Middle entries of a random vector - Conditional expectation and covariance matrix of normal distribution P(X2|X1, X3)
Find conditional expectation from a 3-dimensional random vector?
Highlight whole TreeViewItem line in WPF Does Qt have a C interface? I have added a background image to the world settings. However, it has disappeared from my 3D view and only shows when I view the scene from the camera. Does anyone know if there is something I can do to make it reappear? We have the matrix: $$\left[\begin{array}{rrr} 0.8&amp;0&amp;0\\ 0&amp;1.2&amp;0\\ 0&amp;0&amp;1\\\end{array}\right]$$ Finding the eigenvalues and eigenvectors are easy. But finding ${\bf M}^n$ when $n \rightarrow \infty$ is the hard part. Any suggestions? As we know a first order theory $T$ is finitely axiomatizable if there is a finite set $F\subseteq T$ of axioms such that $F\vdash \sigma$ for every $\sigma \in T$.   How we can prove if $\mathsf{ZF}$ is consistent, then $\mathsf{ZF}$ is not finitely axiomatizable? By using the Reflection Theorem or any other if we could use? What would be the value of $\sum\limits_{n=0}^\infty \frac{1}{an^2+bn+c}$ I would like to evaluate the sum  $$\sum_{n=0}^\infty \frac{1}{an^2+bn+c}$$  Here is my attempt:    Letting    $$f(z)=\frac{1}{az^2+bz+c}$$  The poles of $f(z)$ are located at   $$z_0 = \frac{-b+\sqrt{b^2-4ac}}{2a}$$  and  $$z_1 = \frac{-b-\sqrt{b^2-4ac}}{2a}$$  Then  $$ b_0=\operatorname*{Res}_{z=z_0}\,\pi \cot (\pi z)f(z)= \lim_{z \to z_0} \frac{(z-z_0)\pi\cot (\pi z)}{az^2+bz+c}= \lim_{z \to z_0} \frac{\pi\cot (\pi z)+(z_0-z)\pi^2\csc^2 (\pi z)}{2az+b} $$  Using L'Hopital's rule. Continuing, we have the limit is  $$ \lim_{z \to z_0} \frac{\pi\cot (\pi z)+(z_0-z)\pi^2\csc^2 (\pi z)}{2az+b}= \frac{\pi\cot (\pi z_0)}{2az_0+b} $$  For $z_0 \ne 0$  Similarly, we find  $$b_1=\operatorname*{Res}_{z=z_1}\,\pi \cot (\pi z)f(z)=\frac{\pi\cot (\pi z_1)}{2az_1+b}$$  Then   $$\sum_{n=-\infty}^\infty \frac{1}{an^2+bn+c} = -(b_0+b_1)=\\ -\pi\left( \frac{\cot (\pi z_0)}{2az_0+b} + \frac{\cot (\pi z_1)}{2az_1+b}\right)=  -\pi\left( \frac{\cot (\pi z_0)}{\sqrt{b^2-4ac}} + \frac{\cot (\pi z_1)}{-\sqrt{b^2-4ac}}\right)= \frac{-\pi(\cot (\pi z_0)-\cot (\pi z_1))}{\sqrt{b^2-4ac}}= \frac{\pi(\cot (\pi z_1)-\cot (\pi z_0))}{\sqrt{b^2-4ac}} $$  Then we have  $$\sum_{n=0}^\infty \frac{1}{an^2+bn+c} = \frac{\pi(\cot (\pi z_1)-\cot (\pi z_0))}{2\sqrt{b^2-4ac}}$$    Is this correct?  I feel like I made a mistake somewhere.  Could someone correct me?  Is there an easier way to evaluate this sum? Out-of-control starship results in some crew members landing in a rebirthed universe? How is acceleration measured if there is change only in the direction of velocity & not magnitude? The formal definition of acceleration is      It is the measure of how fast the velocity changes with respect to time.   Now, when the magnitude of velocity changes, acceleration $$\vec{a} = \frac{d \vec{v}}{dt}$$ and its unit is $m.s^{-2}$ .  But what about the case where the velocity only changes its direction keeping its magnitude constant? The definition can then be      Acceleration is the measure of how fast the velocity changes its direction with respect to time.   But how can I calculate the acceleration then?? What should be the unit then??? Please help. Calculate Azimuth from polygon in GeoPandas Is there a simple way / ready-to-go function to calculate the azimuth of 2D-polygons in a GeoPandas DataFrame? My goal is to eventually compute azimuth angles of building footprints. Most of the functions I have found explicitly calculate the azimuth for a line, which is not what I am looking for. I have been thinking along the following lines to calculate the azimuth of a polygon using its normal vector (2 dimensions). Is my approach going into the right directions? I am a bit unsure how to adjust for the quadrants, also my results seem a bit odd. azimuth = 90 - math.degrees(math.atan2(normal[1], normal[0])) if azimuth &gt;= 360.0:     azimuth -= 360.0 elif azimuth &lt; 0.0:     azimuth += 360.0 Windows Explorer sort method I'm looking for an algorithm that sorts strings similar to the way files (and folders) are sorted in Windows Explorer. It seems that numeric values in strings are taken into account when sorted which results in something like  name 1, name 2, name 10   instead of  name 1, name 10, name 2   which you get with a regular string comparison.  I was about to start writing this myself but wanted to check if anyone had done this before and was willing to share some code or insights. The way I would approach this would be to add leading zeros to the numeric values in the name before comparing them. This would result in something like  name 00001, name 00010, name 00002   which when sorted with a regular string sort would give me the correct result.  Any ideas? Experiment to find structure of water Who first determined the structure of water (two hydrogen atoms stuck to an oxygen atom at approx 105 degrees), and, more importantly, how was this done? Why isn't it $E \approx 27.642 \times mc^2$? Why is it that many of the most important physical equations don't have ugly numbers (i.e., "arbitrary" irrational factors) to line up both sides?   