The aim of the study was to create a mathematical model useful for monitoring the release of bioactive aldehydes covalently bonded to the chitosan by reversible imine linkage, considered as a polymer–drug system. For this purpose, two hydrogels were prepared by the acid condensation reaction of chitosan with the antifungal 2-formyl-phenyl-boronic acid and their particularities; influencing the release of the antifungal aldehyde by shifting the imination equilibrium to the reagents was considered, i.e., the supramolecular nature of the hydrogels was highlighted by polarized light microscopy, while scanning electron microscopy showed their microporous morphology. Furthermore, the in vitro fungicidal activity was investigated on two fungal strains and the in vitro release curves of the antifungal aldehyde triggered by the pH stimulus were drawn. The theoretical model was developed starting from the hypothesis that the imine-chitosan system, both structurally and functionally, can be assimilated, from a mathematical point of view, with a multifractal object, and its dynamics were analyzed in the framework of the Scale Relativity Theory. Thus, through Riccati-type gauges, two synchronous dynamics, one in the scale space, associated with the fungicidal activity, and the other in the usual space, associated with the antifungal aldehyde release, become operational. Their synchronicity, reducible to the isomorphism of two SL(2R)-type groups, implies, by means of its joint invariant functions, bioactive aldehyde compound release dynamics in the form of “kink–antikink pairs” dynamics of a multifractal type. Finally, the theoretical model was validated through the experimental data.

In the last few decades, a series of earthquakes were recorded which pointed out several deficiencies regarding the ductile seismic response of MR RC frame structures. Thus, the research problem centres around the failure mechanisms registered by the structures, which differ from the general notions of seismic response commonly found in current design standards and norms regarding seismic actions. In these conditions, in the present paper—by using comparative methods—the analytical validation of the solution of plastic hinge concentration and seismic energy dissipation in the marginal beam areas is proposed. Therefore, the RC beam sections were reduced (weakened) in the marginal areas which exhibit a plastic deformation potential, as well as in the corner areas of concrete slabs with vertical rectangular holes. The significant outcomes of this research imply the partial “guiding” of plastic hinges in the zones adjacent to beam ends. Furthermore, a reduction of both the negative effects of horizontal rigidization of the beams and the cracking and plastic deformation effects of beam-column frame joints was observed. With these technical implications, a complex mechanism of plastic deformation of MR RC frame models is registered in which all lateral elements (including RC columns) participate in the dissipation of seismic energy, without the occurrence of the “weak storey” mechanism for any of the analytical RC frame models. Furthermore, it is possible to observe the partial formation of the global plastic mechanism “Strong Columns—Weak Beams” (SCWB) for some of the structural models. Finally, the analytically studied innovative element regarding the improvement of the seismic response of pure MR RC frame structures is successfully validated.

By assimilating shape memory alloys with mathematical multifractal-type objects, a theoretical model based on Scale Relativity Theory in the form of The Multifractal Theory of Motion, in order to explain the mechanical behavior of such material, is proposed. The model is validated by analyzing the mechanical behavior of Cu–Al–Zn shape memory alloy with various chemical compositions. More precisely, the multifractal tunnel effect can “mime” the mechanical hysteresis of such a material, a situation in which a direct correspondence for several mechanical properties of Cu–Al–Zn is highlighted (the chemical composition can be correlated with the shapes of the curves controlled through the multifractality degree, while the areas delimited by the same curves can be correlated with the multifractal specific potential, as a measure of the mechanical memory degree).

In the first part of the current study, the effectiveness of the transversal cross-section reduction method for RC beams in marginal areas (by means of mechanical drilling) was validated. The said method “encourages” the formation of plastic hinges at the beam ends and, at the same time, allows for taking into account the bending stiffness of RC slabs, which is exerted upon the RC beams. In these conditions, the second part of the current research study (i.e., the current manuscript) highlights the real mode of reducing the lateral stiffness of the slabs upon the RC beams. These elements form a common body, together with the beam–column frame node. The same method as in the first part of the study—“weakening” the plates in the corner area through vertical drilling, without affecting the integrity of the reinforcing elements—was used. The analytical MR RC frame model, studied by means of the comparative method, highlights the efficiency of the transversal cross-section reduction method for RC slabs. Basically, the directing of the plastic deformations from the weakened slab areas towards the marginal areas of the reinforced concrete beams takes place. The beams rotate as far as the weakened slab areas allow its plastic deformation, thus being possible to observe the partial conservation effect of the beam–column frame joint. Furthermore, for the analytical model with the maximum number of vertical holes in the corner areas of the concrete plate, minimal plastic deformations are recorded for the marginal areas of the concrete columns. A partial conservation of the formation mechanism of the “beam-slab-frame node” common rigid block is also noted. Consequently, the dissipation of the seismic energy is made in a partially controlled and directed manner, in the “desired” areas, according to the “Strong Columns—Weak Beams” (SCWB) ductile mechanism of the lateral behavior to seismic actions for reinforced concrete frame structures. The mechanism is specified in current design norms for RC frame systems. The effectiveness of the method for reducing the transversal section of the RC plates in the corner areas by means of transversal drilling is highlighted and validated from the perspective of the local and global ductile seismic response of reinforced concrete frame structures. A significant reduction in the bending stiffness of the slabs upon the beams and a real development of the plastic hinges in the marginal areas of the beams (together with partial implications and plastic deformations) were observed.

Assimilating a complex fluid with a fractal object, non-differentiable behaviors in its dynamics are analyzed. Complex fluid dynamics in the form of hydrodynamic-type fractal regimes imply “holographic implementations” through velocity fields at non-differentiable scale resolution, via fractal solitons, fractal solitons–fractal kinks, and fractal minimal vortices. Complex fluid dynamics in the form of Schrödinger type fractal regimes imply “holographic implementations”, through the formalism of Airy functions of fractal type. Then, the in-phase coherence of the dynamics of the complex fluid structural units induces various operational procedures in the description of such dynamics: special cubics with SL(2R)-type group invariance, special differential geometry of Riemann type associated to such cubics, special apolar transport of cubics, special harmonic mapping principle, etc. In such a manner, a possible scenario toward chaos (a period-doubling scenario), without concluding in chaos (nonmanifest chaos), can be mimed.