I have created a blog in blogspot. What I need to do is to write equation in the blog so adding  would be great to add in the blogspot.
I want to add  on my blog hosted on Blogger. The only solution I found is , but it requires a hosting server. Any other option?
Are you allowed to use numbers as table names in MySQL? I'm thinking of having a program that dynamically creates new tables as the need arises. Can I have tables named with just numbers in MySQL? How to determine which app is causing vibration? I have a Moto G6 Play that vibrates like it is getting notifications but it's not. The vibration is inconsistent and seems random so I would like to know if there is a logcat or something I can run to log which app is calling the vibration so that I can shut it off? The same thing happens on my wife's phone, same model, and it does my head in. The device is running Android 9. How to design a product table for many kinds of product where each product has many parameters I am going through Shankar's Principles of Quantum Mechanics and am having trouble finding the free particle propagator $U(t)$ that satisfies $$\lvert\psi (t)\rangle = U(t)\lvert\psi (0)\rangle$$ due to the degeneracy of the $E$ eigenkets. Shankar says that if the Hamiltonian has degenerate eigenvalues, we change the propagator equation from $$U(t) = \sum_{E} \lvert E\rangle\langle E\rvert e^{-iEt/{\hbar}}\tag{4.3.13}$$ to $$U(t) = \sum_{E} \sum_\alpha \lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar} $$ where $\lvert E, \alpha \rangle $ are the orthonormal eigenkets for the $E$ eigenspace. This makes sense to me, because it seems like whenever we deal with degeneracy to use a projection. He also says for a Hamiltonian with no degeneracy to change the sum into an integral, so I imagine that for a Hamiltonian with a continuous, degenerate spectrum we would have this propagator: $$ U(t) = \sum_\alpha\int_\Re dE\lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar}$$  I tried applying this to the free particle and had some trouble. In the book, after solving for the eigenkets and eigenvalues, $\lvert E\rangle = \lvert p\rangle$ and $p = \pm\sqrt{2mE}$, he chooses the propagator as a function of the momentum eigenvalues $p$ so that $$U(t) = \int_\Re dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m).\tag{5.1.9}$$ I understand how he got to this, but I'm having some trouble because this doesn't appear to be the same expression that I have above. Splitting the integral up, we have $$U(t) = \int_{-\infty}^0dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m) + \int_0^\infty dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m)$$ The left integral can changed to an integral over $-p$, and the bounds switch from $0$ to $\infty$ such that: $$U(t) =  \int_{0}^\infty d(-p)\lvert -p\rangle\langle -p\rvert \exp(-ip^2t/2m) + \int_0^\infty dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m)$$ Now we can substitute and interchange $\lvert p\rangle = \lvert E, + \rangle$,  $\lvert -p\rangle = \lvert E, - \rangle$, $E = p^2/2m$, and $dp = \pm mdE/\sqrt{2mE}$ to get: $$U(t) = \sum_{\alpha = \pm} \int_0^\infty dEm/\sqrt{2mE}\lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar}. $$ This is the result that we are asked to prove in Exercise 5.1.1. Why is this not the same as the first expression for the propagator? Where did the extra $ m/\sqrt{2mE} $ come from? Am I just wrong in assuming that that is the general expression for a Hamiltonian with continuous and degenerate eigenvalues? Let $f : \mathbb{R}^n → \mathbb{R}_∞$ be a function and let $C ⊂ dom f$ be a convex set.  $$**Part I**$$ Prove that $f$ is a convex function if and only if $f$ is convex over every line $L_{v,x_0}$ where $L_{v,x_0} := \{x_0 + tv : t \in \mathbb{R}\}$.  i.e. f is convex iff when restricted to any line, it be convex on that certain line.  So essentially to handle this proof I need to show that convexity for $f$ holds when its convex over any arbitrary line, and on the converse it does not hold when $f$ is not convex over that arbitrary line. If $f$ is convex then,  $$\forall x,y \in C, a \in [0,1]$$ it holds that $$f(ax+(1-a)y) \leq af(x) + (1-a)f(y) \space \space\space\space\space\space\space\space\space\space\space\space (1)$$  EDIT: $$**Part II**$$  Let $f : \mathbb{R}^n \to \mathbb{R}_∞$ be a function.   I want to prove that $f$ is convex over the line $L_{v,x_0}$ iff $\psi : \mathbb{R} \to \mathbb{R}_∞$  $\psi(t) := f (x_0 + tv)$, is convex over $I (x_0, v )$.  