Prove the inequality $u^{\alpha}v^{1-\alpha} \leq \alpha u + (1-\alpha)v $
Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$
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? When speaking precisely or technically, one would say that "Homo erectus and homo sapiens are two species of hominid" rather than "Homo erectus and homo sapiens are two species of hominids." The hominid here should be singular because we are speaking about instances of a single class ("class" being used here in its broader sense, not in the sense of taxonomic grouping).  Now let's consider more common parallel constructions replacing the word species with other words.      Tide and Wisk are two brands of detergent.       Tide and Wisk are two brands of detergents.      Cats and dogs are two types of pet you can buy at Pets-R-Us.       Cats and dogs are two types of pets you can buy at Pets-R-Us.   I feel that using the singular noun to specify the class is more grammatically correct, but sounds stilted in conversation. Almost everyone I hear making these constructions uses the plural form. What are your thoughts about singular vs. plural here?  Edit  Here's a more extreme example that may help clarify the issue. Consider the following sentence:     It was my first attempt at calming an angry crowd, and I just        stood there while people were hurling all kinds of insults at me.   Here I think the singular insult sounds strange, if not downright wrong.  Further Edit  I just noticed in A Treasury for Word Lovers (Morton S. Freeman, 1983) a section entitled "Types of Errors." This is a book about (American) English usage by an English professor and editor, which purports to be a "practical guide for serious writers and readers." I wish I could say this discovery satisfies the question, but in fact I now feel somewhat farther from the truth, if there is any single "truth" in this matter. But at least I don't have to worry about being wrong when using the plural form of the class in these constructions. Is there a way to see how the markup of a per-site meta answer is formed? I sometimes see a useful markup feature in a per-site meta answer. However, I cannot edit it to see how one did it, since editing (or suggesting edits) is disabled for users with less than 2k reputation on per-site metas.  Is there a way to see the source code of a per-site meta answer? Is doing two PhDs a good path? I am studying a combined bachelor of engineering (electrical) and bachelor of mathematics; I just started this year and will graduate in 2018. The reason why I am doing double degrees and not a single degree is because I love both electrical engineering and mathematics and I could not ignore any of them. So with this in mind, I am thinking of doing two PhDs when I graduate (one in electrical engineering and one in mathematics). Is this a good path or I should concentrate on only one of them? I need to add 30 minutes to values in a Oracle date column.  I do this in my SELECT statement by specifying   to_char(date_and_time + (.000694 * 31)  which works fine most of the time.  But not when the time is on the AM/PM border.  For example, adding 30 minutes to 12:30 [which is  PM] returns 1:00 which is AM. The answer I expect is 13:00.  What's the correct way to do this? How much web can Spider-Man produce? Especially if he is not using his  but .   Also what happens to the web that he leaves behind? Does it decay after certain time or lose it stickiness and drops off? I have just upgraded to Ubuntu 17.10 and found out there is no option to have 2 by 2 workspaces. I was able to configure 4 "static" workspaces using , but I would like to have them in 2 rows by 2. Any ideas? Big Parenthesis in an Equation I have an equation contained inside \[...\], which automatically makes a \sum with sub- and superscripts turn big--so that the summation sign looks awkward inside parenthesis.  Any idea how to make the parenthesis completely enclose the whole summation?  \documentclass{article} \begin{document} We have: \[ \sum_{i=1}^n i = (\sum_{i=1}^{n-1} i) + n = \frac{(n-1)(n)}{2} + n = \frac{n(n+1)}{2} \] \end{document} Is there evidence that adding salt to water prior to boiling can damage a stainless steel pan? Two different solutions to integral Given the very simple integral  \begin{equation} \int -\frac{1}{2x} dx \end{equation}  The obvious solution is  \begin{equation} \int -\frac{1}{2x} dx = -\frac{1}{2} \int \frac{1}{x} dx = -\frac{1}{2} \ln{|x|} + C \end{equation}  However, by the following integration rule \begin{equation} \int \frac{1}{ax + b} dx = \frac{1}{a} \ln{|ax + b|} + C \end{equation}  the following solution is obtained \begin{equation} \int -\frac{1}{2x} dx = -\frac{1}{2}\ln{|-2x|} + C \end{equation}  Why are these solutions different? Which is correct?   The second solution can be simplified \begin{equation} -\frac{1}{2}\ln{|-2x|} + C = -\frac{1}{2}\ln{|-2|} -\frac{1}{2}\ln{|x|} + C= -\ln{\frac{1}{\sqrt{2}}} - \frac{1}{2}\ln{|x|} + C \end{equation} but they still differ. How to share files through the local network? I've taken multivariate calculus and am wondering if I can see a specific function where the order of taking the partial derivative matters. I've been told that there are some exceptions where $ \dfrac{\partial ^2 f}{\partial x \partial y} \ne \dfrac{\partial ^2 f}{\partial y \partial x} $, so I'm curious to see what this looks like.  EDIT: And why would this true? recover data after using dd command I never want my PNGs to be interlaced. So I wish I could set a bunch of option defaults in the application so that I am saving time when outputting a lot of images as PNG over the course of a day working in Photoshop.  Does anyone know if this is possible in Photoshop without using Actions OR the Save-for-Web option? Why is the result of a raster calculation integer when it should be decimal? I'm trying to calculate the NDVI using raster calculator, but the result is integer instead of decimal.  NDVI = (NIR - RED)/(NIR + RED)   The result is normally in the range -1 to 1, but when i tried to calculate de result is -1, 0 and 1, and it does not have have any number in between like, -0.9 or 0.1. And i need those values so i can classify the result in term of soil occupation Kubuntu - can not enter wifi credentials after adding external wifi card My ethernet connection is quite unreliable (It uses powerline adapters) so I bought an external wifi adapter. I have plugged it in and run the command sudo ip address add 192.168.0.39/24 brd + dev enp7s0, and KDE has detected it. Unfortunately, it doesn't seem to realise it is a wireless network and therefore needs credentials. Where I could normally find a place to enter credentials (the network widget, system settings) there are none. KDE reports that the device is connected in the UI. How do I make it work out that it is wireless and therefore needs credentials? Edit: With ethernet plugged out, (USB) adapter plugged in: $ ip address show 1: lo: &lt;LOOPBACK,UP,LOWER_UP&gt; mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000     link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00     inet 127.0.0.1/8 scope host lo        valid_lft forever preferred_lft forever     inet6 ::1/128 scope host         valid_lft forever preferred_lft forever 2: enp7s0: &lt;NO-CARRIER,BROADCAST,MULTICAST,UP&gt; mtu 1500 qdisc fq_codel state DOWN group default qlen 1000     link/ether 04:d9:f5:8a:f5:e0 brd ff:ff:ff:ff:ff:ff  With ethernet plugged in, adapter plugged in: 1: lo: &lt;LOOPBACK,UP,LOWER_UP&gt; mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000     link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00     inet 127.0.0.1/8 scope host lo        valid_lft forever preferred_lft forever     inet6 ::1/128 scope host         valid_lft forever preferred_lft forever 2: enp7s0: &lt;BROADCAST,MULTICAST,UP,LOWER_UP&gt; mtu 1500 qdisc fq_codel state UP group default qlen 1000     link/ether 04:d9:f5:8a:f5:e0 brd ff:ff:ff:ff:ff:ff     inet 192.168.0.51/24 brd 192.168.0.255 scope global dynamic noprefixroute enp7s0        valid_lft 863988sec preferred_lft 863988sec     inet6 fe80::7eb1:eca2:96a0:fe5c/64 scope link noprefixroute         valid_lft forever preferred_lft forever  With just ethernet: 1: lo: &lt;LOOPBACK,UP,LOWER_UP&gt; mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000     link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00     inet 127.0.0.1/8 scope host lo        valid_lft forever preferred_lft forever     inet6 ::1/128 scope host         valid_lft forever preferred_lft forever 2: enp7s0: &lt;BROADCAST,MULTICAST,UP,LOWER_UP&gt; mtu 1500 qdisc fq_codel state UP group default qlen 1000     link/ether 04:d9:f5:8a:f5:e0 brd ff:ff:ff:ff:ff:ff     inet 192.168.0.51/24 brd 192.168.0.255 scope global dynamic