The first operation, addition, is abelian and so is multiplication. However, the next operation, exponentiation is not! Why is this? I understand that $2^3$ and $3^2$ are not equal but why do we suddenly lose this property after adding one other operation? Is there a property that we lose for tetration? Or Pentration (the next operation)? If so what property is lost at the nth operation?   Edit: I there anything lost when going from addition to multiplication?
We know that the addition and multiplication operators are both commutative, and the exponentiation operator is not.  My question is why.  As background there are plenty of mathematical schemes that can be used to define these operators. One of these is   where $H_0(a,b) = b+1$   (successor op) $H_1(a,b) = a+b$   (addition op) $H_2(a,b) = ab $   (multiplication op) $H_3(a,b) = a^b$   (exponentiation op) $H_4(a,b) = a\uparrow \uparrow b$   (tetration op: $a^{(a^{(...a)})}$ nested $b$ times ) etc.    Here it is not obvious to me why $H_1(a,b)=H_1(b,a)$ and $H_2(a,b)=H_2(b,a)$ but not $H_3(a,b)=H_3(b,a)$  Can anyone explain why this symmetry breaks, in a reasonably intuitive fashion?  Thanks.
Prevent Windows 10 installer from using the preinstalled serial key without disabling UEFI My laptop (a Lenovo Flex 2 15) came preinstalled with Windows 8 Standard (with the serial key preinstalled into the BIOS), and I've since installed an SSD (a Samsung Evo 840 120GB) &amp; upgraded to Windows 10.  When the laptop still had its hard drive, I upgraded it to Professional so I could use the resources provided by the domain on my homeserver; because of this, I can only use the Professional editions of Windows (Home can't join domains).  I bought Windows 10 Professional &amp; completed a fresh-install (complete reinstallation) of Windows, only that it installed Windows 10 Home instead of Professional. I'm guessing this is due to the Windows 8 Standard Edition serial key preinstalled in the BIOS.  I then downloaded an All-In-One ISO image of Windows 10 from MSDN (of which I have limited access to), transferred the installer onto a USB stick, &amp; reinstalled (a clean install again). However it's still installing Windows 10 Home.  If I try to install using the AIO image from within Windows, the installer does not present me with a list of options for the edition I want to install; instead, it assumes that I want to install Home Edition:    I am able to get Windows 10 Professional to install if I disable UEFI &amp; use Legacy instead, but I feel that this is stepping around the issue rather than resolving it.  Is it possible to have the Windows 10 installer ignore any serial keys preinstalled in the BIOS/UEFI? If so, how? If not, is using Legacy mode really the only way around this? Is it possible to tell a Valyrian steel sword from a very well crafted steel sword, by sight alone I'm sure in Westeros there are some incredibly well crafted steel swords, but could someone instantly spot a Valyrian steel sword over one of these?  Does Valyrian steel have any visual indications that it is Valyrian steel and not just a very well crafted steel sword?  I'm wondering, in canon, has anyone seen a sword and instantly recognised it as Valyrian steel.  For the purpose of this question, no one is allowed pick up or use the sword, so they cant feel it's weight, balance, sharpness, etc... I have a question that says "Show that there is a bounded sequence $x_n$ which is not convergent but has the property that $x_n - x_{n+1} \to 0$ as $n \to 0$.  What does this mean? Do I need to come up with an example or does the problem actually want me to prove such proposition?  By the way, I see that this sequence looks like Cauchy because of $x_n - x_{n+1} \to 0$ as $n \to 0$, but it is obviously not. Why is fundamental group insufficient to classify manifolds? Algorithms for automatic model selection I would like to implement an algorithm for automatic model selection.  I am thinking of doing stepwise regression but anything will do (it has to be based on linear regressions though).   My problem is that I am unable to find a methodology, or an open source implementation (I am woking in java). The methodology I have in mind would be something like:   calculate the correlation matrix of all the factors pick the factors that have a low correlation to each other remove the factors that have a low t-stat add other factors (still based on the low correlation factor found in 2.). reiterate several times until some criterion (e.g AIC) is over a certain threshold or cannot or we can't find a larger value.   