where $L_{v,x_0}:= \{x_0 + tv : t ∈ \mathbb{R}\}$, and $I(x_0,v):=\{t \in \mathbb{R} : x_0 + tv \in C\}$  Any help is very much appreciated There are a number of times in Lord of the Rings where the "east" is mentioned -- the pirate ships, the slave-driven farms. And of course all of the peoples subservient to Sauron who come to fight Gondor and the West. Are there any maps of the East of Middle-earth, and what is out there? Consider these two sentences: "I gave him a pencil," and, "I gave a pencil to him."  Is it correct that the important part of the sentence is placed at the end? When we want to emphasize the pencil that I gave him, must we say, "I gave him a pencil?" When we want to emphasize that it is him to whom I gave a pencil, must we say, "I gave a pencil to him?" I'm looking for a short story whose title I can't remember.  In it a wealthy man practices cloning and consciousness transfer to achieve immortality, often parading older bodies around his estate naked (not a nice guy). The story follows a final transplant, which doesn't go as expected. The man is left trapped with his memories and no ability to speak, while the clone walks off behaving exactly as he would. This code won't compile:  \documentclass{article} \usepackage{listings} \lstset{     language=C,     basicstyle=\small }  \begin{document} lorem ipsum\footnote{% \begin{lstlisting} // some C code or whatever... printf("Blah blah\n"); \end{lstlisting}} \end{document}   It says Illegal parameter number in definition of \lst@insertargs. I know listings is really picky about where it goes, but there must be a way to cram a listing into a footnote, right?  (This is a MWE and my tex files are a bit more complex; my actual error is Paragraph ended before \lst@next was complete. but they are hopefully related; I couldn't get this error with the MWE.) Warning: Total photoshop newb!  I recently got a cool image of my fiancee and I holding hands and showing off her engagment ring. The backdrop is the beach:    I tried to trace an image using a combo of the laso and magic wand tools to select my target. Then I used that to pull out the target as another layer.  I then pasted in a background gradient.     Ended up with a really lame photoshoped looking picture. Notice the white lines where I didn't get the arms close enough and the choppy spots where I got too close.    So what's my main problem here?   Is their a way to bet a better cutout? Is their a blending tool that will make the background look more natural? both   I tried a dozen or so tutorials I found in google searches but they just don't blend these types of images together in any of them. Thanks in advance. Capture characters from standard input without waiting for enter to be pressed I can never remember how I do this because it comes up so infrequently for me. But in C or C++, what is the best way to read a character from standard input without waiting for a newline (press enter).  Also ideally it wouldn't echo the input character to the screen. I just want to capture keystrokes with out effecting the console screen. The limit of a convergent Gaussian random variable sequence is still a Gaussian random variable I'm trying to prove this conclusion but have some problems with one of the steps.  Assume $X_1,\ldots,X_n,\ldots$ is a sequence of Gaussian random variables, converging almost surely to $X$, prove that $X$ is Gaussian.  We use characteristics function here. Since $|\phi_{X_n}(t)|\leq 1$, by dominated convergent theorem, we have for any $t$  $$ \lim_{n\rightarrow\infty}e^{it\mu_n-t^2\sigma_n^2/2}=\lim_{n\rightarrow \infty}\phi_{X_n}(t) = \lim_{n\rightarrow \infty}\mathbb{E}\left[e^{itX_n}\right] = \mathbb{E}\left[e^{itX}\right] = \phi_X(t) $$  this is the step that I cannot figure out: $e^{it\mu_n-t^2\sigma_n^2/2}$ converges for any $t$ if and only if $\mu_n$ and $\sigma_n$ converges.   Let $\mu=\lim_n \mu_n$, and $\sigma=\lim_n\sigma_n$, then $\phi_X(t)=e^{it\mu-t^2\sigma^2/2}$, which proves that $X$ is a Gaussian random variable.  Why can we get that $\mu_n$ and $\sigma_n$ converge? This looks intuitive for me, but I cannot make a rigorous prove. What is sudo apt-get -f install? what is the usage of apt-get -f install . when I run sudo apt-get update I got some errors at the end It says try apt-get -f install. What is this command?  the last part of my error is :   . . .   going to be installed   E: Unmet dependencies. Try 'apt-get -f install' with no packages (or specify a solution). If you have an array of Java objects which have a primitive type (for example Byte, Integer, Char, etc).  Is there a neat way I can convert it into an array of the primitive type?  In particular can this be done without having to create a new array and loop through the contents.  