If we have that $\tau \leq \nu$ are stopping times and that $X_n$ is a submartingale, how to show $E(X_{\tau \wedge n}) \leq E(X_{\nu \wedge n})$
Prove previsibility and $E[X_{T \wedge n}] \le E[X_{S \wedge n}]$
Choosing coordinate system to use in South Africa? What are the principal datums and projections used in South Africa including the names of geodetic datums and their projections? Extracting only the BibTex records relevant to the paper My colleague asked me to write a page for a paper he writes. I wrote the page in LyX, compiled it to LaTeX, and sent him the TeX files. Then he reminded me that I forgot to send the .bib file with all the references.  But, my .bib file is huge - it contains over 1000 references (I have a global BiBTeX file for all my papers). The page I wrote uses only about 20 of them.  Is there an automatic way to extract from the bib file, only the records relevant to the current paper? This is probably a dumb question. I guess what I'm trying to ask is if radio waves travel the same speed as gamma rays, how do gamma photons carry more energy than radio photons? Do they spin faster? What other energy sources could they carry if they are moving the same speed through space? Using Moq to determine if a method is called It is my understanding that I can test that a method call will occur if I call a higher level method, i.e.:  public abstract class SomeClass() {         public void SomeMehod()     {         SomeOtherMethod();     }      internal abstract void SomeOtherMethod(); }   I want to test that if I call SomeMethod() then I expect that SomeOtherMethod() will be called.   Am I right in thinking this sort of test is available in a mocking framework? Are non-citizens of the Capitol permitted to permanently immigrate? Reading  got me thinking about permitted citizenship.  Are citizens of other districts allowed to permanently move to, or become citizens of the Capitol?   My gut says, "No." The citizens of the other districts are being punished, so it makes little sense to allow them a true, legal escape from their punishments. Even victors are only given new homes in their own districts, keeping them there. I am currently running Ubuntu 11.10 off of a SanDisk 16GB USB. I installed it using a Live USB with the following partition configuration:  6GB Primary /dos   FAT32 5GB Logical /      ext4 5GB Logical /home  ext4   I don't have a hard disk, and don't see myself getting one anytime soon.   I rely solely on this 16GB, and two other 4GB USBs, one of which I used as the LiveUSB. I bring the USBs around, and even use the install at work.   I previously used an install that used a swap file. It functioned fine for the most part, save for a few slow moments, but I came across , and it got me thinking about my USB's life, so I reinstalled with the current config.   My problem now is that it is slower. Applications like Firefox would hang more often. In my previous setup (the automatically partitioned setup), Firefox would start hanging if I was running an unzip or install task on the same partition as /.   Now however, it would hang if I had another window open i.e. the system settings window. My guess is that it may have something to do with the swap file or the install being on a Logical partition rather than a Primary partition, but I don't know.   Any insight as to why it has slowed down? Should I use the past tense with did? When should I fudge dice rolls? When I started my first D&amp;D campaign as a DM, I accidentally rolled and confirmed a crit on a PC from a goblin. It was an instant kill as our campaign had no cleric. That time I decided to fudge the dice. Now, there's a similar problem, but this time the PCs are at 3rd level and they are dealing with a very important NPC. If they fail this one diplomacy check, the entire game world will fall apart, as the king would then send his entire army against the PC's homeland and will destroy it, plunging my entire game world into chaos, at which time drow that I wanted to save for a later adventure will have to come out, which will instigate a wizard's guild to release their secret ultra-deadly virus, and everything in my game world, including the PCs, will die. So should I fudge the dice roll? When is it good to fudge it? I am stuck with the proof of below expression;  "If F is an infinite field, then no infinite subgroup of F* (the multiplicative group of F) is cyclic."  anyone can help? How can one derive Schrödinger equation? Why was M87 targeted for the Event Horizon Telescope instead of Sagittarius A*? The first image of a black hole has been released today, April 10th, 2019. The team targeted .  Why didn't the team target  at the center of our own galaxy? Intuitively, it would seem to be a better target as it is closer to us. Why my DAC works without GND and VCC? This is the most interesting thing I have seen! I have built a  that sends data (D0-D7) to a DAC (ADV2175). The design worked fine and I just have a low frequency alias noise (). This noise was present on all my ground planes (This is in prototyping stage and that's not unexpected from the test PCB).  I started tracing the noise path by disconnecting wires to see when the noise is gone. Surprisingly even after disconnecting all VCC and GND connections (+ all VREF and logics gone to 0 or 1) the noise was present yet. The more unexpected behavior is when the DAC is just connected to DATA port+ CLK pin (all coming from the source FPGA, NO VCC or GND), it continues working and a sine wave is produced (but with the same noise)!!!  First I thought it may receive its VCC/GND form the logic 1/0 fed into it but if it was correct, it should not have produce 0x00 and 0xFF levels, but it makes those levels correctly (I changed 0xFF data stage to another level and a notch appeared on top of the waves. this shows it can build oxFF correctly).    This is a big challenge for my PCB design, as if this device really takes its GND from the data port, instead of the dedicated pins, I will have a hard time for designing a reliable ground plane for it (the return voltage should go back to a logic noisy environment and analog-digital ground plane isolation may be impossible).  