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We investigate the oscillatory regimes of the Langmuir probe current recorded for the electrical characterization of femtosecond laser-produced plasma. The influence of metallic target biasing, which can perturb the ambipolar electric field generated through the charge separation at early stages of the expansion, is also studied. Two distinct behaviors are evidenced, and they point to the existence of two plasma structures: a fast one consisting in promptly ejected highly charged particles, and a slow “tail” of thermalized particles. Theoretically, a non-differentiable model which involves two scale resolutions is developed. Assuming the principle of scale resolution superposition, the current density at global scale is expressed as the sum of the current density at Coulomb and the thermal scale resolutions, in agreement with the experimental data. Interestingly, the theoretical estimation of oscillation frequencies shows a correspondence with the filling factor hierarchy in the fractional quantum Hall effect.

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.

We present a short overview of studies performed in our research groups over the last decade on the characterization of transient plasma plumes generated by laser ablation in various temporal regimes, from nanosecond to femtosecond. New results are also presented along with this overview, both being placed in the context of similar studies performed by other investigators. Optical (fast gate intensified CCD camera imaging and space- and time-resolved emission spectroscopy) and electrical (mainly Langmuir probe) methods have been applied to experimentally explore the dynamics of the plasma plume and its constituents. Peculiar effects as plume splitting and sharpening or oscillations onset have been evidenced in vacuum at high laser fluence. New theoretical approaches have been developed to account for the experimental observations.

Langmuir probe measurements were performed on transient plasmas generated by femtosecond laser ablation of several metallic targets (Al, Cu, Mn, Ni, In, Te, W). The analysis of current-voltage characteristics at various delays after the laser pulse gave access to the temporal evolution of ion density, electron temperature and plasma potential. The time-of-flight profile of the current recorded by the probe was also discussed in terms of a shifted Maxwell-Boltzmann velocity distribution, considering both thermal and drift velocities. The plasma parameters derived by these approaches were correlated with the electrical conductivity of the investigated metals. Assuming a direct dependence between the probe ionic signal and the charge carrier mobility in the target, a logarithmic fit was proposed for the plasma potential variation with electrical conductivity, whereas a derivative of this function was applied for the electron temperature. The saturation charge derived from the time-integrated probe ionic signals was influenced by the electrical conductivity of the target and also by the atomic weight of the metal. A steep increase of the thermal and drift velocities with conductivity was observed.

The acoustic human comfort includes the protection from noise and it is closely related to the sound absorption concept. The paper presents an experimental study on sound absorption coefficients, including transmission, for some combined materials: waste materials (rubber particles, polypropylene, crumbled plastic, wood flour, jute and cord fabrics) with different backing plates (plasterboard, oriented strand board - OSB, polystyrene). The tests were realized following the impedance tube method (ISO 10534-1:1998 standard), except for that the end of the tube was inserted into an anechoic room. The effect of this “backing room” on the samples sound absorption performances is discussed. Also, the experimental results emphasize the influence of the combined materials placed in series on the sound absorption performances and promote the idea of environmentally friendly solutions in their improvement.

Experimental and theoretical investigations were performed on transient plasmas generated by ns-laser ablation on ten mineral samples. Space-and time-resolved optical emission spectroscopy and ICCD fast camera imaging were implemented to understand the kinetics of the ablated particles and how the samples physical properties can be reflected by those of the plasma plume. Strong correlations were found between the global and individual plasma velocity and the melting temperature of the geomaterials. The thermal energy of the plasma expressed through the excitation temperature of the ejected particle is correlated with the same target physical parameter, results corelated with the presence of strong thermal ejection mechanisms. To better understand the global empirical laws, we implemented the fractal model to interpret the results. The model manages to emulate the target-plasma correlations using a multifractal hydrodynamic approach and adequately calibrating the theoretical model on the empirical tools used to describe the laser produced plasmas.

Better understanding of complex relationships between laser-produced plasma dynamics and ablation target material properties is key for optimization of numerous technological and analytical applications. Continuing a previous work [Irimiciuc et al., Appl. Surf. Sci. 417 (2017) 108–118] which proposed empirical laws for some of these relationships (in femtosecond irradiation regime), we present here a systematic study on several metallic targets (Al, Cu, Mn, Ni, Ti, Zn) irradiated by nanosecond, picosecond and femtosecond laser pulses. The experimental approach is based on complementary diagnostics, with the implementation of both electrical and optical techniques for transient plasma characterization. The empirical equations previously reported for femtosecond ablation pulses are now extended to the picosecond and nanosecond regimes.

