How to find remainder when $ 975^{40153}$ is divided by $14$?
Modular exponentiation by hand ($a^b\bmod c$)
How to compute the topological space of fibered product of schemes? Trouble downloading packages list due to a "Hash sum mismatch" error When I check for updates, I get a "Failed To Download Repository Information" error.   This is what comes up under details:   W: Failed to fetch gzip:/var/lib/apt/lists/partial/us.archive.ubuntu.com_ubuntu_dists_natty_main_source_Sources  Hash Sum mismatch,  E: Some index files failed to download. They have been ignored, or old ones used instead. This option seems to have gone away after upgrading from Ubuntu 16.04. I tried installing nautilus-open-terminal but it is unable to locate the package for 18.04. Also trying to install nautilus-actions gives:  Package nautilus-actions is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source     How to fix this? Many spells have "At higher levels" that provides for more targets or more damage.  For example Magic Missile can be cast as a second level spell and shoot 4 missiles instead of it's 3 as a first level spell.  Can a spell without that verbiage, such as Firestorm be cast in higher level slots for the same effect as the lower level spell? Photoelectric Effect - Dependence of current on frequency We all know that the amount of current flowing b/w the plates is independent of the frequency. If I were to ask why doesn't it depend then you'd probably say that it depends on the number of electrons rather than the frequency or you might show me a current vs frequency graph.  But here is my theoretical argument. Electric current is defined as the amount of charge flowing through a point/space/wire per unit time. Let I be current due to photo electrons ejected by photons of some frequency. Now if I would increase the frequency, the kinetic energy of the ejected electron would increases, therefore it will move faster and therefore contribute more current (charge will through the same point many times per unit time) . Thereby, giving a net current greater than I.  What is wrong with my argument?  After thinking a lot, I came up with a possible reason. However fast the electrons may move, but when they hit the metal plate, they would lose their velocity and start moving in drift velocity. Now there is a problem with this argument. If this were to be true then I should expect to see a big chunk of extra electrons getting stuck at the anode.  If my reasoning was correct, then why can't I see a big charge buildup at the anode. Let's have 2 numbers, N and K, where K divides N. The number of K-combinations from a given set S of N elements is a well known formula. Let's concatenate N/K groups (resulting in N elements) such as the resulting set is N. How many possibilities are there, i.e. what's the formula?  For instance:    N=4, K=2, C(4, 2) = 6 {1,2},{1,3},{1,4},{2,3},{2,4},{3,4}   The 3 possibilities are:     {1,2},{3,4}   {1,3},{2,4}   {1,4},{2,3}   I generated these combinations and I think the number goes like this:  (4, 2): 3 (6, 3): 10 (6, 2): 15 (10, 5): 126 (9, 3): 280 (10, 2): 945 (14, 7): 1716 (12, 4): 5775 (15, 5): 126126 (15, 3): 1401400   Apparently, the result always divides with (N-1). linux find command to move files How do I prevent directories mounted with 'bind' from appearing on 'Devices' on nautilus? I have these lines in the fstab   # binds /media/DataNtfs/Music           /home/can/Music                  none    rw,bind  /media/DataNtfs/Pictures        /home/can/Pictures               none    rw,bind  /media/DataNtfs/Downloads       /home/can/Downloads              none    rw,bind  /media/DataNtfs/Documents       /home/can/Documents              none    rw,bind  /media/DataNtfs/Backups         /home/can/Backups                none    rw,bind  /media/DataNtfs/Notes           /home/can/Notes                  none    rw,bind  /media/DataNtfs/Other           /home/can/Other                  none    rw,bind  /media/DataNtfs/Packages        /home/can/Packages               none    rw,bind  /media/DataNtfs/Photos          /home/can/Photos                 none    rw,bind  /media/DataNtfs/Videos          /home/can/Videos                 none    rw,bind  /media/DataNtfs/WorkSpace       /home/can/WorkSpace              none    rw,bind If $G$ has no proper subgroup, prove that $G$ is cyclic of order $p$, where $p$ is a prime number.     I know that since $G $is a group with no proper subgroups, $g \in G$ is not just the identity. I don't know where to go from there. Clustering on the output of t-SNE I've got an application where it'd be handy to cluster a noisy dataset before looking for subgroup effects within the clusters.  I first looked at PCA, but it takes ~30 components to get to 90% of the variability, so clustering on just a couple of PC's will throw away a lot of information.    I then tried t-SNE (for the first time), which gives me an odd shape in two dimensions that is very amenable to clustering via k-means.  What's more, running a random forest on the data with the cluster assignment as the outcome shows that the clusters have a fairly sensible interpretation given the context of the problem, in terms of the variables that make up the raw data.  But if I'm going to report on these clusters, how do I describe them?  K-means clusters on principal components reveal individuals who are nearby to one another in terms of the derived variables that comprise X% of the variance in the dataset.  