Can anyone explain what is happening there? How to join a LAN / IPX game on Diablo 1? Recently some of my friends and I grew nostalgic about the original Diablo and realized we've never truly tried the multiplayer aspect of that game. So we set up a modern VPN LAN solution via tuungle (using the appropriate IPX Wrappers) but the problem is, while any of us can manage to create session in Diablo FULL, others cannot find a way to join my IPX/LAN game.  There are no "join" buttons in the game itself.  I've read somewhere that Spawn version is a trial version but can be used to join Multiplayer games. If they can, is it possible for them to play the entire game on LAN or is trial restricted from choosing some classes/accessing some areas?  Or are there some other ways to join? Php string not working in Wordpress Functions.php (trying to fetch 1st category for each blog that post appears in the sidebar) I am trying to fetch 1st category for each blog post that appears in the sidebar of any blogpost landing page. I tried multiple options to add the 2nd block code to the 1st block at the line &quot;.wp_get_post_categories ($post_id).&quot; but I could not find any solution. Please help me with that and if you need closer to the code just let me know. 1st Block $related_blog .=&quot;&lt;div class='related-blog'&gt;                             &lt;div class='row'&gt;                                 &lt;a href=&quot;.get_the_permalink().&quot;&gt;                                                                  &lt;div class='col-lg-4 col-md-4 col-xs-4 blog-image'&gt;                                         &lt;img src=&quot;.$image_related[0].&quot;&gt;                                     &lt;/div&gt;                                      &lt;div class='col-lg-8 col-md-8 col-xs-8 blog-text'&gt;                                         &lt;strong&gt;&lt;p class='related-blog-title'&gt;&quot;.get_the_title().&quot;&lt;/p&gt;&lt;/strong&gt;                                 &lt;/a&gt;                                 &lt;span class='related-blog-sidebar-content '&gt;&quot;.substr(get_the_content(),0,250).&quot;...&lt;/span&gt;                                                                  &lt;p class='related-blog-sidebar-category'&gt;                                 &quot;.wp_get_post_categories ($post_id).&quot;                                                                &lt;/p&gt;                                                                  &lt;a class='related-blog-sidebar-link' href=&quot;.get_the_permalink().&quot;&gt;Read More                                 &lt;/a&gt;                                      &lt;/div&gt;                             &lt;/div&gt;                         &lt;/div&gt;&quot;;  2nd Block &lt;?php $category = get_the_category(); ?&gt; &lt;a  href=&quot;&lt;?php echo get_category_link($category[0]-&gt;cat_ID); ?&gt; &quot;&gt; &lt;?php echo $category[0]-&gt;cat_name; ?&gt; &lt;/a&gt; Identical creatures entering the battlefield Do the 3 Doctors remember that they saved Gallifrey after 'Day of the Doctor'? I've read several posts and I've seen one that says before it was Eleven's turn to take part in the events of 'Day of the Doctor', Gallifrey was most certainly destroyed, and when it came to his turn he re-wrote history and saved Gallifrey. But I've rewatched 'Day of the Doctor' recently and the War Doctor asks if he'll remember saving Gallifrey and the Doctor says no. So I'm a bit confused, I don't want to think that the War Doctor or the 10th Doctor have lost their memories after saving Gallifrey and their life of saying they destroyed Gallifrey was a lie. So did they really remember? How do you pluralize the acronym "POC" ("proof of concept")? Possible Duplicate:          What's the plural form of the acronym , short for proof of concept?     ...for his contributions to many POCs   or      ...for his contributions to many POC      is not a duplicate as it is focused on cases like "ATMs" where the expanded form "automated teller machines" ends in a noun with a regular plural form ending in "s." In "proof of concept," the noun is in the middle of the abbreviated phrase. If we pluralize the uncontracted phrase, we get "proofs of concept," which has an "s" in the middle rather than at the end. It's unclear from the linked answers how abbreviations with this structure should be pluralized. (POC, POCs, PsOC?) I have two exchange accounts on my iPhone 4. One that connects to my gmail, the other connects to my work's Lotus Notes. There are no overlapping contacts in the two accounts. There are also no contacts saved locally on my phone.   When I enable contacts in both accounts, I get two identical entries for each of the contacts in my gmail. If I disable the Notes exchange account, then there are no more duplicates. If I have Notes contacts enabled, and gmail contacts disabled then I have no contacts at all.  Does anyone know what is going on? Is this a problem with having multiple exchange accounts or is it a Lotus Traveler problem? (I tend to blame Notes for all the trouble in the world, but that is just my bias)  Thanks all. Consider a complete metric space $(X,d)$ and $T\colon X\to X$. Suppose there exists $n\in\mathbb{N}$ such that the n-th power of $T$ is $q$-contractive. Show that then $T$ has exactly one fixed point $\overline{x}\in X$.      The n-th power is defined inductively: $$ T^{n+1}(x):=T(T^{n}(x)), n\in\mathbb{N}. $$ And for a function $T\colon X\to X$, $q$-contractive means  $$ \exists 0\leq q&lt;1~\forall~x,y\in X: d(T(x),T(y))\leq q d(x,y). $$    Now to the proof.  I think I have to show, that $T$ is q-contractive, because then it follows with Banach, that $T$ does have exactly one fixed point. So I have to show, that there exists a $0\leq q&lt;1$ so that for all $x,y\in X$ it is $$ d(T(x),T(y))\leq q\cdot d(x,y). $$ And most likely I have to use the q-contractivity of $T^n$, i.e. that there exists a $0\leq q &lt;1$ so that for all $x,y\in X$ it is $$ d(T^n(x),T^n(y))\leq q\cdot d(x,y). $$  Can you help me? Is there an equivalent in mathematical language to the modulo (or mod) function in computing?