Dynamic tensile modulus and X-ray intensity under applied electric field were measured to investigate frequency-temperature dependence of the modulus by Joule heat in relation to the electron transfer mechanisms.
Theorem 3.3 Let f be a modulus function and ( X, g ) be a paranormed space.
If f is a bounded modulus function and X k → ξ ( S θ F ( Δ m ) ), then X k → ξ ( N θ F ( Δ ( p ) m, f ) ). Proof Easy, so omitted.
In addition, |x| and arg(x) represent respectively the modulus and the argument of x. (mathbb {E}[cdot ]) denotes statistical expectation.
To show that the strict inclusion may occur, let f be a modulus and consider the sequence (x =(x_{k})) defined by x_{k}= textstylebegin{cases} 1 &mbox{if }k = n^{2}, 0 &mbox{if }k neq n^{2}, end{cases}displaystyle quad n = 1,2,3, ldots.
In order to obtain rate of convergence in terms of modulus of continuity (omega(f delta)), we assume that, for any (f in C_{B}[0,infty)) and (x geq0), the modulus of continuity of f is given by omega(f delta)=max_{substack {|t-x|leqdelta t,xin[0,infty)}}big|f(t -f(x)big|. (3.7) t -f it implies for any (delta> 0) big|f(x big|)big| leqomega(f;delta) biggl( frac{|x-y|}{delta}+1 biggr). (3.7).
For x, y ∈ C n, their complex scalar product by 〈 x, y 〉 c and the modulus of x by ∥ x ∥ = 〈 x, x 〉 c.
The coating characterization included optical microscopy and SEM of metallographically prepared cross-sections, hardness measurements, determination of the Young's modulus and phase composition by X-ray diffraction.
These two effects have been shown to be competitive in determining the crystal energy of the Gd2(Zr1−xTix)2O7 series and result in a minimum value of the Young's modulus at x = 0.3 and a maximum value of the coefficient of thermal expansion at x = 0.2.
A correlation between the bulk modulus of Mg X alloys and the nearest-neighbor distance between Mg and X is shown.
