Complex matrices are similar over $R$ iff they and thier conjugates are similar over $C$.
Similar matrices and field extensions
Pluralizing abbreviations where the noun is not the last word For abbreviations - both acronyms and initialisms - where the last word in the abbreviation is the noun, I'm accustomed to adding an 's' with no apostrophe (e.g. ATMs), as described in the answer to this question:   But how do you pluralize abbreviations where the noun is not the last word? E.g. 'Point of Presence' (POP)  I can see how 'POPs' could read 'Point of Presences', when the intention is 'Points of Presence.' PreparedStatement IN clause alternatives? What are the best workarounds for using a SQL IN clause with instances of java.sql.PreparedStatement, which is not supported for multiple values due to SQL injection attack security issues: One ? placeholder represents one value, rather than a list of values.  Consider the following SQL statement:  SELECT my_column FROM my_table where search_column IN (?)   Using preparedStatement.setString( 1, "'A', 'B', 'C'" ); is essentially a non-working attempt at a workaround of the reasons for using ? in the first place.    What workarounds are available? Conjectures that have been disproved with extremely large counterexamples? I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.  I'm sure that everyone here is familiar with it; it describes an operation on a natural number – $n/2$ if it is even, $3n+1$ if it is odd.  The conjecture states that if this operation is repeated, all numbers will eventually wind up at $1$ (or rather, in an infinite loop of $1-4-2-1-4-2-1$).  I fired up Python and ran a quick test on this for all numbers up to $5.76 \times 10^{18}$ (using the powers of cloud computing and dynamic programming magic).  Which is millions of millions of millions. And all of them eventually ended up at $1$.  Surely I am close to testing every natural number? How many natural numbers could there be?  Surely not much more than millions of millions of millions. (I kid.)  I explained this to my friend, who told me, "Why would numbers suddenly get different at a certain point?  Wouldn't they all be expected to behave the same?"  To which I said, "No, you are wrong!  In fact, I am sure there are many conjectures which have been disproved by counterexamples that are extremely large!"  And he said, "It is my conjecture that there are none! (and if any, they are rare)".  Please help me, smart math people.  Can you provide a counterexample to his conjecture?  Perhaps, more convincingly, several?  I've only managed to find one! (Polya's conjecture). One, out of the many thousands (I presume) of conjectures. It's also one that is hard to explain the finer points to the layman. Are there any more famous or accessible examples? The Punjab is a rich state.   Is it correct to use the before Punjab? Define a relation and find its equivalence classes. SOQL: Comparing two custom relationship fields in a nested query I'm going to mount my 47" LCD above my fireplace. I'm not looking for arguments against this decision. I'm looking for wiring solutions. The diagram below should help explain the situation.   _________________________________________________ |     |     |    | |   |   |   | |    |     |     | |     |     |    | |   |   |   | |    |     |     | |     |     |    | |   |   |   | |    |     |     | |     |     |    | | p |   | m | |    |     |     | |     |     |    | |   |   |   | |    |     |     | |     |     |    #################    |     |     | |     |     |    ##             ##    |     |     | |     |     |    ##             ##    |     |     | |    P|     |    ##             ##  m |     |P C  | |_____|_____|____##_____________##____|_____|_____|    | = stud # = fireplace brick P = existing power outlet C = existing cable outlet p = new power outlet m = new media outlet   Essentially, there is an existing power outlet on either side of the fireplace and a cable outlet on the far right side of the fireplace. I'm intending to put my non-TV media equipment on the right side of the wall and the television above the fireplace. This means I need two things above the fireplace: power and HDMI. I don't want to just run an HDMI cable in the wall. I'd prefer to have some sort of conduit to simplify replacing the cable or having a second (with one of these on each end: ). I only need to drill through two studs that are 6" apart above the right side of the fireplace in order to access the vertical space where I want the other end. For the power, I need to drill through two 6"-spaced studs above the left-side of the fireplace, but then another 2 full-height studs.  There are a couple of caveats:   This is an externally facing wall, which means there's insulation. I'm assuming this will make things more difficult. I'd like to refrain from cutting more holes in the wall than is necessary for the outlets (to reduce drywall damage/repair).   I've considered a couple of options:   Cut the openings for the outlets above the fireplace, and use a flexible drill bit to drill through the studs above the fireplace. This minimizes dry-wall damage, but how difficult is this with insulation in the wall, how can I feed the conduit through those holes, and how would I drill through the full-height studs on the left for power? Cut a wide section of dry-wall (4" high?) from the desired media outlet to the right until reaching the gap above the desired lower outlet. This results in some dry-wall damage, but would allow more control for drilling holes in the studs and running conduit. This still doesn't solve the problem of getting through the full-height stud for the power. For the power, I would need to extend the cut-out two studs further.   