Let V be a finite set and let $\Omega \subseteq 2^{V}$ be a nested family of odd subsets of V, where subset $A\subseteq V$ is odd if |V(A)| is odd (if A has an odd amount of nodes).  Question: How do I show that $|\Omega| \leq \frac{3}{2}\cdot|V|$?  I have no clue on how I should tackle this problem, thanks in advance!
Hi I have to proof that for $\Omega$ a nested family of odd subsets of $V$ with $V$ finite, we have that $|\Omega|\leq\frac{3}{2}|V|$. I can prove that for certain constructions of $\Omega$, it is satisfies $|\Omega|\leq \frac{3}{2} |V|$, the problem I encounter is that I have to proof that my construction of $\Omega$ is optimal. (Since I have to prove that this holds for every family of odd subsets) Can anyone give me a hint on how to proof that every $\Omega$ satisfies this equation.    $\Omega$ is nested if for all $A,B\in\Omega$ either $A\subseteq B$, $B\subseteq A$, or $A\cap B=\varnothing$; a set is odd if its cardinality is odd.
How to flash a BIOS when only a Windows binary is provided? Why is it disadvantageous in graduate admissions to have a higher degree? indicates that holders of a master’s degree are held to higher standards when it comes to graduate admissions. The stated requirements for some graduate programs () also outright exclude applicants with doctoral degrees. These seem to be saying that holding a higher degree prior to application is a disadvantage (a fatal one in the case of a PhD). This also applies to private universities: Stanford  (at least in computer science); .  I don't understand why:   If there are two applicants, Alice and Bob, who have achieved largely similar things, it seems logical that both should have a roughly equal chance of being admitted. However, based on JeffE's answer, if Alice has a Masters degree while Bob does not, it's Bob that gets admitted instead of Alice. JeffE's answer even indicates that if Alice has achieved slightly more than Bob, it's still Bob that gets admitted.  If graduate admissions is looking for students that are capable of succeeding in its program, then Alice should be more likely to succeed than Bob since she's already had some experience of graduate study. This should be even more the case if Alice already has a PhD – if she can do it once, she should be able to do it again.    JeffE's answer indicates that since Alice has had more opportunities than Bob, she should also be held to higher standard. However it should also be true that Alice has already achieved what Bob has yet to and in fact might not be able to achieve. Further, JeffE's answer seems to be saying that Alice should do the academic equivalent of financial engineering and pass her graduate-level courses &amp; thesis without getting a master’s degree even if she qualifies for it, which sounds silly.  As for applicants with PhD degrees, my first guess would be that this is because , and the university is acting in the applicant's best interests. However if this is the case, then this is effectively telling the applicant "I know what you want better than you do", which also sounds silly. UC Berkeley's requirements also seem to be saying that if someone with a CV worthy of a tenured position were to apply for a PhD student position, the university will say "no" even though it's clearly better in their best interests (in terms of research output) to say "yes".   Presumably I'm misunderstanding something because if this behavior were irrational, it would already have been changed. What am I missing?  EDIT: I wonder if things are different in programs where the student is involved in research directly without spending time on coursework? Doom-style 2.5D Movement Animations I've started working on a demo for my 2.5D game. For a basic scene I figured I would just use the good old fashioned doom sprites textured to a double sided plane. Simple enough. But I've been scratching my head as to how to display the proper sprite animation relative to the player viewing the sprite...if that makes since.  For example suppose I have three players in the scene Players A, B, and C as such:    Player C should see player A as: Here's an example of what I think player A should look like from C's perspective:    While player B would see player A as follows:    My current idea is that for any given player, I will need to know the current direction of each other player, and where they are located/moving to relativly. From this, I could calculate which sprite image is required.  Is this correct? What improvements/optimizations could be made?  ** EDIT ** Total number of directions will be 8 as depicted below:    For each direction there are two sprites (to show walking movement, sprites for the right hand directions are mirrors of the left):    ** EDIT 2 ** Got this working and have a simple javascript/threejs example for anyone interested: Importing installed package from script raises "AttributeError: module has no attribute" or "ImportError: cannot import name" How to address a professor in person at US universities? At US universities, how should one address a professor while talking to him or her? In my home country, we usually address the professors as either "sir" or "madam" while talking about our queries in person.   What's the norm at US universities? My current idea is to use the dual graph, and prove that that is a bipartite graph, and so proving the statement. But I can't seem to figure out how even degree for all verticies means that it is bipartite.  Any help is appreciated. Thanks! Is there some way to PUSH data from web server to browser? Of course I am aware of Ajax, but the problem with Ajax is that the browser should poll the server frequently to find whether there is new data. This increases server load.  Is there any better method (even using Ajax) other than polling the server frequently? MySQL Database User: Which Privileges are needed? The short installation instruction for WordPress () state that:     Create a database for WordPress on your web server, as well as a MySQL user who has all privileges for accessing and modifying it.   While setting up a new blog professionally I was wondering how that maps to what the MySQL database user privileges/permissions configuration offers me:   Data: SELECT, INSERT, UPDATE, DELETE Definition: CREATE, ALTER, DROP Extra: INDEX More:  LOCK TABLES REFERENCES CREATE TEMPORARY TABLES CREATE VIEW SHOW VIEW CREATE ROUTINE EXECUTE ALTER ROUTINE    I'm pretty sure for the first three groups, I named them Data, Definition and Extra here. But what about the others below the More entry? Normally I would say, those are not needed, but I would like to get a second opinion. Why doesn't the T-800 run? I just re-watched the first two films in the Terminator series. The T1000 (Robert Patrick) does a lot of running, and is quite fast.  I didn't see many scenes where the T800 (Arnie) ran. Is it because he's really heavy? Is he mainly designed as a heavy "stand your ground" machine?  The only scene I could see him running is in T2 in the mental hostpital where the T1000 morphs through the metal gate and his gun gets caught. Then the T800 does a little jog to get into the elevator following John and Sarah. (anybody know of any other running/jogging scenes?) (EDIT Arnie also breaks into a run while pursuing Reece and Sarah down the alley after the NOIR nightclub attack.)  If he can at least jog though, why wouldn't he do this throughout the entire film? You would think that from a "chance of survival" point of view, if he could get a John even the slightest bit faster then he'd do so.  Of course, maybe I am overthinking this... Arnie walking around slowly is a lot more badass than Arnie jogging around.  Edit: Just watched T3 again, the T800 also ran towards the end when the big hangar door was closing and he ran over to slow it down. Interested in other examples of the T800 running too. Comprehensive list of activation functions in neural networks with pros/cons Are there any reference document(s) that give a comprehensive list of activation functions in neural networks along with their pros/cons (and ideally some pointers to publications where they were successful or not so successful)? I'm trying to use push with AppInbox/Cloud Pages. I want the Cloud Page that is linked to the push message to be able to display dynamic content based on an attribute for that contact. This page -  - says that it is possible to personalise Cloud Page content but doesn't say how. I've already had a Salesforce support case opened and closed with no luck - apparently it's not their job to assist me with how to do something even if they say something is possible but back it up with no documentation for how it might be done. I've tried the normal way you'd do it from an email, using both the _subscriberkey personalisation string as well as the RequestParameter function with fields from the entry source DE, but neither of these work. Given you're selecting a static Cloud Page when you set up the send and there's no visible way to add attributes, how do you actually get a Cloud Page to pick up these values? Finding the volume of the tetrahedron. Find the volume of the tetrahedron with the vertices $P(1,1,1)$, $Q(1, 2, 3)$, $R(3, 1, 2)$, and $S(2, 3, 1)$. How to execute .sh file on Windows? When I am trying to execute a file(name.sh) in the command line by the command  ./name.sh  , I am getting the error that:     "." is not recognized as an internal   or external command, operable or batch   file   please help me execute the .sh file One of the most commonly asked questions of beginners. How do ranks work in Go for amateurs?  