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We analyze polymer dynamics in a fractal paradigm. Then, it is shown that polymer dynamics in the form of Schrödinger - type regimes imply synchronization processes of the polymers� structural units, through joint invariant function of two simultaneous isomorphic groups of SL(2R) - type, as solutions of Stoka equations. In this context, period doubling, damped oscillations, self - modulation and chaotic regimes emerge as natural behaviors in the polymer dynamics. The present model can also be applied to a large class of materials, such as biomaterials, biocomposites and other advanced materials.

The aim of this study is to offer a comprehensive view on drug release from hydrogel, from both an experimental and a theoretical point of view. Aiming to benefit cyclodextrins’ properties (not irritant; stable; able to modify the physical, chemical and biological properties of active compounds; accessible at low prices) and those of carrageenan polysaccharide (antitumor, immunomodulatory, antihyperlipidemic, anticoagulant, biocompatibility, biodegradability), original hydrogel films based on beta cyclodextrin and kappa carrageenan using epichlorohydrin as crosslinking agent were prepared and characterized from morphological and physical/chemical points of view. The results (morphology, the swelling degree, and the loading/release capacity) proved their potential as carriers for different types of drugs. Further, a new theoretical model, from a multifractal paradigm of motion, was proposed for the drug release from hydrogel films, starting from the fundaments of its evolution at a microscopic level, and aiming to obtain information on system evolution, at both the spatial and temporal scales, inapproachable by quantitative measurements.

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.

Using Maxwell’s equations for the propagation of the electric component of the bioelectromagnetic field, a statistical interpretation of the classical action is established. In such context, correlations between the distribution function, specific to this statistics, and particular dynamics of biostructures with a non-linear growth can be found.

In the motion fractal theory, the scale relativity dynamics of any complex system are described through various Schr&ouml, dinger or hydrodynamic type fractal &ldquo, regimes&rdquo, In the one dimensional stationary case of Schr&ouml, dinger type fractal &ldquo, synchronizations of complex system entities implies a joint invariant function with the simultaneous action of two isomorphic groups of the S L ( 2 R ) type as solutions of Stoka type equations. Among these joint invariant functions, Gaussians become in the Jeans&rsquo, s sense, probability density (i.e., stochasticity) whenever the information on the complex system analyzed is fragmentary. In the two-dimensional case of hydrodynamic type fractal &ldquo, at a non-differentiable scale, the soliton and soliton-kink of fractal type of the velocity field generate the minimal vortex of fractal type that becomes the source of all turbulences in the complex systems dynamics. Some correlations of our model to experimental data were also achieved.

The properties of pulsed laser deposited of Ni60Ti40 shape memory thin films generated in various deposition conditions were investigated. In-situ plasma monitoring was implemented by means of space- and time-resolved optical emission spectroscopy, and ICCD fast camera imaging. Structural and chemical analyses were performed on the thin films using SEM, AFM, EDS, and XRD equipment. The deposition parameters influence on the chemical composition of the thin films was investigated. The peeled layer presented on DSC a solid-state transformation in a different transformation domain compared to the target properties. A fractal model was used to describe the dynamics of laser produced plasma through various non-differentiable functionalities. Through hydrodynamic type regimes, space-time homographic transformations were correlated with the global dynamics of the ablation plasmas. Spatial simultaneity of homographic transformation through a special SL(2R) invariance implies the description of plasma dynamics through Riccati type equations, establishing correlations with the optical emission spectroscopy measurements.