So for example, given  Integer[] array   what is the neatest way to convert this into  int[] intArray   Unfortunately, this is something we have to do quite frequently when interfacing between Hibernate and some third party libraries over which we have no control.  It seems this would be a quite common operation so I would be surprised if there's no shortcut.  Thanks for your help! Can't move/snap Nautilus window to occupy left or right half of the 1366x768 screen - is it just me? I'm running Ubuntu 18.04 LTS with GNOME Shell. Nautilus does not want to occupy 50% (left or right) of the screen sized at 1366x768.    Other applications (such as GNOME Terminal, Eye of GNOME, Gedit) are normally snapping to the left/right half of the screen.  But Nautilus can be resized to the width lower than half of screen (I can get 650 px, which is smaller than 1366/2 = 683 px).  The first usable screen resolution seems to be 1400x1050, then goes 1440x900. But large amount of current devices .  Is it just me?     Updates: 1. standard shortcut Super+← or Super+→ do not work too. 2. I reported  which has linked upstream . 3. It does not work on GNOME FlashBack session too. Abstract algebra book recommendations for beginners. I am taking abstract algebra in this semester. I find that my lecturer didn't provide enough examples for understanding. So I would like to ask what are the recommended books which has lots of solved problems and exercises? The lecturer said in this semester, he will cover group theory, ring theory and a bit of field theory. Which book contains extensive information in these three fields? What developer text editors are available for Ubuntu? An example to show that convergence of Cesaro sum to $0$ does not imply the original sequence converges to $0$. I'd like construct a non-negative sequence $\{a_{k}\}_{k=0}^{\infty}$ with $a_{0}=0$ such that the Cesaro sum $\frac{1}{n}\sum_{k=0}^{n-1}a_{k}\longrightarrow 0$ but $a_{n}$ does not converge to $0$.   I have a really bad attempt:  Define $a_{0}=a_{1}=0$, and $a_{k}:=\log(\log(k))$ for $k\geq 2$, then $a_{k}\longrightarrow\infty$. But $$\dfrac{1}{n}\sum_{k=0}^{n-1}a_{k}=\dfrac{\log\Big(\prod_{k=2}^{n-1}\log(k)\Big)}{n}$$ and Wolframalpha told me that this series converges to $0$ as $n\longrightarrow\infty$.  However, I have no idea about how to show this convergence to $0$. Also, I wish to have a sequence as simple as possible, since at some stage I need too how $f(k):=a_{k}$ is positive semi-definite.   Is there any simpler example?   Thank you! I was trying to install Linux on my MacBook for class when I bricked my computer, which now in an unbootable state.   I made my partition too small, so I was going to delete and create a larger one. However, Disk Utility did not allow me to delete the small partition. I followed the instructions from   However, not only was I still unable to delete it, but upon restarting I realized my primary partition (located on the same disk) changed into FFFFFFFF-FFFF-FFFF-FFFF-FFFFFFFFFFFF, and I had no options to boot.  (I am sure I performed the command on the correct disk and partition)  I saw some similar threads, but my main bootable drive is the one that got corrupt.   I don't want to lose any data. I'm not the most terminal savvy, so ANY help is appreciated, and I will kindly provide any additional information!  (Disk 0, partition 2 is my main partition, and that only one I care about)  Below are the results of  $ diskutil list and $ gpt -r show disk0.        UPDATE: :  I have followed @klanomath 's instructions. At the last step I received the following error:  Checking prerequisites Checking the partition list Checking for an EFI system partition Checking the EFI system partition's size Checking the EFI system partition's file system Problems were found with the partition map which might prevent booting Error: -69766: The partition map needs to be repaired because there's a problem with the EFI system partition's file system Underlying error: 8: POSIX reports: Exec format error    FINAL UPDATE: SUCCESS!  I just wanted thank @klanomath for helping me through this entire process. He spent 11 hours TeamViewing with me until not only my files were accessible, but until I could access my computer as could before this whole debacle (and this was after 5 hours of email correspondence).   To any future reader with this problem: PLEASE don't give up, it can be fixed! I have an HTML string representing an element: '&lt;li&gt;text&lt;/li&gt;'. I'd like to append it to an element in the DOM (a ul in my case). How can I do this with Prototype or with DOM methods?  (I know i could do this easily in jQuery, but unfortunately we're not using jQuery.)