Assuming that the particle movements of fluid–solid mixture occur along continuous but non-differentiable curves, in the framework of scale relativity, the separation processes of the solid components from heterogeneous mixtures are analysed. By means of a numerical simulation of “fuzzy” type in the dissipative approximation of motion of the scale relativity in its non-differentiable hydrodynamic version, it is shown that the separation processes imply both what we call the “relevant dimensions” of the solid components from heterogeneous mixtures and their positions in the fluid–solid mixture velocity field. The above-mentioned phenomena occur in the turbulence regimes of the fluid–solid mixtures. In the scale relativity dispersive approximation of motion, by means of space–time cnoidal oscillations modes of the fluid–mixture velocity field, it is shown that the separation process is controlled through the turbulence regimes of the fluid–solid mixtures. In such a situation three scenarios of transition to turbulence via chaos (quasi-periodicity, sub-harmonic bifurcations, and intermittences) are comparatively given from a theoretical and experimental point of view both for a plasma with complex structures (in the form of double layers), assimilated with fluid–solid mixtures.

In the present paper some generalized physical models are established using differential and integral elements geometry of the homographic group. The generalization of the hyperbolic motions (with constant acceleration) on the Minkowskian space-time and classical Kepler problems is analyzed using a variational principle of Matzner–Misner type. This way the Skyrme theory is placed in an inherent continuity with respect to the Newtonian natural philosophy.

It is shown that Onicescu's informational energy can be correlated by means of Shannon's maximum informational entropy variational principle with SL(2, ℝ) invariance of the Kepler type motions. For constant value of Onicescu's informational energy, the uncertainty relations are obtained.

In the framework of the Scale Relativity Theory, non-linear effects in complex fluids and, implicitly, the separation control of the solid components from heterogeneous mixtures are analysed. Assuming that the movements of the complex fluid entities occur on continuous but non-differentiable curves, the specific momentum and the local energy density conservation law equations are obtained. For potential movements at a fractal scale, as well as for non-potential ones at a differentiable scale, the non-differentiable hydrodynamic model is established. In such context, for different curve motions at various resolution scales, the non-differentiable hydrodynamic model is reduced to either a quantum hydrodynamic model, a standard hydrodynamic model, a random walk model, or a Stokes model. By numerical simulations using the non-differentiable hydrodynamic equations and the internal energy density conservation law with adequate initial and boundary conditions, some non-linear effects are obtained. Moreover, eliminating the time between the viscosity stress tension type and temperature field, for various given positions, thermal hysteresis type effects can be obtained. For each of the models mentioned above, the separation control processes are discussed.

The paper presents a theoretical study of the source of the spinning movement of the solid particles flowing in a moving fluid and the influence of the resulting Magnus force on the particles' trajectories along the stream lines, based on interactions occurring in the boundary layers. The subject is important for technological applications, like aerodynamic separation process of a mixture of solid particles. First, it is shown that the boundary layer equations generate local soliton-type, kink-type and soliton-kink-type nonlinear solutions for the velocity field. Using Prandtl's equations for boundary layer, nonlinear solutions of the velocity field are obtained. It was found that through the interaction on the boundary layers, the transition from the movement on continuous and differentiable curves (stream lines) to the movement on continuous and non-differentiable curves (fractal curves) occurs. This last characteristic can be used in the separation process of the solid components from a heterogeneous mixture.

We present arguments for the existence of new higher dimensional black holes with a Ricci flat horizon in Einstein–Yang–Mills theory with negative cosmological constant. The SU ( 2 ) non-Abelian gauge fields are purely magnetic and live on a three-dimensional subspace. A discussion of the main properties of the black holes and the differences with respect to the four-dimensional case is presented. We find that the mass of these solutions always diverges. A regularized action and mass of the d = 5 , 6 non-Abelian configurations is found by using a suitable boundary counterterm prescription.

