Analysis of Harmonic Oscillator In griffiths,   Why was the lowest chosen like that while in the ladder analogy, the lower energies are like $a_-^n\psi$? That is, why didnt he chose the lowest rung to be  $$a_-^n\psi$$
How does one determine ladder operators systematically? In textbooks, the ladder operators are always defined," and shown to 'raise' the state of a system, but they are never actually derived.  Does one find them simply by trial and error? Or is there a more systematic method/approach to obtain them?
Why is main() in java void? In the case of languages with a C-like syntax, we declare the main() method to return an int or float value (or void).  Is it possible to declare a non-void return type from main() in Java?  If not, then why not?  Does this mean that a Java program doesn't return any value to the OS? I'm looking for a good way to perform multi-row inserts into an Oracle 9 database.  The following works in MySQL but doesn't seem to be supported in Oracle.  INSERT INTO TMP_DIM_EXCH_RT  (EXCH_WH_KEY,   EXCH_NAT_KEY,   EXCH_DATE, EXCH_RATE,   FROM_CURCY_CD,   TO_CURCY_CD,   EXCH_EFF_DATE,   EXCH_EFF_END_DATE,   EXCH_LAST_UPDATED_DATE)  VALUES     (1, 1, '28-AUG-2008', 109.49, 'USD', 'JPY', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (2, 1, '28-AUG-2008', .54, 'USD', 'GBP', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (3, 1, '28-AUG-2008', 1.05, 'USD', 'CAD', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (4, 1, '28-AUG-2008', .68, 'USD', 'EUR', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (5, 1, '28-AUG-2008', 1.16, 'USD', 'AUD', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'),     (6, 1, '28-AUG-2008', 7.81, 'USD', 'HKD', '28-AUG-2008', '28-AUG-2008', '28-AUG-2008'); Please add TeX rendering in titles on the iOS app Proving that this function is continuous on $G\times G$ Crossbow Expert Feat... for Spells? I was thinking about creating a wizard character. Reviewing the spellcasting rules, I saw this:  Most spells that require attack rolls involve ranged attacks. Remember that you have disadvantage on a ranged attack roll if you are within 5 feet of a hostile creature that can see you and that isn't incapacitated (see chapter 9).  There are a lot of occasions that might arise that would make this a problem, or at least a bit of a hindrance. So as a D&amp;D player normally would when they wanted to break a rule without actually cheating, I went to a list of feats and found Crossbow Expert:  Thanks to extensive practice with the crossbow, you gain the following benefits: You ignore the loading quality of crossbows with which you are proficient. Being within 5 feet of a hostile creature doesn't impose disadvantage on your ranged attack rolls. When you use the Attack action and attack with a one handed weapon, you can use a bonus action to attack with a hand crossbow you are holding.  I may be (am) overthinking things, but does this apply to magic attacks, such as cantrips like Eldritch Blast or Fire Bolt? The literal meaning of the feat says that I can, but it could also imply by the title and also the effects of the feat that this only applies to crossbows, or any other ranged weapon, excluding magic. The strongest evidence that it might not affect ability with ranged magic attacks is:  Thanks to extensive practice with the crossbow  With a crossbow, not with magic. Expectation inequality for a set and its subset. Given that a function $g(x)$ is a monotone increasing function. Its domain is an interval [c,d]. There are two sets $a_j \in A, j= 1,...,J$ and $b_k \in B,k=1,...,K$  on this interval, which satisfy $c_z \in C = B$\A and $c_z \geq a_j$ for all $a_j \in A$. For example, the interval $[c,d]=[0,8]$,$A=\{0,1,2,3,4,5\}$ and $B=\{0,1,2,3,4,5,6,7\}$. In this case, does the following inequality hold? If yes, could you give me some hints to prove that?  $$\frac{1}{K} \sum_{k=1}^{K}g(b_k) \geq \frac{1}{J} \sum_{j=1}^{J}g(a_j)$$ When to use “that” and when to use “which”, especially in relative clauses When is it appropriate to use that as opposed to which with relative clauses? Combining two covariance matrices I'm calculating the covariance of a distribution in parallel and I need to combine the distributed results into on singular Gaussian. How do I combine the two?  Linearly interpolating between the two almost works, if they are similarly distributed and sized.   provides a forumla at the bottom for combination but it doesn't seem right; two identically distributed distributions should have the same covariance, but the formula at the bottom of the page doubles the covariance.  Is there a way to combine two matrices? How to delete a file with a weird name? I wrote a program that uses POSIX memory-mapping function (mmap)  The program takes a file (a.dat) and memory-maps it for reading/writing.  Due to errors in the program, every time I run the program a file with some weird names (e.g., ?d?P?^z??d?P?^z?) is created. The error is resolved but I am not able to delete the files.  