In unbiased data, RMSE and standard deviation is same?
RMSE vs Standard deviation in population
If a linear operator has an adjoint operator, it is bounded Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$. Is the following true? Let $x_n$ be a sequence with the following property: Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.  I guess that it is true but I am not sure how to prove this. In the Stack Exchange Android app, Achievements section I can see only Today's achievements. No Yesterday, Last 7 days, Last 30 days, ... details.  Previously in the 1.0.85 version itself I saw all the achievements.   I  refreshed the page many time, but still seeing Today's achievement only.  Screenshot for reference:     App Version: 1.0.85 Device Manufacturer: Motorola Device Model: XT1033 OS Version: 5.1 (2) I have a mid-2007 MacBook Pro, the one with the trackpad that had the physical button on along the bottom edge. The trackpad is not well and using it can cause it to stick and freak out OS X.  Is there a way to disable the trackpad that's a little more low-level than ? A BIOS setting perhaps? I'd like the trackpad to never be useable on the unit, whether an external mouse is connected to it or not. In AVR datasheets under the Electrical Characteristics section you will typically find a graph like this (this one is from the ATMega328):    I've seen designs that seem to "work" but operate outside the shaded envelope. Specifically, I've seen 3.3V (Arduino) designs that run the clock from an external 16MHz crystal. Clearly, this is out of spec. What are the practical negative consequences of running outside this envelope? Suppose I have a python object x and a string s, how do I set the attribute s on x? So:  &gt;&gt;&gt; x = SomeObject() &gt;&gt;&gt; attr = 'myAttr' &gt;&gt;&gt; # magic goes here &gt;&gt;&gt; x.myAttr 'magic'   What's the magic? The goal of this, incidentally, is to cache calls to x.__getattr__(). What causes a "Java Hotspot 64-Bit Server VM Warning"? I get this warning:     Java HotSpot(TM) 64-Bit Server VM warning: Using incremental CMS is deprecated and will likely be removed in a future release   What does it mean? How are connections differentiated from behind NAT routers? Can I travel to the UK unaccompanied on an accompanied child visa now that I am over 18? I got a visa with my mother when I was 17 years old. On it was stated "Not valid unless accompanied by my mother".    Can I travel to the UK alone now that I am 18? Why does Maximum Likelihood estimation maximize probability density instead of probability I am trying to understand Maximum likelihood estimation but it looks like I am missing something rather elementary.          suppose we have an iid random sample $X_1, X_2,..., X_n$ for which the probability density function of each $X_i$ is $f(x_i; \theta)$ where $\theta$ is an unknown parameter. Then, the joint probability density function of $X_1, X_2,..., X_n$ is given by:          $f(X_1=x_1,X_2=x_2,\cdots,X_n=x_n)=\prod_{i=1}^{n} f(x_i, \theta)$      In Maximum Likelihood estimation, we try to maximize $f$ as a function of $\theta$.  Question: Why do we maximize probability density instead of probability? In what way does it make sense?  Edit: As explained in the answers, the probability of choosing a finite number of sample points from a continuous probability distribution is zero, so maximizing probability doesn't make sense, but how does maximizing probability density make sense? Proving a sequence converges by defining the sequence recursively. Let $a &gt; 0$. Choose $x_1 &gt;$ $\sqrt a$. Define a sequence {$x_n$} recursively as $x_{n+1} = 1/2(x_n + a/x_n)$ for $n &gt; 1$.   Prove that lim$x_n = \sqrt a$.  I think I first want to prove that {$x_n$} is bounded below by $\sqrt a$. But, I am having trouble with this. Could someone help me out a little? I was browsing PhD position offers in Europe on different websites and I was quite surprised to find that several offers contained sentences such as     University of X value equality and diversity. We strongly encourage women and BAME applicants for this position.    where BAME stands for Black, Asian and Minority Ethnicities; or     The research group aims to increase the number of women in scientific positions. Female candidates are therefore encouraged to apply.   Is it considered normal in the academic environment to discriminate between  candidates according to their race and/or gender for "egalitarian" reasons?   