4. Noah Snyder - March 31, 2009

Thanks Scott!

So what’s the homology at the monoid spot. Extending the table upward you have Finite Sets -> Finite Rank Monoids -> Commutative Finite Rank Monoids where the first functor is look at all endomorphisms and the second functor is the usual center. (Or maybe I should be requiring some sort of duals? In which case perhaps monoid should be replaced by group?) So we’re looking at finite rank monoids with no center modulo the endomorphisms of sets. This doesn’t look like anything I’ve seen before. How do you express it in terms of some sort of Hochshild homology?

5. Scott Carnahan - March 31, 2009

Concerning your second question, extending one row up means considering E_0-algebras, which are just pointed objects. This is because the nth space in the E_0 operad is homotopy equivalent to configurations of n distinct points in R^0. Taking E_0 Hochschild cohomology amounts to taking pointed endomorphisms. If you’re working with vector spaces, the E_1 center is then the center of a matrix ring.

I think both (-1)-categories are pointed, and taking the E_0-center takes both to $\Delta^0$.

[Edit: Upon further reflection, I’m pretty sure the empty (-1)-category $\Delta^{-1}$ is not pointed.]

6. Scott Carnahan - March 31, 2009

I really think you should use monoids instead of groups, since the abelianness of S_2 would make the sequence no longer a complex. I’m a little confused about taking the homology of the complex, and I think it’s because I don’t know what sort of equivalence relation you want to use when you say “modulo”.

If O is an operad, the O-Hochschild cohomology (aka O-center) of an O-algebra A is the A-O module homomorphisms from A to A (this should be viewed as an object in your category). The O-Hochschild homology is given by tensoring with the universal enveloping algebra of A. This is all in some sort of derived setting. There is a conjecture of Kontsevich asserting that the E_n center of an E_n-algebra (in some symmetric category which I think is unspecified) has an E_{n+1}-algebra structure.

[Edit: I should have said, “tensoring A with itself over the universal enveloping algebra.”]

7. rwbarton - March 31, 2009

Surely we can agree that in the second column, if we consider categories which happen to be discrete (no nonidentity 1-morphisms), then Z_1 is the center in the usual sense and Z_2 is the identity. So at this level of generality, Z_2 Z_1 is not “zero” in any sense. I assume then that you at least want to work with tensor categories, not just monoidal categories.

In that case, should the first column really be {monoids} -> {abelian monoids}? Wouldn’t it be more appropriate to take something like {rings} -> {commutative rings}?

8. David Ben-Zvi - March 31, 2009

First the Kontsevich conjecture about the E_n center being E_{n+1} is a
theorem in various contexts, see papers by Tamarkin and Hu-Kriz-Voronov (the n=1 case, the Deligne conjecture, has zillions of proofs at this point). Also in general I think it is quite formal from Jacob’s general theory (this is the idea of the Hu-Kriz-Voronov proof above, but I think the formalism makes it work very generally) — the E_n center has an E_n structure (as an internal Hom object) and a compatible E_1 structure (as endomorphisms), and together these make E_{n+1}..(see Jacob’s TFT preprint, section 4.1)

Second, the statement that Z_2(Z_1) seems to me suspect, in that I think it’s false once you leave the discrete setting of semisimple abelian categories where Mueger’s theorem applies. In particular if you take as your algebra the monoidal oo-category D(G) of D-modules or perverse sheaves on a group, its E_1 center is D(G/G), “class sheaves” on the group. This is braided (E_2) but I don’t think it’s modular (by which let’s mean has trivial E_2 center) when G is an algebraic group (rather than a finite group) — for example the work of Boyarchenko and Drinfeld for G a unipotent group over a finite field writes this as a family of modular categories (“L-packets”) labeled by the E_2 center, which is nontrivial..

