linear algebra – $lim_{epsilonto 0^+}int_{0}^{2pi}log|f(epsilon e^{itheta})|dtheta=2pilog|f(0)| $ where $f$ is analytic in an open set.

Denote $mathbb{D}={zmid |z|<1}$.
Let $f(z)$ be analytic in the slit disc $mathbb{D}setminus(0,1)$ and $f(0)=lim_{zto 0}f(z)$.

Question Prove that for $epsilon>0$ arbitrarily small,
$$lim_{epsilonto 0^+}int_{0}^{2pi}log|f(epsilon e^{itheta})|dtheta=2pilog|f(0)| $$
My try:
Since $f$ is analytic in the slit disc $mathbb{D}setminus(0,1)$ (which is open set), so by Jensen’s formula,
$$frac{1}{2pi}int_{0}^{2pi}log|f(epsilon e^{itheta})|dtheta=log|f(0)|+sum_{|alpha|<epsilon,f(alpha)=0}logfrac{epsilon}{|alpha|}$$
As $epsilonto 0^+$, the set ${alphamid|alpha|<epsilon,f(alpha)=0}$ is empty and hence we have,
$$lim_{epsilonto 0^+}int_{0}^{2pi}log|f(epsilon e^{itheta})|dtheta=2pilog|f(0)| $$
Any approach (like mean value theorem for harmonic function or Lebesgue’s dominated convergence theorem) would be appreciated.

set theory – Which countable ordinals are “Barwise compact” for $mathcal{L}_{infty,omega_1}$?

Barwise compactness says (as a special case) that whenever $alpha$ is countable and admissible, $Tsubseteqmathcal{L}_{infty,omega}cap L_alpha$ is $alpha$-c.e., and every subset of $T$ which is an element of $L_alpha$ is satisfiable, then $T$ itself is satisfiable.

Now say that a countable ordinal $alpha$ is $omega_1$-Barwise iff it satisfies the analogue of the above property with $mathcal{L}_{infty,omega}$ replaced by $mathcal{L}_{infty,omega_1}$. This is a nontrivial notion:

  • $omega_1^{CK}$ is not $omega_1$-Barwise. Take the usual computable infinitary theory used in the construction of a Harrison order via Barwise compactness and add a sentence asserting well-orderability. In fact this gives a fairly large lower bound on the smallest $omega_1$-Barwise ordinal, although I have no idea whether this bound is at all good (and I suspect it’s actually quite bad).

  • There exist $omega_1$-Barwise ordinals. Let $(T_i)_{i<omega_1}$ be an enumeration of the $omega_1$-c.e. $mathcal{L}_{infty,omega_1}$-theories such that for each admissible $alpha<omega_1$ the sequence $(T_icap L_alpha)_{i<alpha}$ enumerates the $alpha$-c.e. $mathcal{L}_{infty,omega_1}$-theories. For each $i<omega_1$ let $rho(i)=i$ if $T_i$ is countably satisfiable and $min{i+alpha: T_icap L_alphamodelsperp}$ if $T_i$ is not countably satisfiable. There is then some $theta<omega_1$ such that $rho(i)<theta$ for all $i<theta$; this $theta$ is $omega_1$-Barwise. (Note that this is extremely coarse: the same argument works for basically any logic at all, and we can moreover replace “$Sigma_1$” with “definable” by restricting attention to elementary initial segments of $L_{omega_1}$ so that “unsatisfiability is preserved” as we progress up $omega_1$.)

I’d love an exact characterization of $omega_1$-Barwiseness, but I suspect that’s rather difficult. Instead, the following seems more approachable:

What is the smallest $omega_1$-Barwise ordinal?

(Of course in some sense we have the answer already: it’s the smallest $omega_1$-Barwise ordinal. I’m looking for a characterization in terms of more standard notions – e.g. restricted elementary substructures and Mostowski collapses, levels of $L$ satisfying some first-order theory or having natural “superstructures” satisfying some first-order theory, supremum of ordinals computable in some higher-type functional, etc.)

How do I set up a views block that filters contextually on a field included in a paragraph?

I have two content types (ContentA, ContentB) which both use the same paragraph (MyParagraph) as field type “Entity reference revisions.”

MyParagraph consists of two “List (text)” fields: ParagraphFieldA, ParagraphFieldB.

The values for ParagraphFieldA and ParagraphFieldB come in both cases from allowed values lists which have a key and label.

ContentA is limited to one MyParagraph reference but ContentB can have multiple MyParagraph references.

I have been trying to set up a Views Block that will be shown with any ContentA node. I want it to list titles of all ContentB nodes that have the same ParagraphFieldA value as the ContentA node.