An increasing number of studies are beginning to show that both low-density lipoprotein and high-density lipoprotein cholesterol can constitute risk factors for myocardial infarction. Such a behaviour has been called by experts in the field the “chameleonic effect” of cholesterol. In the present paper, a fractal/multifractal model for low-density lipoprotein and high-density lipoprotein cholesterol dynamics is proposed. In such a context, a fractal/multifractal tunneling effect for systems with spontaneous symmetry breaking is analyzed so that if the spontaneous symmetry breaking is assimilated to an inflammation (in the form of a specific scalar potential), then a coupling between two fractal/multifractal states can be observed. These two states, which have been associated to biological structures such as low-density lipoprotein and high-density lipoprotein, transfer their states through a fractal/multifractal tunneling effect. Moreover, in our opinion, the widely used notions of “good” and “bad” cholesterol must be redefined as two different states (low-density lipoprotein and high-density lipoprotein) of the same biological structure named “cholesterol.” In our work, for the first time in the specialized literature, low-density lipoprotein and high-density lipoprotein have been regarded as two different states of the same biological structure (named “cholesterol”), such as in nuclear physics, the neutron and proton are two different states of the same particle named nucleon.

In this chapter, we propose a mathematical-physical model, starting from the morphological-functional assumption of the fractal brain, by activating brain non-differentiable dynamics through the determinism-nondeterminism inference of the responsible mechanisms.

The problem of plasma self-structuring during the pulse laser deposition of multicomponent materials is discussed from both theoretical and experimental perspectives. A non-differential mathematical model was used to build the set of equation to characterize a complex fluid, assimilated with the laser produced plasmas in the fractal paradigm. Numerical simulations regarding the flow of a fractal fluid were performed reflecting real multi-component plasmas. These revealed that the structuring of a fluid is strongly related to its fractality and thus to individual properties of its components. The theoretical results were confronted with experimental data regarding the flow of multi-component plasma in a fluid representation. The laser produced plasma was investigated by means of ICCD fast camera imaging and optical emission spectroscopy. The experimental results showcase a splitting process on the two main directions coupled with a separation of the ejected particles with respect to their excitation energy. Both theoretical and experimental data are in good agreement.

In the materials area there are many theoretical and experimental investigations concerning their sound absorption behaviour, covering a wide range of applications. An alternative approach consists in identifying the ensemble acoustic field-propagation environment with a fractal gives its functionality in the form of structure parameters that is dependent on scale resolution. These fractal parameters can be matched with “classical” ones typical for sound absorption experiments in various materials. The mathematical methodology presented here implies the substitution of a dynamics with constraints on continuous but differentiable curves, in an Euclidian space, with “synchronous” dynamics, free from any constraints, on continuous but non-differentiable curves with various fractal dimensions but on a fractal space (i.e. the geodesics of that space). In a very special representation, the external constraints select the fractal geodesics type. In our study this “type selection” refers to the geodesics tunnel effect of fractal (acoustic) type for which we calculate the reflectance and the transparency of an external fractal barrier. An experimental procedure, using a modified impedance tube technique, to determine the sound absorption coefficients for various composite materials was conducted. The procedure uses an anechoic room and the measured sound absorption coefficients also include the sound transmission. The fractal approach of the acoustic behaviour, through the fractal parameters determined (transparency and reflectance) matched the experimental results, in terms of sound absorption, emphasizing the high degree of generality of the fractal theory in the dynamics of physical processes.

Understanding the fundamental mechanisms behind industrial laser-based technologies represents one the cornerstones of the development and tailoring of new materials. With the thin-film deposition using laser ablation being one of the most important techniques for obtaining complex materials with controllable stoichiometry, there is a high request for both experimental and theoretical studies towards understanding the behavior of multi-component alloys under high-power laser irradiation. Here we investigate the laser ablation process on two ternary metallic alloys (Cu–Mn–Al and Fe–Mn–Si) by means of space-and time-resolved optical emission spectroscopy and fast camera imaging with the focus being on the spatial distribution of each composing element. Information regarding the kinetic and thermal energy of the ejected particles is extracted and discussed in the framework of an inner structuring of the laser-produced plasmas based on a mass and energy distribution. The hypothesis is then verified by implementing a fractal analysis to the multi-component plasmas. The theoretical fractal approach offers results in good agreement with the experimental data gives important insight in the inner dynamics of complex laser-produced plasmas.

