In order to develop certain fractional probabilistic analogues of Taylor’s theorem and mean value theorem, we introduce the nth-order fractional equilibrium distribution in terms of the Weyl fractional integral and investigate its main properties. Specifically, we show a characterization result by which the nth-order fractional equilibrium distribution is identical to the starting distribution ...

Computational vision often needs to deal with derivatives of digital images. Derivatives are not intrinsic properties of a digital image; a paradigm is required to make them well-deened. Normally, a linear ltering is applied. This can be formulated in terms of scale space, functional minimization or edge detection lters. In this paper, we take regularization (or functional minimization) as a st...

The nth order differential equation x + c (t )x + ƒ( t,x) = e(t),n>3 is considered. Using the Leray-Schauder principle, it is shown that under certain conditions on the functions involved, this equation possesses a periodic solution.

The paper investigates and explains a new simple analytical tuning of proportional-integrative-derivative (PID) controllers. In combination with nth order series binomial low-pass filters, they are to be applied the double-integrator-plus-dead-time (DIPDT) plant models. With respect use derivatives, it should understood that design appropriate filters is not only an implementation problem. Rath...

Theorem 1. The nth derivative of the function 1/r along n constant (translation-invariant) vector fields in R coincides on the sphere with a spherical function of degree n. Any nonzero spherical function of degree n can be obtained by this construction from some n-tuple of nonzero vector fields. These n fields are uniquely defined by the function (up to multiplication by nonzero constants and p...