I realize there is an R implementation for this (stepAIC), but I find the code quite hard to understand. Also I have not been able to find articles describing the stepwise regression. Plot showing discontinuity where it shouldn't How to make a figure span on two columns in a scientific paper? More than one optional argument for newcommand The \newcommand macro allows to use an optional argument for the first parameter #1 with:  \newcommand{\mycommand}[3][defaultfor1]{blah blah blah}   Is it possible to have more than one option with \newcommand? Is there a proof of existence of time? It seems to me that there is no such thing as time. There  is only movement in the universe and we compare our own movement to a different object to have a sense of time. It can be a clock or a atomic vibration.  Does this view of time work within the current framework of phsics?  Do physicists have an explanation/proof about time's existence? LSTM : shape of tensors? I'm trying to understand LSTM, using for instance   I get the overall idea, I guess. But I'm not quite sure I get the maths.  I'll set a very simple problem : I have a sequence of numbers and want to predict next number.   So x_t is of shape (1), and as I understand, h_t will be the prediction, so it should also be of shape (1). (I'm just here ignoring batch size)  Now, the equation producing h_t, using * operation should then have two operands of the same size as the result; that is, C_t and o_t should also be of shape (1).  Following on that idea, equation producing C_t forces also shape of (1) for f_t, i_t and ~C_t.  So... everything reduced to scalar real numbers in that case ?  what am I getting wrong ? because this would just not be able to learn much, would it ? When using Alt+Tab, I can't seem to get back to Skyrim. When clicking its icon or Alt+Tab to get back, I just see an empty window. Windowed mode doesn't allow me to use it in a Windowed (Fullscreen) manner as seen with other games, so that doesn't seem to be an option either. Is there any way to fix this behavior? What is the status icon that looks like a house? There is a strange icon that shows up in my status bar only when I'm at a certain restaurant. It looks like a cross between a house and wifi.   I suspect it might be a free wifi hotspot or something related to a nearby phone store, but I can't find any information about it in manual for my phone.  See the following screenshot: Not enough space in /var/cache/apt/archives/ How to shrink Windows 7 boot partition with unmovable files I have just bought an HP laptop with Windows&nbsp;7 (64 bit). It has a 500&nbsp;GB HDD with three partitions: a small hidden system partition, a 12&nbsp;GiB HP recovery partition, and a 450&nbsp;GiB C: boot partition.  I would like to split this large C: partition into two partitions, leaving only 100&nbsp;GiB for the system, and giving the rest to a new data partition.  Although the Windows built-in Disk Management utility has an option to shrink the bootable partition, it only allows me to shrink it roughly by half, even though only 20&nbsp;GiB on the partition is used. As far as I understand, system unmovable files lie in the middle of the partition, preventing Disk Management utility to do what I want.    And since new HP laptops don't come with OS installation disks (they only allow you to create recovery disks yourself), I can't just repartition HDD and then reinstall OS.  So, is there a way to shrink the C: bootable partition and preserve Windows&nbsp;7 working?  P.S.: I have tried to use the third-party  utility, and after shrinking the partition, Windows&nbsp;7 stopped booting, with a . System recovery didn't work, and I had to do factory recover. Since this is a long process, I would like to avoid doing it again :) So, please, suggest only proven solutions. Getting "'file' is different from the same file on the system" when I attempt to update through synaptic I haven't found a similar question on here, though I suspect the question may be rather well-covered.   I want to find a conformal map from the vertical strip $\{z:-1&lt;Re(z)&lt;1\}$ onto the unit disc.  Under the exponential map the region is taken to the annulus with radii $e$ and $e^{-1}$, but I'm not sure how useful this will be.  Can anyone advise on what the conformal map may be?  Thanks. Is something wrong with StackOverflow's spam filter? Recently there have been huge numbers of spam topics related to football streaming. They have all been spam flagged and deleted but there are more getting posted all the time, I just flagged another 6 or so a few minutes ago. This has been going on for around 3 days now with these topics getting constantly posted over and over again in the  tag. How can I remove gpg key that I added using apt-key add -? I don't need the key in my server's keyring anymore. Is it possible to remove it? I added the key using this command:   curl http://repo.varnish-cache.org/debian/GPG-key.txt | apt-key add -   Thanks for helping I'm too spoiled by Windows utilities that take a digital snapshot of your entire drive, which you can then restore from in the event of a drive crash.  (e.g. like Time Machine for Mac OS X).  Is there a similar way of doing this in Linux? To prove $\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$ for some $M>0$