Different Higgs models and the number of Higgs particles

In this short paper we discuss some interesting Higgs models. It is concluded that the most likely scheme for the Higgs particles consists of five physical Higgs particles. These are two charged H⁺, H⁻ and three neutrals h⁰, H⁰, A⁰. Further more the most probably total number of elementary particles for each model is calculated [El Naschie MS. Experimental and theoretical arguments for the number of the mass of the Higgs particles. Chaos, Solitons & Fractals 2005;23:1091–8; El Naschie MS. Determining the mass of the Higgs and the electroweak bosons. Chaos, Solitons & Fractals 2005;24:899–905; El Naschie MS. On 366 kissing spheres in 10 dimensions, 528 P-Brane states in 11 dimensions and the 60 elementary particles of the standard model. Chaos, Solitons & Fractals 2005;24:447–57].     

3-Adic Cantor function on local fields and its p-adic derivative

The problem of “rate of change” for fractal functions is a very important one in the study of local fields. In 1992, Su Weiyi has given a definition of derivative by virtue of pseudo-differential operators [Su W. Pseudo-differential operators and derivatives on locally compact Vilenkin groups. Sci China [series A] 1992;35(7A):826–36. Su W. Gibbs–Butzer derivatives and the applications. Numer Funct Anal Optimiz 1995;16(5&6):805–24. [2] and [3]]. In Qiu Hua and Su Weiyi [Weierstrass-like functions on local fields and their p-adic derivatives. Chaos, Solitons & Fractals 2006;28(4):958–65. [8]], we have introduced a kind of Weierstrass-like functions in p-series local fields and discussed their p-adic derivatives. In this paper, the 3-adic Cantor function on 3-series field is constructed, and its 3-adic derivative is evaluated, it has at most order. Moreover, we introduce the definition of the Hausdorff dimension [Falconer KJ. Fractal geometry: mathematical foundations and applications. New York: Wiley; 1990. [1]] of the image of a complex function defined on local fields. Then we conclude that the Hausdorff dimensions of the 3-adic Cantor function and its derivatives and integrals on 3-series field are all equal to 1.There are various applications of Cantor sets in mechanics and physics. For instance, E-infinity theory [El Naschie MS. A guide to the mathematics of E-infinity Cantorian spacetime theory. Chaos, Solitons & Fractals 2005;25(5):955–64. El Naschie MS. Dimensions and Cantor spectra. Chaos, Solitons & Fractals 1994;4(11):2121–32. El Naschie MS. Einstein’s dream and fractal geometry. Chaos, Solitons & Fractals 2005;24(1):1–5. El Naschie MS. The concepts of E infinity: an elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals 2004;22(2):495–511. [9], [10], [11] and [12]] is based on random Cantor set which takes the golden mean dimension as shown by El Naschie.     

Critical fluctuation in daily incidence of acute myocardial infarction

Continuous periodogram power spectral analysis of daily incidence of acute myocardial infarction (AMI) reported at a hospital for cardiology in Pune, India for the two-year period June 1992–May 1994 show that the power spectra follow the universal and unique inverse power law form of the statistical normal distribution. The same time inverse power law form for power spectra of space-time fluctuations are also ubiquitous to dynamical systems in nature and have been identified as signatures of self-organized criticality. The unique quantification for self-organized criticality presented in this paper is shown to be intrinsic to quantumlike mechanics governing fractal space-time fluctuation patterns in dynamical systems and suggest a possibly fruitful relation and analogy between different subject such as chaos, diffusion and quantum physics. The results found which mimic those obtained in quantum physics by El Naschie using the concept of Cantorian space ε^(∞) suggest that, that tools developed in some of these areas may be used advantageously in the medical field as pioneered by A.T. Winfree [Int. J. Bifurcation and Chaos 7 (3) (1997) 487–526] and A.V. Holden [Chaos, Solitons and Fractals 5 (3/4) (1995) 691–704; Int. J. Bifurcation and Chaos 7 (9) (1997) 2075–2104].     

An Analytical Study of Bifurcations Generated by Some Iterated Function Systems

In a recent paper Bahar [Chaos, Solitons + Fractals, 1996, 7(1), 41] described bifurcation from a fixed point generated by iterated function systems. An analytical study of it, by using Banach theorem, was proposed by us in Chaos, Solitons + Fractals, 1998, 9(3), 449. In this paper we present an extension of our previous study and we prove that by a special transformation, the considered two-dimensional map can be reduced to two distinctive one-dimensional maps, such that each one determines the behavior of the entire system.© 1999 Elsevier Science Ltd. All rights reserved.     

Comment on “Reconnection scenarios…” Chaos Solitons and Fractals 2002;14(1):117–127

We point out that Proposition 3.1 in [E. Petrisor. Reconnection scenarios and the threshold of reconnection in the dynamics of non-twist maps. Chaos Solitons Fractals 2002;14(1):117–27] is, strictly speaking, false. On the other hand, we suggest that for near integrable mappings, the results of [E. Petrisor. Reconnection scenarios and the threshold of reconnection in the dynamics of non-twist maps. Chaos Solitons Fractals 2002;14(1):117–27] are qualitatively correct and quantitatively very approximate.     

On a new class of hyperbolic functions

This article presents the results of some new research on a new class of hyperbolic functions that unite the characteristics of the classical hyperbolic functions and the recurring Fibonacci and Lucas series. The hyperbolic Fibonacci and Lucas functions, which are the being extension of Binet's formulas for the Fibonacci and Lucas numbers in continuous domain, transform the Fibonacci numbers theory into “continuous” theory because every identity for the hyperbolic Fibonacci and Lucas functions has its discrete analogy in the framework of the Fibonacci and Lucas numbers. