Show that for any reals $a; b; x; y \ge 0, (ax + by)^2 \le (a^2 + b^2)(x^2 + y^2)$. (Hint: Use Cauchy-Schwarz inequality in $\mathbb{R}^{2}$.)
Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$.
Title of 80's movie: future-era timetraveling humans go extinct and rescue/kidnap todays air crash victims What is the title of a 80's (i believe?) movie where humans in the far future are going extinct, and are aquiring new humans by going back in time to todays world, kidnapping air travelers that are known to have died that day, from the flying passenger aircraft just before its crashing.  A key visual:    Inside a hangar, a time portal opened, and a tail of a (flying) passenger aircraft extruded into the hangar where a stairs provided access to the back door of the aircraft.  The kidnappers made sure the surviving passengers remain seated and would forget about the whole ordeal. Kidnapped humans got replaced by lookalike dummies, including things they had on them. They tried hard not to alter the timeline. Why and how do "bigger pixels" make a better picture? This is going off of today's iPhone 5S announcement, when Phil Schiller said that "bigger pixels" is the key to a better picture.      What does this mean? Is it true or just a marketing ploy? Prove that there are no positive integers $x$ and $y$ such that $x^3 + y^3 = 10^3$. Prove that there are no positive integers $x$ and $y$ such that $x^3 + y^3 = 10^3$.   This is a homework question, and I understand that its part of Fermat's Last Theorem, but when I looked that up to try to figure out the homework, I realized that it is way further than what we've learned in class so far. We have to prove this, but the only way I can think to do it is by exhaustion, which would be extremely lengthy. Any little shove in the right direction would be appreciated. Thanks! How can I tell what year my bike was made? This is a canonical question that will hopefully encompass all of the questions we get asking us to determine what year a bicycle was manufactured (how old is my bike, how old is my frame, etc.). Each "answer" should address a different way to determine the manufacturing year of a bicycle. To some extent, these will also help you narrow down the model as well as it will tell you what distinguishing features to look at.  The span of answers would include things such as:      Skiptooth cranks Cartridge bottom brackets (BB30/90 etc) Elliptical chainring Quick releases versus contemporary through-axles How the steerer attaches to the fork (quill vs. threadless)  Braking system (especially with mountain bikes: v-brake, direct pull, etc.)   Shifting ramps and pins on cassettes/sprockets and chainrings (versus flat sprockets) Frame material (e.g., shift from tubular steel to use of CF and hydroformed aluminium)  Frame tubing type (e.g. Reynolds 531 tubing introduced around 1934, still used today; Reynolds 501 tubing introduced around 1983). So a frame with an original 501 decal can't be from the 70', 531 decal doesn't help as much because of the range of years produced. Also, see Decals.   *Decals: style/design, amount, placement, color, etc. can help not only determine models, but also year of manufacture. Many changed decal design for the same model bike often yearly, with special/limited editions even more specific. This applies only if decals are original or have been replaced with the identical design. Tubing decals (Reynolds, Columbus, etc.) also changed designs for same tubing made in different years, although the changes aren't made as often, and manufacturers didn't always put tubing Brand decals on every model. Earlier bikes tended to have less fancy, less colorful, less quantity of decals.   Headbadges: most early bikes had actual "badges", often quite detailed and fancy, made of metal, then plastic, and finally just using decals on the headtube. Most bikes lost their metal badges in the 60's and early 70's, although some brands still have actual badges to this day, but in general a real badge indicates an earlier model bike. Dropout style (including the later inclusion of lawyers lips) Presence or absence of CPSC reflectors (USA only) Manufacturing method (shift from lugs to TIG welding, for example)  Number of gears (shift from 2x5 ten-speed to 3x8, 2x9, 1x11, etc.)  