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Considering that the movements of complex system entities take place on continuous, but non-differentiable, curves, concepts, like non-differentiable entropy, informational non-differentiable entropy and informational non-differentiable energy, are introduced.First of all, the dynamics equations of the complex system entities (Schrödinger-type or fractal hydrodynamic-type) are obtained. The last one gives a specific fractal potential, which generates uncertainty relations through non-differentiable entropy. Next, the correlation between informational non-differentiable entropy and informational non-differentiable energy implies specific uncertainty relations through a maximization principle of the informational non-differentiable entropy and for a constant value of the informational non-differentiable energy. Finally, for a harmonic oscillator, the constant value of the informational non-differentiable energy is equivalent to a quantification condition. Keywords:non-differentiable entropy; informational non-differentiable entropy; informational non-differentiable energy; uncertainty relations Entropy 2014, 16 6043

Correlating Shannon's maximum informational entropy variational principle with the constant value of Onicescu's informational energy, the uncertainty relations for canonical systems with SL(2R) invariance are obtained. The constant value of Onicescu's informational energy corresponds, through transitivity manifolds of the SL(2R) group, to the ergodic theorem and, in particular case of a linear oscillator, to a quantification condition. Specifying de Broglie's idea by a periodic field in a complex coordinate, it is proved that the oscillator synchronization group of the same ensemble is still an achievement of the SL(2R) (Barbilian's group). Integrally invoking the invariant functions through the simultaneous action of the two isomorphic SL(2R) groups, the uncertainty relations and Stoler group of synchronization among oscillators from different ensembles (i.e., the second quantification when the annihilation and creation operators refer to a linear oscillator) are obtained. Barbilian's group parameters are interpreted by means of a variational principle as field amplitudes of a stationary vacuum metric.

We study Riemann-Lebesgue integrability for interval-valued multifunctions relative to an interval-valued set multifunction. Some classic properties of the RL integral, such as monotonicity, order continuity, bounded variation, convergence are obtained. An application of interval-valued multifunctions to image processing is given for the purpose of illustration; an example is given in case of fractal image coding for image compression, and for edge detection algorithm. In these contexts, the image modelization as an interval valued multifunction is crucial since allows to take into account the presence of quantization errors (such as the so-called round-off error) in the discretization process of a real world analogue visual signal into a digital discrete one.

Abstract.A comparison between a set-valued Gould type and simple Birkhoff integrals of bf (X)-valued multifunctions with respect to a nonnegative set functionis given. Relationships among them and Mc Shane multivalued integrability is given under suitable assumptions.