Let $a_1, a_2, ..., a_k$ are real numbers greater thah $1$. Prove that   $$\frac1{1+a_1}+\frac1{1+a_2}+...+\frac1{1+a_k}\ge\frac k{1+\sqrt[k]{a_1a_2...a_k}}$$   I have this inequality proved by Sturm's method. Is there another way? May be  Jensen's inequality.
If $x_1,x_2,\ldots,x_n$ are real numbers larger than $1$, prove that $$\dfrac{1}{1+x_1}+\dfrac{1}{1+x_2}+\cdots+\dfrac{1}{1+x_n} \geq \dfrac{n}{\sqrt[n]{x_1x_2\cdots x_n}+1}$$   Attempt  AM-GM doesn't work here since we will get an upper bound. I don't see Cauchy-Schwarz working either. Thus, I think a substitution might work, but I am unsure of which one to use.
Let's say I have a generic member in a class or method, like so: public class Foo&lt;T&gt; {     public List&lt;T&gt; Bar { get; set; }          public void Baz()     {         // get type of T     }    }  When I instantiate the class, the T becomes MyTypeObject1, so the class has a generic list property: List&lt;MyTypeObject1&gt;. The same applies to a generic method in a non-generic class: public class Foo {     public void Bar&lt;T&gt;()     {         var baz = new List&lt;T&gt;();                  // get type of T     } }  I would like to know what type of objects the list of my class contains. So what type of T does the list property called Bar or the local variable baz contain? I cannot do Bar[0].GetType(), because the list might contain zero elements. How can I do it? Why does this function return a different value every time? I am looking for a way to replace a string in a file with a string that contains a slash by using sed.  connect="192.168.100.61/foo" srcText="foo.bar=XPLACEHOLDERX" echo $srcText | sed "s/XPLACEHOLDERX/$connect"   The result is:  sed: -e Expression #1, Character 32: Unknown option for `s' How can I take a site offline using Drush? What does echelon mean? When you solve a system of linear equations, you write down the augmented matrix and reduce this to reduced row echelon form. What is the meaning of the word echelon? I'm having some difficulty with understanding why KVL (Kirchhoff voltage law) is valid when considering an AC circuit. I mean, the system is by definition not in steady state, right? That in turn means that the change in current causes a change in the magnetic field, which is manifested within the electric potential. I realize the potential is still defined and thus a line integral over a crossed curve of ALL of it's component should be zero.   However, what I often see (for, say, RLC circuits with an AC emf) is total disregard to the time varying components of the electric potential. Only the emf of the different components are considered and their sum is said to be zero. How can that be? What about the "second" part of the potentials? Interaction between Leyline of Anticipation and Teferi, Time Raveler With the reprint of  in Core Set 2020, the interaction with  came to mind and I couldn't figure out what is right.  If I have the Leyline and my opponent has Teferi, may I cast spells during my opponent's turn? In the Law of Reflection, the angle of incidence is equal to angle of reflection. Why is this true? This is clearly true experimentally, but how does one prove this true mathematically? Has anyone used PostgreSQL with PostGIS successfully?   I am a semi-experienced ArcGIS user and want to build out a database and connect it to ArcCatalog.   I've scoured the internet for resources but nobody has really given a good walkthrough. How to use exclamation marks in Minecraft commands? I know that you can use exclamation marks in Minecraft commands to check for an entity that does not have a specific tag. For example, you can use /kill @e[type=!Player] to kill all entities except for the player. I have, however, encountered two problems with using exclamation marks that I can't find a solution for.  For my first problem, I've set up a series of command blocks to summon shootable fireballs above the player’s head when a certain item is being held;  /scoreboard players add @a Fireball 1 {SelectedItem:{tag:{display:{Name:"Summon Fireball"}}}} /execute @a[score_Fireball_min=1,score_Fireball=1] ~ ~ ~ /summon Fireball ~ ~2.5 ~ {direction:[0.0,0.0,0.0]} /scoreboard players set @a Fireball 0   I used a comparator and a redstone torch so that the last command would activate when the first command would deactivate, that is, when the player is not holding the item.  The problem is, that when I move far away from the setup, the redstone stops working, leaving me unable to summon any new fireballs. Instead, I'm trying to use a command block to check when anything except for item is being held;  /scoreboard players set @a Fireball 0 {SelectedItem:{tag:{display:{Name:!"Summon Fireball"}}}}   I’ve either put it at the wrong place, or you simply can’t use exclamation marks in NBT tags. Could someone tell me where I should put the exclamation mark, or in the latter case, how I can check for anything except for a specific item?  Further, I’ve tried using exclamation marks to set the score of all entities, including the ones with a blank score, but except for the ones with a score of 2 or higher, to 0;  /scoreboard players set @e[score_arrows_min!