noprefixroute enp7s0        valid_lft 863960sec preferred_lft 863960sec     inet6 fe80::7eb1:eca2:96a0:fe5c/64 scope link noprefixroute         valid_lft forever preferred_lft forever  With nothing: 1: lo: &lt;LOOPBACK,UP,LOWER_UP&gt; mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000     link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00     inet 127.0.0.1/8 scope host lo        valid_lft forever preferred_lft forever     inet6 ::1/128 scope host         valid_lft forever preferred_lft forever 2: enp7s0: &lt;NO-CARRIER,BROADCAST,MULTICAST,UP&gt; mtu 1500 qdisc fq_codel state DOWN group default qlen 1000     link/ether 04:d9:f5:8a:f5:e0 brd ff:ff:ff:ff:ff:ff  lsusb, adapter plugged in: Bus 004 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub Bus 003 Device 002: ID 046d:c31c Logitech, Inc. Keyboard K120 Bus 003 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub Bus 002 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub Bus 001 Device 003: ID 1d57:32da Xenta 2.4GHz Receiver (Keyboard and Mouse) Bus 001 Device 004: ID 0bda:b812 Realtek Semiconductor Corp. 802.11ac NIC Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub  I'm not adding it with it plugged out because it's obviously the Bus 001 Device 004: ID 0bda:b812 Realtek Semiconductor Corp. 802.11ac NIC. There were no restarts between any of these changes. Natbib -- change author conjunction in citation call-out from 'and' to '&' I'm using natbib, german babel and natbib creates output of the form   Foo and Bar (1337)   I would like to get   Foo &amp; Bar (1337) for citet and (Foo &amp; Bar 1337) respectively for citep. As of patch 1.5, I am completely unable to see underwater in Skyrim.  It's not even that it's murky or dark - it's just a solid grey or green screen.  I can still skim across the surface and see into deep waters, but if I go any deeper than that, nothing is visible.  I can still access items like chests underwater if I figure out their location first (by skimming across the surface) and then point my character there and hope really hard that it gets me to the right place.  I was never able to see much underwater, but I at least used to be able to see small things like wood floating near a shipwreck.  Now, there's just nothing. I am running the Steam version of Skyrim with graphics on Ultra High, with the high res textures DLC.  I've tried disabling the high res textures (along with all my other mods) and it still happens.  In case it's relevant, my GPU is a GTX 560 Ti.  Does anyone have any solutions to this? What is the difference between the Circle-K and Lawson Power Cube? JSON and escaped double quote I'm currently making a callout to a 3rd party vendor and I'm getting a huge response. What is currently messing me up is this section:   "MessageSeq":13,"Description":"config file: config-qa1.txt\"","MessageTS":"2014-02-17T14:46:58","MessageLevel":4,"IsSystemMessage":false   As you can see here, the portion at the end of the Description value is ended with backslash double quote. If I attempt to parse this via the JSONParser I get an exception. Does anyone have an recommendations on how to resolve this?  Exception Message:  System.JSONException: Unexpected character ('"' (code 34)): was expecting comma to separate OBJECT entries at [line:1, column:3699]  Updated with sample Code   public class ResponseWrapper{     // fields from JSON to auto map     public Integer messageSeq;     public String Description;     public String messageTS;     public Integer messageLevel;     public Boolean isSystemMessage;     // parsing fields     private ResponseWrapper theResponse;     private JSONParser parser;      public void parse(String jsonString){         parser = JSON.createParser(jsonString);         parseJSON();     }      private void parseJSON(){            while (parser.nextToken() != null) {             if (parser.getCurrentToken() == JSONToken.START_OBJECT) {                 theResponse = (ResponseWrapper)parser.readValueAs(ResponseWrapper.class);                            }         }     } }  String responseString = '{"MessageSeq":13,"Description":"config file: config-qa1.txt\"","MessageTS":"2014-02-17T14:46:58","MessageLevel":4,"IsSystemMessage":false}'; system.debug('\n\nBEFORE\n' + responseString); ****Place code in here to test such as ReplaceALL*** ResponseWrapper parser = new ResponseWrapper(); parser.parse(responseString);