If we can see only visual light then why we can't see other electromagnetic radiation as they also a type of light?
Why can't we see infrared light?
proving of inequality $(n!)^2 \leq n^n\cdot n!\leq (2n!)$, where $n\in \mathbb{N}$ How can we prove the inequality $(n!)^2 \leq n^n\cdot n!\leq (2n!)$, where $n\in \mathbb{N}$  $\bf{My \; Try}$::  $1\leq n$  $2\leq n$  $3\leq n$  ....  ....  $n\leq n$  So $1\cdot 2 \cdot 3 ..n \leq n^n$  So $n!\leq n^n\Rightarrow (n!)^2 \leq n^n\cdot n!$  But  did not understand how can we prove $n^n\cdot n! \leq (2n!)$  Help me  Thanks Let $f \colon \mathbb{R} \rightarrow  \mathbb{R}$ be a smooth function such that $$f'(x) = f(1+x)$$ How can we find the general form of $f$? I thought of some differential equations, but not sure how to use them here. Thanks. I have been using this command successfully, which changes a variable in a config file and then executes a Python script within a loop: for((i=114;i&lt;=255;i+=1)); do echo $i &gt; numbers.txt; python DoMyScript.py; done  As each DoMyScript.py instance takes about 30 seconds to run before terminating, I'd like to relegate them to the background while the next one can be spawned. I have tried what I am familiar with, by adding in an ampersand as below: for((i=114;i&lt;=255;i+=1)); do echo $i &gt; numbers.txt; python DoMyScript.py &amp;; done  However, this results in the below error: -bash: syntax error near unexpected token `;' Of Half-elven and mortal life When half-elvens choose mortal life, how would their bodies behave?  Half-elven are always elves by nature. Ever since they're born, their physical body and behaviour would be elvish - they have keen senses, they don't get physically weary, no illness, and so on.  Half-elvens are then offered a choice - Immortal elvish lives or mortal fate of men. When they choose the latter, like Elros and Arwen did, what would happen to their elvish nature? Would they lose it? I don't think they would lose their pointy ears, but what about the other elvish features that I mentioned before?  Are there any canonical references? Prove that for any piecewise smooth curve it is possible to find the parametrisation In thinking about Sturm-Liouville theory a bit I see I have no actual idea what is going on.  The first issue I have is that my book began with the statement that given  $$L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  the problem $L[y] \ = \ f$ can be re-cast in the form $L[y] \ = \ \lambda y$.  Now it could be a typo on their part but I see no justification for the way you can just do that!  More importantly though is the motivation for Sturm-Liouville theory in the first place. The story as I know it is as follows:  Given a linear second order ode   $$F(x,y,y',y'') = L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  it is an exact equation if it is derivable from a differential equation of one order lower, i.e.  $$F(x,y,y',y'') = \frac{d}{dx}g(x,y,y').$$  The equation is exact iff  $$a''(x) - b'(x) + c(x) = 0. $$  If $F$ is not exact it can be made exact on multiplication by a suitable integrating factor $\alpha(x)$.  This equation is exact iff  $$(\alpha(x)a(x))'' - (\alpha(x)b(x))' + \alpha(x)c(x) = 0 $$  If you expand this out you get the Adjoint operator  $$L^*[\alpha(x)] \ = \ (\alpha(x)a(x))'' \ - \ (\alpha(x)b(x))' \ + \ \alpha(x)c(x) \ = 0 $$  If you expand $L^*$ you see that we can satisfy $L \ = \ L^*$ if $a'(x) \ = \ b(x)$ &amp; $a''(x) \ = \ b'(x)$ which then turns $L[y]$ into something of the form  $$L[y] \ = \ \frac{d}{dx}[a(x)y'] \ + \ c(x)y \ = \ f(x).$$  Thus we seek an integratiing factor $\alpha(x)$ so that we can satisfy this &amp; the condition this will hold is that $\alpha(x) \ = \ \frac{1}{a(x)}e^{\int\frac{b(x)}{a(x)}dx}$  Then we're dealing with:  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \alpha(x)f(x)$$  But again, by what my book said they magically re-cast this problem as  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \lambda \alpha(x) y(x)$$  Then calling  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ ( \alpha(x)c(x)y \ - \ \lambda \alpha(x) )y(x) \ = \ 0$$  a Sturm-Liouville problem.  My question is, how can I make sense of everything I wrote above? How can I clean it up &amp; interpret it, like at one stage I thought we were turning our 2nd order ode into something so that it reduces to the derivative of a first order ode so we can easily find first integrals then the next moment we're pulling out eigenvalues &amp; finding full solutions - what's going on? I want to be able to look at $a(x)y'' \ + \ b(x)y' \ + \ c(x)y \ = \ f(x)$ &amp; know how &amp; why we're turning this into a Sturm-Liouville problem in a way that makes sense of exactness &amp; integrating factors, thanks for reading! Difference between logit and probit models Make locking posts more granular Which of those functions are PRFs? I just wanted to know how to add "create node form" in a block  when i  copy the source code and make a block out of that code, only public could post from that. when i am posting as a user it says timeout and tells us to refresh the page. plz someone give me solution.  