What equivalent statement can be made about t-SNE clusters?  Perhaps something to the effect of:     t-SNE reveals approximate contiguity in an underlying high-dimensional manifold, so clusters on the low-dimensional representation of the high-dimensional space maximize the "likelihood" that contiguous individuals will not be in the same cluster   Can anyone propose a better blurb than that? I've seen this Kuratowski definition for ordered pairs, but can't fathom why it implies an order to $x$ and $y$  $(x,y):=\{\{x\}, \{x,y\}\}$  As I understand sets, $\{\{x\}, \{x,y\}\}$ is also $\{\{x,y\}, \{x\}\}$. Only when I think about the Axiom of Union does $\{\{x\}, \{x,y\}\}$ "collapse down" to $S = \{x, y\}$, but that doesn't give me much either. All I can see is some as yet hidden message in the set saying "I am the set $\{x,y\}$ and my order of $x$ first is indicated by having $\{x\}$ along for the ride." Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ Possible Duplicate:          On Wikipedia, it says that:     Matrix representation of complex numbers   Complex numbers $z=a+ib$ can also be represented by $2\times2$ matrices that have the following form: $$\pmatrix{a&amp;-b\\b&amp;a}$$   I don't understand why they can be represented by these matrices or where these matrices come from. Calling Python in PHP I have a Python script I recently wrote that I call using the command line with some options. I now want a very thin web interface to call this script locally on my Mac.  I don't want to go through the minor trouble of installing mod_python or mod_wsgi on my Mac, so I was just going to do a system() or popen() from PHP to call the Python script.  Any better ideas? Thanks in advance! Is a black hole singularity a single point? General relativity is expressed in terms of differential geometry, which allows you to do interesting things with the coordinates: multiple coordinates may refer to a single point, eg. the  has whole lines at the top and the bottom that correspond to the North and South poles, respectively; or a single coordinate may refer to multiple points, for example by using inverse geometry the origin refers to all the points infinitely far away.  So, is the singularity just a single point in the curved spacetime, or can it be a more extended object, described by a single coordinate? How can magnets be used to pick up pieces of metal when the force from a magnetic field does no work? I learned that the force from a magnetic field does no work. However I was wondering how magnets can be used to pick up pieces of metal like small paperclips and stuff. I also was wondering how magnets can stick to pieces of metal like a refrigerator? How to get things off your head in Minecraft Mo' Creatures mod I installed the Mo'Creatures mod into Minecraft, then made a fish bowl. After catching a fish I went to put it on top of a bookshelf in my room, but instead of placing it, the bowl just dropped at my feet. When I tried to pick it up my character put it on his head. Now I'm stuck with a fish bowl on my head and I can't find any mention anywhere of how to get it off. There are a number of other animals from the mod that you can put on your head for some effect, but no mention of how to remove them pretty much anywhere. I'm getting the following error when trying to update a taxonomy field value via REST api:  {   "error": {     "code": "-1, Microsoft.SharePoint.Client.ClientServiceException",     "message": {       "lang": "en-US",       "value": "Cannot deserialize data for type Microsoft.SharePoint.Taxonomy.TaxonomyFieldValueCollection."     }   } }   The request:  https://server/_api/web/GetFileByServerRelativeUrl('/Shared Documents/file.txt')/ListItemAllFields  {   "__metadata": {     "type": "SP.Data.Shared_x0020_DocumentsItem"   },   "Title": "some title",   "Project": {     "results": [       {         "Label": "Term1",         "TermGuid": "897bdf7f-08d6-44b9-a71a-ddc4cf131664",         "WssId": 5       }     ]   } } past simple or present perfect Please have a look at this sentence:     We haven't gotten a lot of Finnish groups that we liked here at Bananas but I'm pretty happy with this four tracker from Helsinki, apparently recorded in the same place where the Ramones played in the 70's.     The first verb is in present perfect (up to now) and the second is in simple past tense. I don't understand the simple past for the second verb.  Why "like" is not in its present perfect form as the first verb is? What is the fundamental group of $(\mathbb{C} \setminus {\{0\}})~/~\{e,a\}$, where $e$ is the identity homeomorphism and $az = -\bar{z}$?  Clearly this is homeomorphic to the half cylinder , which is homotopy equivalent to the half circle.  But what is the fundamental group of the half circle? I've had a very strange Finer error caused by recent macOS High Sierra betas that affected Macs running macOS 10.13 as well as macOS 10.12 Sierra. This error causes an entire Finder window to show visual artifacts when opening certain folders on either an APFS or HFS+ volume.  After moving the cursor or the affected Finder window, the window may look something like this;    Attempting to use Disk Utility to repair the affected volume does not resolve this issue, nor does restarting the affected Mac (even in Safe Mode).  Without formatting this disk, how can I prevent Finder from exhibiting this graphical error when opening particular folders?