Inspection of lug design: pantographs, cut-out, stampings, custom brazing/filing, chrome, fork crow n sweep, etc. Frame/Fork Braze-ons. Race bikes tended have less braze-ons the earlier the year of manufacture. Most 60's racers and earlier had few or none with cable guides/stops, shifters, bidons, etc. being clamp-on. By the 80's if it was clamp-on, it was a sign of very cheap models and most frames had braze-ons for all cables, with multiple bottle and accessory bosses depending on style and use.   and so forth. Please edit this question if you have an answer that isn't in the index above.   A couple of caveats:   A previous owner could have replaced or upgraded components on a bicycle, so this guide only applies to original equipment There is a delay between the introduction of a technology (such as indexed shifting) and when manufacturers start selling products that use it. Low-end manufacturers may continue using technology that is severely outdated (such as one-piece cranks, freewheels, or quill stems) because it is cheaper to do so. A number of bikes are built in a Retro style, but with modern components.  Dutch bikes, trendy coffee cruisers, and beach cruisers may be modern but appear to be 1950s styling.  These tend to have modern rims and brakes as a giveaway.     Note that bicycles.stackexchange does not do valuations of bicycles. How much your bicycle is worth depends on the location of the buyer and seller, how much the buyer wants the bike, etc. etc. etc. Velocity Maxwell-Boltzmann distribution for dummies I have a volume with $N$ molecules; I need to assign to each particle a velocity vector: $$|\mathbf{v}_{i}|=[v_{x}, v_{y}, v_{z}]^{T}$$ for the $i$-th molecule; the velocities must follow the Maxwell-Boltzmann distribution; here is the question: I know the definition of the M.B. distribution, how do I extract the velocities that I have to assign to each molecule? Basically what I want to know is how to calculate $$v_{x}=?$$ $$v_{y}=?$$ $$v_{z}=?$$ for each molecule. I am trying to create the following trigger:      CREATE TRIGGER update_customers AFTER INSERT ON wp_usermeta     FOR EACH ROW      BEGIN             if new.wp_metakey = first_name then insert into test_customer(first_name) values (new.wp_meta_value);     END   I keep getting a 1064 error that says, "You have an error in your SQL syntax; check the manual that corresopons to your MySQL server version for the right syntaxt to use near " at line 5"  What am I doing wrong? Why do you need to know who I know on Google? I just created a SE account using my Google account. It said that it needs to "Know who you are on Google" (I guess my connections). As far as I know, SE is not a social network. I can't even follow other people (seeing when they post questions and answers).  What is the reason you need to know my Google connections?  This was on code review. Actually I have an account there, but for some reason it asked me to reconnect. This is the same on MSO; one sees this explained as "This app is requesting permission to associate you with your public Google profile": If a spell is cast on a moveable creature/object and lasts longer than a round, and the creature/object moves, does the spell move with it? Apache accepting requests to other servers? I'm having problems with apache server (CentOS VPS at Amazon). When apache starts, it starts to receive hundreds of requests, this is an example of the log:  173.208.216.165 - - [25/Jan/2015:18:23:11 +0000] "GET http://go.padstm.com/resources/img/iebt.png HTTP/1.0" 200 36023 "http://go.padstm.com/?id=173374&amp;t=iframe&amp;var=33110" "Mozilla/5.0 (Windows; U; Windows NT 6.1; ru; rv:1.9.2.3) Gecko/20100401 Firefox/4.0 (.NET CLR 3.5.30729)"   The path /resources/img/iebt.png is in my server, but not go.padstm.com. The strangest thing is that apache is accepting all the requests.  