For $\alpha &gt; 0$, we want to prove that $$ \left| \sum_{k=1}^n a_k b_k \right| \leq \dfrac{1}{\alpha}  \sum_{k=1}^n a_k^2 + \dfrac{\alpha}{4} \sum_{k=1}^n b_k^2 $$  attempt We may rewrite our inequality as $$ \sum \alpha a_k b_k \leq \sum a_k^2 + \sum \left( \frac{b_k \alpha }{2} \right)^2 $$ Now, we may apply $a^2 + b^2 \geq 2ab $ as follows: $$ \alpha a_k b_k = 2 a_k \dfrac{ b_k \alpha}{2} \leq a_k^2 + \left( \dfrac{b_k \alpha }{2} \right)^2$$ Now, adding up from $k=1$ to $n$ proves our result. Is my approach correct?
Let $X_1,X_2,\ldots,X_n$ and $A_1,A_2,\ldots,A_n$ be real numbers taken at random, and $\alpha\gt0$. Show that: $$\sum_{k=1}^n |X_kA_k| \le \frac{1}{\alpha}\sum_{i=1}^n X_k^2 + \frac{\alpha}{4}\sum_{k=1}^n A_k^2$$
What is the difference between Fact and Truth? I'm curious about the difference between Fact and Truth. I was searching on the internet if I could find it. But still I'm confused about the exact meaning. I first read the forum discussion here  where an author has given two examples for each like below     A fact is a reality that cannot be logically disputed or rejected. If   I say "fire is hot," I don't care how great your reasoning skills are,   if you touch fire your skin will burn (and don't give me that "but   people can walk on hot coals!" bull. There's a difference between the   transfer of heat through conduction and training one's body to deal   with the agonizing pain of said conduction). Now when I say this, I am   not speaking a truth, I am speaking a fact. If you say "fire is not   hot," you are not lying, you are incorrect. Facts are concrete   realities that no amount of reasoning will change. When one   acknowledges a fact, they are doing just that. Facts are not   discovered, facts are not created, facts are simply acknowledged.      A truth on the other hand, is almost the opposite. Truths are those   things that are not simply acknowledged, but must be discovered, or   created. If I say "God exists," and I possess strong reasoning for the   affirmative of that statement, then God really does exist, that is a   reality. However, if another individual possesses strong reasoning for   the negative, and because of this reasoning they believe that God does   not exist, then that is also a reality. If we were to debate our   ideologies, and my reasoning appeared stronger than theirs, they may   choose to adopt my belief that God does exist. If they do, then the   existence of God is just as true as the nonexistence of God which they   believed a week ago. Truths, as opposed to fact, are much more fluid   and malleable than their empirical counterparts.   and followed by further discussion.  Then I found this . Article from above link says like below:     Facts are notes and lyrics on sheet music.  Truth is what the singer   gives to the listener when she’s brave enough to open up and sing from   her heart.   But still curious about the difference between both of them.  In our daily life, in general conversation, we generally use these both terms interchangeably. Then what is the difference? Are they synonym or have specific difference? Difference between "I'm home" vs "I'm at home" I was trying to accomplish this through a trigger, but its not really working. Is there a way to get this done through Process Builder? If so, what is that process?  I'm trying to have ProductB being added to the opportunity automatically once ProductA is manually added to the opportunity.  Also, is there a way to have the $ amount of ProductB being 0.5% of ProductA?  Trigger that I have so far:  trigger Attempt1 on OpportunityLineItem (after insert) { List&lt;OpportunityLineItem&gt; oliList = new List&lt;OpportunityLineItem&gt;();  for( OpportunityLineItem item: [SELECT  OpportunityId,                                          Name                                  FROM    OpportunityLineItem                                 WHERE   Id = :Trigger.new AND                                          Name = 'NAME OF PRODUCTA'])  {   oliList.add(new OpportunityLineItem(     OpportunityId = item.OpportunityId,      PricebookEntryId ='01uA000000QPDldEAt',     Quantity = 1,     UnitPrice = 25)   );  }  Insert oliList; } So if I'm correct, when you chose your starter gear and wealth you can either choose to roll some dice depending on your class and buy everything from that or you can choose the starter gear from your class and add the background items (including some gold). Here are my 2 questions:   If I choose the gear from my class, do I have to take all the equipment from only one of the (a) or (b) choices, or can I freely mix-and-match choices from the two sets? Can you spend the extra gold you get from your background on extra equipment (e.g. some soap and some fishing equipment) or do you have to start with that gold in its pouch?   