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 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 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.     

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).     

On an optimal stopping problem of Gusein-Zade

We study the problem of selecting one of the r best of n rankable individuals arriving in random order, in which selection must be made with a stopping rule based only on the relative ranks of the successive arrivals. For each r up to r=25, we give the limiting (as n→∞) optimal risk (probability of not selecting one of the r best) and the limiting optimal proportion of individuals to let go by before being willing to stop. (The complete limiting form of the optimal stopping rule is presented for each r up to r=10, and for r=15, 20 and 25.) We show that, for large n and r, the optical risk is approximately (1−t^*)^r, where t^*≈0.2834 is obtained as the roof of a function which is the solution to a certain differential equation. The optimal stopping rule τ_r,n lets approximately t^*n arrivals go by and then stops ‘almost immediately’, in the sense that τ_r,n/n→t^* in probability as n→∞, r→∞     

Quantum decoherence and El Naschie’s complex temporality

It is well-known that the Schrödinger equation reduces to a classical diffusion equation by means of Wick rotation (t → it), suggesting a correspondence between quantum and classical mechanics. Nonetheless, this result does not admit a clear conceptual interpretation. In the framework of his fractal space-time theory, El Naschie showed that great conceptual advantage could be achieved by extending the imaginary time, it, to a perfectly symmetric, complex conjugate time 0 ± it. In this note we show through a simple analysis, involving formal analytic continuation (t → 0 ± it), that El Naschie’s time complexification provides the basis for a physical interpretation of the correspondence between quantum and classical mechanics in terms of quantum decoherence. We find that decoherent states inevitably arise due to time symmetry breaking as we go from the micro Cantorian space-time, where the two symmetric times, 0 + it and 0 − it, coexist to our 4-dimensional smooth space-time, where t is the only time.     

The maximum number of elementary particles in a super symmetric extension of the standard model

In a series of papers over the last few years El Naschie addressed the question of the minimum and maximum number of elementary particles which a mathematically consistent and a physically meaning full extended standard model should contain. El Naschie’s minimum is 62 particles namely 60 believed to have been already discovered in addition to one Higgs boson and one graviton which are theoretically needed but are not jet experimentally conformed. By contrast the maximum number of 69 particles is although consistent with many quantum field theories based models as well as a classical result by Dyson may not be the only possibility. In the present work we show that a larger number of 72 or even 84 particles are easily shown to be consistent with super string theory and super symmetry. Our work consists of two parts. The first part is a reappraisal of El Naschie’s results and the second is a derivation of the proposed possibility of an upper bound of 72 or 84 elementary particles.     

Full quadrature sums for pth powers of polynomials with Freud weights

In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r> 0, b (− ∞, 2]. We establish a full quadrature sum estimate

1 p < ∞, for every polynomial P of degree at most n + rn^1/3, where W² is a Freud weight such as exp(−¦x¦), > 1, λ_jn are the Christoffel numbers, x_jn are the zeros of the orthonormal polynomials for the weight W², and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree M = m(n) if m(n) = n + ξ_nn^1/3, where ξ_n → ∞ as n → ∞. Previous estimates could sum only over those x_jn with ¦x_jn¦ σx_1n, some fixed 0 < σ < 1.     

On the relation between interior critical points and parameters for a class of nonlinear problems with Neumann–Robin boundary conditions

We consider boundary value problem

where   0, λ > 0 are parameters and f  C²[0, ∞) such that f(0) < 0. In this paper we study for the cases p  (0, β) and p  (β, θ) (p is the value of the solution at x = 0 and β, θ are such that f(β) = 0, , the relation between λ and the number of interior critical points of the positive solutions of the above system.     

On the extreme eigenvalues of hermitian (block) toeplitz matrices

We are concerned with the behavior of the minimum (maximum) eigenvalue λ₀⁽ⁿ⁾ (λ_n⁽ⁿ⁾) of an (n + 1) × (n + 1) Hermitian Toeplitz matrix T_n(ƒ) where ƒ is an integrable real-valued function. Kac, Murdoch, and Szegö, Widom, Parter, and R. H. Chan obtained that λ₀⁽ⁿ⁾ — min ƒ = O(1/n^2k) in the case where ƒ C^2k, at least locally, and ƒ — inf ƒ has a zero of order 2k. We obtain the same result under the second hypothesis alone. Moreover we develop a new tool in order to estimate the extreme eigenvalues of the mentioned matrices, proving that the rate of convergence of λ₀⁽ⁿ⁾ to inf ƒ depends only on the order ρ (not necessarily even or integer or finite) of the zero of ƒ — inf ƒ. With the help of this tool, we derive an absolute lower bound for the minimal eigenvalues of Toeplitz matrices generated by nonnegative L¹ functions and also an upper bound for the associated Euclidean condition numbers. Finally, these results are extended to the case of Hermitian block Toeplitz matrices with Toeplitz blocks generated by a bivariate integrable function ƒ.     

More on θ-compact fuzzy topological spaces

Recently, El-Naschie has shown that the notion of fuzzy topology may be relevant to quantum particle physics in connection with string theory and ε^∞ theory. In 2005, Caldas and Jafari have introduced θ-compact fuzzy topological spaces. The purpose of this paper is to investigate further properties of θ-compact fuzzy topological spaces. Moreover, the notion of θ-closed fuzzy topological spaces is introduced and properties of it are obtained.     

