I am trying to understand the answer to  of a .    The vertical coordinate (“height") of an object in free fall is described by an equation of the form $x(t) = \theta _0 + \theta _1t + \theta _2 t^2,$  We assume that $\theta_0$ is a known constant. We also assume that $\theta_2$ is known. We view $\theta_1$ as the realized value of a continuous random variable $\Theta _1$.  In this case, finding the MAP estimate of $\Theta _1$ involves the minimization of $$\theta _1^2+ \frac{1}{\sigma ^2}\sum _{i=1}^ n(y_ i-\theta _0-\theta _1 t_ i -\theta _2 t_ i^2)^2$$ with respect to $\theta_1$.  1)$$\hat{\theta }_1 = \frac{\sum ^ n_{i=1}t_ i(y_ i-\theta _0-\theta _2t^2_ i)}{\sigma ^2+\sum _{i=1}^ n t_ i^2}$$  2) Let $\sigma =1$, and consider the special case of only two observations ($n=2$). Write down a formula for the mean squared error ${\bf E}[(\hat\Theta _1-\Theta _1)^2]$    Here in 2, it is stated that:  $${\bf E}[(\hat\Theta _1-\Theta _1)^2] = {\bf E}[(\hat\theta _1-\Theta _1)^2] = var(\Theta)$$  For a normal distribution, we recognize that the variance is determined by the coefficient that comes next to a term of the form $\theta _1^2$. So what we're looking for is the constant that sits next to the $\theta _1^2$ term. We collect all of them, it is $1/2*(1+t _1^2+t_2^2)$. So the final answer is $$1/(1+t_1^2+t_2^2)$$    I don't quite follow and understand what he is saying.  Can someone please clarify?  Thank you.    According to this paper on , $\Theta = (\Theta_0, \Theta_1, \Theta_2)$, as can be seen here:      I don't quite follow and understand what he is saying.  Can someone please clarify?  Thank you.
From this  on Bayesian Least Mean Square estimation:  $$\mathbf E\left[(\Theta -\hat\theta)^2\right] = \rm{var}(\Theta -\hat\theta) + \left(\mathbf E[\Theta -\hat\theta]\right)^2 = \rm{var}(\Theta) + \left(\mathbf E[\Theta -\hat\theta]\right)^2$$  which is to say that $\rm{var}(\Theta -\hat\theta) = \rm{var}(\Theta)$.  I would like to ask how that comes about?  Thanks.  $\Theta$ is the random variable being estimated, and $\hat\theta$ is a constant estimate.
Let $G$ be a graph of minimum degree $k&gt;1$. Show that $G$ has a cycle of length at least $k+1$.  How would I show this? I understand that a cycle is a sequence of non-repeated vertices and the degree of a graph is the number of neighbors the vertex has. Is it possible to have the dock in different positions in different desktop? Is it possible to have the dock in different positions in different desktops? In particular I'd like to change the settings between a normal desktop and a desktop where there's an app at full screen. In this way I'd keep the dock in lower side of the desktop when there's an app at full screen and vertically on the left side during normal use.  I have this problem that, sicne I usually have the dock vertically on the left, it pops out everytime i go to the far left in an app I'm using with full screen, and using this app I have to move the mouse to the left quite often. I have a population density raster and would like to create 5 polygons from the 5 natural break (jenks) classification ranges. 1 polygon for each classification range. I'm using ArcMap10 and have all of the Extensions enabled in the ToolBox to my disposal.   Thanks! Assigning values to divergent series I have been looking at divergent series on wikipedia and other sources and it seems people give finite "values" to specific ones. I understand that these values sometimes reflect the algebraic properties of the series in question, but do not actually represent what the series converges to, which is infinity. Why is it usefull to assign values to divergent series?  The only theory I could come up with, is this:  Say you have 2 divergent series, series' A and B, and you assign each a value,   Series ($A= \sum_{n=0}^\infty a_n$), which I assigned the value Q  and series ($B= \sum_{n=0}^\infty b_n$ ), which I assigned the value P  But it just so happens that series $C=A-B= \sum_{n=0}^\infty (a_n-b_n)$ converges. Could that imply that the actual value of series $C$ is the difference of the two assigned values to $A$ and $B$, that is $\sum_{n=0}^\infty (a_n-b_n)=Q-P$ ?  If so, then that would make some sense to me, as to why people sometimes assign values to divergent series. I am aware that the "Eye of Sauron" in Peter Jackson's movies was not Sauron's actual physical form.  