An innovative theoretical model is developed on the backbone of a classical Lorenz system. A mathematical representation of a differential Lorenz system is transposed into a fractal space and reduced to an integral form. In such a conjecture, the Lorenz variables will operate simultaneously on two manifolds, generating two transformation groups, one corresponding to the space coordinates transformation and another one to the scale resolution transformation. Since these groups are isomorphs various types isometries become functional. The Lorenz system was further adapted to describe the dynamics of ejected particles as a result of laser matter interaction in a fractal paradigm. The simulations were focused on the dynamics of charged particles, and showcase the presence of current oscillations, a heterogenous velocity distribution and multi-structuring at different interaction scales. The theoretical predictions were compared with the experimental data acquired with noninvasive diagnostic techniques. The experimental data confirm the multi-structure scenario and the oscillatory behavior predicted by the mathematical model.

The mathematical concept of minimal atomicity is extended to fractal minimal atomicity, based on the nondifferentiability of the motion curves of physical system entities on a fractal manifold. For this purpose, firstly, different results concerning minimal atomicity from the mathematical procedure of the Quantum Measure Theory and also several physical implications are obtained. Further, an inverse method with respect to the common developments concerning the minimal atomicity concept has been used, showing that Quantum Mechanics is identified as a particular case of Fractal Mechanics at a given scale resolution. More precisely, for fractality through Markov type stochastic processes, i.e., fractalization through stochasticization, the standard Schrödinger equation is identified with the geodesics of a fractal space for motions of the physical system entities on nondifferentiable curves on fractal dimension two at Compton scale resolution. In the one-dimensional stationary case of the fractal Schrödinger type geodesics, a special symmetry induced by the homographic group in Barbilian’s form “makes possible the synchronicity” of all entities of a given physical system. The integral and differential properties of this group under the restriction of defining a parallelism of directions in Levi-Civita’s sense impose correspondences with the “dynamics” of the hyperbolic plane so that harmonic mappings between the ordinary flat space and the hyperbolic one generate (by means of a variational principle) a priori probabilities in Jaynes’ sense. The explicitation of such situation specifies the fact that the hydrodynamical variant of a Fractal Mechanics is more easily approached and, from this, the fact that Quantum Measure Theory can be a particular case of a possible Fractal Measure Theory.

Since the non-differentiability becomes a fundamental property of the motions space [1, 2, 3, 4], a correspondence between the interaction processes and multifractality of the motion trajectories can be established. Then, for all scale resolutions, the geodesics equations (in the form of the Schrodinger equation of fractal type) and some applications (similarities between dynamics at atomic and cosmic scales) are obtained.

A theoretical model was developed, able to explain complex phenomena that take place at the interaction between two complex space charge structures in low-temperature discharge plasma, leading to the self-modulation of the discharge current. The model is based on the presumption of non-differentiability of both the physical variables and the dynamics equations, which impose their dependence on the scale resolution. The dynamics of the electrons is described by a forced damped oscillating system, being analyzed the response of the global discharge current to different changes in the scale resolution, forcing frequency and damping coefficient. Experimental results were presented, which are in good agreement with the results provided by the theoretical model.

Assuming that the Q-machine magnetized plasma particles are moving on continuous and non-differentiable curves (fractal curves), a theoretical model was developed in the frame of the scale relativity theory. The model is able to explain some characteristics of the potential relaxation instability and the electrostatic ion-cyclotron instability, as well as the interaction between these two instabilities which leads to the amplitude and frequency modulation of the second instability by the first one. Experimental result are shown, which are in agreement with the theoretical model predictions.

Considering that the structural units of the biological structures (assimilated to complex systems) move on continuous but non-differentiable curves, new mathematical procedures in the form of the Extended Scale Relativity Model are presented. Then, the equivalence between the formalism of the fractal hydrodynamics and the one of the Schrodinger type equation was established. The advantage of such model is that it can be applied to all scale resolution, for example from nano to the mesoscopic scales.

In classical concepts, theoretical models are built assuming that the dynamics of the complex system’s stuctural units occur on continuous and differentiable motion variables. In reality, the dynamics of the natural complex systems are much more complicated. These difficulties can be overcome in a complementary approach, using the fractal concept and the corresponding non-differentiable theoretical model, such as the scale relativity theory or the extended scale relativity theory. Thus, using the last theory, fractal entropy through non-differentiable Lie groups was established and, moreover, the pairs generating mechanisms through fractal entanglement states were explained. Our model has implications in the dynamics of biological structures, in the form of the “chameleon-like” behavior of cholesterol.

In the Weyl-Dirac non-relativistic hydrodynamics approach, the non-linear interaction between sub-quantum level and particle gives non-differentiable properties to the space. Therefore, the movement trajectories are fractal curves, the dynamics are described by a complex speed field and the equation of motion is identified with the geodesics of a fractal space which corresponds to a Schrodinger non-linear equation. The real part of the complex speed field assures, through a quantification condition, the compatibility between the Weyl-Dirac non-elativistic hydrodynamic model and the wave mechanics. The mean value of the fractal speed potential, identifies with the Shanon informational energy, specifies, by a maximization principle, that the sub-quantum level “stores” and “transfers” the informational energy in the form of force. The wave-particle duality is achieved by means of cnoidal oscillations modes of the state density, the dominance of one of the characters, wave or particle, being put into correspondence with two flow regimes (non-quasi-autonomous and quasi-autonomous) of the Weyl-Dirac fluid. All these show a direct connection between the fractal structure of space and holographic principle.

In this paper, we prove that many parallel communication topologies and several parallel algorithms have fractal properties, which leads to the idea that there is a strong connection between the classical parallel computing and a plausible computing on a fractal medium. Furthermore, we show that some parallel algorithms are suitable for incursive and hyperincursive computation, which links parallel computing to quantum computing and a possible fractal computing.

Considering that the motions of the complex system structural units take place on continuous, but non-differentiable curves, in the frame of the extended scale relativity model (in its Schrödinger-type variant), it is proven that the imaginary part of a scalar potential of velocities can be correlated with the fractal information and, implicitly, with a tensor of “tensions”, which is fundamental in the construction of the constitutive laws of material. In this way, a specific differential geometry based on a Poincaré-type metric of the Lobachevsky plane (which is invariant to the homographic group of transformations) and also a specific variational principle (whose field equations represent an harmonic map from the usual space into the Lobachevsky plane) are generated. Moreover, fractal information (which is made explicit at any scale resolution) is produced, so that the field variables define a gravitational field. This latter situation is specific to a variational principle in the sense of Matzner–Misner and to certain Ernst-type field equations, the fractal information being contained in the material structure and, thus, in its own space associated with it.

Inconsistencies of some standard quantum mechanical models (Madelung’s, de Broglie’s models) are eliminated as- suming the micro particle movements on continuous, but non-differentiable curves (fractal curves). This hypothesis, named by us the fractal approximation of motion, will allow an unitary approach of the phenomena in quantum me-chanics (separation of the physical motion of objects in wave and particle components depending on the scale of resolution, correlated motions of the wave and particle, i.e. wave-particle duality, the mechanisms of duality, by means of both phase wave-particle coherence and wave-particle incoherence, the particle as a clock, particle incorporation into the wave and the implications of such a process). Moreover, correspondences with standard gravitational models (Einstein’s model, string theory) can be also distinguished.

Considering that the motions of the particles take place on fractals, a non-differentiable mechanical model is built. Only if the spatial coordinates are fractal functions, the Galilean version of our model is obtained: the geodesics satisfy a Navier-Stokes-type of equation with an imaginary viscosity coefficient for a complex speed field or respectively, a Schrodinger-type of equation or hydrodynamic equations, in the case of irrotational movements. Moreover, in this approach, the analysis of the fractal fluid dynamics generates conductive properties in the case of movements synchronization both on differentiable and fractal scales, and convective properties in the absence of synchronization (e.g. laser ablation plasma is analyzed). On the other hand, if both the spatial and temporal coordinates are fractal functions, it results that, the geodesics satisfy a Klein-Gordon-type of equation on a Minkowskian manifold.