What should I do? (Why) Has Kohonen-style SOM fallen out of favor? How did Dumbledore gain the loyalty of Fawkes? How did Dumbledore find Fawkes. Did Dumbledore have Fawkes even when working as a Transfiguration teacher or acquired it after having Headmaster position. They both have a very strong bond between them. When did Dumbledore gain this. Update: Dumbledore says that Ollivander wrote to him when Harry's wand was purchased; if Dumbledore did not have Fawkes before 1938, he would not know about the two feathers being in the wands, so Fawkes and Dumbledore must have met and bonded before 1938  “My wand's feather came from Fawkes?” Harry said, amazed. &quot;Yes,” said Dumbledore. “Mr. Ollivander wrote to tell me you had bought the second wand, the moment you left his shop four years ago.” -Harry Potter and the Goblet Of Fire, Chapter 36-The Parting Of the Ways.  Update 2: JKR says that Dumbledore was the only owner of Fawkes. So it is to be believed that Fawkes was a wild Phoenix before its bonding with Dumbledore.  Peter Humphreys for BBC Newsround: Who did Fawkes previously belong to and will he play a vital role in the next book? JK Rowling: I am not going to answer about the role in the next books, which probably gives you a big clue, and he has never been owned by anyone but Dumbledore. You will notice that when Harry goes back in the Pensieve in this book, Fawkes is never there, and ­­ no, I am sorry, not in this book, I take that back. When Harry has previously seen the study with a different headmaster he saw it with Dippet and Fawkes was not there then. Fawkes is Dumbledore's possession, not a Hogwarts possession. I want to find out the number of possible combinations of $x$ numbers that sum to $y$. For example, I want to calculate all combination of 5 numbers, which their sum equals to 10.  An asymptotic approixmation is also useful. This question seems to be very close to number partitioning, with the difference that a number can be 0. See:    All possible partitions for sum 10 and 3 positions that can be zero, are 63 possiblities: (numbers shown as 3 digits)  019 028 037 046 055 064 073 082 091 109 118 127 136 145 154 163 172 181 190 208 217 226 235 244 253 262 271 280 307 316 325 334 343 352 361 370 406 415 424 433 442 451 460 505 514 523 532 541 550 604 613 622 631 640 703 712 721 730 802 811 820 901 910 How can I produce an encrypted Swap Partition for 14.04 ? All the advices I saw here are for versions &lt;14.04 and do not work for me.  I tried a lot but nothing lasted longer than a reboot.  On installation I decided to encrypt my home and 9 GiB partition for swap but this swap was never used by Kubuntu. Gparted finds an "unknown filesystem".  I many times formated it as swap, edited fstab and crypttab and used all the tricks mentioned in different answers here. The often mentioned PunyGeek letter does not work for me probably because /etc/initramfs-tools/conf.d/resume does not even exist on my system. List has No Rows for Assignment to SObject: Error How to clear the Skype chat history how clear the skype chat history ? Is it possible to access a static resource of a managed package in your local environment?  Thanks I have scripts that expect specific drive letters for hard drives. Windows sometimes assigns the different drive letters (e.g. I expected my Maxtor drive to be E:, but it's now F:, therefore my scripts will fail), depending on the order of how the drives are plugged into my laptop.  In Windows 7, how do I assign a permanent drive letter to a drive? How to include audio file in PDF find and rsync? I want to be able to search for files over 14 days and over 10k and than rsync those found files to a destination.  Is there a way to combine these two commands?  find ./ -mtime +14 -size +10k  rsync --remove-sent-files -avz /src /dest I am trying to print a large quantity (several megabytes) of UTF-8 encoded text which consists of Chinese and Latin characters (and maybe a sprinkling of others). I would like to print it in several columns per page, in a very small, condensed font, preferably with control over line spacing. I'd quite like inter-column lines, but I can live without them. The aim is to print to PDF for transfer, as well as to paper.  I have tried enscript and a2ps, but neither of these support Unicode.  I have also tried paps, but this produces bitmapped outputs which cannot be PDF'd effectively, and also look terrible.  Is there a modern way to print UTF-8 text like this without resorting to something like constructing it manually in Python? How to initiate a Full Reboot/Shutdown on Windows 10 Fall Creators Update Environment: Windows 10 Fall Creators Update / 1709  Background: There is a new feature in the Fall Creators update where post reboot applications and browser windows are reopened to the state they were in before the reboot.  Issue: The only time i really reboot is when I no longer want the applications to be open. Its driving me crazy that i cant fully shutdown/reboot the machine.   Solutions attempted I always reboot by using Win+R and running "Shutdown -r -t 0". the "-r" option should do a full reboot but that isnt working for me.   Any other suggestions or hacks to control via commandline on how to do a full reboot vs. this Zombie reboote (Which is a feature i like when i did not want to reboot and had to) How to shift EXIF date/time created by time in days, hours, minutes? I have two camera bodies, I was shooting while on vacation, when I downloaded the photos to lightroom, one body was set to the correct date/time, the other was off by 722 days, 4 hours, 32 minutes.  Is there a tool available that can help me fix the date/time so that they appear in sequence while browsing my lightroom library?  I'd like for them to be as close to the right time.     I was able to adjust created year, month, date, but it doesn't appear lightroom will allow me to adjust increments (only hour increments).  Many thanks if you can help me in this task!