In MCMC simulation, how to deal with very small likelihood values that couldn't be represented by computer?
Computation of likelihood when $n$ is very large, so likelihood gets very small?
Strictly Upper Triangular $n\times n$ Matrices form a Nilpotent Lie Algebra over $\mathbb{C}$ for $n\geq 2$ I have been attempting this problem for a while, it is an assignment problem so I don't want somebody to just post the answer, I'm just looking for hints.  Let $\mathfrak{u}(n,\mathbb{C})$ be the Lie algebra of strictly upper triangular $n\times n$ matrices over $\mathbb{C}$.  I am trying to show that $\mathfrak{u}(n,\mathbb{C})$ is a nilpotent Lie algebra for $n\geq 2$.  This is what I have looked at so far.  I have done some explicit calculations (I shall omit the bulk of them and just summarise below) for $n=2,3,4$ and have come up with some ideas.  Let $L=\mathfrak{u}(n, \mathbb{C})$.  If $n=2$,  $L^1 = [L, L] = &lt;0&gt;_{\mathbb{C}}$  So $L$ is nilpotent.  If $n=3$,  $L^1 = [L, L] = &lt;e_{13}&gt;_{\mathbb{C}}$  $L^2 = [L, L^1] = &lt;0&gt;_{\mathbb{C}} = \left\{ 0 \right\}$  So, $L$ is nilpotent.  If $n=3$  $L^1 = [L,L] = &lt;e_{13},e_{14},e_{24}&gt;_\mathbb{C}$  $L^2 = [L,L^1] = &lt;e_{14}&gt;_{\mathbb{C}}$  $L^3 = [L, L^2] = &lt;0&gt;_{\mathbb{C}} = \left\{ 0 \right\}$  So, $L$ is nilpotent.  So my hypothesis so far is that for an arbitrary $m\in\mathbb{N}$, $L^{m-1}=\{0\}$  My instinct is to do this by induction. My base case, $n=2$ has already been shown. Now I assume that $\mathfrak{u}(n,\mathbb{C})$ is nilpotent for $n=k$ ($k\in\mathbb{N}$) and consider the case for n=k+1.  However, I have no idea how to link these two cases together. I have read in Introduction to Lie Algebras and Representation Theory - J.E. Humphreys that $L^k$ should be spanned by basis vectors $e_{ij}$ where $j-i=k+1$ - however in my calculations for $L^1$, I obtain an $e_{14}$ which does not satisfy 4-1=2.  Am I missing something completely obvious? Or have I made a mistake somewhere? (I have checked my calculations numerous times, but cannot find any errors!)  Thanks,  Andy. Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n \mathbb Z$. Prove that either every element of $H$ is... Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n\mathbb Z$. Prove that either every element of $H$ is even or exactly half of its elements are even. (Gallian, Contemporary Abstract Algebra, Exercise 24, Chapter 3.) found a 1994 uncashed profit sharing retirement plan check I found an old un-cashed check dated 1994 for $487 from my Xerox Retirement plan with no expiration date. Can I still cash it? Copy a DVD in Windows 7 Is there any built-in way of copying DVDs in Windows 7 or do I need to install a third party tool? Is Cesaro convergence still weaker in measure? I've got about a billion rows of data in a table with a name and an integer in the range 1-288.  For a given name, every int is unique, and not every possible integer in the range is present--so there are gaps.  This query generates an example case:  --what I have: SELECT * FROM ( VALUES ('foo', 2),               ('foo', 3),               ('foo', 4),               ('foo', 10),               ('foo', 11),               ('foo', 13),               ('bar', 1),               ('bar', 2),               ('bar', 3)      ) AS baz ("name", "int")   I'd like to generate a lookup table with a row for each name and sequence of contiguous integers.  Each such row would contain:  name   --  the value of the name column start  --  the first integer in the contiguous sequence end    --  the final value in the contiguous sequence span   --  end - start + 1  This query generates example output for the above example:  --what I need: SELECT *  FROM ( VALUES ('foo', 2, 4, 3),               ('foo', 10, 11, 2),               ('foo', 13, 13, 1),               ('bar', 1, 3, 3)      ) AS contiguous_ranges ("name", "start", "end", span)   Because I have so many rows, more efficient is better.  That said, I only have to run this query once, so it isn't an absolute requirement.  Thanks in advance!  Edit:  I should add that PL/pgSQL solutions are welcome (please explain any Fancy Tricks--I'm still new to PL/pgSQL). Did this character even qualify for being Azor Ahai or The One Who Was Promised? In the books and to some degree in the show as well there is a prophesy about the legendary figure Azor Ahai. But in the latest Game of Thrones episode (S08E03) we see Arya      killing the Night King.   However in a  we discussed​ all possible candidates but nobody thought about Arya. Does she even fulfill the prophecy? If she does, how? Prevent mail being marked as spam Why didn't (or did) I or some other user get a hat although the requirements were (or weren't) fulfilled? During Winter Bash, users can earn hats. But sometimes it seems that a hat was awarded even though its requirements weren't (yet) met, or vice versa.  