Considering that the motions of the particles take place on continuous but non-differentiable curves, i.e. on fractals with constant fractal dimension, an extended scale relativity model in its hydrodynamic version is built. In this approach, static (particle in a box and harmonic oscillator) and time-dependent (free particle etc.) systems are analyzed. The static systems can be associated with a coherent fractal fluid (of superconductor or of super-fluid types behavior), whose particles are moving on stationary trajectories. The complex speed field of the fractal fluid proves to be essential: the zero value of the real (differentiable) part specifies the coherence of the fractal fluid, while the non-zero value of the imaginary (non-differentiable or fractal) part selects, through some “quantization” relations, the “stationary” trajectories (that may correspond to the observables from quantum mechanics) of the fractal fluid particles. Moreover, the momentum transfer in the fractal fluid is achieved only through the fractal component of the complex speed field. The free time-dependent systems can be associated with an incoherent fractal fluid, and both the differentiable and fractal components of complex speed field are inhomogeneous in fractal coordinates due to the action of a fractal potential. It exist momentum transfer on both speed components and the “observable” in the form of an uniform motion is generated through a specific mechanism of “vacuum” polarization induced by the same fractal potential. The analysis on the fractal fluid specifies conductive properties in the case of movements synchronization both on differentiable and fractal scales, and convective properties in the absence of synchronization.

Effects of nanoparticle clustering on the heat transfer in nanofluids using the scale relativity theory in the topological dimension D T = 3 are analyzed. In the one-dimensional differentiable case, the clustering morphogenesis process is achieved by cnoidal oscillation modes of the speed field. In such conjecture, a non-autonomous regime implies a relation between the radius and growth speed of the cluster while, a quasi-autonomous regime requires El Naschie’s e (∞) theory through the cluster–cluster coherence ( El Naschie global coherence ). Moreover, these two regimes are separated by the golden mean. In the one-dimensional non-differentiable case, the fractal kink spontaneously breaks the ‘vacuum symmetry’ of the fluid by tunneling and generates coherent structures. This mechanism is similar to the one of superconductivity. Thus, the fractal potential acts as an energy accumulator while, the fractal soliton, implies El Naschie’s e (∞) theory ( El Naschie local coherence ). Since all the properties of the speed field are transferred to the thermal one, for a certain conditions of an external load (e.g. for a certain value of thermal gradient) the soliton and fractal one breaks down (blows up) and release energy. As result, the thermal conductibility in nanofluids unexpectedly increases. Here, El Naschie’s e (∞) theory interferes through El Naschie global and local coherences.

The scale relativity model was extended for the motions on fractal curves of fractal dimension D F and third order terms in the equation of motion of a complex speed field. It results that, in a fractal fluid, the convection, dissipation and dispersion are compensating at any scale (differentiable or non-differentiable), whereas a generalized Schrodinger type equation is obtained for an irrotational movement of the fractal fluid. For D F = 2 and the dissipative approximation of the motions, the fractal model of atom is build: the real part of the complex speed field describes the electron motion on stationary orbits according to a quantification condition, while the imaginary part of the complex speed field gives the electron energy quantification. For D F = 3 and the dispersive approximation of motions, some properties of the matter are explained: at the differentiable scale the flowing regimes (non-quasi-autonomous and quasi-autonomous) of the fractal fluids are separated by the experimental “0.7 structure”, while for the non-differentiable scale the fractal potential acts as an energy accumulator and controls through coherence the transport phenomena. Moreover, the compatibility between the differentiable and non-differentiable scales implies a Cantor space-time, and consequently a fractal at any scale. Thus, some properties of the matter (the anomaly of nano-fluids thermal conductivity, the superconductivity etc.) can be explained by this model.

