Principal Component Analysis, computing the degree of freedom Consider data $x ∈ R^3$ that is represented in a 3D space, i.e., using 3 coordinates $(x^1, x^2, x^3)$. Consider that we observe a large dataset $x_1, x_2, · · · , x_n$, where n = 100. Suppose each datum $x_i$ is known to be such that its coordinates exhibit a functional relationship such that $ax_1^i + bx_2^i + cx_3^i + d = 0$, for the same $a, b, c, d ∈ R$. Thus, although the data lies in a 3D space, the underlying degrees of freedom in the variability of the data is lower than 3. Your goal is to represent each datum using just as many coordinates as the number of degrees of freedom, without any loss of information, i.e., the mapping from the 3D space to the lower-dimensional space should be linear and distance preserving (Euclidean distance between any 2 points in the original 3D space should equal Euclidean distance between the mapped points in the lower-dimensional space).  • How many degrees of freedom are present in the variability in the dataset ?  • Give a detailed algorithm to find the lower-dimensional representation of the 3D data.  I understand that the algorithm is PCA algorithm. But how to actually find the dimension of the lower subspace using the given constraint?
Testing for linear dependence among the columns of a matrix I have a correlation matrix of security returns whose determinant is zero. (This is a bit surprising since the sample correlation matrix and the corresponding covariance matrix should theoretically be positive definite.)  My hypothesis is that at least one security is linearly dependent on other securities. Is there a function in R that sequentially tests each column a matrix for linear dependence?  For example, one approach would be to build up a correlation matrix one security at a time and calculate the determinant at each step. When the determinant = 0 then stop as you have identified the security who is a linear combination of other securities.  Any other techniques to identify linear dependence in such a matrix are appreciated.
How to animate noise texture "seed"? It's possible to animate a change noise texture by animating the texture coordinates with a mapping node or by animating the texture space, however this only works on one axis at a time (on the other axes the texture will appear to be sliding across the object's surface).  For example, the top face has the desired effect but the other faces appear to be moving/sliding upwards:    Even with all three axes location and rotation animated it still appears to be sliding rather than statically changing.    Is there some way to make each face appear like the top face in the first gif but still be seamlessly blended with the other faces? This should work for other shapes as well.  In other words, I'm looking for a way to animate the "seed" of the texture. Windows update giving error 80070002 and manual update not working how to update? I have been trying to update my pc from 20h1 to 20h2 but the windows update shows error message 80007002 and when I manually install the update using windows 10 update assistant it downloads completely and then even starts install and gets stuck at 87 percent on the screen where it says do not turn off. What to do? This is the log file. Matching Profile found: FindRollbackFailure - 3A43C9B5-05B3-4F7C-A955-88F991BB5A48 SetupDiag version: 1.6.0.0 System Information:     Machine Name =      Manufacturer =      Model =      HostOSArchitecture = 1033     FirmwareType = UEFI     BiosReleaseDate = 20160513000000.000000+000     BiosVendor = A12     BiosVersion = A12     HostOSVersion = 10.0.19041     HostOSBuildString = 19041.1.amd64fre.vb_release.191206-1406     TargetOSBuildString = 10.0.19041.621 (vb_release_svc_prod1.201014-1758)     HostOSLanguageId =      HostOSEdition = CoreSingleLanguage     RegisteredAV = Quick Heal Internet Security     FilterDrivers = FileInfo     UpgradeStartTime = 12/20/2020 2:54:38 PM     UpgradeEndTime = 12/20/2020 5:30:40 PM     UpgradeElapsedTime = 02:36:02     CV = Pi3IdU3SlEqpFx93     ReportId =    Error: SetupDiag reports rollback failure found. Last Phase = Pre OOBE Boot Last Operation = OOBE boot apply Error = 0xC1900101-0x4000D LogEntry:   Refer to &quot;https://docs.microsoft.com/en-us/windows/desktop/Debug/system-error-codes&quot; for error information.  Last Setup Phase: Phase Name: Pre OOBE Boot     Phase Started: 12/20/2020 4:37:50 PM     Phase Ended: 1/1/0001 12:00:00 AM     Phase Time Delta: 00:00:00     Completed Successfully? False   Last Setup Operation: Operation Name: OOBE boot apply     Operation Started: 12/20/2020 4:37:50 PM     Operation Ended: 1/1/0001 12:00:00 AM     Operation Time Delta: 0:00:00:00.0000000     Completed Successfully? False I was trying to fix my audio problems using suggestions from . One of the comments suggested I execute this and only this:     killall pulseaudio   So I did, and I restarted my computer just in case, but all it did was remove my soundcard and I still couldn't hear anything. Now I don't know how to get my soundcard back.  