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Systems biologist.


Nov
12
awarded  Enlightened
Nov
12
awarded  Nice Answer
Nov
12
comment What is the integral of $x(1-x)^8$?
Why $x=(x-1)+1$? Because I want to treat $x-1$ as sort of a unit, which makes the calculation easier.
Nov
12
comment What is the integral of $x(1-x)^8$?
$(x-1+1)(x-1)^8=(x-1)\cdot (x-1)^8+1\cdot (x-1)^8$ follows from the distributive law.
Nov
12
revised What is the integral of $x(1-x)^8$?
added 112 characters in body
Nov
12
answered What is the integral of $x(1-x)^8$?
Nov
11
comment Finding pairs of integers such that $x^2+3y$ and $y^2+3x$ are both perfect squares
Symmetry ($x^2+3y$ and $y^2+3x$) suggests that without loss of generality we can assume $y\ge x$. Also notice that if $x\sim y$, $y^2+3x$ is not much larger than $y^2$, so it may be equal to $(y+1)^2$ or $(y+2)^2$... To discuss this rigorously you need to assume $y\ge x$.
Nov
10
comment Show that the value of a definite integral is unity
As Dinesh points out, $g(x)=\frac{f(x)}{f(x)+f(6-x)}$ satisfies $g(x)+g(6-x)=1$ (more generally, $g(x)+g(a-x)=b$ where $a$ and $b$ are constants); this is the condition where you can use this integration trick.
Nov
10
comment Is $ x^2+ax+a$ irreducible over ring $\mathbb{Z}$ of integers?
@pedja I think in that case, the same method as posted by Bruno Joyal works (i.e. think in $\mathbb{F}_2[x]$).
Nov
10
answered Is $ x^2+ax+a$ irreducible over ring $\mathbb{Z}$ of integers?
Nov
10
comment Must every event have a probability?
The probability that you can find life on Mars (or a Nessie in Loch Ness). I don't think it is "impossible to define any probability", but I think it is mathematically ill-defined (is it?).
Nov
10
revised Show that the value of a definite integral is unity
edited body
Nov
10
revised Show that the value of a definite integral is unity
added 337 characters in body
Nov
10
comment Show that the value of a definite integral is unity
Because $y$ is a dummy variable; please see my edited comment.
Nov
10
revised Show that the value of a definite integral is unity
added 111 characters in body
Nov
10
comment Show that the value of a definite integral is unity
I posted an answer below.
Nov
10
answered Show that the value of a definite integral is unity
Nov
10
comment Show that the value of a definite integral is unity
In general: $\int_2^4 \frac{f(x)}{f(x)+f(6-x)}\,dx=1$.
Nov
9
awarded  Enlightened
Nov
9
revised Equivalent conditions to $0\leq x+\frac{1}{2}x(1-x)a\leq 1$?
added 12 characters in body