195 Nanofluid is a new kind of heat transfer medium containing nanoparticles which are uniformly and stable distributed in a base fluid. Experiments on nanofluids have demonstrated that the thermal conductivity increases with decreasing grain size [1-4]. Keblinski et al [4] have examined four possible mechanisms for the anomalous enhancement observed in nanofluids: Brownian motion of the nanoparticles [5-7], molecular-level layering of the liquid at the liquid-nanoparticles interface [8], the effects of nanoparticle clustering [9], and ballistic phonons transport [4]. Recently, the increasing of the heat transfer in nanofluids was related to the fractal effects [10, 11]. Moreover, Wang et al. [12] reported that the modified fractal model agreed well with the experimental data obtained for the SiO2/ ethanol nanofluid. In such conjecture, the fractal theories (particularly the scale relativity theory (SRT) [13,14] is a new approach to understand quantum mechanics, and furthermore physical domains involving scale laws, such as the nanosystems [15,16, 17]. It is based on a generalization of Einstein’s principle of relativity to scale transformations. Namely, one redefines space-time resolutions as characterizing the state of scale of reference systems, in the same way as velocity characterizes their state of motion. Then one requires that the laws of physics apply whatever the state of the reference system, of motion (principle of motion-relativity) and of scale (principle of SRT). The principle of SRT is mathematically achieved by the principle of scale-covariance, requiring that the equations of physics keep their simplest form under transformations of resolution. For example, considering that the motion of micro-particles take place on continuous but non-differentiable curves, i.e. on fractals [13, 14], it was demonstrated that, in the topological dimension [18] DT=2, the geodesics of the fractal space-time are given by a Schrodinger’s type equation. In the present paper, using the SRT, we analyzed the effects of nanoparticle clustering on the heat transfer in nanofluids. Effects of Nanoparticle Clustering on the Heat Transport in Nanofluids Through Fractal Theories

Using the finite differences method it is shown that a diffusion equation can generate but not maintain a double layer (DL). Instead of these, a reaction–diffusion equation system induces a Koch-type fractal which leads to a self-organization scenario of plasma–plasma interface as a DL. Using the scale relativity theory (SRT) it was shown that a plasma–plasma interface behaves as a junction of Josephson type: a negative differential resistance is its self-structuring condition, it has memory through hysterezis and work in two oscillation regimes. The correspondence with El Naschie’s e (∞) space time is achieved, through the generation of the harmonics of Cantorian fractal type, as well as through the separation of the oscillation regimes, i.e. the intermittent self-organization. In the linear regime of oscillation, the plasma–plasma interface works as a string of Cantorian-fractal type (El Naschie’s string). The model was verified by means of our experimental data too.

Assuming that the motions of a complex system structural units take place on continuous, but non-differentiable curves of a space-time manifold, the scale relativity model with arbitrary constant fractal dimension (the hydrodynamic and wave function versions) is built. For non-differentiability through stochastic processes of the Markov type, the non-differentiable entropy concept on a space-time manifold in the hydrodynamic version and its correspondence with motion variables (energy, momentum, etc.) are established. Moreover, for the same non-differentiability type, through a scale resolution dependence of a fundamental length and wave function independence with respect to the proper time, a non-differentiable Klein–Gordon-type equation in the wave function version is obtained. For a phase-amplitude functional dependence on the wave function, the non-differentiable spontaneous symmetry breaking mechanism implies pattern generation in the form of Cooper non-differentiable-type pairs, while its non-differentiable topology implies some fractal logic elements (fractal bit, fractal gates, etc.).

New results in fractal space–time theory are established: the fractal operator for the fractal dimension D = 2 implies a generalized Schrodinger equation in the Nottale’s scale relativity theory, while the fractal operator for the fractal dimension D = 3 implies the Korteweg–de Vries equation in the one-dimensional case. The connection with El Naschie’s e(∞) theory result by means of the Cooper-type pairs or through wave-particle duality.

It is shown that the gravitational instanton becomes source of energy-momentum tensor in the Weyl–Dirac hydrodynamical model. For null fluctuating mass in ϕ, ϕ2, ϕ3, ϕ4 Cantorian sequences this tensor has a fractal structure.

