In Cohen forcing $\mathrm{Fn}(\kappa\times\omega, 2)$ prove $(\lambda^\vartheta)^{M[G]} = ((\max\{\lambda, \kappa\})^\vartheta)^M$
Powers of infinite cardinals in the generic extension
I'm bugged by the following that's summarized on p. 109 of .     False theorem: All horses are the same color.      Proof by induction:      $\fbox{$P(n)$ is the statement: In every set of horses of size $n$, all $n$ horses are the same color.}$   $\fbox{Base Case or $P(1)$:}$ One horse is the same color as itself. This is true by inspection.   $\fbox{Induction Step:}$ Assume $P(k)$ for some $k \geq 1$.   $\fbox{Proof of $P(k + 1) :$}$   Since $\{H_1, H_2, ... , H_n\}$ is a set of $n$ horses, the induction hypothesis applies to this set. Thus, all the horses in this set are the same color.   Since $\{H_2, H_3, ... , H_{n+1}\}$ is also a set of $n$ horses, the induction step likewise holds for this set. Thus, all the horses in this set are the same color too.         Therefore, all $n +1$ horses in $\{H_1, H_2, H_3, ... , H_n , H_{n+1}\}$ are the same color. QED.   The issue the instructor was trying to point out is clearly valid. For the case $n = 1$, there is only ${H_1}$. So this case says nothing about possible overlapping elements of each set of $(n + 1)$, for instance $H_2$ in the above proof.    But it was proposed in the class discussion that this was the only problem. Had you been able to prove $P(2)$ true, then a proof of the above format would have been fine.    My interpretation is that yes, you could prove all horses are the same color, if you can prove that any set of two horses will be the same color. But this format would not work.  Why not? The problem I see is that the above proof is for the existence of at least one particular pair of sets of horses of sizes $n$ and $n + 1$, such that in each set, all horses are the same color.  Particularly when the set of size $n$ is a subset of the set of size $n + 1$.  In order to prove the induction step, don't you need to prove that sets of sizes $n$ and $n + 1$, do not necessarily contain the same, overlapping elements?  You could prove that any horse can be added to a set of 2 horses.  Take the last two, and they must be the same color, and so on.  Wwhatever color the first two happen to be, all other horses must thus be the same color.    Am I misinterpreting the example, or am I making a logical error? Thanks in advance. Avoiding Mixed DML (setup/non-setup) Approach I need an approach to allow delegated admins (i.e., non-Administators) to add members to Public Groups and Collaboration Groups (the GroupMember and CollaborationGroupMember objects).  I added a custom object (UserGroupRequest__c) to capture which Users should be added/removed to which Groups.    I've also built a batchable class to process the requests.  However, GroupMember is a Setup object, and CollaborationGroupMember is not.  So this doesn't work if I combine the operations in a single batchable class.  Should I instead be using 2 batch classes, calling one from the finish method of the other?    Any recommendations on how best to proceed?  Am I way off base?    Would a trigger on the Request object perhaps be better? I'm guessing that there may be a permission problem if this is attempted via a trigger as the delegated admins won't be able to add users to public Groups.  Also, I'm guessing the trigger approach would run into the same mixed DML issue - on Insert of the Custom object, I would need to Insert/Delete from GroupMember and from CollaborationGroupMember, and then perform a 2nd update on the Custom object to indicate that processing was completed... How to override Drupal.ajax.prototype.beforeSend? I need to add my own little if's and checks in 3 functions inside   misc/ajax.js (   Drupal.ajax.prototype.beforeSend,   Drupal.ajax.prototype.success,   Drupal.ajax.prototype.error )   How can I override them in my module? I can't find a manual and it's not in Examples for Developers. Sorry if it seems a dumb question, but I'm a php developer - I can work with js, but I'm not so fluent with it.   Can I simply write it in my module's js file? If 2 or 3 modules needs to modify them, how to avoid conflicts? I have a sword that currently rotates around a pivot point thats off of the actual object. Im new to blender and im not sure how i would go about changing it so that the sword rotates around the handle. Any help would be appreciated! Where can I ask questions regarding Electronics? I am currently doing some electronics and microcontroller programming, and I have a couple of questions to ask the people here.    Most of the questions are regarding wiring on breadboards and components.    On which of the websites (Stack Overflow, Server Fault, or Super User) should I ask these particular questions? Sci-fi series: humans are immune to telepathy, become soldiers for an alien group fighting against invertebrate telepaths I have an armature set up, but the bones aren't moving the mesh properly  The mesh and armature:    The vertex group of the bone I am going to move:    As you can see, the circled region doesn't belong to the vertex group. But, when I move the bone look what happens:    That part of the mesh has moved with the bone.. Why? What defines the maximum size for a command single argument? I was under the impression that the maximum length of a single argument was not the problem here so much as the total size of the overall argument array plus the size of the environment, which is limited to ARG_MAX. Thus I thought that something like the following would succeed:  env_size=$(cat /proc/$$/environ | wc -c) (( arg_size = $(getconf ARG_MAX) - $env_size - 100 )) /bin/echo $(tr -dc [:alnum:] &lt;/dev/urandom | head -c $arg_size) &gt;/dev/null   With the - 100 being more than enough to account for the difference between the size of the environment in the shell and the echo process. Instead I got the error:  bash: /bin/echo: Argument list too long   After playing around for a while, I found that the maximum  was a full hex order of magnitude smaller:  /bin/echo \   $(tr -dc [:alnum:] &lt;/dev/urandom | head -c $(($(getconf ARG_MAX)/16-1))) \   &gt;/dev/null   When the minus one is removed, the error returns. Seemingly the maximum for a single argument is actually ARG_MAX/16 and the -1 accounts for the null byte placed at the end of the string in the argument array.  Another issue is that when the argument is repeated, the total size of the argument array can be closer to ARG_MAX, but still not quite there:  args=( $(tr -dc [:alnum:] &lt;/dev/urandom | head -c $(($(getconf ARG_MAX)/16-1))) ) for x in {1..14}; do   args+=( ${args[0]} ) done  /bin/echo "${args[@]}" "${args[0]:6534}" &gt;/dev/null   Using "${args[0]:6533}" here makes the last argument 1 byte longer and gives the Argument list too long error. This difference is unlikely to be accounted for by the size of the environment given:  $ cat /proc/$$/environ | wc -c 1045   Questions:   Is this correct behaviour, or is there a bug somewhere? If not, is this behaviour documented anywhere? Is there another parameter which defines the maximum for a single argument? Is this behaviour limited to Linux (or even particular versions of such)? 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