9. David Ben-Zvi - April 1, 2009

To elaborate on my last comment, if G is a complex semisimple or reductive group, let D(G) be the derived (or rather oo-)category of D-modules on G with convolution – the “D-group algebra” of G. The Drinfeld center of D(G) is D(G/G) – “D-class functions”, which is E_2 (braided). However such a class function has a “central character” – you can “decompose” D-modules according to their monodromy along a torus inside G. In some (weak) sense, D(G/G) is a family of modular categories – the categories of Lusztig character sheaves – labeled by this central character, living in the dual Cartan algebra mod W.
They correspond to extended three-dimensional TFTs (really “2 1/2 dimensional” – not defined on all 3-manifolds) called character theories which are studied in a paper of David Nadler and myself which should appear later this week. But in any case the picture is very similar to what Boyarchenko and Drinfeld have – the Drinfeld double is not itself modular, but there’s a central parameter (roughly the E_3 center) over which the theory decomposes into modular pieces. This seems to contradict the exactness idea in the homotopical setting..

10. Toby Bartels - April 7, 2009

Regarding extending the table upwards and to the left:

See the table at http://ncatlab.org/nlab/show/k-tuply+monoidal+n-category and the argument in the appendix to the last reference there. In particular, a 0-tuply monoidal n-category is really a *pointed* n-category. As a (-1)-category is (in the best understanding) a truth value, a 0-tuply monoidal (-1)-category is a *true* truth value, so there is only one.

This means that the (-1)-categorial column stabilises early, but in fact you must take the pointed approach to get stabilisation to work correctly at all! As Tom Leinster (first?) pointed out, a weak 2-category (bicategory) with exactly (one object and) one morphism has a bit more structure than an abelian monoid, if you classify up to equivalence of bicategories, and there is even more structure (so *not* stabilised) in a weak 3-category with exactly one 2-morphism. But if you classify up to equivalence of *pointed* n-categories (that is, requiring equivalences and higher equivalences to preserve the basepoint in a certain sense), then you get the desired result.

Anyway, people have been talking about whether things are pointed, so I figured that I’d better mention that.

Sorry comments are closed for this entry


text:
License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2017.58
URN: urn:nbn:de:0030-drops-81944
URL: https://drops.dagstuhl.de/opus/volltexte/2017/8194/
 Go to the corresponding LIPIcs Volume Portal

### Metatheorems for Dynamic Weighted Matching

 pdf-format:

### Abstract

We consider the maximum weight matching (MWM) problem in dynamic graphs. We provide two reductions. The first reduces the dynamic MWM problem on m-edge, n-node graphs with weights bounded by N to the problem with weights bounded by (n/eps)^2, so that if the MWM problem can be alpha-approximated with update time t(m,n,N), then it can also be (1+eps)alpha-approximated with update time O(t(m,n,(n/eps)^2)log^2 n+log n loglog N)). The second reduction reduces the dynamic MWM problem to the dynamic maximum cardinality matching (MCM) problem in which the graph is unweighted. This reduction shows that if there is an \alpha-approximation algorithm for MCM with update time t(m,n) in m-edge n-node graphs, then there is also a (2+eps)alpha-approximation algorithm for MWM with update time O(t(m,n)eps^{-2}log^2 N). We also obtain better bounds in our reductions if the ratio between the largest and the smallest edge weight is small. Combined with recent work on MCM, these two reductions substantially improve upon the state-of-the-art of dynamic MWM algorithms.

### BibTeX - Entry

@InProceedings{stubbs_et_al:LIPIcs:2017:8194,
author =	{Daniel Stubbs and Virginia Vassilevska Williams},
title =	{{Metatheorems for Dynamic Weighted Matching}},
booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
pages =	{58:1--58:14},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-029-3},
ISSN =	{1868-8969},
year =	{2017},
volume =	{67},
editor =	{Christos H. Papadimitriou},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/8194},
URN =		{urn:nbn:de:0030-drops-81944},
doi =		{10.4230/LIPIcs.ITCS.2017.58},
annote =	{Keywords: dynamic algorithms, maximum matching, maximum weight matching}
}

 Keywords: dynamic algorithms, maximum matching, maximum weight matching Collection: 8th Innovations in Theoretical Computer Science Conference (ITCS 2017) Issue Date: 2017 Date of publication: 28.11.2017

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# Build

Algebra Level 3

Bobby can build a house in 24 days. Danny can build the same house in 36 days. Andy can build the same house in 27 days. If Bobby and Danny work together for 8 days, and will be continued by Andy, how long does Andy take to complete the remaining work?

