prove by induction that (2n)!≥2^n(n!)^2
Use mathematical induction to prove $ (2n)!\geq 2^n(n!)^2$ for $n \in \mathbb{N}$
Why does my dock keep moving back to my other monitor? I have a larger, external monitor connected to my MacBook Pro and have arranged it such that the dock is in the external monitor (via System Preferences > Displays > Arrangement, by dragging the white bar to the larger screen). This works fine, except for the fact that every now and then, the dock will suddenly shift back to my MacBook screen, but when I open up Displays to change this back, the white bar is still on the external monitor. So I have to drag it to the smaller screen, and then drag it back to the external monitor.   It's a minor issue, obviously, but I've noticed that it has been happening more and more frequently and it's getting quite annoying. Perhaps there's something I've been doing unintentionally for this to happen?   I'm using Mavericks, 10.9.5. How can I serialize an object to JSON without properties that are null? To communicate with a REST interface on a private system, I need to send a JSON message from a trigger. I have a message object that has some fields, lists and a child object:  public class MerchantConfig {  public class MIDConfig {     public String mid;     public String currencyCode;     public String cardTemplateName;     public String paymentPanelRestriction;     public List&lt;String&gt; restrictedCountries;      public String paymentPanelRestrictionForCountries; }  public String merchantName; public String sopgTemplateName; public Integer dispositionTimeWindowInMinutes; public List&lt;String&gt; features; public Boolean activateImmediately; public List&lt;MIDConfig&gt; midConfiguration; }   I can serialize it to JSON using the built-in serializer:  String body = JSON.serializePretty([MerchantConfig instance]);   The result:  {   "sopgTemplateName" : null,   "midConfiguration" : [ {     "restrictionCountries" : [ "DE", "AT" ],     "paymentPanelRestrictionForCountries" : "MANDATORY",     "paymentPanelRestriction" : "OPTIONAL",     "mid" : "1000000001",     "currencyCode" : "EUR",     "cardTemplateName" : null   }, {     "restrictionCountries" : [ ],     "paymentPanelRestrictionForCountries" : "",     "paymentPanelRestriction" : "OPTIONAL",     "mid" : "1000000002",     "currencyCode" : "USD",     "cardTemplateName" : null   }, {     "restrictionCountries" : [ ],     "paymentPanelRestrictionForCountries" : "",     "paymentPanelRestriction" : "PROHIBITED",     "mid" : "1000000003",     "currencyCode" : "AUD",     "cardTemplateName" : null   } ],   "merchantName" : "Test Merchant",   "features" : [ XYZ ],   "dispositionTimeWindowInMinutes" : 10,   "activateImmediately" : true }   How can I serialize this object so that all properties that are null don't show up in the JSON string?   In this example, "sopgTemplateName" and all "cardTemplateName" properties are null and should not appear in the JSON string. The empty list "restrictionCountries" should still appear as an empty list, though. Customizing the output of Compass sprites Navigating lookup relationships with dot notations, to retrieve field values I need to access the value from a roll-up field on my Opportunities, from it's related Opportunity Product.  When I use oli.Opportunity.Software_Line_Item__c the value retrieved is always null.  So I've also tried oli.OpportunityId.Software_Line_Item__c which causes an error in the compiler     Invalid foreign key relationship: OpportunityLineItem.OpportunityId   &amp; just to make sure that it's not the roll-up field calculation that's not working as expected, I've tested oli.Opportunity.Name which is also returning null.  I know this is basic but I don't have Eclipse installed so I can't check the correct reference to use. Could someone please let me know how I can navigate this relationship, to retrieve the value from the Software_Line_Item__c field? How to make `sudo` preserve $PATH? Why was Count Dooku not given the title "Darth"? I don't know the exact significance of the word "Darth" in the Sith world, but I think it was a word of honor for powerful Sith. That's why Anakin was given this word in his new name.  All powerful Sith in canon had this word in their names: Darth Plagueis, Darth Bane, Darth Sidious, Darth Vader, Darth Maul, etc. As such name similarity can't be names given by their parents, I can assume they were given such words in names as a honor.   Now, look at Count Dooku... he was a powerful Sith too. So, why wasn't he given this title?    Darth word to address Dooku hasn't been used in the movies, as far as I know. If you are claiming that Darth and Lord words are same, give citations because my vocabulary rejects it. What is the coefficient of the term $a^i b^j c^k$ in the expansion of $(a+b+c)^n$? What is the coefficient of the term $a^i b^j c^k$  in the expansion of   $(a+b+c)^n$ ?   Given solution:     Before collecting like terms, each term in the expansion is a product   of $n$ factors each  of which is $a, b,$ or $c$.  Those terms which   have exactly i a’s, j b’s and k c’s can be combined into   the single term $a^i b^j c^k$ and the number of such terms will be   equal to the number of n letter strings containing i a’s, j b’s and k   c’s, which by the Mississippi rule is $\frac{n!}{(i!\times j! \times  k!)}$.   I understand that this problem is actually about the derivation of the coefficient of the multinomial theorem,but I am not sure about the combinatoric reasoning,what I mean is that I understand the analogy problem but what I couldn't is how the analogy holds here,Could anybody explain this to me? I am looking to replicate black &amp; white maps in QGis, using hatching patterns and greyscales to indicate different categories of a continuous variable (example below).  It is possible/straightforward to do so in QGis? At first sight I only see color gradients and symbol fills.   ). ASP.NET MVC JsonResult Date Format I have a controller action that effectively simply returns a JsonResult of my model. So, in my method I have something like the following:  return new JsonResult(myModel);   This works well, except for one problem. There is a date property in the model and this appears to be returned in the Json result like so:  "\/Date(1239018869048)\/"   How should I be dealing with dates so they are returned in the format I require? Or how do I handle this format above in script? I have a doubt on something. I have a theorem that says that if $R$ is a ring, then $R/I$ is a domain iff $I$ is prime. Since $X^2-1$ is note prime, $\mathbb Z[X]/(X^2-1)$ is not a domain. But, $X^2-1=(X-1)(X+1)$, and since $X-1$ and $X+1$ are prime in $\mathbb Z[X]$, there are coprime, and thus, by the chinese remainder $$\mathbb Z[X]/(X^2-1)\cong \mathbb Z[X]/(X-1)\times \mathbb Z[X]/(X+1),$$ since $\mathbb Z[X]/(X-1)\times \mathbb Z[X]/(X+1)$ is a domain, then so is $\mathbb Z[X/(X^2-1)$. What is the mistakes here ? In The Force Awakens, Finn and Poe steal and escape in a TIE wearing no personal life support apparatus. TIEs have been upgraded with shields, hyperdrives (for "/sf"), weapons etc, but as far as I know they weren't upgraded with built in life support.   Finn and Poe should have frozen and/or suffocated after hitting space in the TIE. In the Rebels series, Sabine did the same thing. I've looked at what Wookieepedia states about the TIE as well as the old video game Star Wars "TIE Fighter".  Does the movie or any other work explain how Poe and Finn survived? Chinese Passport Holder Transit in South Korea A Chinese passport holder with a US visa has a flight scheduled from Beijing to Los Angeles in mid-November.  This person wants to go to South Korea for a 3 day conference; the conference is one week prior to the PEK-LAX flight.  If this person buys a round-trip ticket from China to Korea, will the transit visa be sufficient? Or will they need a regular visa?    Extra info:  What if the original ticket were modified to make it ICN-PEK-LAX?  . It says "Holding a confirmed onward flight ticket for departure within 30 days." Would a return ticket to China than a flight to LAX a week later be sufficiently "onward"? Cannot translate vertice/edge will duplicate Most of the time when I try to move a vertex or an edge, instead of translating that, it extrudes it, it creates another face or, vertiex or whatever. I do not really get why it behaves like that. I hope I explained it in depth. I am sorry if this is a duplicate post but I could not find anything! Set origin to bottom center of multiple objects I have a lot of 3D objects of various sizes, shapes in blender 2.76b and I would like to set the origin to the bottom center of the objects bounds.  I cannot select the bottom edge and snap to cursor etc as each shapes edges are in different X/Y positions.  Is there a script or tool that will allow me to set the origin to the lowest edge z position? A bob hanging in an accelerating train moves backward. What is the force moving it backward? When the train accelerates, the bob would also accelerate with the same magnitude and direction as the train. From the free body diagram, only tension and weight are exerted on the bob. I understand how to relate the "horizontal component" of the tension to the acceleration of the bob. It moves "backward" and in fact its vertical position is different from when the train is not accelerating.  I am not sure whether inertia could explain this situation. What is the force moving it backward/ lifting with the height? For example:  javac Foo.java Note: Foo.java uses unchecked or unsafe operations. Note: Recompile with -Xlint:unchecked for details. Someone flagged my question as already answered, but it's not Prove if f(z) is entire function s.t $|f(z)|\leq M|z|$ then $f(z)$ Is polynomial of degree $1$ Let $f(z)$ be an entire function s.t $|f(z)|\leq M|z|$ ; $M&gt;0$ is a constant then $f(z)$ Is polynomial of degree 1   My try $f(z)$ is entire so it can be written as a $\sum a_n z^n$  $$|\sum a_n z^n|\leq \sum |a_n||z|^n\leq M|z|$$  So $n=1$  Seems that the move $\sum |a_n||z|^n\leq M|z|$ is problematic How do I make my textures show in cycles? I am admittedly new to Blender, and of course cycles. I am working on a scene and I have textures set to my objects that render with internal engine just fine. However, when I go into cycles, everything is greyish. I have two light sources that should be sufficient (I think), I just don't know what I'm doing wrong. This is a screenshot of the cycles result. number of distinct solution of Integral equation Total number of distinct $0\leq x\leq 1$ for which $\displaystyle \int_{0}^{x}\frac{t^2}{1+t^4}dt = 2x-1$   $\bf{My\; Try::}$ Given $\displaystyle \underbrace{\int_{0}^{x}\frac{t^2}{1+t^4}dt}_{\geq 0} = 2x-1\;,$ Where $0\leq x\leq 1$  So we get $\displaystyle x\geq \frac{1}{2}\;\;\;,$ Now  $\displaystyle \frac{d}{dx}\left\{\int_{0}^{x}\frac{t^2}{1+t^4}dt\right\} =\frac{d}{dx}(2x-1)$  So $$\frac{x^2}{1+x^4}=2\Rightarrow \underbrace{x^2+\frac{1}{x^2}}_{\geq 2} = \frac{1}{2}$$  So no real values of $x$  Is my solution is Right, If not then how can i solve it some short way, Thanks