I am not able to delete it either using command line or by select/deleting from window manager.  So how should I delete it? I'm using Ubuntu 11.04. Do there Exist Proper Classes that aren't "Too Big" Some proper classes are "too big" to be a set in the sense that they have a subclass that can be put in bijection with $\alpha$ for every cardinal $\alpha$. It is implied in  post that every proper class is "too big" to be a set in this sense, however I have been unable to prove it. It's true if every two proper classes are in bijection, but it's consistent with ZFC for there to be a pair of non-bijective classes.  So, is the following true in ZFC:     For all proper classes, $C$, and $\alpha\in\mathbf{Card}$, $\exists S\subset C$ such that $|S|=\alpha$?   If not, is there something reasonable similar that preserves the intuition about classes that are "too big to be sets"? Show that $$\int_{0}^{\infty} \frac{x - \sin(x)}{x^3} \, dx = \frac{\pi}{4}$$   My attempt is as follows: Let $$f(z) = \frac{z - i e^{iz}}{z^3}$$ and consider the contour on $[\epsilon, R]$ followed by a semicircular arc in the counter clockwise direction, then on $[-R, -\epsilon]$, then the semicircular clockwise contour avoiding the origin. We have, then, that  $$0 = \int_{\Gamma} f(z) dz = \int_{[\epsilon, R]} f(t) dt + \int_{C_R}f(Re^{it})Rie^{it}dt + \int_{[-R, -\epsilon]}f(t)dt + \int_{C_{\epsilon}}f(\epsilon e^{-it})\epsilon i e^{-it} dt$$  Then the first and third integrals ( $I_1$ and $I_3$) combine so that  $$I_1 + I_3 = 2\int_{\epsilon}^R \frac{t - \sin{t}}{t^3}\,dt$$  Further,   $$|I_{C_R}| \leq \int_0^\pi \left|\frac{Re^{it} - ie^{-R\sin{t}}e^{iRcos{t}}}{R^2 e^{2it}} \right|dt \rightarrow 0 \text{ as } R\rightarrow \infty$$  (I've omitted the details, it isn't too bad to bound)  However, I'm having trouble computing the limit  $$\lim_{\epsilon \rightarrow 0}\int_{C_{\epsilon}} f(\epsilon e^{-it})\epsilon i e^{-it} dt$$  No matter which way I look at it, it seems like this limit does not exist. Perhaps I'm seeing something wrong or have I chosen a bad $f(z)$? Best compiler warning level for C/C++ compilers? Why is $\frac{\operatorname dy'}{\operatorname dy}$ zero, since $y'$ depends on $y$? I know that $\frac{dy'(x)}{dy}=0$ (where $y'=\frac{dy(x)}{dx}$). The reason explained is that $y'$  does not depend explicitly on $y$. But intuitively, $y'$ depends on $y$, since if you vary $y$ you will modify $y'$. Why is my reasoning wrong (my reasoning sounds like it's related to functional calculus, instead of standard calculus)?  I tried to write  $\frac{dy'(x)}{dy}=\frac{d}{dy}\frac{dy}{dx}=\frac{d}{dx}\frac{dy}{dy}=\frac{d}{dx}1=0$, but this proof doesn't convince me.  I think that other way to see this is saying that if a function is of the form $f(y)$, it will not dependend on the variable $y'$.  But the same way you write $f(y)=y^2$, you could write $f(y)=\frac{d}{dx}y$, which clearly depends on $y'$. So I don't know if there are some types of operations which are restricted (for example, taking limits):  Note: The problem raises in the context of Classical Mechanics, where: $\frac{\partial L(\dot{x},t)}{\partial x}=0$. What's the best way make an "augmented" coefficient matrix? When studying systems of linear equations, it's nice to remind people that the last column of the coefficient matrix holds the constants.  This is often done in books by putting a vertical line between the last column and the next to last column.  What is a good way to do this in LaTex? How to prevent page breaking for lstinputlistings? How can I get a shapefile from a postgis query? Number formation from given digits I was asked this question in a recent interview. You are given two digits $0$ and $9$. Using only these two digits you must form the least positive number that is a multiple of a given '$n$' . The value of $n$ will be provided before hand. For example if we are given $n$ as $7$ then the answer will be $9009$ .  I tried using some basic divisibility principles but was not able to come up with a correct solution and code it up. Probably I need some help in number theory I think that  is pretty good prima-facia evidence that  is just facilitating the deposition of crap. Block apps from accessing the Internet on Android device In Android, is it possible to block apps from accessing the Internet? Many apps will often connect to remote servers in the background, including Google's own apps or system services. Over time this can eat away at a user's data limit set by a mobile carrier. Does Android allow an app to act as a firewall whereby the user can select which apps have access to the Internet? Or does the device have to be rooted to have that ability? Are submodules of finitely generated free modules always free? I don't believe it's true that all submodules of a finitely generated free module are free just based on results from google. Is there a canonical simple example of a finitely generated free module with a submodule that is not free?