A vector space is defined as a set $V$ together with a field $F$, such that $(V,+)$ is an abelian group, and $V$ has scalar multiplication that is compatible with $+$. But why is it that in this definition, we require $F$ to be a field, and not any commutative ring?  To illustrate, consider $V=\mathbb Z^n$, the set of all $n$ dimensional vectors with integer coordinates. Why doesn't it make sense to extend the definition of vector spaces such that we can say $V$ is a vector space over $\mathbb Z$? All the familiar properties remain: we certainly have closure under scalar multiplication, and the other axioms remain. In particular, the axioms say nothing about needing to have inverses, which is the only thing fields add on top of being a commutative ring. So why wouldn't ordinary commutative rings suffice?
I was reviewing the  recently, and it occurred to me that one could allow for only scalar multiplication by the integers and still satisfy all of the requirements of a vector space.   Take for example the set of all ordered pairs of integers. Allow for scalar multiplication over the integers and componentwise vector addition as usual. It seems to me that this is a perfectly well-defined vector space.  The integers do not form a , which begs the question: Is there any reason that the "field" over which a vector space is defined must be a mathematical Field? If so, what is wrong with the vector field I attempted to define above? If not, what are the requirements for the scalars? (For instance, do they have to be a  - Abelian or otherwise?)
Change Folder Permissions I started off with changing to the folder I want to change permissions for, and that is the opt folder.  $ cd /opt/ test@testVM:/opt$   So I tried changing the permissions for this folder now using:  sudo chmod 775   And that hasn't worked. It showed this message:  Try 'chmod --help' for more information.   There is something I am forgetting or leaving out.  Please can you show me what I am doing wrong?  Thanks in advance. I have to find the limit of this sum: $$\sum_{r=1}^{\infty}\cot ^{-1}(r^2+\frac{3}{4})$$  I tried using sandwich theorem , observing:  $$\cot ^{-1}(r^3)\leq\cot ^{-1}(r^2+\frac{3}{4})\leq\cot ^{-1}(r^2)$$  Now when I was calculating the limit of left hand expression, I convert it to $\tan^{-1}$, by using:  $$\tan^{-1}\frac{1}{x} = \cot^{-1}x$$  but couldn't sum up the terms of arctan series.     How can I proceed?   Is there any better way ? Is it possible to bake dupliverts into one mesh for the BGE? (I have a large ground landscape that I use as the parent to duplicate trees, and the BGE doesn't support most modifiers.) An idiom for deriving pleasure from another's suffering I believe it is what the Germans call "". English itself has no such equivalent word. (Although it has been adopted as a .)  Does an idiom exist that describes it? There's a way to safely try KDE on top of an Ubuntu installation, keeping the possibility to revert the changes in case I dislike how it runs? Device : Moto X Play ( Rooted, Stock, Android 6.0.1, msm8916 Board, XT1562)  Why do I want to limit charging ?  ( Let's leave out whether my reasons are valid or not and accept that I want to, if you disagree with my reasons)  To improve the longevity of battery , I wish to limit charging when it reaches 90%, as explained here    Is it possible?  Yes, at least on my previous device Honor 6, as brought out here , with two different ways of doing it  I have tried  unsuccessfully to identify the file responsible in which changing values would limit charging  I am looking for the system file responsible for stopping charging, with it's true path (without symlinks)  (as I saw on on my previous device, locating the path of file was a big challenge after identifying it) Can Ubuntu 32-bit access 8GB RAM? I need to use a large software package (Xojo) that requires the 32-bit architecture to run. Try as I might, I can't seem to install the 32-bit libraries on my 64-bit installation of 15.04.  I don't have a problem running a 32-bit version of Ubuntu (I'm only using this computer for Xojo) but the computer has 8GB of RAM and I'd like to be able to use it all. Is there a way to utilise all 8GB of RAM with a 32-bit Ubuntu install? Does Ubuntu do this out of the box? I've read the , and I don't get it. I am really interested, but I've read the explanation there a couple of times and I still don't get it how the Duff's device works.  