Please note that I am extremely inexperienced when it comes to Ubuntu. nginx open reverse proxy? Somehow, and I'm not sure how, we have configured an nginx reverse proxy that is acting as an open proxy.  The log files tells us the matching requests are going against the following conf file:  map $http_upgrade $connection_upgrade {         default upgrade;         ''      close; }  access_log      /var/log/nginx/access.p.log; error_log       /var/log/nginx/error.p.log;  proxy_next_upstream     error timeout invalid_header http_500 http_502 http_503 http_504; proxy_buffers           8 32k; proxy_buffer_size       64k; proxy_http_version      1.1; proxy_set_header        Host            $host; proxy_set_header        X-Real-IP       $remote_addr; proxy_set_header        X-Forwarded-For $proxy_add_x_forwarded_for; proxy_redirect          off; proxy_set_header        Upgrade         $http_upgrade; proxy_set_header        Connection      $connection_upgrade;  server {         listen          80;         server_name     ~first.*\.second\.com;          location / {                 resolver                172.16.1.1 172.16.1.2;                 proxy_pass              http://$host;                 proxy_read_timeout      7200s;         } }  server {         listen          443;         server_name     ~first.*\.second\.com;         ssl             on;          location / {                 resolver                172.16.1.1 172.16.1.2;                 proxy_pass              https://$host;                 proxy_read_timeout      7200s;         } }   The top-level conf file looks like this:  user nginx; worker_processes 8; pid /var/run/nginx.pid;  events {         worker_connections 4096;         # multi_accept on; }  http {      ##     # Basic Settings     ##      sendfile on;     tcp_nopush on;     tcp_nodelay on;     keepalive_timeout 65;     types_hash_max_size 2048;     # server_tokens off;      # server_names_hash_bucket_size 64;     # server_name_in_redirect off;      include /etc/nginx/mime.types;     default_type application/octet-stream;      ##     # Logging Settings     ##      access_log /var/log/nginx/access.log;     error_log /var/log/nginx/error.log;      ##     # Gzip Settings     ##      gzip on;     gzip_disable "msie6";      # gzip_vary on;     # gzip_proxied any;     # gzip_comp_level 6;     # gzip_buffers 16 8k;     # gzip_http_version 1.1;     # gzip_types text/plain text/css application/json application/x-javascript text/xml application/xml application/xml+rss text/javascript;      ##     # nginx-naxsi config     ##     # Uncomment it if you installed nginx-naxsi     ##      #include /etc/nginx/naxsi_core.rules;      ##     # nginx-passenger config     ##     # Uncomment it if you installed nginx-passenger     ##      #passenger_root /usr;     #passenger_ruby /usr/bin/ruby;      ##      include /etc/nginx/conf.d/*.conf;     #include /etc/nginx/sites-enabled/*;   }  what here is causing this? $a,b,c$ are real numbers $>0$. If $a+b+c=1$, show that $(a+\frac{1}{a})^2+(b+\frac{1}{b})^2+(c+\frac{1}{c})^2\ge\frac{100}{3}$ How do I avoid the "S to Skip" message on boot? Do you know a good way to compare PDF files side-by-side and show the modifications between the two?  I'm looking for Windows software to accomplish this. It would be great if you can post both free and not-free products. What is the smallest positive integer $n$, such that there exist positive integers $a$ and $b$, with $b$ obtained from $a$ by a rearrangement of its digits, so that   $a – b = 11\dots 1$ (The number of '$1$'s equal to $n$)?   Is there any example that can satisfy the equation? How do you reset timers? I tried out the clock setting cheat on a throw-away vault but somehow it also messed up my real one so now they both have 3000+ hours on radio station and 70d+ on returning wastelanders..   Any way to reset all the timers to default?  For some reason my phone refuses to change time at all anymore.. How do I check OS with a preprocessor directive? When is a review task "completed" or "invalidated"? To make Latin-letter variables bold I can use e.g. \mathbf{a}, but while putting Greek letters or symbols such as \nabla inside \mathbf doesn't cause any errors or warnings, it also doesn't do anything else.  What is the best way to make bold math symbols, in particular Greek letters and \nabla? Which manga chapter correspond to the end of episode 12 of season 2 of the anime? Rough Smart objects problem in Photoshop I am using Adobe CC(2015) latest version and the behaviour of the smart objects isn't normal. When I paste a vector from Illustrator to Photoshop, the vector looks low quality or rough.   