What could be the reason for this? I'm trying to  when I run into this:     "You have already deleted 5 of your own posts today; further deletes are blocked"      I found conflicting information on this very annoying limit:   This post from May 25, 2010: . This is marked . If this is no longer the case, please edit that question and flag it as . Then , indicates that the problem has risen from the dead -- despite :     Each time that I've seen it (pruning subpar answers), it's been a user who is deleting their own redundant zero-upvote answers. Can you tweak the algorithm slightly so answers with less than one upvote don't count towards tripping this flag?     (Emphasis added.) :     I don't mind it when         cleaning up redundant answers on questions that have multiple alternate good answers   cleaning up old, "less awesome" answers that are no longer reflective of the user's current skill level         (Emphasis added.)     So why do we have barriers in place, to making SO answers less, "less awesome"?      Proposed Solution:  Please let us (¿once again?) delete an unlimited number of our own answers, when they have zero up-votes and are not marked as the answer.  Otherwise, please edit the linked questions to clear up the confusion. Ill implement the some UI but its not working in the visualpage , can u know how implement jquery ui in viusalpage ,  Kindly tell the procedure ,     Im using this url its not working ,,,,, MacBook Pro stuck in gray screen when booting I've got a Late-2011 15" MacBook Pro stuck in the gray screen when booting AFTER login. I've tried removing the drive (it's non-retina) and booting from a working bootable Yosemite installer, no luck. (it sees the startup USB drive, the progress bar progresses, but stuck at the gray screen again even without a drive.  Tried NVRAM reset (cmd option P R), it has reset (the login sound's volume changed) but nothing changed.  Tried holding D on startup, AHT doesn't start, instead, it tries to boot into the OS.  Tried Cmd V, it sometimes works and displays the verbose output, sometimes, it acts like I haven't even pressed. Again, stuck after login.  Tried booting again from USB, I select USB (after seeing both drives correctly), but no luck, it's also stuck.  I've stumbled upon this:    I've tried booting into single user mode with Cmd S, it worked ONCE, I've disabled the graphics extensions just as shown at that post, restarted, but again, nothing changed.  More weirdly, now the Cmd S doesn't work either. It just takes me to login screen without showing the terminal, just as I haven't pressed anything.  Why would this happen? Why would a Mac refuse to enter hardware test mode, single user mode, or refuse to boot from a working bootable Yosemite drive both with or without the main drive installed?  UPDATE: I've run the AHT and got an error:  4MEM/66/40000000: 0x84e0ec98  It looks like a memory issue, but I'm not sure.   UPDATE 2: I've tried reinserting the RAMs and the the problem is not reported anymore. However, the Mac still won't boot. Stars and pins are retained when chat messages are moved! In the chat system, stars and pins are retained when moved to another room. This is probably a bug. Here a picture:  When a room owner pins a message, they can move it to another room, and they cannot unpin it. However when a starred message is moved to another room, the user can unstar it in the room it was moved to. Conversation source:  * is irrelevant and does not have to do with this bug report. Many thanks to . Series as an integral (sophomore's dream) Why doesn't my flash fire from the hot shoe trigger? I have a Yongnuo YN-560 III and a pair of RF-603II (C3) triggers. I have a Canon Rebel T5i.   I put one trigger on the hot shoe.  When this trigger is in TX mode I get a blue comm light and I can fire a test flash using the large test button on the trigger. But when I press the shutter button to take a picture, there's no flash.  The trigger and flash are on Channel 1. The trigger on the hot shoe is set to TRX. TX is fine, TRX gives me no flash. The flash and camera are in M mode.  I don't have a trigger on the flash because the Mark III has a built-in trigger The problem seems to be the camera.  The trigger can talk to the flash but the camera is not sending a flash command through the trigger in the hot shoe to the flash.  Grateful for anything you could suggest. I find I always have trouble scaling the amount of time the eggs should be boiled, cooled, etc. for different-sized batches of hard-boiled eggs.  I've heard a variety of "folk lore"-type rules for how it should be done, but what's really the right way? I want to run a daily incremental backup and append the day of the week to the file name so I end up with:  backup_mon.bak backup_tue.bak  etc.  Not really bothered about what exactly is appended - 0 - 6 is fine.  How do I go about this? I will soon buy a new iPhone. If I log in with my Apple ID in my new phone could I still access my Clash of Clans village without starting from scratch? How can I get a complete list of fields for a standard object? I upgraded from 16.04 to 18.04. When I press Ctrl+Alt+F3 I get the terminal which prompts for the username. When I type my username and press Enter, it doesn't wait for my password (its as if I pressed Enter without entering anything). Even if I enter something, the text shown in plaintext and then it keeps on saying my password is wrong. After few times, the screen is cleared and it prompts for my username again and the same loop goes on.