What would a more detailed explanation be? Taskbar > File Explorer > Context Menu failing me So I have the latest version of Windows 10, and this is a new PC.  File Explorer is in the taskbar.  When I open the context menu (right-click) on it, I get the list of folders as seen in the picture.  When I right click on any of them, I get the blank black box you see in the picture.  If I click on any of the folders, such as Downloads or Update.  Nothing happens.  They don't open.  The only way to use it is to click on the actual File Explorer icon and then navigate from there.  I've tried uninstalling things (not that I have much installed since it's a fresh install) and I've tried scans, etc.  This only happens with File Explorer.  Everything else works fine.  I can't figure it out. How can I install the Google Apps Package (Play Store, ...) on my Android device? How to use a decimal range() step value? Is there a way to step between 0 and 1 by 0.1?   I thought I could do it like the following, but it failed:  for i in range(0, 1, 0.1):     print i   Instead, it says that the step argument cannot be zero, which I did not expect. Apport-gpu-error-intel.py crash Objects, shadows and reflections not rendering in EEVEE How does the pull down resistor work in this circuit? What happens when a caster targets an object that looks like a creature with a spell that targets only creatures? There are many spells which explicitly target creatures:     Choose one creature within range   The target creature is normally supposed to make a saving throw in this case.  What happens when the caster chooses a creature, that is actually an object? For instance, a scarecrow, a mannequin, an illusion, or a corpse.  Examples:   An evil mage is hiding in the bushes. He casts an illusion of himself on the road, using the Silent Image spell. A PC sorcerer walks down the road, sees the illusion and casts Magic Missile on it. A cleric is standing near a pile of dead bodies. He has suspicions that a few corpses are actually undead creatures. He casts Sacred Flame on every corpse, see which if them succeed, hence, reveals the undead. A wizard casts Acid Splash on a mannequin, trying to damage it with acid.   Possible outcomes I can think of are:   The caster cannot even try to cast the spell, regardless of their information about what the target actually is. The caster tries to cast the spell, the casting fails, not expending the spell slot. The caster casts the spell, it expends the spell slot, then fails. The caster casts the spell, but it hits the original and not the illusion. The caster casts the spell, it flies to the target illusion but deals no damage to it (passes through it).   References in the rules and Sage advice  I was trying to investigate, but it didn't make things clearer:    has an answer, that Magic Missile won't hit. It says nothing about if it can be targeted at all, it leaves it up to the DM. Magic Missile automatically finds the original, even when the caster targets an illusion. At least . Now . This supports the counter-intuitive option 4.   on the other hand   , if there were no perceivable effects (let's say you can try to cast the Command spell on an illusion). That assumes that casters can waste their spells on illusion, which means they at least can target it. This supports options 3 and 5. How to compute   $$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}\ ?$$   My Working :   $$\prod^{\infty}_{n=2} \frac{n^3-1}{n^3+1}= 1 - \prod^{\infty}_{n=2}\frac{2}{n^3+1} =  1-0 = 1$$  Is it correct I have problems with getting an Asus PCE-N53 11n N600 PCI-E Adapter card to work on my desktop computer.  As far as I can tell no drivers are installed for the card. I know I can manually download the drivers directly from Asus, but I would rather not go that route.  If there are anyone that knows about any packages or other things I can do to make this work would be much appreciated.  