The question asks to prove that -  $$\frac {\sin x} {\cos 3x} + \frac {\sin 3x} {\cos 9x} + \frac {\sin 9x} {\cos 27x}  = \frac 12 (\tan 27x - \tan x) $$  I tried combining the first two or the last two fractions on the L.H.S to allow me to use the double angle formula and get $\sin 6x$ or $\sin 18x$ but that did not help at all.   I'm pretty sure that if I express everything in terms of $x$, the answer will ultimately appear but I'm also certain that there must be another simpler way.
Show that  $$\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$$
I often read that in case of Deep Learning models the usual practice is to apply mini batches (generally a small one, 32/64) over several training epochs. I cannot really fathom the reason behind this.  Unless I'm mistaken, the batch size is the number of training instances let seen by the model during a training iteration; and epoch is a full turn when each of the training instances have been seen by the model. If so, I cannot see the advantage of iterate over an almost insignificant subset of the training instances several times in contrast with applying a "max batch" by expose all the available training instances in each turn to the model (assuming, of course, enough the memory). What is the advantage of this approach? I have a dictionary of values read from two fields in a database: a string field and a numeric field. The string field is unique, so that is the key of the dictionary.  I can sort on the keys, but how can I sort based on the values?  Note: I have read Stack Overflow question here  and probably could change my code to have a list of dictionaries, but since I do not really need a list of dictionaries I wanted to know if there is a simpler solution to sort either in ascending or descending order. How to make a cyclorama I can create a collection of pictures giving a 360-degree view from a certain spot, just taking a bunch of photographs from the same spot but in a different direction.  However, what software could I use (on Windows) to stitch these images together to get one big image?  I prefer something that would stitch them together without showing any seams. Are these solutions of $2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$ correct? This is a question I've been asked several times by students and I tend to have a hard time phrasing it in terms they can understand. This is a natural question to ask and it is not usually well covered in textbooks, so I would like to know of various perspectives and explanations that I can use when teaching.  The question comes up naturally in what is usually students' second course in quantum physics / quantum mechanics. At that stage one is fairly comfortable with the concept of wavefunctions and with the Schrödinger equation, and has had some limited exposure to operators. One common case, for example, is to explain that some operators commute and that this means the corresponding observables are 'compatible' and that there exists a mutual eigenbasis; the commutation relation is usually expressed as $[A,B]=0$ but no more is said about that object.  This naturally leaves students wondering     what is, exactly, the physical significance of the object $[A,B]$ itself?   and this is not an easy question. I would like answers to address this directly, ideally at a variety of levels of abstraction and required background. Note also that I'm much more interested in the object $[A,B]$ itself than what the consequences and interpretations are when it is zero, as those are far easier and explored in much more depth in most resources.    One reason this is a hard question (and that commutators are such confusing objects for students) is that they serve a variety of purposes, with only thin connecting threads between them (at least as seen from the bottom-up perspective).   Commutation relations are usually expressed in the form $[A,B]=0$ even though, a priori, there appears to be little motivation for the introduction of such terminology. A lot of stock is placed behind the canonical commutation relation $[x,p]=i \hbar$, though it is not always clear what it means.  (In my view, the fundamental principle that this encodes is essentially de Broglie's relation $\lambda=h/p$; this is made rigorous by the  but that's quite a bit to expect a student to grasp at a first go.) From this there is a natural extension to the Heisenberg Uncertainty Principle, which in its general form includes a commutator (and an anticommutator, to make things worse). Canonically-conjugate pairs of observables are often introduced, and this is often aided by observations on commutators. (On the other hand, the energy-time and angle-angular momentum conjugacy relations cannot be expressed in terms of commutators, making things even fuzzier.) Commutators are used very frequently, for example, when studying the angular momentum algebra of quantum mechanics. It is clear they play a big role in encoding symmetries in quantum mechanics but it is hardly made clear how and why, and particularly why the combination $AB-BA$ should be important for symmetry considerations.  This becomes even more important in more rigorous treatments of quantum mechanics, where the specifics of the Hilbert space become less important and the algebra of observable operators takes centre stage. The commutator is the central operation of that algebra, but again it's not very clear why that combination should be special. An analogy is occasionally made to the Poisson brackets of hamiltonian mechanics, but this hardly helps - Poisson brackets are equally mysterious. This also ties the commutator in with time evolution, both on the classical side and via the Heisenberg equation of motion.   I can't think of any more at the moment but they are a huge number of opposing directions which can make everything very confusing, and there is rarely a uniting thread. So: what are commutators, exactly, and why are they so important? We know that Accio spell (aka ) has some limits:   It can be blocked (e.g. Harry couldn't Accio the Diadem in RoR in Deathly Hallows). Many more failures are listed on Wikia.   But what are the exact full rules and limitations governing the spell in Harry Potter world?   Distance?  Line-of-sight?  Knowing exact or approximate location?  Specificity of item (e.g. "Accio Broom" seemed to work for Weasleys in OotP to bring their locked-up brooms - why not just a random broom?) Types of Accio-able stuff? Can you Accio water? Lump of earth? Gases/liquids/etc...? Can you Accio based on jQuery selector (or an SQL where clause), for lack of a better description? E.g. "Accio the book that has the word 'Flammel'"? "Accio biggest green ball from this room"? Can you Accio ALL items of a kind? (the reason I thought of this whole question was seeing "Accio all guns" comment in ). Can you Accio vague stuff? ("Accio meal"). Other rules that can be inferred.   NOTE: While Wikia article has some of the info, it is clearly very lacking in details on specific rules based on specific instances in canon. I'm looking for a fully comprehensive, organized ruleset for the charm, not merely 1-2 tidbits in Wikia article. Why my tweet is not appearing on the timeline of the person I am following? I have tried tweeting someone I follow couple of times.  My tweets appear on my profile but when I look at the other person's timeline, my tweet is not there. I am following him but he is not following me. File.Copy is giving exception Prove: $|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$ $$|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$$   I have tried: $$|a\sin x+b \cos x|\leq |a+b|\leq \sqrt{a^2+b^2}$$  enough to prove: $$|a+b|\leq \sqrt{a^2+b^2}$$  But I can find how to continue from here How do I install an R package from source? I tried running Android 4.1.1 in VirtualBox, but it was too slow and sluggish to do anything. I came across Windroy which runs Android as a Windows app, but it won't run Google Play.  I want to run this on my laptop.  Any help would be greatly appreciated. How can I serialize an object to JSON in JavaScript? Can't insert code in my beamer slide I want to put some PHP code in my slide with the package listings but I have this error message:  Argument of \frame has an extra }.&lt;inserted text&gt;\par }   My latex code :   \documentclass{beamer}  \usepackage[utf8]{inputenc}  \usepackage[T1]{fontenc} \usepackage[frenchb]{babel} \usepackage{setspace} \usepackage{color} \usepackage{listings}  \usetheme{CambridgeUS}   \begin{frame} \begin{large}     Contexte : \newline \pause \end{large}  \begin{itemize}     \item Avertir Drupal \end{itemize} \begin{lstlisting}  // Gestion du contexte ete2013 // Utilisation d'un nouveau template if (%variables['ctpage'] == "ete2013") {     variables['template_files']=array('page-ete'); } \end{lstlisting} \end{frame} How to prove $D^n/S^{n-1}\cong S^n$? In my textbook it is said that the quotient space $D^n/S^{n-1}$ is homeomorphic to $S^n$. I can imagine it for $n=2$, but fail to make a mathematical proof for any dimension.  Can anyone provide a rigorous proof? What is batch size in neural network? I'm using Python Keras package for neural network. This is the . Is batch_size equals to number of test samples? From Wikipedia we have  information:     However, in other cases, evaluating the sum-gradient may require   expensive evaluations of the gradients from all summand functions.   When the training set is enormous and no simple formulas exist,   evaluating the sums of gradients becomes very expensive, because   evaluating the gradient requires evaluating all the summand functions'   gradients. To economize on the computational cost at every iteration,   stochastic gradient descent samples a subset of summand functions at   every step. This is very effective in the case of large-scale machine   learning problems.   Above information is describing test data? Is this same as batch_size in keras (Number of samples per gradient update)? Python: pop from empty list I know empirically that is the case.  I have just developed models that run into this conundrum.  I also suspect it is not necessarily a yes/no answer. I mean by that if both A and B are correlated with C, this may have some implication regarding the correlation between A and B.  But, this implication may be weak.  It may be just a sign direction and nothing else.    Here is what I mean...  Let's say A and B both have a 0.5 correlation with C.  Given that, the correlation between A and B could well be 1.0.  I think it also could be 0.5 or even lower.  But, I think it is unlikely that it would be negative.  Do you agree with that?    Also, is there an implication if you are considering the standard Pearson Correlation Coefficient or instead the Spearman (rank) Correlation Coefficient?  My recent empirical observations were associated with the Spearman Correlation Coefficient. Is there a subtle difference between "somebody" and "someone", "anybody" and "anyone"? After the upgrade of salesforce to Spring 15, I am getting this error "Error: You have exceeded the maximum number (5) of Apex scheduled jobs." when scheduling more than 5 batches. I have this issue in several environment. I used to more than 20 batches scheduled at once.   On the link below, it says that I can schedule up to 100 batches but I am being restricted to 5 only.   I can't have more than 5 jobs here. Understanding DC-DC boost converters I’m currently working on a solar powered boat with a modest sized team of people. For our competition we are only allowed to have 3 12v batteries on board but we are allowed to have up to 52v at any point in our setup. We are hoping to kick up the speed of our outboard motor for the sprint portion of the completion and are looking into using a boost converter for this.   After doing a little research I’m a bit confused about how they fundamentally work. From what I can tell they take in a voltage and current and put out a higher voltage with a lower current with some level of efficiency. But it seems to be independent from the load( in this case the motor) which doesn’t make much sense to me because I thought the current would be determined by whatever my load is. How do loads actually interact with boost converters?( The simpler the better.)  For reference we are using the Saietta 119r motor system and I’m considering using the MPS MPQ3425 boost converter.