I have a Views Block with Fields: Content: Title; and Filter Criteria: Content: Content type (=ContentB).

It has relationships to Paragraph referenced from MyParagraph and Content using MyParagraph.

I have added a contextual filter to (MyParagraph: Paragraph) Paragraph: ParagraphFieldA with “Content ID from URL” selected under “Provide default value.” I expected this to filter on the ParagraphFieldA value from the displayed ContentA node, but no results are returned.

If I put a key value from ParagraphFieldA in the preview box, I get a list of the relevant ContentB titles, but the view is not getting this automatically from the URL.

I have searched high and low for a solution and have tried various permutations without success. Clearly I am making a basic mistake here but I can’t see what it is.

Can anyone help?

dnd 5e – Are the Material Planes comprising the various campaign settings surrounded by a shared set of outer planes?

The spell Dream of the Blue Veil from Tasha’s Cauldron of Everything explicitly sets all worlds as being in the same material plane:

You and up to eight willing creatures within range fall unconscious for the spell’s duration and experience visions of another world on the Material Plane, such as Oerth, Toril, Krynn, or Eberron. (…)

Beyond this spell, we have other evidence for this being the case. D&D 5e is set in a multiverse by default, which contains many worlds. This is explicitly called out in the Gods of the Multiverse appendix of the Players Handbook (PHB):

Religion is an important part of life in the worlds of the D&D multiverse.


Many people in the worlds of D&D worship different gods at different times and circumstances.

Other worlds are explicitly called out in the D&D Pantheons section of this appendix:

Each world in the D&D multiverse has its own pantheons of deities, ranging in size from the teeming pantheons of the Forgotten Realms and Greyhawk to the more focused religions of Eberron and Dragonlance. (…)

The Dungeon Masters Guide (DMG) also explicitly calls out that there are multiple worlds in the Material Plane in the Making a Multiverse chapter:

(…) In this context, the Material Plane is the nexus where all these philosophical and elemental forces collide in the jumbled existence of mortal life and matter. The worlds of D&D exist within the Material Plane, making it the starting point for most campaigns and adventures. The rest of the multiverse is defined in relation to the Material Plane. (…)

When talking about the Material Plane, this chapter also has a section called Known Worlds of the Material Plane which states (along with more in depth per world descriptions):

Worlds of the Material Plane are infinitely diverse. The most widely known worlds are the ones that have been published as official campaign settings for the D&D game over the years. If your campaign takes place on one of these worlds, that world belongs to you in your campaign. Your version of the world can diverge wildly from what’s in print.

Chapter 11 of the DMG A World of your Own it reinforces this idea:

This book, the Player’s Handbook, and the Monster Manual present the default assumptions for how the worlds of D&D work. Among the established settings of D&D, the Forgotten Realms, Greyhawk, Dragonlance, and Mystara don’t stray very far from those assumptions. Settings such as Dark Sun, Eberron, Ravenloft, Spelljammer, and Planescape venture further away from that baseline. As you create your own world, it’s up to you to decide where on the spectrum you want your world to fall.

If you needed more evidence of this, the adventure Dungeon of the Mad Mage (set on Faerun) makes named worlds of other campaign settings explicitly set within the same multiverse:

In room 15b on Level 9 of the Dungeon, the following treasure is found (emphasis mine):

The arcanaloth is an avid reader and has collected countless books from across the multiverse. Most of the books cover mundane subjects such as etiquette, oratory, and poetry. Twenty of the books are treatises on the Outer Planes and chronicles of historical events on various Material Plane worlds, including Toril, Oerth, Athas, and others; these tomes are worth 100 gp each to an interested buyer. A character who spends 1 hour searching can find one of these rare tomes.

How do you set sharepoint user field to allow multiple users with powershell

In order to add a multi-choice user field, you need to use the Add-PnPFieldFromXML method.

Also, the multi-choice user field is a not a simple property. In fact, it’s a totally different field type: Type=”UserMulti”:

$fieldXml = '<Field Type="UserMulti" DisplayName="Demo Presentors" List="UserInfo" Required="FALSE" ID="{bd66d0d0-c441-4ce3-909a-204ff01cdcb7}" ShowField="EMail" UserSelectionMode="PeopleOnly" StaticName="DemoPresentors" Name="DemoPresentors" Mult="TRUE" />'

Add-PnPFieldFromXml -List $RecordListName -FieldXml $fieldXml

WordPress media library attachment-filter: Different set of mime types are appearing in List and Grid view modes

I have an array of mime_types like the following:

$mime_map = array(
    /** Data types */
        'name' => 'Comma separated (csv)',
        'mime' => 'text/csv',
        'ext'  => 'csv',
        'name' => 'Spreadsheet',
        'mime' => 'application/excel',
        'ext'  => 'xl',
        'name' => 'Microsoft Excel',
        'mime' => 'application/',
        'ext'  => 'xlsx',
        'name' => 'Microsoft Excel (xls)',
        'mime' => 'application/x-xls',
        'ext'  => 'xls',
        'name' => 'Spreadsheet (odf)',
        'mime' => 'application/vnd.openxmlformats-officedocument.spreadsheetml.sheet',
        'ext'  => 'odf',
    /** Image types */
        'name' => 'Image (gif)',
        'mime' => 'image/gif',
        'ext'  => 'gif',
        'name' => 'Image (png)',
        'mime' => 'image/png',
        'ext'  => 'png',
        'name' => 'Image (jpg)',
        'mime' => 'image/jpeg',
        'ext'  => 'jpg',
        'name' => 'Image (jpeg)',
        'mime' => 'image/pjpeg',
        'ext'  => 'jpeg',
        'name' => 'Image (svg)',
        'mime' => 'image/svg+xml',
        'ext'  => 'svg',
        'name' => 'Image (WebP)',
        'mime' => 'image/webp',
        'ext'  => 'webp',
    /** Video types */
        'name' => 'Video (3gp)',
        'mime' => 'video/3gp',
        'ext'  => '3gp',
        'name' => 'Video (3gpp)',
        'mime' => 'video/3gpp',
        'ext'  => '3gpp',
        'name' => 'Video (avi)',
        'mime' => 'video/avi',
        'ext'  => 'avi',
        'name' => 'Video (mp4)',
        'mime' => 'video/mp4',
        'ext'  => 'mp4v',
    /** Audio types */
        'name' => 'Audio (mp4)',
        'mime' => 'audio/mp4',
        'ext'  => 'mp4',
        'name' => 'Audio (mpeg)',
        'mime' => 'audio/mpeg',
        'ext'  => 'm4a',
        'name' => 'Audio (mp3)',
        'mime' => 'audio/mp3',
        'ext'  => 'mp3',
    /** Document types */
        'name' => 'Photoshop Document',
        'mime' => 'application/octet-stream',
        'ext'  => 'psd',
        'name' => 'PDF Document',
        'mime' => 'application/pdf',
        'ext'  => 'pdf',
        'name' => 'Office Word',
        'mime' => 'application/vnd.openxmlformats-officedocument.wordprocessingml.document',
        'ext'  => 'docx',
        'name' => 'MS Word Document',
        'mime' => 'application/msword',
        'ext'  => 'doc',
        'name' => 'MS PowerPoint',
        'mime' => 'application/',
        'ext'  => 'ppt',
        'name' => 'Office PowerPoint',
        'mime' => 'application/vnd.openxmlformats-officedocument.presentationml.presentation',
        'ext'  => 'pptx',
        'name' => 'Document (odt)',
        'mime' => 'application/vnd.oasis.opendocument.text',
        'ext'  => 'odt',
    /** Text types */
        'name' => 'Text (Plain)',
        'mime' => 'text/plain',
        'ext'  => 'txt',
        'name' => 'Text (RTF)',
        'mime' => 'text/rtf',
        'ext'  => 'rtf',
    /** Archive types */
        'name' => 'Archive (gzip)',
        'mime' => 'application/x-gzip',
        'ext'  => 'gz',
        'name' => 'Archive (7zip)',
        'mime' => 'application/x-7z-compressed',
        'ext'  => '7zip',
        'name' => 'Archive (zip)',
        'mime' => 'application/zip',
        'ext'  => 'zip',
        'name' => 'Archive (rar)',
        'mime' => 'application/rar',
        'ext'  => 'rar',
        'name' => 'Archive (tar)',
        'mime' => 'application/x-tar',
        'ext'  => 'tar',

I am using the above array to populate attachment-filter dropdown with specific file types for better filtration. The filter works well in both list and grid view modes but the list of mime-types that appear in the said modes are different!

In list view, correct mime-types are appearing, i.e. only those which have at least one media was uploaded to. But in grid view, all available mime-types in the above array are appearing.

add_filter( 'post_mime_types', array( $this, 'wpmf_custom_mime_types' ), 10, 1 );
function wpmf_custom_mime_types( $post_mime_types ) {
    $media_mime_types       = new Media_Filter_Mime_Types();
    $types                  = $media_mime_types->media_types;
    foreach( $types as $key => $mime ) {
        $post_mime_types($mime('mime')) = array( __( $mime('name'), 'text-domain' ) );
    return $post_mime_types;

Screenshot (List view mode)
enter image description here

Screenshot (Grid view mode: more items are appearing)
enter image description here

Update #1
I added the following in the array

    'name' => 'How about this',
    'mime' => 'text/hath',
    'ext'  => 'hat',

I tried to upload a file (how-about-this.hat), which as expected, failed to upload and this mime-type is not visible in the filter for list view either. But it does appear in grid view mode.