SO i tried to add the following code in page.tpl.php  $form = node_add('nodetype'); print drupal_render($form); Set of branch points isn't discrete, but branch points are isolated? I refer to , which I understand says that the branch points of a nonconstant holomorphic map $F: X \to Y$ between Riemann surfaces $X$ and $Y$, both of which are not necessarily compact, form a discrete subset of range $Y$ of the map $F$. (Not sure if 'connected' is part of the definition of Riemann surfaces in other textbooks, but it is here.) Question 1: Do I understand correctly, and is this indeed true? What else I understand:  $A$ is a discrete subspace/subset (I just assume everything here is the usual subspace topology) of $B$ if and only if every singleton subset of $A$ is open in $A$ if and only if every point of $A$ is isolated.  This overflow question () seems to suggest this is not true but is true if $F$ is proper. However, I'm not sure the definitions of Riemann surface are the same (maybe some answers use definitions where Riemann surfaces are not necessarily connected or something).  2.1. I know $F$ is 'discrete' (meaning that its fibres are discrete; here, i refer to a ): see definition before Lemma 3.1, on p. 7), by Proposition II.3.12, and open, by open mapping theorem, which is Proposition II.3.8, but I'm not sure these help show $F$ is proper or are otherwise helpful. Ostensibly 'discrete' maps don't map discrete subsets to discrete subsets.   This stackexchange question () seems to suggest this is true.  I think this is true if $F$ is injective or at least something like locally injective. (Please don't make me type up all the stuff I did for this part.) However based on the question in (3) and based on the , which I think uses the term 'branch points' to mean the same thing as what Miranda means by 'ramification points' (and thus is different from Miranda's 'branch points'), it seems that no neighbourhood $U$ of a ramification point $p \in U \subseteq X$ of $F$ is such that the restriction $F|_U$ is injective.  Kinda related:    This page () says    Now, the ramification and branch points must form a discrete set  I don't think the page assumes anything like $F$ is proper.  The  text in (2.1) actually seems to assume proper based on the '3.7. Proper holomorphic maps' part, but said part refers to 'Lemma 3.17', so I think proper can be replaced with closed. Maybe $F$ is closed or something.  Kinda related (see Question 2 below):     says    If $F$ is proper then the image $\Delta = F(R)$ is discrete in $Y$.   This seems to have all the same assumptions as with Miranda. Ostensibly, either Miranda is wrong or there's some higher level machinery that allows us to omit $F$ proper.  Question 2: To possibly generalise this, what are the minimum requirements on $F, X$ and $Y$ to make $F$ map discrete subsets to discrete subsets? I mean, I'm not sure we really need 'holomorphic' here. I have a feeling this applies perhaps to maps that are just open/closed, continuous and discrete and spaces that are just Hausdorff and locally compact or something.  (again) Kinda related: Movement arrow in gloss Can I use B1 visa (Visitor-Business) to travel to the Canada without a business purpose? I got a B1 visa (Visitor-Business) for Canada in 2012 when I traveled to Canada for a business trip from my company. It expires in 2015 and I'd love to visit the Canada again, but for no other purpose than traveling, visiting family.  I want to know, If I can legally travel to Canada with B1 visa without a real business need (assuming I'm honest);  I live in India, and I have B1/B2 from US and used same Visa for Business &amp; Personal visits multiple times. Work is not done by the magnetic field? Software for text search in files I am looking for a software that will search text in files from a folder similar to .  Is there something similar? Dirac equation as Hamiltonian system Let us consider Dirac equation $$(i\gamma^\mu\partial_\mu -m)\psi ~=~0$$ as a classical field equation. Is it possible to introduce Poisson bracket on the space of spinors $\psi$ in such a way that Dirac equation becomes Hamiltonian equation   $$\dot{\psi}~=~\{ \psi,H\}_{PB}~?$$  Of course, such Poisson bracket would be graded (super Poisson bracket), but if it exists this would explain on classical level why $\frac{1}{2}$-spinors correspond to fermions. How to convert from high poly to low poly I'm making a game in the unity engine and for my models i'm using blender 2.6. The thing is that i'm pretty new to graphics design (modeling, sculpting, etc) and by following some tutorials i've managed to make some very high poly meshes for my characters with the sculpt editor, the meshes have about 1.5M - 2M faces and of course a game character should be less than 30K faces so the game runs smoothly with good optimization on an average pc.  my question is how do i translate the details of the high poly stuff to low poly meshes (7k - 30k faces), i've heard about uv unwrapping and stuff but i'm not sure what is the process to follow.  i don't mind if the recommendation is very hard and for advanced users because the primary thing i want to do with this, is to learn. What is the mechanism behind the slowdown of light/photons in a transparent medium? PHP import Excel into database (xls & xlsx) How can I prevent my mail from being classified as spam?