Consider the function $$f: \mathbb{R} \rightarrow \mathbb{R}$$ defined by $$f(x)=\frac{2 x}{\sqrt{1+9 x^2}}$$. If the composition of $$f, \underbrace{(f \circ f \circ f \circ \cdots \circ f)}_{10 \text { times }}(x)=\frac{2^{10} x}{\sqrt{1+9 \alpha x^2}}$$, then the value of $$\sqrt{3 \alpha+1}$$ is equal to _______.

Let $$\mathrm{A}=\{1,2,3, \ldots, 7\}$$ and let $$\mathrm{P}(\mathrm{A})$$ denote the power set of $$\mathrm{A}$$. If the number of functions $$f: \mathrm{A} \rightarrow \mathrm{P}(\mathrm{A})$$ such that $$\mathrm{a} \in f(\mathrm{a}), \forall \mathrm{a} \in \mathrm{A}$$ is $$\mathrm{m}^{\mathrm{n}}, \mathrm{m}$$ and $$\mathrm{n} \in \mathrm{N}$$ and $$\mathrm{m}$$ is least, then $$\mathrm{m}+\mathrm{n}$$ is equal to _________.

Let $$\mathrm{A}=\{1,2,3,4,5\}$$ and $$\mathrm{B}=\{1,2,3,4,5,6\}$$. Then the number of functions $$f: \mathrm{A} \rightarrow \mathrm{B}$$ satisfying $$f(1)+f(2)=f(4)-1$$ is equal to __________.

Let $$\mathrm{R}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}\}$$ and $$\mathrm{S}=\{1,2,3,4\}$$. Total number of onto functions $$f: \mathrm{R} \rightarrow \mathrm{S}$$ such that $$f(\mathrm{a}) \neq 1$$, is equal to ______________.