The reason I ask is that I have recently created a level 1 wood elf druid and I mixed-and-matched the gear from the class, so that I chose a shortbow as a "(b) simple weapon" for the first point and a "(a) Scimitar" at the second point. My DM wasn't so happy about this, as he can't understand why I'm mixing (a) and (b).  He also wasn't so happy with me spending my 10gp from my outlander background. I decided that my druid wouldn't lay so much worth in 10gp compared to a bunch of stuff I got for that (e.g. some soap and some fishing equipment and some other useful things) and he seems to think that I'm supposed to just start with the gold instead. I am looking for an option to require a passcode to power off my iPhone. That way, if the phone is lost, the person who finds it cannot simply turn it off, rendering Find My iPhone useless. Is there a way to accomplish this? In thinking about Sturm-Liouville theory a bit I see I have no actual idea what is going on.  The first issue I have is that my book began with the statement that given  $$L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  the problem $L[y] \ = \ f$ can be re-cast in the form $L[y] \ = \ \lambda y$.  Now it could be a typo on their part but I see no justification for the way you can just do that!  More importantly though is the motivation for Sturm-Liouville theory in the first place. The story as I know it is as follows:  Given a linear second order ode   $$F(x,y,y',y'') = L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  it is an exact equation if it is derivable from a differential equation of one order lower, i.e.  $$F(x,y,y',y'') = \frac{d}{dx}g(x,y,y').$$  The equation is exact iff  $$a''(x) - b'(x) + c(x) = 0. $$  If $F$ is not exact it can be made exact on multiplication by a suitable integrating factor $\alpha(x)$.  This equation is exact iff  $$(\alpha(x)a(x))'' - (\alpha(x)b(x))' + \alpha(x)c(x) = 0 $$  If you expand this out you get the Adjoint operator  $$L^*[\alpha(x)] \ = \ (\alpha(x)a(x))'' \ - \ (\alpha(x)b(x))' \ + \ \alpha(x)c(x) \ = 0 $$  If you expand $L^*$ you see that we can satisfy $L \ = \ L^*$ if $a'(x) \ = \ b(x)$ &amp; $a''(x) \ = \ b'(x)$ which then turns $L[y]$ into something of the form  $$L[y] \ = \ \frac{d}{dx}[a(x)y'] \ + \ c(x)y \ = \ f(x).$$  Thus we seek an integratiing factor $\alpha(x)$ so that we can satisfy this &amp; the condition this will hold is that $\alpha(x) \ = \ \frac{1}{a(x)}e^{\int\frac{b(x)}{a(x)}dx}$  Then we're dealing with:  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \alpha(x)f(x)$$  But again, by what my book said they magically re-cast this problem as  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \lambda \alpha(x) y(x)$$  Then calling  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ ( \alpha(x)c(x)y \ - \ \lambda \alpha(x) )y(x) \ = \ 0$$  a Sturm-Liouville problem.  My question is, how can I make sense of everything I wrote above? How can I clean it up &amp; interpret it, like at one stage I thought we were turning our 2nd order ode into something so that it reduces to the derivative of a first order ode so we can easily find first integrals then the next moment we're pulling out eigenvalues &amp; finding full solutions - what's going on? I want to be able to look at $a(x)y'' \ + \ b(x)y' \ + \ c(x)y \ = \ f(x)$ &amp; know how &amp; why we're turning this into a Sturm-Liouville problem in a way that makes sense of exactness &amp; integrating factors, thanks for reading! writing js objects to files (including methods) in nodejs? I know that if (p) is maximal then Z/(p) is a field which implies its an integral domain which then implies that (p) is a prime ideal and p is prime. I'm having trouble going the forward direction assuming that p is prime and then showing it's a prime ideal. How do I unit test code that includes a callout? This is a canonical question and answer developed by the community to help address common questions. If you've been directed here, or your question has been closed as a duplicate, please look through the resources here and use them to shape more specific questions. To browse all canonical questions and answers, including more unit test resources, navigate to the  tag.  