What the answers at '' don't indicate is, despite this, what abilities of vision Sauron may have had.  Did he have the ability to see from afar? If so, what are his limitations or details of this power? Is it some kind of Elvish-sight like Legolas but stronger? Was it a magical power? Or perhaps with the help of the Palantir (I'm not sure of this last one because I dont remember if the Palantir could only see what other Palantíri were seeing or anything he wanted)? How to change password using usermod? I am superadmin of a server and like to change password of an existing user . How can I do that ?  I tried   usermod -p 'new-password' john   but it didn't worked ? Group cohomology is one thing I don't seem to get my head around.  I understand $H^0$ as being the "fixed points" but when it comes to anything higher I have no idea what the notions are meant to be capturing...if anything.  For example I can see that 1-cocycles are homomorphisms with a specific property but that property doesn't seem to mean anything to me...and I think it should!  Is there an intuition or example that explains the meaning or is it just something that "stuck" and seems to work in lots of places? I believe it's a common practice in some communities to name children after their grandparents or parents or relatives and sometimes even friends and popular people. Places, roads, streets etc., get named after famous people, too. Many inventions and discoveries have been named after people who invented or discovered them.  But I am not yet aware of a term that refers to the person whose name is given to people, places or objects this way. Is there a suitable word to fill in the blank below?     They named their son after the famous football player, David Beckham.      David Beckham is .......... of their son. In SQL Server, how do I generate a CREATE TABLE statement for a given table? I've spent a good amount of time coming up with solution to this problem, so in the spirit of , I'm posting it here, since I think it might be useful to others.   If anyone has a better script, or anything to add, please post it.  Edit: Yes guys, I know how to do it in Management Studio - but I needed to be able to do it from within another application. How do I test asynchronous Apex? 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.  I'm testing asynchronous code - like Batch Apex, Queueable Apex, @future methods, or Scheduled Apex - and I'm getting unexpected results. It seems like portions of my code aren't running, or I'm getting confusing errors about batch invocations. How do I effectively test this kind of code? $(\frac 3p)$ is the Legendre symbol here, not a fraction, if that wasn't clear.  This is what I have so far:  We know by Gauss's Lemma that $(\frac qp) = (-1)^v$ where $$v = \#\left\{1 \le a \le \frac{p-1}{2} \mid aq \equiv k_a \pmod{12}),\space -p/2 &lt; k_a &lt; 0\right\}$$  And we also know that multiples of $(\dfrac{p-1}{2}) \times 3$ must be between $0$ and $p/2$, between $p/2$ and $p$, or between $p$ and $3p/2$.   Representing $p$ as $12k + r$ seems like the next step, but I'm not sure where to go from there. 14.04 --> 16.04 failed; apt (>= 1.0.1ubuntu2.13)' is not installed (This question was not resolved. Due to time constraints Computer was restored to factory settings.)  I was attempting to upgrade from 14.04 to 16.04, but apt threw this error:  The required dependency 'apt (&gt;= 1.0.1ubuntu2.13)' is not installed.   What does this mean, and how can I fix it?  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Here's my confusion   In the applications/overview I can see Search and Indexing. This seems to be a configuration tool. But if this is a server, I can't find a client. In applications I find Nepomuk File Indexing -- I think this is a KDE thing? Ditto, no client? I've installed tracker. This is a strange mix of powerful and unreliable/limited. e.g. it doesn't find files like locate does (returns no results where locate returns 96 results), but is will unpack every tag I've added to my photos! And you can't limit by folder for a quick search within a folder. The old Gnome 2 Search for files... thing is not in the Accessories menu anymore.   I feel that this is basic/core-level functionality and I can't help feeling I'm missing something obvious. 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