Some systems details:  $ lsb_release -a No LSB modules are available. Distributor ID: Ubuntu Description:    Ubuntu 12.04.1 LTS Release:    12.04 Codename:   precise   $ sudo lshw -C network   *-network                       description: Ethernet interface        product: RTL8111/8168B PCI Express Gigabit Ethernet controller        vendor: Realtek Semiconductor Co., Ltd.        physical id: 0        bus info: pci@0000:03:00.0        logical name: eth0        version: 06        serial: d4:3d:7e:03:b9:1d        size: 100Mbit/s        capacity: 1Gbit/s        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress msix vpd bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd 1000bt 1000bt-fd autonegotiation        configuration: autonegotiation=on broadcast=yes driver=r8169 driverversion=2.3LK-NAPI duplex=full firmware=rtl8168e-3_0.0.4 03/27/12 ip=192.168.0.173 latency=0 link=yes multicast=yes port=MII speed=100Mbit/s        resources: irq:43 ioport:d000(size=256) memory:f2104000-f2104fff memory:f2100000-f2103fff  *-network UNCLAIMED        description: Network controller        product: Ralink corp.        vendor: Ralink corp.        physical id: 0        bus info: pci@0000:04:00.0        version: 00        width: 32 bits        clock: 33MHz        capabilities: pm msi pciexpress bus_master cap_list        configuration: latency=0        resources: memory:f7100000-f710ffff  $ lsmod Module                  Size  Used by nvidia              12319264  51  vesafb                 13844  1  snd_hda_codec_hdmi     32474  1  joydev                 17693  0  bnep                   18281  2  rfcomm                 47604  0  bluetooth             180104  10 bnep,rfcomm snd_hda_codec_realtek   224173  1  snd_seq_midi           13324  0  ppdev                  17113  0  snd_rawmidi            30748  1 snd_seq_midi usbhid                 47199  0  hid                    99559  1 usbhid nouveau               774641  0  parport_pc             32866  1  snd_hda_intel          33773  5  ttm                    76949  1 nouveau snd_hda_codec         127706  3 snd_hda_codec_hdmi,snd_hda_codec_realtek,snd_hda_intel drm_kms_helper         46978  1 nouveau drm                   242038  3 nouveau,ttm,drm_kms_helper snd_seq_midi_event     14899  1 snd_seq_midi snd_hwdep              13668  1 snd_hda_codec snd_seq                61896  2 snd_seq_midi,snd_seq_midi_event i2c_algo_bit           13423  1 nouveau mxm_wmi                12979  1 nouveau wmi                    19256  1 mxm_wmi mac_hid                13253  0  snd_pcm                97188  3 snd_hda_codec_hdmi,snd_hda_intel,snd_hda_codec psmouse                97362  0  video                  19596  1 nouveau snd_timer              29990  2 snd_seq,snd_pcm snd_seq_device         14540  3 snd_seq_midi,snd_rawmidi,snd_seq snd                    78855  20 snd_hda_codec_hdmi,snd_hda_codec_realtek,snd_rawmidi,snd_hda_intel,snd_hda_codec,snd_hwdep,snd_seq,snd_pcm,snd_timer,snd_seq_device serio_raw              13211  0  soundcore              15091  1 snd snd_page_alloc         18529  2 snd_hda_intel,snd_pcm mei                    41616  0  lp                     17799  0  parport                46562  3 ppdev,parport_pc,lp r8169                  62099  0 function a () {     return "foo"; }  a.b = function () {     return "bar"; }  function c () { }; c.prototype = a;  var d = new c(); d.b(); // returns "bar" d(); // throws exception, d is not a function   Is there some way for d to be a function, and yet still inherit properties from a? Is there a general formula for solving 4th degree equations (quartic)? There is a general formula for solving quadratic equations, namely the Quadratic Formula:  $$ x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a } $$  For third degree equations of the form $ax^3+bx^2+cx+d=0$, there is a set of three equations: one for each root. Is there a general formula for solving equations of the following form? $$ ax^4 + bx^3 + cx^2 + dx + e = 0 $$ How about for higher degrees? If not, why not? Is there any lower or upper limit on the energy of a photon? i.e. does the mathematical framework we currently use to study photons blow up when a photon surpasses a certain upper limit of energy? (or the same on the opposite side?)  My thoughts: If I let the energy of a photon tend to infinity, its wavelength would be tending to zero, and since it is thought that we cannot really distinguish things when they are on the scale of a Planck Length, would the photon have its maximum energy when its wavelength is equal to the Planck length? (Feel free to correct me, I feel I might not be.) and for the opposite end of the spectrum a photon with the least energy would have zero wavelength, implying no photon, which is a trivial case.