Update #2
I emptied the function body to look like this:

function wpmf_custom_mime_types( $post_mime_types ) {
    // Nothing here

And, as expected, the attachment-filter in both view modes become synchronized. But whenever I am putting the code back in the function body, the said problem starts to appear.

Is there anything I am missing or something specific is required to be done to synchronize the items because I am adding them through code?

ms office – Set the list separator in Mac

In Windows, List separator (, as shown in my system) is a parameter that we could set:

enter image description here

I’m trying to find this setting in my Mac, but I could not find it. Here is the only thing I could find:

enter image description here

It seems that this separator is used to separate arguments in Excel functions.

Does anyone know how to set the list separator in Mac?

set theory – Detecting uncountable cardinals in $(mathbb{R};+,times,mathbb{N})$

For a structure $mathcal{X}=(X;…)$, say that a cardinal $kappa$ is $mathcal{X}$-detectable iff there is some sentence $varphi$ in the language of $mathcal{X}$ together with a fresh unary predicate symbol $U$ such that an expansion of $mathcal{X}$ gotten by interpreting $U$ as $Asubseteq X$ satisfies $varphi$ iff $vert Avertgekappa$.

For example, $omega_1$ is $(omega_1;<)$-detectable since a subset of $omega_1$ is countable iff it is bounded above. By contrast, it turns out that $omega_1$ is not $mathcal{R}=(mathbb{R};+,times)$-detectable.

I’m interested in the expansion $mathcal{R}_mathbb{N}:=(mathbb{R};+,times,mathbb{N})$ of $mathcal{R}$ gotten by adding a predicate naming the natural numbers (equivalently, adding all projective functions and relations). Since we can talk about one real enumerating a list of other reals, $omega_1$ is $mathcal{R}_mathbb{N}$-detectable (“there is no real enumerating all elements of $U$“). More pathologically, if $mathfrak{c}=2^omega$ is regular and there is a projective well-ordering of the continuum of length $mathfrak{c}$ then $mathfrak{c}$ is $mathcal{R}_mathbb{N}$-detectable. So for example it is consistent with $mathsf{ZFC}$ that $omega_2$ is $mathcal{R}_mathbb{N}$-detectable.

I’m curious whether this type of situation is the only way to get $mathcal{R}_mathbb{N}$-detectability past $omega_1$. There are multiple ways to make this precise, of course. At present the following two seem most natural to me:

  • Is it consistent with $mathsf{ZFC}$ that there are at least two distinct regular cardinals $>omega_1$ which are $mathcal{R}_mathbb{N}$-detectable?

  • Is it consistent with $mathsf{ZFC}$ that there is a singular cardinal which is $mathcal{R}_mathbb{N}$-detectable?

Note that an affirmative answer to either question requires a large continuum, namely $geomega_3$ and $geomega_{omega+1}$ respectively. Although my primary interest is in first-order definability, I’d also be interested in answers for other logics which aren’t too powerful (e.g. $mathcal{L}_{omega_1,omega}$).

Windows Server 2016 schedule automatic updates not working as set in Group Policy

I have a Windows Server 2016 VPS where a web app is works fine but when windows install updates,nothing works because,in this case CPU usage is 99% and no user can access the app

I’ve setup automatic updates in Group Policy this way:

  • in Computer Configuration/Administrative Templates/Windows Components/Windows Update
    I put Enable for “Configure Automatic Updates”
  • then value 4 for auto download/install and time for every Monday at 3am;not check the automatic maintenance box
  • restart system

but Saturday on 11am no client can work on the web app because of automatic updates(CPU 99%) and I had to disable automatic update and everything went fine

how to fix schedule update to work as set in Group Policy?


redhat – Needs to set nodev option on DB directories in linux

I have created two directory for new build linux server & each directory is 200GB in size and will be used for DB . security team did scan they found vulnerability as ” No nodev option set to directory”. I tried with “”mount -o remount,nodev /mountpoint”” but nodev is not being added in /etc/fstab .

Is it good to add nodev option to DB folder and how I can do this ??

Will unmount the directory and append the line with nodev in /etc/fstab file and remount again will work.
I am new to linux and need help on this topic

thanks in advance