We report on stochastic resonance and vibrational resonance in an electric charge double layer configuration as usually found in electrical discharges, biological cell membranes, chemical systems and nanostructures. The experiment and numerical computation show the existence of a barrier expressible in terms of the golden mean above which the two phenomena do not take place. We consider this as new evidence for the importance of the golden mean criticality in the oscillatory dynamics, in agreement with El Naschie’s E-infinity theory. In our experiment, the dynamics of a charge double layer generated in the inter-anode space of a twin electrical discharge is investigated under noise–harmonic and harmonic–harmonic perturbations. In the first case, a Gaussian noise can enhance the response of the system to a weak injected periodic signal, a clear mark of stochastic resonance. In the second case, similar enhancement can appear if the noise is replaced by a harmonic perturbation with a frequency much higher than the frequency of the weak oscillation. The amplitude of the low frequency oscillation shows a maximum versus the amplitude of the high frequency perturbation demonstrating vibrational resonance. In order to model these dynamics, we derived an excitable system by modifying a biased van der Pol oscillator. The computational study considers the behaviour of this system under the same types of perturbation as in the experimental investigations and is found to give consistent results in both situations.

Experimental investigations and theoretical analysis on the dynamics of a metal–oxide semiconductor heterostructure used as nonlinear capacity in a series RLC electric circuit are presented. A harmonic voltage perturbation can induce various nonlinear behaviours, particularly evolution to chaos by period doubling and torus destabilization. In this work we focus on the change in dynamics induced by a sinusoidal driving with constant frequency and variable amplitude. Theoretical treatment based on the microscopic mechanisms involved led us to a dynamic system with a piecewise behaviour. Consequently, a model consisting of a nonlinear oscillator described by a piecewise second order ordinary differential equation is proposed. This kind of treatment is required by the asymmetry in the behaviour of the metal–oxide semiconductor with respect to the polarization of the perturbing voltage. The dynamics of the theoretical model is in good agreement with the experimental results. A connection with El Naschie’s E-infinity space-time is established based on the interpretation of our experimental results as evidence of the importance of the golden mean criticality in the microscopic world.

A n -dimensional generalized Cantor’s discontinuum and its correspondence with El Naschie’s sets are established.

Considering that the motion of the micro-particles takes place on continuous but non-differentiable curves, in the topological dimension D T = 1, a theoretical approach of the transport mechanisms in nanostructures is established: generalized Euler’s type equations, Schrodinger’s type equation as an irrotational motion of the Euler’s fluid, Josephson type effect, and hydrodynamic model with the current expressions and conductance quantization. The correspondence with El Naschie’s e (∞) space–time is given by means of some examples (the heat transfer in nanofluids, the compatibility of the acoustic regime of the phononic spectrum with the optical one, etc.).

A scale relativity theory in the topological dimension D = 3 is developed and its corresponding hydrodynamic model is discussed. Some possibilities of the theory and the connection to El Naschie’s e (∞) space–time by means of a Korteweg–de Vries type equation for an incompressible fluid are established.

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.

A generalization of the Nottale’s scale relativity theory is elaborated: the generalized Schrodinger equation results as an irrotational movement of Navier–Stokes type fluids having an imaginary viscosity coefficient. Then ψ simultaneously becomes wave-function and speed potential. In the hydrodynamic formulation of scale relativity theory, some implications in the gravitational morphogenesis of structures are analyzed: planetary motion quantizations, Saturn’s rings motion quantizations, redshift quantization in binary galaxies, global redshift quantization etc. The correspondence with El Naschie’s e (∞) space–time implies a special type of superconductivity (El Naschie’s superconductivity) and Cantorian-fractal sequences in the quantification of the Universe.

One obtain various interesting results if one introduces the hydrodynamic formulation of scale relativity theory (SRT), for a coherent quantum fluid, into the time dependent Ginzburg–Landau (TDGL) equation, which describes superconductivity. The London gauge and the zero momentum of the Copper pairs (i.e. the London equations) arise naturally from the imaginary parts of the computed system. One obtains a particular relation between the diffusion coefficient, the (dimensionless) friction coefficient and the (dimensionless) Ginzburg–Landau parameter which yields a new natural gauge for the TDGL equation (as postulated by some authors). If the value of the real velocity of the Copper pairs tends to zero, the imaginary velocity of the pairs becomes real. The subquantum potential is proportional to the density of the Copper pairs. Moreover, under special circumstances, the superconductor acts as a subquantum medium energy accumulator. In this context, we discuss a certain special coherence of the Cooper-pairs by means of e(∞) space–time (El Naschie’s superconductivity).

