WPF: More efficient way of displaying quickly-changing images?
How to display quick-updating images without large memory allocation?
Prove that $(a+1)(a+2)...(a+b)$ is divisible by $b!$ The problem is following, prove that:  $$(a+1)(a+2)...(a+b)\text{ is divisible by } b!\text{ for every positive integer a,b}$$    I've tried solving this problem using mathematical induction, but I don't think that i did it correctly. Here's what i've done  $1.\ b=1\ (Basis)$  $$b!| (a+1)(a+2)...(a+b)$$ $$1 | a+1 \text{, which is true}$$  $2.\ b=k\ (Induction\ Hypothesis)$  $$k! | (a+1)(a+2)...(a+k)\text{, we assume it's true}$$ $$k!*n = (a+1)(a+2)...(a+k)\text{, n is some positive integer}$$  $3.\ b=k+1\ (Inductive\ Step)$  In order to prove I should get:  $$(k+1)!*m = (a+1)(a+2)...(a+k)(a+k+1)\text{, where m is some positive integer}$$ $$(a+1)(a+2)...(a+k)(a+k+1) = k!*n*(a+k+1)$$ $$(a+1)(a+2)...(a+k)(a+k+1) = (k+1)!*n + k!*n*a$$ $$(a+1)(a+2)...(a+k)(a+k+1) - a(a+1)(a+2)...(a+k) = (k+1)!*n$$ $$(a+1)(a+2)...(a+k)(k+1) = (k+1)!*n$$  And as you can see I'm returning at the beginning.  Also I've tried to use the fact that in b consecutive numbers, there must be at least one that divides b, but since i eliminate that number (because I couldn't be sure that the quotient is divisible with (b-1) or (b-2) or ... or 2. And after this i can't continue with (b-1) using this method, because it's not necessary for the other (b-1) to be consecutive. How to set a maximum execution time for a mysql query? Does an Indian citizen need a transit visa in Australia? Indian passport holder. Do I need transit visa in Sydney? Where did dedicated.abac.net. come from in my rDNS and PTR records? So I followed this guide to the letter:   I did so for 3 domains.  When I "test" the records and enter: host 64.150.176.81 on my linux machine, it shows me 81.172.150.64.in-addr.arpa domain name pointer 64-150-172-81.dedicated.abac.net.  when it should be showing me 81.176.150.64.in-addr.arpa domain name pointer 64-150-172-81.example.com.  Has anybody encountered a similar problem?  I am hosting with codero, not abac.net btw. Get total height of webView's content using Javascript What do repeating '^@' symbols in a logfile mean? A few times I have come across this in a log file:  Sep 17 09:35:57 00187d378926 ovpn-00187d378926[1701]: /sbin/ifconfig tun0 0.0.0.0 Sep 17 09:35:57 00187d378926 kernel: Kernel logging (proc) stopped. Sep 17 09:35:57 00187d378926 rsyslogd: [origin software="rsyslogd" swVersion="4.6.4" x-pid="1707" x-info="http://www.rsyslog.com"] exiting on signal 15. ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@ ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@ ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@   What does it mean?  Sometimes it is a single line, other times it is lots of lines.  I usually see it happen on Debian 6 machines. This question relates to a discussion on . Euclid's proof of the infinitude of primes is an indirect proof (a.k.a. proof by contradiction, reductio ad absurdum, modus tollens). My understanding is that Intuitionists reject such proofs because they rely on the Law of the Excluded Middle, which they don't accept. Does there exist a direct and constructive proof of the infinitude of primes? Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist? find the number of analytic functions which vanish only on $S$? Financial requirements to sponsor relatives visiting the Schengen area [Germany]? Solving 2-degree equations in 3 variables. We are given 3 equations: $x^2+\sqrt3 xy + y^2 = 25$ $y^2 + z^2 = 9$ $x^2 +xz+ z^2 = 16$. $x,y,z$ are positive real numbers. Then we have to find value of $xy + 2yz + \sqrt3 xz$. Ring of rational-coefficient power series defining entire functions How do I add a directory to the $PATH in Ubuntu and make the changes permanent? Is there anybody who can help me show the following?      $$ \sum_{x=0}^{n-1} \cos\left(k +x{2\pi\over n}\right) =0  \qquad\hbox{and}\qquad  \sum_{x=0}^{n-1} \sin\left(k +x{2\pi\over n}\right) =0 $$   I can only prove this claim for the sine expression when $k=0$, by grouping the output values such that they all cancel out (for both $n$ even and odd). I have thought about using complex numbers in exponential form to prove it, but I can't do it ... :P   I would really appreciate an algerbraic proof. Mention a specific potential adviser and project in the PhD statement of purpose I am writing my statement of purpose for my Ph.D. and I want to work in a specific field with a specific professor in the department that I am going to apply for and I have a very good experience in this field. I sent an email for that professor asking if he/she is accepting Ph.D. students and he/she did not respond. I built my whole statement of purpose based on my previous achievements and future plan for that specific area. I also mentioned that I am willing to work with that professor. My statement of purpose tells that I am very passionate about this field and I am only willing to work with that professor (if there were more than one professor working on the same area, I would mention all of them). if that professor doesn't accept new Ph.D. students, will I risk my admission to that school?   Will the committee response be something like this:  "the applicant really knows what he/she is going to do and there is a specific future plan and he/she also got a good experience in this field, but he/she sounds very strict to his plan and doesn't show any flexibility...REJECTED"? Interactive / Real time code compilation for stackoverflow.com. That will improve the answer correctness of the stackoverflow.com.  When adding a new code block, let the user try to compile the code and see the result. It will be useful to demonstrate class designs, algorithms, and logics. What happens to global variables declared in a DLL? Let's say I write a DLL in C++, and declare a global object of a class with a non-trivial destructor.  Will the destructor be called when the DLL is unloaded? In RSA, why is it important to choose e so that it is coprime to φ(n)? When choosing the public exponent e, it is stressed that $e$ must be coprime to $\phi(n)$, i.e. $\gcd(\phi(n), e) = 1$.   I know that a common choice is to have $e = 3$ (which requires a good padding scheme) or $e=65537$, which is slower but safer.  I also know that for two primes $p,q$, we have $\phi(pq) = (p - 1) (q - 1)$  Now, let me give a (simple) example:  Say I choose $e = 3$, and two random primes $p = 5$ and $q = 13$.  I can now compute $\gcd(3, \phi(5 \cdot 13)) = 3$.  This reveals that $3$ and $\phi(n)$ are not coprime. I assume this could also happen for large values of $p$ and $q$, and likewise for another $e$. I therefore assume that the RSA algorithm must check that $\gcd(e, \phi(pq)) = 1$. But let's assume it doesn't.  How does RSA become vulnerable if $\gcd(e, \phi(pq)) \neq 1$? Prove that $ \int _a^b\:f\left(x\right)\:\:dx\:=\:\int _a^b\:f\left(a+b-x\right)\:\:dx $ This is the prove- $y=a+b-x$. Then $dy/dx=-1$ and thus $$\int_a^bf(a+b-x)dx= \int_b^a-f(y)dy = \int_a^bf(x)dx$$  But I dont understand the last part of this - Why ? $$\int_b^a-f(y)dy = \int_a^bf(x)dx$$  I know that $\int_b^a-f(y)dy = \int_a^bf(y)dy $. But something is not clear to me. Why are my photos not crisp?