The α-order fractional derivative of the Dirichlet η function is computed in order to investigate the behavior of the fractional derivative of the Riemann zeta function ζ(α) on the critical strip. The convergence of η(α) is studied. In particular, its half-plane of convergence gives the possibility to better understand the ζ(α) and its critical strip. As an application, two signal processing networks, corresponding to η(α) and to its Fourier transform respectively, are shortly described.

In this paper, a new operational matrix of variable-order fractional derivative (OMV-FD) is derived for the second kind Chebyshev wavelets (SKCWs). Moreover, a new optimization wavelet method based on SKCWs is proposed to solve multi variable-order fractional differential equations (MV-FDEs). In th e proposed method, the solution of the problem under consideration is expanded in terms of SKCWs. Then, the residual function and its errors in 2-norm are employed for converting the problem under study to an optimization one, which optimally chooses the unknown coefficients. Finally, the method of constrained extremum is applied, which consists of adjoining the constraint equations obtained from the given initial conditions to the object function obtained from residual function by a set of unknown Lagrange multipliers. The main advantage of this approach is that it reduces such problems to those optimization problems, which greatly simplifies them and also leads to obtain a good approximate solution for them.

Dealing with simplicial decompositions which are dimension independent allows for the convergence of disparate viewpoints from computer graphics, solid and geometric modeling.In this framework it is possible to treat in a unified manner several geometric problems, such as solid modeling of articulated objects, simplicial approximation of curved manifolds, motion encoding and interference detection, free configuration space computation, and graphical representation of multidimensional data.In the paper the authora describe the winged scheme, a simple representation based on simplicial decompositions, which can be used for linear polyhedra of any dimension and which allows for "solid" approximation of curved manifolds when combined with curved maps (e.g., NURBS).Various operators and algorithms are discussed, including boundary evaluation, linear and screw extrusion, grid generation, simplicial maps, and set operations.A simple manipulation language is also introduced, and some nontrivial examples are discussed.