How is the choice of a candidate based on their gender and/or race considered "egalitarian"? Specific system of differential equations I have the following system of equations:  \begin{eqnarray}\frac{dx}{dt} = x(1 - x^2 - y^2) \\ \frac{dy}{dt} = y(4 - x^2 - y^2) \end{eqnarray}  I want to prove that if a solutions starts (at time $t = 0$) in the region   $$ G = \{(x, y) \in \mathbb{R}^2 : x &gt; 0, y &gt; 0 \text{ and } 1 &lt; x^2 + y^2  &lt; 4\} $$  that it remains there for all $t \geq 0$. I have already proven that, from $G$, a solution cannot cross the $x$-axis where $0 &lt; x &lt; 2$ or the $y$-axis where $0 &lt; y &lt; 2$ (since those line segments are orbits), but the pieces of the circles with radius 1 and 2 (where $x &gt; 0, y &gt; 0$) are the lines I haven't done yet. In this question we can focus on just the circle with radius 1 since the proof for the other circle is completely analogous to it.  In my book, there is a similar problem where they show that if a solution crosses a certain line, that solution attains its maximum or minimum value for $x$ and $y$, and then they derive a contradiction. However, in this case, for $n \geq 1$ the $n$-th derivative of $x$ equals 0 on the circle with radius 1. (If the second derivative were negative, $x$ would attain its maximum on the circle, a contradiction since $\frac{dx}{dt} &lt; 0$ in $G$, so $x$ is decreasing.)  I don't see why a solution can't cross the circle at all. On the circle, $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} &gt; 0$, so after crossing the circle, the solution would go back up to $G$.  Is the circle itself an orbit?  EDIT I think I have found the answer. Using polar coordinates, I found $$ \frac{dr}{dt} = r - r^3 + 3r \sin^2(\theta) $$ Now, $$ \frac{dr}{dt} &gt; 0 \Leftrightarrow 0 &lt; r - r^3 + 3r \sin^2(\theta) \leq r - r^3 + 3r = 4r - r^3 $$ since $0 \leq \sin^2(\theta) \leq 1$. Note that $0 &lt; r \leq 1$ satisfies $4r - r^3 &gt; 0$, and thus $\frac{dr}{dt} &gt; 0$. Therefore, if a solution goes from $G$ to the edge of the unit circle, $r$ must, on the edge, either be decreasing or constant (i.e., its derivative is less than or equal to 0), contradicting the fact that the derivative is positive.  Can someone tell me if this is correct? The difference between GPU and CPU distribution upgrade 14.04 to 15.04 halted due to multiple errors What is Electromotive force (EMF)? How is it related to potential difference? What is  (EMF)? How is it related to the potential difference? Is it created by the potential difference in any conductor? Is it a process? Why is it called force?  Does writing EMF instead of voltage make any difference in AC circuits? (Studying alternating current I found some books using EMF while others used voltage.) Switching window manager/desktop environments? In general what is the approach to switch the desktop environment or window manager on Ubuntu? The main difference between stock Ubuntu, Kubuntu and Lubuntu is supposedly the use of Unity vs. KDE vs. LXDE.  However, in order to try various desktop environments and window managers I don't want to have to reinstall the respective flavor from scratch just to get a particular desktop environments or window manager. There probably is an easier method than that.  The question: How can I easily test different desktop environments and respective compatible window managers (say KDE, LXDE, Unity, GNOME, XFCE and StumpWM)? Calculate text width with JavaScript Most people say I wish I could, I wish you would.    Can we use I wish I can, I wish you will?    I'd like to know what the main differences between the usage of can/will and could/would are when wishing.  Is it related to the subjunctive? Because one says “I wish I were”, not “I wish I am”, so are would and could the subjunctive forms of can and will?  Or is it something else entirely? In the two following snippets, is the first one safe or must you do the second one?  By safe I mean is each thread guaranteed to call the method on the Foo from the same loop iteration in which the thread was created?  Or must you copy the reference to a new variable "local" to each iteration of the loop?  var threads = new List&lt;Thread&gt;(); foreach (Foo f in ListOfFoo) {           Thread thread = new Thread(() =&gt; f.DoSomething());     threads.Add(thread);     thread.Start(); }   -  var threads = new List&lt;Thread&gt;(); foreach (Foo f in ListOfFoo) {           Foo f2 = f;     Thread thread = new Thread(() =&gt; f2.DoSomething());     threads.Add(thread);     thread.Start(); }   Update: As pointed out in Jon Skeet's answer, this doesn't have anything specifically to do with threading.