How do I unit test code that includes a SOAP or REST-based callout, or indirectly calls code that makes a callout? I'm getting an error saying that callouts are not supported. How many physical degrees of freedom does the $\mathrm{SU(N)}$ Yang-Mills theory have? The $\mathrm{U(1)}$ QED case has two physical degrees of freedom, which is easy to understand because the free electromagnetic field must be transverse to the direction of propagation. But what are the physical degrees of freedom the $\mathrm{SU(N)}$ Yang-Mills theory?   I believe that the $\mathrm{SU(N)}$ Y.M. theory has four degrees of freedom, and it is easy to see that by gauge fixing (such as the Lorentz condition or the Coulomb condition) that we can always remove one redundant degree of freedom. But then we still haven't completely fixed the redundant degrees of freedom. Thus my question is:     How many physical degrees and how many are redundant degrees of freedom does the Y.M. theory have?    I would be interested to understand how we can determine this mathematically, and also to understand what the physical intuition behind this is? Why upload rates are lower than download rates in normal internet connections? For example, in a novel, if someone is so powerful that no one can stop him, something like this would be said:     He [insert word] in the continent [or some other location].   "Moved unhinderedly" somewhat fits it, but it doesn't really seem like the right term.  The word I'm looking for describes a few things:   Should have an implication that the person himself is powerful That person can go wherever he wishes to, and cannot be stopped because of his strength That person can do whatever he wishes to, and like above, cannot be stopped because of his strength   The other word I have is "swept", i.e.:     He swept through the continent.   I used "swept" because when you sweep the floor, nothing stops you and everything is swept away, similar to the example. However, the dictionary definitions don't match up, so that isn't the word either.  Any suggestions? How do I change a close vote to a duplicate? Voted to close a question based on "lack of minimal understanding".  Later realized the question was a duplicate.  How to I change vote to mark as duplicate instead? I'm trying to figure out a recursive formula for the number of possible distinct binary trees at any depth d. I haven't been able to find any sort of sources on this.  basically, at depth 0, the only binary tree is the empty one so 0, 1  then at depth 1 the only possibility is a single node so 1, 1  at depth = 2 there are 3 possibilities, and at depth 3 I counted 21 possibilities.  I haven't been able to figure out some recursive reasoning for what 4 could be and so on. The only thing I have in place is that there will always be 1 for the full tree. Derivative of a factorial What is ${\partial\over \partial x_i}(x_i !)$ where $x_i$ is a discrete variable?  Do you consider $(x_i!)=(x_i)(x_i-1)...1$ and do product rule on each term, or something else? Thanks. XmlInclude : List and array center text in div I would like to run  something &gt; file   on a remote system through ssh, but if I run  ssh host something &gt; file   the redirection is executed locally as ssh etc &gt; file  I've tried it with ' or '' or dd or with a pipe | instead, but I can't get it to work. How can this be done? A 256GB SSD has Windows-10 and 16.04 installed.  BIOS not UEFI.  A charity I work with requires me to use Quickbooks Desktop on Windows.  The plan is to ghost the existing system to a 512GB or larger SSD and create a third bootable Windows 7  partition for performing charity activities.   Is this possible to have a third partition? Is there a good example you have used as a guide? Is triple booting with two Windows partitions different from 2 Linux partitions?   If you have done this before ( with GRUB) please indicate this in your response: lessons-learned are appreciated.  Thank you I have a Redstone Power loop that connects to a Command Block that is set to kill all entities except for myself (/kill @e[type=!Player]), because I'm building a creative map, but don't want slimes all over the place. Problem is if I want to place an Item Frame than the Command Block kills it.  So, is there anyway to exclude items from getting deleted, or make it so it will only target mobs?