Using the concept of ‘combined field’, an electrodynamics of polarizable media on a fractal space–time is constructed. In this context, using the scale relativity theory, the permanent electric moment, the induced electric moment, the vacuum fluctuations, the paraelectrics, the diaelectrics, the electric Zeeman-type effect, the electric Einstein–de Haas-type effect, the electric Aharonov–Bohm-type effect, the superconductors in the ‘combined field’, the double layers as coherent structures, the magnetic Aharonov–Casher-type effect, are analyzed. Correspondence with the e (∞) space–time is accomplished either by admitting an anomal electric Zeeman-type effect, or through a fractal string as in the case of a superconductor in ‘combined field’, or, by phase coherence of the electron–ion pairs from the electric double layers (El Naschie’s coherence). Moreover, the electric double layer or multiple layer may be considered as two-dimensional projections of the same El Naschie’s fractal strings (higher-dimensional strings in e (∞) space–time).

One builds the vacuum metrics of the stationary electromagnetic field through the complex potential model. There are thus emphasized both a variational principle, independent on the Ricci tensor, and some internal symmetries of the vacuum solutions. One shows that similar results may be obtained using the Barbiliant’s group. By analytical continuation of a Barbilian transformation the link between the fixed points of the modular groups of the vacuum and the golden mean Φ=(1/(1+Φ))=( 5 −1)/2 of e(∞) space–time is established. Finally, a Cantorian fractal axiomatic model of the space–time is presented. The model is explained using a set of coupled equations which may describe the self organizing processes at the solid–liquid, plasma–plasma, and superconductor–superconductor interfaces.

One shows that the dendritic morphogenesis is induced by one-dimensional cnoidal thermal oscillation modes of the solid-liquid interface. Particularly, the one-dimensional l-thermal antisoliton - one-dimensional l-thermalsoliton modes imposes laws characterizing dendritic branches growth. One deduces a relation between the dendritic growth speed and the dendritic tip radius that depends on the supercooling degree. Particularly one obtains the Oldsfield's relation. The variation of the fractal dimension with the supercooling degree specifies that the dendritic growth are two-dimensional projections of a higher-dimensional fractal.

Some abilities of the SRT theory in studying the polarization gravitational field are analyzed. Thus, one builds a set of Maxwell-type equations for the polarization gravitational field and one studies the behaviour of a gravitomagnetic charge in such fields on a fractalic space-time. One finds that the interaction between the gravitomagnetic charge and the polarization gravitational field reduces to the Van der Waals gravitational type dipole–dipole interaction. From the study of GMHD wave, on a fractal cosmological background it results that the speeds in the planetary and galactic structures are discrete and the Cantorian structure is induced by means of a Tifft and Cocke Cantorian effect (at least for the galaxy pairs NGC 4294–NGC 4299, NGC 4085–NGC 4088). By an iterated map, one gets an object which may be identified with a cosmic fractal string, whose 2D projection corresponds to a cosmic string.

The fractal dimension analysis of the Weyl–Dirac type object shows that the wave and the corpuscle are two-dimensional projections of a higher-dimensional string in a Cantorian fractal space–time (Cantorian fractal string).

We prove that the fractal space-time interpretation of Nelson’s stochastic quantum mechanics may be put into a one-to-one correspondence to the hydrodynamic model of quantum mechanics. Some implications of this correspondence are analyzed: Navier–Stokes type equations, uncertainty relation and type II superconductivity by means of a higher-dimensional fractal string. The two-dimensional (2D) projections of the higher fractal string corresponds to the anyons and 1D projections to the Cooper pair. The Cantorian-fractal structures occur, either by assimilating the Cooper pair with a two coupled oscillators (a simple mechanical model for e (∞) theory), or by El Naschie’s filling factor ν ≡ φ 3 =0.23606 (Cantorian-fractal quantum Hall effect).

