El Naschie’s coherence on the subquantum medium

In the hydrodynamic formulation of the Scale Relativity theory one shows that a stable vortices distribution of bipolaron type induces superconducting pairs by means of the quantum potential. One builds the superconducting fractal by an iterated map and demonstrates that the superconducting pairs results as projections of this fractal. Thus, usual mechanisms (as example the exchange interaction used in the bipolaron theory) are reduced to the coherence on the subquantum medium in a ε^(∞) space (El Naschie’s coherence).     

El Naschie’s structures in the electrodynamics of polarizable media

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 ε^(∞) 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 ε^(∞) space–time).     

El Naschie’s cantorian strings and duality in Weyl–Dirac theory

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).     

Derivation of the energy–momentum and Klein–Gordon equations from El Naschie’s complex time

In a previous note, we have provided a formal derivation of the transverse Doppler shift of special relativity from the generalization of El Naschie’s complex time. Here, we show that the relativistic energy–momentum equation, and hence the Klein–Gordon equation, are also natural consequences of the complex time generalization.     

El Naschie’s ε theory and effects of nanoparticle clustering on the heat transport in nanofluids

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 ε^(∞) 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 ε^(∞) 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 ε^(∞) theory interferes through El Naschie global and local coherences.     

El Naschie’s ε space–time, hydrodynamic model of scale relativity theory and some applications

A generalization of the Nottale’s scale relativity theory is elaborated: the generalized Schrödinger 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 ε^(∞) space–time implies a special type of superconductivity (El Naschie’s superconductivity) and Cantorian-fractal sequences in the quantification of the Universe.     

Newton’s method’s basins of attraction revisited

In this paper, we revisit the chaotic number of iterations needed by Newton’s method to converge to a root. Here, we consider a simple modified Newton method depending on a parameter. It is demonstrated using polynomiography that even in the simple algorithm the presence and the position of the convergent regions, i.e. regions where the method converges nicely to a root, can be complicatedly a function of the parameter.     

A relative version of Webb’s theorem on the equivariant chain complex of a subgroup complex

We prove a relative version of the theorem of Webb that the augmented chain complex of the p-subgroup complex of a finite group, considered as a complex of modules for the group, is homotopy equivalent to a complex of projectives. This allows us to take into account the group of automorphisms of the group.     

Lyapunov’s inequality on timescales

The purpose of this work is to establish the timescale version of Lyapunov’s inequality as follows: Let $x\left(t\right)$ be a nontrivial solution of ${\left(r\left(t\right)x^{\Delta }\left(t\right)\right)}^{\Delta }+p\left(t\right)x^{\sigma }\left(t\right)=0\text{on }[a,b]$ satisfying $x\left(a\right)=x\left(b\right)=0$. Then, under suitable conditions on $p$, $r$, $a$ and $b$, we have ${\int }_a^b{p}_{+}\left(t\right)\Delta t\ge \{\begin{array}{cc}\frac{r\left(a\right)}{r\left(b\right)}\frac{b-a}{f\left(d\right)}\text{,}& \text{if }r\text{ is increasing}\text{,}\\ \frac{r\left(b\right)}{r\left(a\right)}\frac{b-a}{f\left(d\right)}\text{,}& \text{if }r\text{ is decreasing}\text{,}\end{array}$ where $p_{+}\left(t\right)=max\{p\left(t\right),0\},f\left(t\right)=(t-a)(b-t)$ and $d\in \mathbb{T}$ satisfies $|\frac{a+b}{2}-d|=min\{|\frac{a+b}{2}-s|\mid s\in [a,b]\cap \mathbb{T}\}$ if $\frac{a+b}{2}\in \mathbb{T}$. Here $\mathbb{T}$ is a timescale (see below).     

Horseshoes in chromosome’s attractors

In this paper, we study dynamics of a class of chromosome’s attractors. We show that these chromosome sequences are chaotic by giving a rigorous verification for existence of horseshoes in these systems. We prove that the Poincaré maps derived from these chromosome’s attractors are semi-conjugate to the 2-shift map, and its entropy is no less than log 2. The chaotic behavior is robust in the following sense: chaos exists when one parameter varies from −5.5148 to −5.4988.     

Evaluating the exact infinitesimal values of area of Sierpinski’s carpet and volume of Menger’s sponge

Very often traditional approaches studying dynamics of self-similarity processes are not able to give their quantitative characteristics at infinity and, as a consequence, use limits to overcome this difficulty. For example, it is well known that the limit area of Sierpinski’s carpet and volume of Menger’s sponge are equal to zero. It is shown in this paper that recently introduced infinite and infinitesimal numbers allow us to use exact expressions instead of limits and to calculate exact infinitesimal values of areas and volumes at various points at infinity even if the chosen moment of the observation is infinitely faraway on the time axis from the starting point. It is interesting that traditional results that can be obtained without the usage of infinite and infinitesimal numbers can be produced just as finite approximations of the new ones. The importance of the possibility to have this kind of quantitative characteristics for E-Infinity theory is emphasized.     