Other example: in this case, the entire resource is NOT in my server, but apache returns 200:  198.204.239.250 - - [25/Jan/2015:19:04:39 +0000] "GET http://fstads.com/show.php?z=26&amp;pl=494&amp;j=1&amp;code=1422237874120 HTTP/1.0" 200 10819 "http://financezhen.com/?p=186#respond" "Mozilla/5.0 (Windows NT 6.1; Win64; x64; rv:2.2a1pre) Gecko/20110324 Firefox/4.2a1pre"   I don't know why these requests goes to my server, or why it returns 200...  Also, when apache starts and module prefork is enabled, it launches multiple httpd processes. If worker module is enabled, it launches multiple threads. Ok, but in both cases apache ends up consuming all the CPU and memory of the server, and there is not traffic directed to my server.  More info: ProxyRequests is Off. Also: I'm using ip-based vhosts:  NameVirtualHost xxx.xxx.xxx.xxx:80  &lt;VirtualHost xxx.xxx.xxx.xxx:80&gt;     ServerName mydomain.com     ServerAlias www.mydomain.com     DocumentRoot /var/www/html/w3prod     ServerAdmin admin@localhost     UseCanonicalName Off     CustomLog /var/log/httpd/w3prod.mydomain.com combined     ErrorLog /var/log/httpd/w3prod_error.mydomain.com &lt;/VirtualHost&gt; &lt;VirtualHost xxx.xxx.xxx.xxx:80&gt;     ServerName picfarm.mydomain.com     ServerAlias www.picfarm.mydomain.com     DocumentRoot /var/www/html/picfarm     ServerAdmin admin@localhost     UseCanonicalName Off     CustomLog /var/log/httpd/picfarm.mydomain.com combined     ErrorLog /var/log/httpd/picfarm_error.mydomain.com &lt;/VirtualHost&gt; &lt;VirtualHost xxx.xxx.xxx.xxx:80&gt;     ServerName test.mydomain.com     ServerAlias www.test.mydomain.com     DocumentRoot /var/www/html/engine-test     ServerAdmin admin@localhost     UseCanonicalName Off     CustomLog /var/log/httpd/test.mydomain.com combined     ErrorLog  /var/log/httpd/test_error.mydomain.com     ProxyRequests Off     ProxyPreserveHost On     &lt;Proxy *&gt;             Order deny,allow             Allow from all     &lt;/Proxy&gt;     ProxyPass /bosh http://xxx.xxx.xxx.xxx:7070/http-bind/     ProxyPassReverse /bosh http://xxx.xxx.xxx.xxx:7070/http-bind/     ProxyPass / http://xxx.xxx.xxx.xxx:8080/     ProxyPassReverse / http://xxx.xxx.xxx.xxx:8080/ &lt;/VirtualHost&gt;   Any help would be appreciated. Thanks! Can we get our personal voting data displayed on the Questions lists? There's been several times where I have been surfing the site, and noticed a question that I am not sure I've checked. Even if I've checked the question before, I can't recall off-hand whether or not I've voted on it or its answer.  Is there a way we can have our personal voting history reflected to some degree or another, in the Questions listing? For example, could there be an indicator as to whether or not we have already voted (up or down) on a question, and perhaps maybe an additional indicator as to whether we've up-voted any of its answers?  It would be very convenient to have some of this data available up-front, without actually having to view the question again. New top bar doesn't show hot threads across Stack Exchange sites Right-hand sidebar covers the footer on the "ask question" page and markdown help page Why does the set of all singleton sets not exist? When should I use exposure compensation? When should I use exposure compensation, rather than ISO, shutter speed or aperture? Fastest method to determine my PC's IP address (Windows) This may seem trivial but I'm looking for the quickest method to determine my PC's IP address within my network.   This changes regularly as I connect from one network to the next via DHCP The connection changes from wired to wireless Windows 7 PC (although ideally the perfect solution would work on all versions of Windows) Various adapters installed (e.g. VMWare, Tunnels, etc.)   I need to know this as I often run a local web server where I need to access files over the local network... and since it changes regularly I want the fastest method to retrieve the address.  