Crank–Nicolson least-squares Galerkin procedures for parabolic integro-differential equations

Two Crank–Nicolson least-squares Galerkin finite element schemes are formulated to solve parabolic integro-differential equations. The advantage of this method is that it is not subject to the LBB condition. The convergence analysis shows that the methods yield the approximate solutions with optimal accuracy in H(div; Ω) × H¹(Ω) and (L²(Ω))² × L²(Ω), respectively. Moreover, the two methods both get the approximate solutions with second-order accuracy in time increment.     

A fixed point theorem and a norm inequality for operator means

There is one to one correspondence between positive operator monotone functions on (0, ∞) and operator connections. For a symmetric connection σ, it is proved that the map X → (AσX)σ(BσX) from positive operators on a Hilbert space to itself, has a unique fixed point. Here σ denotes the dual of σ. It is also proved that |||AσB||| |||A|||σ|||B||| for all unitarily invariant norms ||| · ||| and for all positive operators A,B.     

Asymptotics of a certain integral

The asymptotic behaviour for t → ∞ of ^∞₀ exp[tx–c(x)]dx is studied. The function c is positive and ′(x) → ∞ (x → ∞). Sufficient conditions on c are given in order that the method of Laplace is applicable.     

A two-sided estimate in the Hsu-Robbins-Erdös law of large numbers

Let X₁, X₂, … be independent identically distributed random variables. Then, Hsu and Robbins (1947) together with Erdös (1949, 1950) have proved that

,

if and only if E[X²₁] < ∞ and E[X₁] = 0. We prove that there are absolute constants C₁, C₂ (0, ∞) such that if X₁, X₂, … are independent identically distributed mean zero random variables, then

c₁λ⁻² E[X₁²·1_{|X1|λ}]S(λ)C₂λ⁻² E[X₁²·1_{|X1|λ}]

,

for every λ > 0.     

Strong approximation for cross-covariances of linear variables with long-range dependence

Suppose {_k, −∞ < k < ∞} is an independent, not necessarily identically distributed sequence of random variables, and {c_j}^∞_j=0, {d_j}^∞_j=0 are sequences of real numbers such that Σ_jc²_j < ∞, Σ_jd²_j < ∞. Then, under appropriate moment conditions on {_k, −∞ < k < ∞}, y_k Σ^∞_j=0c_jk-j, z_k Σ^∞_j=0d_jk-j exist almost surely and in ⁴ and the question of Gaussian approximation to S_[t]Σ^[t]_k=1 (y_k z_k − E{y_k z_k}) becomes of interest. Prior to this work several related central limit theorems and a weak invariance principle were proven under stationary assumptions. In this note, we demonstrate that an almost sure invariance principle for S_[t], with error bound sharp enough to imply a weak invariance principle, a functional law of the iterated logarithm, and even upper and lower class results, also exists. Moreover, we remove virtually all constraints on _k for “time” k ≤ 0, weaken the stationarity assumptions on {_k, −∞ < k < ∞}, and improve the summability conditions on {c_j}^∞_j=0, {d_j}^∞_j=0 as compared to the existing weak invariance principle. Applications relevant to this work include normal approximation and almost sure fluctuation results in sample covariances (let d_j = c_j-m for j ≥ m and otherwise 0), quadratic forms, Whittle's and Hosoya's estimates, adaptive filtering and stochastic approximation.     

The thermal equilibrium state of semiconductor devices

The thermal equilibrium state of two oppositely charged gases confined to a bounded domain , m = 1,2 or m = 3, is entirely described by the gases' particle densities p, n minimizing the total energy (p, n). it is shown that for given P, N > 0 the energy functional admits a unique minimizer in {(p, n) ε L²(Ω) x L ²(Ω) : p, n ≥ 0, _Ωp = P, _Ωn = N} and that p, n ε C(Ω) ∩ L^∞(Ω).

The analysis is applied to the hydrodynamic semiconductor device equations. These equations in general possess more than one thermal equilibrium solution, but only the unique solution of the corresponding variational problem minimizes the total energy. It is equivalent to prescribe boundary data for electrostatic potential and particle densities satisfying the usual compatibility relations and to prescribe V^e and P, N for the variational problem.     

From greedy to lazy expansions and their driving dynamics

In this paper we study the ergodic properties of non-greedy series expansions to non-integer bases β > 1. It is shown that the so-called ‘lazy’ expansion is isomorphic to the ‘greedy’ expansion. Furthermore, a class of expansions to base β > 1, β , ‘in between’ the lazy and the greedy expansions are introduced and studies. It is shown that these expansions are isomorphic to expansions of the form Tx = βx + (mod 1).     

The W-convexity and normal structure in banach spaces

Let X be a Banach space, S(X) - x ε X : #x02016; = 1 be the unit sphere of X.The parameter, modulus of W^*-convexity, W^*(ε) = inf <(x − y)/2, f_x> : x, y S(X), x − y ≥ ε, f_x Δ_x , where 0 ≤ ε ≤ 2 and Δ_x S(X^*) be the set of norm 1 supporting functionals of S(X) at x, is investigated_ The relationship among uniform nonsquareness, uniform normal structure and the parameter W^*(ε) are studied, and a known result is improved. The main result is that for a Banach space X, if there is ε, where 0 < ε < 1/2, such that W^*(1 + ε) > ε/2 where W^*(1 + ε) = lim_→ε W^* (1 + ), then X has normal structure.     

L2-approximation of real-valued functions with interpolatory constraints

We construct the polynomial p_m,n^* of degree m which interpolates a given real-valued function f L²[a, b] at pre-assigned n distinct nodes and is the best approximant to f in the L₂-sense over all polynomials of degree m with the same interpolatory character. It is shown that the L₂-error p_m,n^* − f → 0 as m → ∞ if f C[a, b].