Try another problem on my set! Let's Practice

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# finite/infinite product $\prod\limits_{i=0}^N (1-i\,\epsilon)$

I fear this may be pedestrian but after consulting Abramowitz and Stegun, as well as wikipedia and wolfram, I could not find the answer, though I'm sure it may be simple for someone more gifted than I in analysis :)

I need to know how to simplify, $$\prod_{i=0}^N\; (1-i\,\epsilon)$$

I also did not see a way of shaping this into either of the trigonometric functions sin or cos...

P.S. $N$ becomes very big, like Avogadro's number big, so I believe I may be able to use either a finite or an infinite product...?

P.P.S. $\epsilon \ll 1$

P.P.P.S $i \neq \sqrt{-1}$ - it is only an index ( a poor choice too )

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I don't think the infinite product converges. –  Daniel Littlewood Jan 19 '13 at 20:23
You might want to use \prod and \ll. –  Did Jan 19 '13 at 23:12

If $N\epsilon \ll 1$, using that $\ln (1+x) \approx x$ for small $x$:

$\prod_{n=0}^N (1-n\epsilon) = \exp(\ln(\prod_{n=0}^N (1-n\epsilon))) = \exp(\sum_{n=0}^N \ln(1-n\epsilon)) \approx \exp(\sum_{n=0}^N (-n\epsilon)) = \exp(-\epsilon N(N+1)/2)$

-
I was wondering how it would change if instead of $1-n\,\epsilon$, it was $V-n\,\epsilon$ where $V >> 1$. I guess it would be $\exp(\sum\,(\ln(V) - n\,v_0\,(d/dV)[\ln(V)\])$ –  nate Jan 19 '13 at 23:24
A related question... of course $1>>n\,\epsilon$, so how can you get rid of it? I know $\ln$ is pretty flat but.... –  nate Jan 19 '13 at 23:33
It does work here. If you're bothered by the derivative, you can always say $\ln(V-n\epsilon)=\ln(V*(1-n\epsilon/V))$ and use the fact that $\ln a*b = \ln a + \ln b$. (But it's essentially the same here) –  Guest 86 Jan 20 '13 at 22:55
actually I had just forgot about that finite sum $\sum_i^N=N(N+1)/2$... Tell the truth I am trying to get the "4" in the reduced volume van der Waal equation from counting microstates ;) ($V-4*v_0$) –  nate Jan 21 '13 at 2:47
Maple says $$\prod_{k=1}^N(1 - k a) = \frac{(-1)^{N} a^{N} \Gamma \Bigl(\frac{N a + a - 1}{a}\Bigr)}{\Gamma \Bigl(\frac{a - 1}{a}\Bigr)}$$ But I don't know if you think the Gamma function is "simple" or not.


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# PowerShell File Management

Ryan (Netwrix)
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Jul 10, 2018 5 Minute Read

Every day, sysadmins have to perform various standard operations on the numerous files and folders on their Windows servers. These tasks often include managing users’ data on shared resources and maintaining backups properly. You can use PowerShell to reduce amount of manual work involved.

• View the objects in a directory
• Create files and folders
• Delete files and folders
• Copy files and folders
• Move files or directories
• Rename files
• Change file extensions
Before you start, make sure your system policy allows running PowerShell scripts as described in “Windows PowerShell Scripting Tutorial for Beginners.”