ORIGINAL  IN PHOTOSHOP   However, if I click over the smart object layer(in Photoshop)and this is opened in Illustrator, the vectors looks as the original one.   VECTOR OPENED IN ILLUSTRATOR FROM PHOTOSHOP SMART OBJECT   How can I solve this? I am trying to understand model vs. interpretation of sentences vs. theories, with respect to a propositional calculus.  Is a model of a wff just a sentence that is true under some interpretation?   Or a set of wffs that are all true under some interpretation?   Or when an interpretation satisfies a wff (or set of wffs)?   Does it make sense to say that an interpretation is a model if all valid propositions are true? Or an interpretation being a model iff all theorems from the theory are still valid under the interpretation? Converting string into datetime variable access outside of if statement In Drupal 8 you can load an entity with:  $node = \Drupal::entityManager()-&gt;getStorage('node')-&gt;load(123);   This looks for the entity definitions and finds that node is defined by Drupal\node\Entity\Node - so (I guess) Drupal\node\NodeStorage will instantiate a new Drupal\node\Entity\Node instance.  What I'd like to achieve is subclassing Drupal\node\Entity\Node and being able to instantiate this subclass when it's appropriate. For example if I have a node bundle article, there would be a class:  namespace Drupal\my_module\Entity\Article; class Article extends Drupal\node\Entity\Node { }   And I'd call:  $node = \Drupal::entityManager()-&gt;getStorage('node_article')-&gt;load(123);   And the return would be my Article subclass.  I can achieve this by creating a new entity type and wiring it back to another existing entity definitions, for example the node-article example would be this class:  namespace Drupal\my_module\Entity; use Drupal\node\Entity\Node; /**  * @ContentEntityType(  *   id = "node_article",  *   label = @Translation("Content"),  *   bundle_label = @Translation("Content type"),  *   handlers = {  *     "storage" = "Drupal\node\NodeStorage",  *     "storage_schema" = "Drupal\node\NodeStorageSchema",  *     "view_builder" = "Drupal\node\NodeViewBuilder",  *     "access" = "Drupal\node\NodeAccessControlHandler",  *     "views_data" = "Drupal\node\NodeViewsData",  *     "form" = {  *       "default" = "Drupal\node\NodeForm",  *       "delete" = "Drupal\node\Form\NodeDeleteForm",  *       "edit" = "Drupal\node\NodeForm"  *     },  *     "route_provider" = {  *       "html" = "Drupal\node\Entity\NodeRouteProvider",  *     },  *     "list_builder" = "Drupal\node\NodeListBuilder",  *     "translation" = "Drupal\node\NodeTranslationHandler"  *   },  *   base_table = "node",  *   data_table = "node_field_data",  *   revision_table = "node_revision",  *   revision_data_table = "node_field_revision",  *   translatable = TRUE,  *   list_cache_contexts = { "user.node_grants:view" },  *   entity_keys = {  *     "id" = "nid",  *     "revision" = "vid",  *     "bundle" = "type",  *     "label" = "title",  *     "langcode" = "langcode",  *     "uuid" = "uuid",  *     "status" = "status",  *     "uid" = "uid",  *   },  *   bundle_entity_type = "node_type",  *   field_ui_base_route = "entity.node_type.edit_form",  *   common_reference_target = TRUE,  *   permission_granularity = "bundle",  *   links = {  *     "canonical" = "/node/{node}",  *     "delete-form" = "/node/{node}/delete",  *     "edit-form" = "/node/{node}/edit",  *     "version-history" = "/node/{node}/revisions",  *     "revision" = "/node/{node}/revisions/{node_revision}/view",  *   }  * )  */ class Article extends Node { }  // Results my Article sub type. $node = \Drupal::entityManager()-&gt;getStorage('node_article')-&gt;load(123);   This works fine (as much as I can see); however, it smells. It adds a new entity type, which is not true, and might cause other troubles in the future.  How do I define a subclass for an entity bundle so that loading the entity would return an object of that class? Is there a contraction for "I was"?    There are contractions for "I am" (I'm), "I will" (I'll), "I have" (I've), "I would" (I'd), and yet the simple past tense seems conspicuously missing. Why is that? Does that reasoning apply to "I did" and "I had" as well? Import PLY doesn't display vertex colors I'm trying to import a .PLY mesh in Blender. I've troubles with colors: the mesh appear in a grayscale palette. Other softwares open it correctly (eg. MeshLab).  This is the header of the PLY file:  ply format binary_big_endian 1.0 comment created by Geomagic Sculpt Exporter. element vertex 182488 property float x property float y property float z property uchar red property uchar green property uchar blue element face 364980 property list uchar int vertex_indices end_header   Any tips? Thanks!