Q:

Generating a fixed number

I have a fixed number $a$. Now using $a$ I need to construct a number $b$ such that $0.99\leq b\leq 1$. Is there any mathematical formulation of such a construction that looks random. The generation of such a number should be deterministic. can somebody hint at any algorithms

A:

The provided value, $a$, is your seed value.
The algorithm you want is a pseudo-random number generator (PRNG).  Note that pseudo-random number generators are explicitly deterministic; they merely appear random.
Depending on how random you need it to look, you may want to use a cryptographically secure PRNG.  Such an algorithm, if truly cryptographically secure, should pretty much look random as far as any human-like observer could tell (though its operation may be plain and obvious to super-human observers, e.g. advanced aliens or deities; can't do much there).
Most PRNG's are designed to produce a bit-array (a bunch of $\left\{0,~1\right\}$ values) given a seed.  If $a$ is real $\left(a\in\mathbb{R}\right)$, then that bit-array may contain arbitrarily many bits.  Typically, the bit-array's defined such that each bit is pseudo-unrelated to the others, such that can be assumed to be mutually independent.
Then, it's just a matter of mapping the bit-array to numeric values.  You might do this by interpreting the bit array as a serialization of a binary real $\in{\left[0.9,1\right]}$.

A:

What about 
$$0.99+\frac1{100(1+a^2)}?$$

A:

For example $\,b=0.99 + \dfrac{\sin^2(\lambda a)}{100}\,$, with $\,\lambda \in \mathbb{R}\,$ adjusted for the range of $\,a\,$.