Some implications of the Cantorian fractal space-time in the study of the inertial motion of one and two material points, de Broglie and field theories, using the external forms and the contraforms are analysed. For a material point the isotropy of space implies the dependence of mass, momentum and energy on the speed and on an arbitrary limit speed. For two material points the spatial isotropy implies the dependence of the momenta and energy on speed, an arbitrary limit speed and on an arbitrary function of the distance between the material points, functions introduced through the interaction mass. In particular, the gravitational interaction mass depends on two diffusion coefficients in Nottale's way (intra- and extragalactic). Through the intragalactic diffusion coefficient the external metrics weakly Lorentz covariant are induced, which verify the “classical” tests of General Relativity. Through the extragalactic diffusion coefficient the internal metrics are induced, the Lorentz transformations are generalized and the Hubble effect is explained as an “appearance” of the momentum conservation law. The “removal” of the repulsive mass induces the intragalactic fractal, the dependence of the extragalactic diffusion coefficient on position implies the extragalactic fractal, and the vanishing of the gravitational charge of the Universe leads to the proportionality of two fractal scales. The proportionality of the undulatory and inertial 1-forms implies the de Broglie theory, and through the momentum and Hamilton–Jacobi equations, the nonlinear theory in the de Broglie–Takabayasi representation. The use of the potential 1-form and of a motion principle leads to the linear theory of the field, and the non-equivalence of the inertial 1-form and the field 1-form to the nonlinear theory, which by means of the energy tensor conservation, leads to the Born–Infeld equations. Contraforms in the non-relativistic formulation of electromagnetism conclude at the end, the entire picture. In all these models, the fractal character of space-time implies the simultaneous existence, either of two inertial 1-forms, as it is the case for Cartan theories, or of two undulatory 1-forms, as it is the case for undulatory theories, or of two potential 1-forms, as it is the case for the field theories. The Cantorization of space-time implies the two-dimensional Toda lattices. In such a context from the analysis of the fractal dimension of a vortex street and of the superconducting pair it results that these physical objects can be considered as 2D projections of a higher-dimensional fractal string.

Using a one-dimensional nonlinear diffusion equation we show that the oscillation modes of a discharge plasma are cnoidal. By their degeneration, we get striations, solitons, etc., the last corresponding to quasi-autonomous structures of a double layer (DL) type. Considering of the DL as a superconducting structure, its inside dynamics is given both by expelling the electrostatic field and by focusing the charge particle beam. In two dimensions, the cnoidal oscillation modes of the discharge plasma DLs assimilated to a superconducting lattice impose the Cantorian structure of space-time. In this context it results in a direct relationship between the high-dimensional fractal string and E (∞) theory [see Solitons & Fractals 12 (2001) 875] via the electromagnetic fine structure constant. The theoretical model is validated by experimental studies of plasma DLs, through electrical and spectral methods.

In a linear gravitation model and on a space-time with a fractal structure, the repulsive component is associated to the gravitational Meissner effect, and the attractive one to the absence of this effect. In this context, by means of a complex time and of a two-dimensional Cantorian fractal time vortex lattice, one shows that there are two mechanisms which induce superconducting properties to matter: one by in-phase oscillations of the vortex lattice (specific to the attraction) and the other by “generation” of anyons (specific to the repulsion). Thus, the assumption of El Naschie that gravitation is generated by the interaction of dipoles (van der Waals-type forces) is confirmed.

Various differentiable physical models are frequently used to describe the dynamics of laser-produced plasma plumes (e.g., kinetic models, two-fluid models, etc.). Given the complexity of all the phenomena involved in the laser-matter interactions, it is required to introduce the laser ablation plasma dynamic variable dependencies both on the space-time coordinates and on the resolution scales. Therefore, an adequate theoretical approach may be the use of non-differentiable physical models (fractal models). Continuing our previous work on the fractal hydrodynamic model for laser ablation plasma dynamics, we propose here a compact version for the analysis of the spatial and temporal evolution of some plasma dynamic variables, such as velocities, currents, number densities, or temperatures. Moreover, the influence of external factors on the ablation plasma dynamics is considered. The predictions of this model are compared with the experimental data obtained by using a Langmuir probe on an Aluminum laser-pro...

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.