Why can so many equations be expressed so neatly with small natural numbers while recycling a relatively small set of physical and mathematical constants?  For example, why is mass–energy equivalence describable by the equation $E = mc^2$ and not something like $E \approx 27.642 \times mc^2$?  Why is time dilation describable by something as neat as $t' = \sqrt{\frac{t}{1 - \frac{v^2}{c^2}}}$ and not something ugly like $t' \approx 672.097 \times 10^{-4} \times \sqrt{\frac{t}{1 - \frac{v^2}{c^2}}}$.  ... and so forth.    I'm not well educated on matters of physics and so I feel a bit sheepish asking this.   Likewise I'm not sure if this is a more philosophical question or one that permits a concrete answer ... or perhaps even the premise of the question itself is flawed ... so I would gratefully consider anything that sheds light on the nature of the question itself as an answer.    EDIT: I just wished to give a little more context as to where I was coming from with this question based on some of the responses:  @Jerry Schirmer comments:      You do have an ugly factor of $2.997458 \times 10^8 m/s$ in front of everything. You just hide the ugliness by calling this number c.   These are not the types of "ugly constants" I'm talking about in that this number is the speed of light. It is not just some constant needed to balance two side of an equation.  @Carl Witthoft answers:      It's all in how you define the units ...   Of course this is true, we could in theory hide all sorts of ugly constants by using different units on the right and the left. But as in the case of $E=mc^2$, I am talking about equations where the units on the left are consistent with the units on the right, irrespective of the units used. As I mentioned on a comment there:     $E=mc^2$  could be defined using units like $m$ in imperial stones ($\textsf{S}$), $c$ in cubits/fortnight ($\textsf{CF}^{−1}$) and $E$ in ... umm ... $\textsf{SC}^2\textsf{F}^{−2}$ ... so long as the units are in the same system, we still don't need a fudge factor.   When the units are consistent in this manner, there's no room for hiding fudge constants. Error sending edittext in java server. See code Does anyone remember this article about the Euclidean Algorithm? In the 70's I had a stack of old Amateur Radio magazines (50s-60s), and for a long time I saved an article about using the  to combine a number of resistors to achieve a specific value. Does anyone recall and have a copy of this article, or know how the Euclidean algorithm is applied to solve this problem? How to prompt for user input and read command-line arguments Quantum Field Theory without LSZ, how is it possible? Can a lingering injury that requires "magical healing of 6th level or higher" be healed by an upcast lower-level spell? The  table in the DMG (p. 272) contains some wounds that can only be fixed by powerful healing magic, for instance:     Horrible Scar. You are disfigured to the extent that the wound cannot be easily concealed. You have disadvantage on Charisma (Persuasion) checks and advantage on Charisma (Intimidation) checks. Magical healing of 6th level or higher, such as heal and regenerate, removes the injury.   Can the necessary magical healing described here be a lower-level spell (such as cure wounds) that is upcast to 6th level? Or must it be a healing spell that is already 6th level or above before upcasting? Using InDesign CS6, how can I simultaneously edit A4 and A5 versions of a poster? I'm on Adobe Creative Cloud, so I have the latest InDesign, and I'm attempting to design a poster for my choir's next concert. Traditionally, I've done this in Photoshop, in two separate files, however I've recently started using InDesign after converting to it to create wedding albums through Blurb.  I've created an A4 document, and created an alternate-layout for it in A5. I've then lined them up side-by-side, and I've added a black frame to the A4 page, to act as the black background for my page (by the way, feel free to advise on better ways to do this).  My assumption is that I've failed to understand how this works, as I was hoping that I'd magically get a black frame on the A5 version, and that I could use liquid layout to get this to automatically scale (along with all other elements), in order to create the two versions simultaneously.  All help/advice would be greatly appreciated! Is there a Java equivalent or methodology for the typedef keyword in C++? Do positive semidefinite matrices have to be symmetric? Can you have a non-symmetric matrix that is positive definite? I can't seem to figure out why you wouldn't be able to have such a matrix, but all my notes specify positive definite matrices as "symmetric $n \times n$ matrices."  Can anyone help me with an example of a non-symmetric positive definite matrix, or some insight into a proof for why it would need to be symmetric should that be the case? Thanks!