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.

In the topological dimension D = 4 of the scale relativity theory, the self-structuring of a coherent quantum fluid implies the Golden mean renormalization group. Then, the transfinite set of El Naschie’s e(∞) space–time becomes the background of a new physics (the transfinite physics).

In this work, we present a novel evidence of the importance of the golden mean criticality of a system of oscillators in agreement with El Naschie’s E-infinity theory. We focus on chaos inhibition in a system of two coupled modified van der Pol oscillators. Depending on the coupling between the two oscillators, the system shows chaotic behavior for different ranges of the coupling parameter. Chaos suppression, as a transition from irregular behavior to a periodical one, is induced by perturbing the system with a harmonic signal with amplitude considerably lower than the value which causes entrainment. The frequency of the perturbation is related to the main frequencies in the spectrum of the freely running system (without perturbation) by the golden mean. We demonstrate that this effect is also obtained for a perturbation with frequency such that the ratio of half the frequency of the first main component in the freely running chaotic spectrum over the frequency of the perturbation is very close (five digits coincidence) to the golden mean. This result is shown to hold for arbitrary values of the coupling parameter in the various ranges of chaotic dynamics of the free running system.

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.

A connection between e (∞) space–time and supergravity theory, by means of the gravitational instantons synchronization, is established.

A geometric formulation of the gravitation theory on a fractal space–time is given. Then, a connection is introduced on a tangent bundle, the connection coefficients, the Riemann curvature tensor and the Einstein field equations are calculated. We can characterize the “relative resolution parameter” as a ‘state scale’ of the coordinate system, as compared with Einstein’s formulation of the relativity principle, in which the relative speed characterize the movement state of the frame. In such a context, it results, by means of a dilation operator, the equivalence of this model with Einstein’s quantum gravity. Through the gravitational instantons synchronization, the correspondence with El Naschie’s gravity is established.

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.

The parabolic motion of a test particle in the Schwarzschild’s gravitational field is assimilated to a field theory with spontaneous symmetry breaking. Then, the space–time of general relativity is substituted with the e ∞ space–time by generating an instanton (El Naschie’s instanton) which tunnelates the Dirac vacuum states.

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).

A discussion on the “∞” (infinite) as ‘a process’ and, not only as a symbol is proposed. This also implies that “0” (zero) is ‘a process’, not only a value like in other algebraic structures. Through the physical meaning of such ‘a process’, the correspondence with El Naschie’s Cantorian space–time is given.

A frame is characterized by a hermitic operator with independent of direction eigenvalues. For a particular type of frames (El Naschie's frame), the hermitic operator eigenvalues may be put in correspondence with a Cantor set. This is the situation when the used coordinates, could be some values directly measurable for instance, via the mass spectrum of high energy particle physics. By using the Cayleyene metric associated to the hermitic operator and to Ernst's equations, the generalized hyperbolic stationary metrics is obtained. When imposing on the metrics the conform Minkowskian character, we obtain both the inertial nature of the Universe expansion and, by the gravitational coupling constant, the correspondence with e (∞) space–time. The quantum mechanics in the form of a Schrodinger type equation with complex eigenvalues appears in the process of frames synchronization.

One shows that the dendritic morphogenesis is induced by cnoidal thermal oscillation modes of the solid–liquid interface. Particularly, the one-dimensional l -thermal antisoliton–one-dimensional l -thermal soliton 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 Cantorian-fractal structure of space-time induces, by means of the dendritic growth velocities, an intrinsic anisotropy and critical wavelengths capable to cause the dendritic tip to undergo a morphological instability, and by means of the energy spectrum, forbidden energy bands. In such a space-time, the El Naschie’s Cantorian strings generate the dendritic morphogenesis.

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.

