What is the difference between loops and recursion and which one is more preferred for solving a problem?
I love recursion. I think it simplifies things a lot. Another may disagree; I think it also makes the code a lot easier to read. However, I've noticed that recursion is not used as much in languages such C# as they are in LISP (which by the way is my favorite language because of the recursion).   Does anybody know if there is any good reasons not use recursion in the languages such as C#? Is it more expensive than iteration?
The question is to show that for any $x,y \in\Re^n$  $||x+y||\cdot||x-y||\le||x||^2+||y||^2$  with equality if and only if $\langle x,y \rangle= 0.$  I have that $||x+y||^2=||x||^2+2 \langle x,y \rangle+||y||^2$ , which helps me prove equality but I'm lost on how to show this and show that the inequality part. Analyzing $\biggl\lfloor{\frac{x}{5}}\bigg\rfloor=\bigg\lfloor{\frac{x}{7}}\bigg\rfloor$ How many non negative integral values of $x$ satisfy the equation :$$\biggl\lfloor{\dfrac{x}{5}}\bigg\rfloor=\bigg\lfloor{\dfrac{x}{7}}\bigg\rfloor$$.   My try:  Writing few numbers and putting in the required equation.  $\underbrace{0,1,2,3,4}_{\implies 0=0},\underbrace{5,6}_{\displaystyle\times},\underbrace{7,8,9}_{\implies 1=1},\underbrace{10,11,12,13}_{\displaystyle\times} ,\underbrace{14}_{\implies 2=2},\underbrace{15,16,17,18,19,20}_{\text{LHS gives 3 but RHS gives 2}},\underbrace{21,22,23,24}_{\text{LHS gives 4 but RHS gives 3}}$.  The point which I wish to make here is that after $14$ we'll get $5's$ multiple before than $7's$, hence $\forall x \geq 15 \implies \biggl\lfloor{\dfrac{x}{5}}\bigg\rfloor&gt;\bigg\lfloor{\dfrac{x}{7}}\bigg\rfloor$  Hence the only solution are: $\{0,1,2,3,4,7,8,9,14\}$. Does these make sense?  I need to know other solutions that don't incorporate such analyzing, means using some properties and framing it to solve above. Which partition is mounted to where? The  (or conjecture), suggests that black holes can be entirely described by their mass, angular momentum and charge. All other details of the BH formation are lost.  Is there a simple way of understanding why you would be able to tell the difference between a BH made from protons and one made from electrons, but not between one made from matter rather than anti-matter?  i.e. What is special about charge, and why aren't lepton number, baryon number, strangeness etc, also black hole properties? implementing functions in a game As a non-native speaker, I feel that it is ridiculous that can and can't could sound so similar in American accent.  Just now, I was just listening to a video in which the speaker with an American accent says     Fat cells can't reproduce themselves.   The speaker dragged the word "can't" longer for emphasis, but perhaps as a result, what I heard at first was     Fat cells can reproduce themselves.   Of course from the context I later figured out what he was really saying, but I felt extremely surprised by how these two words which literally mean the opposite could sound so similar in American accent. How is that not affecting American people's daily conversations? As a non-native speaker, is there a way for me to further clarify them when my listeners are confused? (Like when you say "one-eight" to clarify that you were trying to say "eighteen" as opposed to "eighty".) Why does holding something up cost energy while no work is being done? I read the definition of  as  $$W ~=~ \vec{F} \cdot \vec{d}$$  $$\text{ Work  =  (Force) $\cdot$ (Distance)}.$$ If a book is there on the table, no work is done as no distance is covered. If I hold up a book in my hand and my arm is stretched, if no work is being done, where is my energy going? Understanding GeoTIFF tags? Consistency of an order statistic in exponential distribution Why are there two tensions on my free body diagram? How do you use plural or singular form of an adjective? So I was taught that you should almost always use singular form of an adjective instead of plural form. But recently, I've seen people using plural form and it sounds quite weird and I don't know if it is correct.  For example,   "Salary tax" instead of "Salaries tax" "Office supplies" instead of "Offices supplies" "Product demo" instead of "Product demo"   In the above cases, the word "Salary" is used as an adjective to describe the type of "tax", so it will be in singular form. But I saw people using salaries tax and not sure if it is grammatical.  So my question is if there are any general rules on how to use the singular/plural form of an adjective? What kind of matter are black holes made of? Adobe Illustrator: Lines with stroke < 2.0pt doesn't change color I created lines using the line segment tool in Adobe Illustrator (latest Adobe Cloud version) and I can't change the color of the stroke (through the Appearance panel) if the stroke weight is less than 2.0pt.   If the stroke is 2pt or more, then the color around the line changes but not the line itself.    The whole behavior is odd for me as I've never faced in Illustrator before.  Edit: added screen below with the line selected. An example of a regular function over an open set Division rules for other number systems? How could we make the same division rules for other number systems, like in our decimal system: a number is divisible with 2 if it's last digit is 0,2,4,6,8, by 3 if the sum of digits is divisible with 3, and so on.  Can we do the same rules for g-based number systems too, where g is not 10, and greater than 1? How do I get my C# program to sleep for 50 milliseconds?  This might seem an easy question, but I'm having a temporary brain failure moment! Prove: $\binom{n}{0}F_0+\binom{n}{1}F_1+\binom{n}{2}F_2+\cdots+\binom{n}{n}F_n=F_{2n}$;  I was stuck with this question for a while... Help me please!!! Thanks!!! Uploading an image from the web can leave paste broken in editor How do people come up with difficult math Olympiad questions? My request is simple: Many times, there will be a question where some people feel like it should be closed. At the same time, I'll feel like it's a perfectly legit question, and it should therefore NOT be closed. Right now, I have to wait till it gets enough close votes (which it always does for some reason, as soon as 1 or 2 people vote to close, e/o else decides to vote as well) and then vote to re-open. How about, as soon as there's one vote to close, there should be an option to counter the close vote, something like "vote not to close" or whatever. This would then bring the "Vote to close" number down by one. Only when the vote to close outnumbers the vote not to close by 5, that's when a question is indeed closed. Any thoughts?