=2] Arrows 0 /scoreboard players set @a Arrows 1 {SelectedItem:{tag:{display:{Name:"Rain of Arrows"}}}} /scoreboard players set @e[type=Arrow] Arrows 2 {inGround:1b}   The code does change the scores of entities without scores to 0, but it doesn’t exclude the arrows with a score of 2. Could anyone tell me what I’m doing wrong here? Converting a drupal/drupal site to Work With Composer Easy way to remap non-modifier keys on Mac? I want to remap § to ` and ± to ~ on my Mac keyboard because they are located in a place I am not used to. Is there an easy way to do this? Identify which OrgWideEmailAddress records are valid for current user I'd like to create a popup list on one of my Visualforce pages, for a user to select from a list of OrgWideEmailAddress records.  My goal is to limit the list to only those records that they're allowed to use.  I'm going to save the selection for use in later automatic emails.  Based on what I've read, I won't be able to send emails to certain OrgWideEmailAddresses:   The address hasn't been verified yet.  I don't see any fields in the schema that appear to show verification state. The address is only available to profiles that the user isn't a member of.  I don't see where the profile list is stored for each record.   Some of my problem could just be due to not understanding the standard objects so well, and perhaps there's a way to get at this through UserInfo, or by making my query on the User object which may have a child relationship to OrgWideEmailAddress since it's listed in the schema viewer of the Force.com IDE.  Here's the test code I'm working with:  for (OrgWideEmailAddress addr : [SELECT DisplayName, Address          FROM OrgWideEmailAddress ORDER BY DisplayName]) {     System.debug( addr.DisplayName + ' &lt;' + addr.Address + '&gt;'); }   I tried creating a new Messaging.SingleEmailMessage and then calling setOrgWideEmailAddressId() on it with each OrgWideEmailAddress Id, in hopes that invalid Ids would throw an exception, but to no avail.  My goal is to modify the test code to provide just the subset of records that the current user will be able to use. Find and Replace several several different values all at once I know that there exists at least one solution, because $ax \equiv b \mod m \implies ax - b = qm$  Rearranging terms gives $ax - qm = b$, and since $d | a$ and $d | m$, so $d | b$ it can be said that $b \equiv 0 \mod d$.  I implemented Euclid's extended algorithm and have been able to experimentally verify that the claim holds, but I do not see a pattern or how I can begin to show there are exactly $d$ solutions.  def euclid_extended(x, y, i=0):   if y == 0:     print("iteration {0} returns ({1}, {2}, {3}) - base case hit with ({4}, {5})".format(i, x, 1, 0, x, y))     return (x, 1, 0)   else:     (d, a, b) = euclid_extended(y, x % y, i + 1)     print("iteration {0}. ({4}, {5}) =&gt; ({1}, {2}, {3})".format(i, d, b, a - b * (x//y), x, y))     return (d, b, a - b * (x//y)) I read on  that Ubuntu will no longer use the familiar file size units we all know by now (kB, MB, GB, TB) and switch to a different IEC standard (KiB, MiB, GiB, TiB). If this is true, I would like to know what's the reasoning behind this change, and the impact (if any) this change has, especially with multiplatform applications or applications run with Wine. I am having a problem with Windows 7. When I boot up, the CPU jumps to 100% execution for about 10-15 minutes. During this time, it is really slow, and if I play any music, the songs play slowly and sound distorted.  I tried to scan my computer with different antivirus programs, I used Spybot, HijackThis, and CCleaner, but I didn’t find anything of note.  I watched the process list in Task Manager and saw that there are two processes that are to blame. There are two copies of svchost.exe, one using ~48% CPU, the second (under the Network Service account) using ~37%. For the first 15 minutes, the CPU runs at about 70-80%, then eventually drops to 30-40%.  I don’t know what to do. I could not find a way to stop the services, and if I try to kill them, sometimes the system crashes. This one seems to be a common question in most forums and all over the web, it is asked here in many formats that typically sound like this:  In SQL Server -  What are some reasons the transaction log grows so large? Why is my log file so big? What are some ways to prevent this problem from occurring? What do I do when I get myself on track with the underlying cause and want to put my transaction log file to a healthy size? I want to install Ubuntu on the C: partition where my Windows 7 is, simply to replace the systems. I've also tried to install it on another partition (without formating it), which failed!  All I get is:      No root file system is defined. Please correct this from the partitioning menu.   I want to install Ubuntu without formatting any of my partitions. Is that possible? Python — How can I find the square matrix of a lower triangular numpy matrix? (with a symmetrical upper triangle)