My current flow is:   WindowsKey + R to open the Run dialog "cmd" + Enter to open a command prompt "ipconfig" + Enter to get the diagnostics info Scroll to or resize the window so that I can see the line in my Ethernet adapter Local Area Connection for my ipv4 Address Remember it to type elsewhere... or  Right click > select Mark > highlight the address > press Enter to copy it to the clipboard   This wasn't so bad with Windows XP when I had no additional adapters, tunnels, wireless connections etc. but the amount of data returned with this command makes it hard to pluck out.  Surely there must be a better, faster way! (bonus points if adding it to the clipboard is easily accomplished) Under what conditions is the exponential map on a Lie algebra injective? Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\exp :\mathfrak{g}\rightarrow G$ be the exponential map.  In his blog,  that if a Lie group is not simply-connected, then $\exp$ will not be injective.  Conversely, is it true that if a Lie group is simply-connected, then $\exp$ is injective?  If not, what is a counter-example? C: pointer to struct in the struct definition How can I have a pointer to the next struct in the definition of this struct:  typedef struct A {   int a;   int b;   A*  next; } A;   this is how I first wrote it but it does not work. Show that if $f: \mathbb{R} \to \mathbb{R}$ is a continuous injective map, then it is strictly monotonic.    Could someone give me a proof for this? I have the intuition for why it's true - I'm just having trouble expressing that intuition in a rigorous manner. Basically consider two points $x_1, x_2 \in \mathbb{R}$. By the problem statement, $f$ is continuous on $[x_1, \, x_2]$. WLOG, assume that $f$ is strictly increasing. It there exists a point where it is not increasing, then $f$ hits a value twice, and it's not injective. Can't compile any c/ c++ program, linker problem It's my second post here, i have a huge problem which prevents me to study so i have to fix it as soon as it's possible ! My gcc was working fine and i had to work on FairThreads (in C). I downloaded the following archive :  As you can see, in /src there's a makeFile, i replaced the second line which was : LIB_DIR     = ../lib  by LIB_DIR     =  /usr/bin/ld and i compiled the makeFile by the command &quot;make&quot; And now, i cannot compile any c/c++ program anymore. Even a hello world program. I got the following error message on the terminal for every compiling attempt : /usr/bin/ld: 2: /usr/bin/ld: Syntax error: newline unexpected collect2: error: ld returned 2 exit status  I have found on the net this solution but it doesn't work for me : sudo apt-get autoremove binutils     sudo apt-get install binutils  I also tried to reinstall gcc, when executing : sudo apt-get purge gcc   i got : Des erreurs ont été rencontrées pendant l'exécution :   linux-firmware   linux-image-generic-hwe-18.04   linux-generic-hwe-18.04 E: Sub-process /usr/bin/dpkg returned an error code (1)  but when i trie a gcc --version it tells me that it is not found. So when i execute the command to reintall it : sudo apt-get install gcc   i also get : Des erreurs ont été rencontrées pendant l'exécution :  linux-firmware  linux-image-generic-hwe-18.04  linux-generic-hwe-18.04  linux-image-5.4.0-72-generic  initramfs-tools E: Sub-process /usr/bin/dpkg returned an error code (1)  but gcc --version tellls that it's installed. Finally, when i trie to compile a program, i got : /usr/bin/ld: 2: /usr/bin/ld: Syntax error: newline unexpected collect2: error: ld returned 2 exit status  I hope that i explained well, and i'll get a solution for this :/, as a computer science student, i can't do anything without gcc :( EDIT 1 : When i try to remove bin/ld by executing : sudo rm -fr /usr/bin/ld;sudo apt-get install --reinstall binutils  i got : Des erreurs ont été rencontrées pendant l'exécution :  linux-firmware  linux-image-generic-hwe-18.04  linux-generic-hwe-18.04  linux-image-5.4.0-72-generic  initramfs-tools E: Sub-process /usr/bin/dpkg returned an error code (1)  and this particular error message : WARNING: no ldd around - install libc-bin  EDIT : I have installed ldd by the command : sudo apt-get install libc-bin  still no changes... How do I get started in Astrophotography?