Each frequency band of the blocks is encrypted by modulus function, and then combined each to get the new block.
Let f be an unbounded modulus function and (widetilde{alpha }in ( 0,1 ] ).
[10, 25] Let f be a modulus function and let 0 < δ <1.
Theorem 3.3 Let f be a modulus function and ( X, g ) be a paranormed space.
Let f be a modulus function and α̃ be a positive real number.
Proof (a) Let f be a modulus function, and let ε be a positive number.
We introduce the vector-valued sequence spaces,, and, and, using a sequence of modulus functions and the multiplier sequence of nonzero complex numbers.
His 1914 paper on 'Modulus functions and approximation to π' contains several new innovative empirical formulas and geometrical constructions for approximating π.
Let be a sequence of seminormed spaces such that for each, a sequence of strictly positive real numbers, a sequence of seminorms, a sequence of modulus functions, and any fixed sequence of nonzero complex numbers.
Let f = ( f mn ) be a Musielak-modulus function and q = (q mn ) be a double analytic sequence of strictly positive real numbers; the sequence spaces χ f μ 2 q, d x 1, d x 2, ⋯, d x n − 1 p φ Open image in new windowand Λ f μ 2 q, d x 1, d x 2, ⋯, d x n − 1 p φ Open image in new windoware linear spaces.
Let f = ( f mn ) be a Musielak-modulus function and q = (q mn ) be a double analytic sequence of strictly positive real numbers; the sequence space χ f μ 2 q, d x 1, d x 2, ⋯, d x n − 1 p φ Open image in new windowis a paranormed space with respect to the paranorm defined by g ( x ) = inf f mn μ mn ( x ), d x 1, d x 2, ⋯, d x n − 1 p φ q mn 1 / H ≤ 1, Open image in new window.