Viewing the objects in a directory

To view the content of a directory on a Windows file server, use the Get-ChildItem cmdlet. To show all hidden files, add the -Force parameter. The command below shows all root objects in the Shared folder:

Powershell
Get-ChildItem -Force \\fs\Shared

If you want to also check all subfolders and their content, add -Recurse parameter:

Powershell
Get-ChildItem -Force \\fs\Shared -Recurse

To filter the output, add the Filter, Exclude, Include and Path parameters to the Get-ChildItem cmdlet. For advanced object filtering, use the Where-Object cmdlet. The script below searches for all executable files in the IT folder that were modified after April 1, 2018:

Powershell
Get-ChildItem -Path \\fs\Shared\IT -Recurse -Include *.exe | Where-Object -FilterScript {($_.LastWriteTime -gt '2018-04-01')}  Creating files and folders with PowerShell To create new objects with Windows PowerShell, you can use the New-Item cmdlet and specify the type of item you want to create, such as a directory, file or registry key. For example, this command creates a folder: Powershell New-Item -Path '\\fs\Shared\NewFolder' -ItemType Directory  And this command creates an empty file: Powershell New-Item -Path '\\fs\Shared\NewFolder\newfile.txt' -ItemType File  Creating files and writing data to them There are at least two built-in methods to create a file and write data to it. The first is to use the Out-File cmdlet: Powershell $text = 'Hello World!' | Out-File $text -FilePath C:\data\text.txt  To overwrite an existing file, use the –Force switch parameter. You can also create files using the Export-Csv cmdlet, which exports the output into a csv file that can be opened in Excel: Powershell Get-ADuser -Filter * | Export-Csv -Path C:\data\ADusers.csv  Creating files after checking that they don’t already exist The following script checks whether a specific file (pc.txt) already exists in a particular folder; if not, it generates a list of all AD computers and saves it to a new file named pc.txt: Powershell #create array of text files$files=Get-ChildItem C:\data\*.txt | select -expand fullname
#check if file exists inside the array
$files -match "pc.txt" #if matching return “True” key then exit, if “False” then create a report if($files -eq 'False'){
Get-ADComputer -Filter * | Export-Csv -Path C:\data\pc.txt
}
else{exit}

Deleting files and folders with PowerShell

To delete objects, use the Remove-Item cmdlet. Please note that it requires your confirmation upon execution if the object is not empty. The example below demonstrates how to delete the IT folder and all the subfolders and files inside it:

Powershell
Remove-Item -Path '\\fs\shared\it\'
Confirm
The item at \\pdc\shared\it has children and the Recurse parameter was not
specified. If you continue, all children will be removed with the item. Are you
sure you want to continue?
[Y] Yes  [A] Yes to All  [N] No  [L] No to All  [S] Suspend  [?] Help
(default is "Y"):

If you have already made sure that every object inside the folder should be deleted, you can use the ?Recurse switch to skip the confirmation step:

Powershell
Remove-Item -Path '\\fs\shared\it\' -Recurse

Deleting files and folders older than X days

Sometimes you need to clean up old files from a certain directory. Here’s the way to accomplish that:

Text
$Folder = "C:\Backups" #delete files older than 30 days Get-ChildItem$Folder -Recurse -Force -ea 0 |
? {!$_.PsIsContainer -and$_.LastWriteTime -lt (Get-Date).AddDays(-30)} |
ForEach-Object {
$_ | del -Force$_.FullName | Out-File C:\log\deletedbackups.txt -Append
}

#delete empty folders and subfolders if any exist
Get-ChildItem $Folder -Recurse -Force -ea 0 | ? {$_.PsIsContainer -eq $True} | ? {$_.getfiles().count -eq 0} |
ForEach-Object {
$_ | del -Force$_.FullName | Out-File C:\log\deletedbackups.txt -Append
}

Deleting files after checking they exist

Here’s how to check whether a file exists and delete it if it does:

Powershell
$FileName = 'C:\data\log.txt' If (Test-Path$FileName){
Remove-Item $FileName }  Deleting files from multiple computers in one script To delete files from remote PCs, you must have the appropriate security permissions to access them. Be sure to use UNC paths so the script will correctly resolve the file locations. Powershell $filelist = @(" \c$\Temp", "\c$\Backups") #variable to delete files and folder
$computerlist = Get-Content C:\data\pc.txt #get list of remote pc's foreach ($computer in $computerlist){ foreach ($file in $filelist){$filepath= Join-Path "\\$computer\" "$filelist" #generate unc paths to files or folders
if (Test-Path $filepath) { Remove-Item$filepath -force -recurse -ErrorAction Continue}}}