A new generalization of Aczél’s inequality and its applications to an improvement of Bellman’s inequality

A new generalized version of Aczél’s inequality is proved. This is a unified generalization of some known results. Moreover, the result is applied to the improvement of the well-known Bellman’s inequality.     

An optimization of Chebyshev’s method

From Chebyshev’s method, new third-order multipoint iterations are constructed with their efficiency close to that of Newton’s method and the same region of accessibility.     

Constrained versions of Sauer’s lemma

Let [n]={1,…,n}. For a function h:[n]→{0,1}, x[n] and y{0,1} define by the width ω_h(x,y) of h at x the largest nonnegative integer a such that h(z)=y on x−a≤z≤x+a. We consider finite VC-dimension classes of functions h constrained to have a width ω_h(x_i,y_i) which is larger than N for all points in a sample or a width no larger than N over the whole domain [n]. Extending Sauer’s lemma, a tight upper bound with closed-form estimates is obtained on the cardinality of several such classes.     

Generalizations of Heilbronn’s triangle problem

For given integers d,j≥2 and any positive integers n, distributions of n points in the d-dimensional unit cube [0,1]^d are investigated, where the minimum volume of the convex hull determined by j of these n points is large. In particular, for fixed integers d,k≥2 the existence of a configuration of n points in [0,1]^d is shown, such that, simultaneously for j=2,…,k, the volume of the convex hull of any j points among these n points is Ω(1/n^{(j−1)/(1+|d−j+1|)}). Moreover, a deterministic algorithm is given achieving this lower bound, provided that d+1≤j≤k.     

Fan’s inequality in geodesic spaces

Fan’s minimax inequality is extended to the context of metric spaces with global nonpositive curvature. As a consequence, a much more general result on the existence of a Nash equilibrium is obtained.     

Quantum repulsive Nonlinear Schrödinger models and their ‘Superconductivity’

The fundamental role played by the quantum repulsive Nonlinear Schrödinger (NLS) equation in the evolution of our understanding of the phenomenon of superconductivity in appropriate metals at very low temperatures is surveyed. The first major work was that in 1947 by N. N. Bogoliubov, who studied the very physical 3-space-dimensions problem and super fluidity; and the survey takes the form of an actual dedication to that outstanding scientist who died four years ago. The 3-space-dimensions NLS equation is not integrable either classically or quantum mechanically. But a number of recently discovered closely related lattices in one space dimension (one space plus one time dimension) are integrable as both classical lattices and quantum lattices while their continuum limits are the now well-known fundamental and integrable system the quantum ‘Bose gas’. These models are all examined in this paper in a physical application of recent so-called ‘quantum groups’ theory, itself fundamental to integrability theory. The ‘superfluid’ phase transitions shown by these lattices, as well as by the bose gas, all at zero temperature in 1 + 1 dimensions, are analysed in terms of the behaviour of certain lattice correlation functions which are either quantum or, in the case of the so-called XY-model, classical correlation functions. Although the repulsive NLS models in 1 + 1 are integrable, they do not have actual soliton solutions. Nevertheless the material as surveyed here is a fundamental application of soliton-theory in the broader context of integrability or near-integrability which has had profound effects in the evolution of current understandings in all of modern theoretical physics.     

On Moving Frames and Noether’s Conservation Laws

Noether’s Theorem yields conservation laws for a Lagrangian with a variational symmetry group. The explicit formulae for the laws are well known and the symmetry group is known to act on the linear space generated by the conservation laws. The aim of this paper is to explain the mathematical structure of both the Euler‐Lagrange system and the set of conservation laws, in terms of the differential invariants of the group action and a moving frame. For the examples, we demonstrate, knowledge of this structure allows the Euler‐Lagrange equations to be integrated with relative ease. Our methods take advantage of recent advances in the theory of moving frames by Fels and Olver, and in the symbolic invariant calculus by Hubert. The results here generalize those appearing in Kogan and Olver [ 1 ] and in Mansfield [ 2 ]. In particular, we show results for high‐dimensional problems and classify those for the three inequivalent SL(2) actions in the plane.     

On stability and bifurcation of Chen’s system

This article investigates some subtle characteristics of stability and bifurcation of the chaotic Chen’s system, based on rigorous mathematical analysis and symbolic computations.