Why do we start backwards? And why do we suddenly count upwards again? What is the difference between 9d and 9p? How is this related to rating? What are the basic differences between ranks used in different countries and different servers? Problem working fontspec, babel and datetime with cyrillic I have discovered that fontspec with russian babel settings doesn't work with datetime package. I tried to compile such code:  \documentclass[a4paper,11pt]{article} \usepackage[T2A]{fontenc} \usepackage[utf8]{inputenc} \usepackage[main=russian,english]{babel} \usepackage{datetime} \usepackage[cm-default]{fontspec}  \begin{document} \today \end{document}   with a xelatex and got an error: Command \cyri unavailable in encoding EU1.  Is there any way to handle it?  UPDATE: I really bungle with charset in XeLateX. Thanks to egreg. I have removed fontenc and inputenc and used some cyrillic font, so my MWE now looks this way:  \documentclass[a4paper,11pt]{article} \usepackage[main=russian,english]{babel} \usepackage{fontspec} \setromanfont{PT Sans} \usepackage{datetime}  \begin{document} \today \end{document}   and I've got another error, if I used russian as a main language in babel: Undefined control sequence. \\monthnamerussian ...cyrm \cyra \cyrya \or \cyri  \cyryu \cyrn \cyrya \or \c...  Is it still because of XeLaTeX? How much voltage/current is "dangerous"? Related:        From what I've heard:   110 V (or 220 V; household voltage pretty much) is dangerous (i.e. can kill you) I think there's consensus on this, no need to try :) 60 V (old telephone lines) is supposedly dangerous (never tried, only heard it once... probably won't try)   From what I know first-hand:   9 V is not dangerous (I've put a 9-V battery on my tongue, nbd... actually it kinda hurt!) 1.5 V can indeed be quite shocking with enough current (fell for one of those "Do you want some gum?" tricks back in high school...), but they sometimes do not use 1.5 V with the low amperage levels, some use a DC motor to vibrate and complete the trick.   So I guess there's two parameters here, voltage and current... but are there rough numbers on how much of each (or in combination, which I guess would be power) would be considered hazardous?  No old telephone lines have always been 48vDC well at least since from 1950s, if your skin is wet you can feel it slightly, like on your forearm.  Now the ring voltage is 90-110vAC with a 2 on 4 sec off cycle (USA).  It will ring your bell but good, should you be touching the wires when someone calls. The ring voltage rides on top of the 48vDC, so its present on the same two conductors that the voice voltage(DC) is on. Luckily it's 4 seconds off will give you a chance to get off the conductors with a scream (of pain). Linear Transformations: Proving 1 dimensional subspace goes to 1 dimensional I am having trouble understanding this whole question, and how to prove it.  Let $F:\mathbb{R}^n\to\mathbb{R}^m$ be a linear transformation. Prove that if $L$ is a $1$-dimensional subspace of $\mathbb{R}^n$, then  $F(L)=\{y\in\mathbb{R}^m: y=F(x)\text{ for some }x\in L\}$  the image of $L$ under $F$ is a subspace of $\mathbb{R}^m$ with dimension $1$ or $0$. Hence prove that if the kernel of $F$ is $\{0\}$, then the image of $L$ is a $1$-dimensional subspace of $\mathbb{R}^m$ (and so, in this case, $F$ takes lines in $\mathbb{R}^n$ to lines in $\mathbb{R}^m$). We have officially discontinued the Amazon book affiliate remnant ads, since despite our best efforts they don't perform well for our audience and cannot be made to perform well.  We've often wondered,   I'm thinking this isn't a problem; it has gotten lots of support historically. So we're going to go ahead and do it, so the company Amazon account is still getting some use, and we benefit a little.   To be clear: from this point on, all Amazon links posted on all sites will automatically be rewritten to add our affiliate link. We have a clever affiliate redirect that geolocates the IP address and redirects to the appropriate regional Amazon store (amazon.uk, amazon.ca, et cetera).  If you have any feedback on this, or suggestions, as they say &hellip; speak now or forever hold your peace! Get full path of the files in PowerShell Packages: graphics vs graphicx Different websites suggest using one of those two packages.  \usepackage{graphicx} \usepackage{graphics}   What are differences between them? Which is better?