Copying files and folders with PowerShell

The Copy-Item cmdlet enables you to copy objects from one path to another. The following command creates a backup by copying the file users.xlsx from one remote computer (fs) and saving it to another (fs2) over the network:

Powershell
Copy-Item -Path \\fs\Shared\it\users.xlsx -Destination \\fs2\Backups\it\users.xlsx

If the target file already exists, the copy attempt will fail. To overwrite the existing file, even if it is in Read-Only mode, use the -Force parameter:
Powershell
Copy-Item -Path \\fs\Shared\it\users.xlsx -Destination \\fs2\Backups\it\users.xlsx -Force

Copying files with PowerShell to or from a remote computer

If you’re copying files to or from remote computers, be sure to use UNC paths.

For example, use this command to copy files from a remote file server to the local C: directory:

Powershell
Copy-Item \\fs\c$\temp -Recurse C:\data\  To copy files from your local directory to the remote folder, simply reverse the source and destination locations: Powershell Copy-Item C:\data\ -Recurse \\fs\c$\temp

Copying multiple files from one server to another over the network in one script

You can also copy files from one remote server to another. The following script recursively copies the \\fs\Shared\temp folder to \\fs\Shared\test:

Powershell
Copy-Item \\fs\Shared\temp -Recurse \\fs\Shared\test

Copying only certain types of files

To copy only certain files from the source content to the destination, use the -Filter parameter. For instance, the following command copies only txt files from one folder to another:

Powershell
Copy-Item -Filter *.txt -Path \\fs\Shared\it -Recurse -Destination \\fs2\Shared\text

Copying files using XCOPY and ROBOCOPY commands or COM objects

You can also run XCOPY and ROBOCOPY commands to copy files, or use COM objects as in the example below:

Powershell
(New-Object -ComObject Scripting.FileSystemObject).CopyFile('\\fs\Shared', 'fs2\Backup')

Moving files and directories with PowerShell

The Move-Item cmdlet moves an item, including its properties, contents, and child items, from one location to another. It can also move a file or subdirectory from one directory to another location.

The following command moves a specific backup file from one location to another:

Powershell
Move-Item -Path \\fs\Shared\Backups\1.bak -Destination \\fs2\Backups\archive\1.bak

This script moves the entire Backups folder and its content to another location:

Powershell
Move-Item -Path \\fs\Shared\Backups -Destination \\fs2\Backups\archive

The Backups directory and all its files and subfolders will then appear in the archive directory.

Renaming files with PowerShell

The MRename-Item cmdlet enables you to change the name of an object while leaving its contentintact. It’s not possible to move items with the Rename-Item command; for that functionality, you should use the Move-Item cmdlet as described above.

The following command renames a file:

Powershell
Rename-Item -Path "\\fs\Shared\temp.txt" -NewName "new_temp.txt"

Renaming multiple files

To rename multiple files at once, use a script like this:

Powershell
$files = Get-ChildItem -Path C:\Temp #create list of files foreach ($file in $files) {$newFileName=$file.Name.Replace("A","B") #replace "A" with "B" Rename-Item$file $newFileName }  Changing file extensionswith PowerShell You can also use the Rename-Item to change file extensions. If you want to change the extensions of multiple files at once, use the Rename-Item cmdlet with the Get-ChildItem cmdlet. The following script changes all “txt” file extensions to “bak”. The wildcard character (*)ensures that all text files are included: Powershell Get-ChildItem \\fs\Shared\Logs\*.txt | Rename-Item -NewName {$_.name -Replace '\.txt$','.bak' }  Conclusion Using the information in this article, you can automate a variety of simple operations related to file management on your file storages and save time for more important tasks. Good luck! Difficulty Level: Intermediate Ryan Netwrix This person is a verified professional. Verify your account to enable IT peers to see that you are a professional. Ryan (Netwrix)IT Security Consultant at Netwrix Main Areas of Contribution: ## 6 Comments Habanero Glenn_P This person is a verified professional. Verify your account to enable IT peers to see that you are a professional. This is a very good summary of file handling with Powershell. I'm afraid that it's going to take a while for me to reach for these commands instead of falling back to the old command window. Thai Pepper deanmoncaster This person is a verified professional. Verify your account to enable IT peers to see that you are a professional. This is just what i needed thank you! Chipotle JPanz This person is a verified professional. Verify your account to enable IT peers to see that you are a professional. Nice summary. Thank you. I'm going to bookmark this one. Serrano configur3d Good write up. I enjoyed it. However, the part where you create a file is not correct. Ryan (Netwrix) wrote: Powershell #create array of text files$files=Get-ChildItem C:\data\*.txt | select -expand fullname
#check if file exists inside the array
$files -match "pc.txt" #if matching return “True” key then exit, if “False” then create a report if($files -eq 'False'){
Get-ADComputer -Filter * | Export-Csv -Path C:\data\pc.txt
}
else{exit}