In the weak-field approximation, some implications of duality in the Weyl–Dirac (WD) theory, using the Gregorash–Papini–Wood approach, are investigated. Any particle is in a permanent interaction with the ‘subquantic level’ (Madelung’s fluid) and, as a result of this interaction, the particle acquires the proper fluctuation curvature and the proper fluctuation energy, respectively. By fixing the fluctuations scale, the quantum fluid orders either by means of bright cnoidal oscillation modes inducing causality, or by means of dark cnoidal oscillation modes inducing acausality, and non-linear effects, respectively. The periodic mode is associated with the undulatory characteristic, and the solitonic one with the corpuscular one. By not fixing the fluctuations scale and keeping the symmetry, the quantum fluid orders like a two-dimensional (2D) lattice of vortices, so that the duality needs coherence. In the compatibility between quantum hydrodynamics in the Madelung’s representation and the wave mechanics, the self-gravitational field of the Weyl–Dirac type physical object is generated. El Naschie’s space–time implies, by means of transfinite heterotic string theory, the masses of nucleons, and, by the gravitational fractional quantum Hall effect, the dispersion of the wave-packet on the particle. The analysis of the fractal dimension of the physical object described by the WD theory shows that the waves, and corpuscle, respectively are 2D projections of a higher dimensional special string in El Naschie’s space–time (El Naschie’s string).

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).

In the frame of hydrodynamic formulation of scale relativity theory for a coherent quantum fluid, one shows that some parameters of high-temperature superconductors (critical field, critical current, pair breaking time, fluxoid, state density, gap energy, etc.) come from the non-differentiability of the quantum space–time. Then, the vortex streets (responsible for superconductivity by means of the in-phase lattice oscillations mechanism) and the anyons (responsible for superconductivity by means of the composite fermions mechanism) may be considered as two-dimensional projections of a higher-dimensional fractal string, which implies the existence of just one mechanism inducing type II superconductivity.

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.

In fractal space-time theory we built a superconducting model for generalized fields (electromagnetic and linear gravitation): generalized Maxwell and London's equations, generalized Meissner and shielding effects, oscillation modes in thin cylindrical generalized superconductors (generalized supertrons) and generalized superconductivity as a gauge theory. In such a context the atomic, planetary and double galaxies systems are self-organized as fractal superconducting structures. The Cantorian E (∞) structure of space-time implies a two-dimensional cnoidal distribution of the particles concentration for the solid–liquid interface and allows the evaluation of the kinetic moment of a neutron star.

In the fractal space-time theory the generalized precession, the generalized Stern–Gerlach experiment, the generalized Zeeman effect, the generalized Landau's levels and the generalized fractional quantum Hall effect are analyzed. The Cantorian structure of space-time implies the supplemental decreasing rate of the Ecliptic tilt, the “global” quantization in units of 36 km s−1 and the filling factor ν=1/2n+1.

We show that the de Broglie time for Cooper pairs admits discontinuities for Cantorian sequences 1/3,1/5,1/7,…, and the pair-breaking time depends only on the phase of the pair wave function.

We develop a dynamics of the Cooper pair given by the complex action S =ℏ ln ψ with ψ solution of the two-dimensional Ginzburg–Landau (GL) equation. The ‘motion energy’ admits discontinuities for the Cantorian sequences 1/3,1/5,1/7,…, and corresponds to the energy gap of high-temperature superconductors.

One builds the solution of GL equation in terms of the elliptic cn function of complex argument. The real part of the complex action, S =ℏ ln cn (u) , corresponds to the potential of a vortex lattice, and from here, through the elliptic function degeneration, to the vortex streets. Considering the vortex streets fixed on vacuum by a background magnetic field through pinning, from equating the current density to zero one determines the field structure: the mean value will be roughly equal to BC2, and its flux will be fractional. The fractional flux will be associated to quasi-particles obeying the ‘anyonic’ statistics. At low temperatures and high external magnetic field, the structure of background field will be of Cantorian type.

Maxwell’s and London’s generalized equations, generalized Meissner effect and gravitational shielding in an electromagnetic field are obtained. In such a context, we show that a neutral particle beam in an infinitely thin superconducting cylinder placed in the Earth’s gravitoelectric field is focused, and that the penetration depth increases with the pulsation of the electromagnetic field.

The solutions of Ginzburg–Landau (GL) equations are built in terms of the Jacobian elliptic function sn with modulus s, and by s=s(t) the dependencies of the superconducting parameters on reduced temperature. The correspondence with BCS theory shows that the model can be applied on a wide range of temperatures, low temperatures being included.