After you get the files you place them in the $files variable. Then you check it but you don't save the result into another variable or equate it to existing one to False. In other words,$files will always be a list of files. Never a true or false.
Tabasco
Josh_the_Geek
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Glenn_P wrote:

This is a very good summary of file handling with Powershell. I'm afraid that it's going to take a while for me to reach for these commands instead of falling back to the old command window.

You don't even have to go that far, Majority of CMD commands still work in the Powershell window.

Sonora
Tamizh
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Basic commands, but very useful ones. Thank you


text:
Answered according to Hanafi Fiqh by Muftionline.co.za

Q: Is the following message proven from sahih Hadeeth? If it is then what is the reference?

A traveller once came to the masjid  to see the Prophet Muhammad (Sallallahu Alayhi Wasallam) after greeting the prophet, he was asked where he was from. The traveller replied  that he came from very far just to get a few questions answered. Following is the dialogue between the traveller and the Prophet (Sallallahu Alayhi Wasallam).

Traveller      : I do not want adhaab (punishments) to be written in my account.

*Prophet     : Behave well with your parents *

Traveller      : I want to be known amongst people as an intelligent person.

*Prophet     : Fear Allah (Jalla Jalalahu), always. *

Traveller      : I want to be counted amongst Allah’s favourites.

*Prophet      : Recite Quran every morning and evening. *

Traveller: I want my heart to always be enlightened. ( Munawer )

*Prophet      : Never forget death. *

Traveller: I never want to be away from Allah’s blessing.

*Prophet      : Always treat fellow creatures well. *

Traveller: I never want to be harmed by my enemies.

*Prophet       : Always have faith in only Allah. *

Traveller: I never want to be humiliated.

*Prophet      : Be careful of your actions. *

Traveller: I wish to live long.
*Prophet      : Always do silaa rahmi. (Goodness towards blood Relations) *

Traveller: I want my sustenance to increase.

*Prophet      : Always be in wudhoo. *

Traveller: I wish to stay free of adhaab in the grave.
*Prophet      : Always wear pure (paak) clothes. *

Traveller: I never want to be burned in hell.
*Prophet      : Control your eyes and tongue. *

Traveller: How do I get my sins forgiven?

*Prophet      : Always ask forgiveness from Allah with a lot of humility.

Traveller: I want people to respect me always.
*Prophet      : Never extend your hands of need at people. *

Traveller: I want to always be honoured.
*Prophet      : Never humiliate or put down anyone. *

Traveller: I don’t want to be squeezed by fishare qabr.(Squeezing in the grave)
*Prophet      : Recite Surat Al Mulk (The Dominion) often. *

Traveller: I want my wealth to increase.
*Prophet      : Recite surat el Waqiah (The Inevitable) every night. *

Traveller: I want to be safe and at peace on Day of Judgment.
*Prophet       : Do zikr (Praises) of Allah from dusk to night. *

Traveller: I want to be in full attention & concentration during prayers.
*Prophet       : Always do wudhoo with concentration and attention. *

A: We have not come across this dialogue.

And Allah Ta’ala (الله تعالى) knows best.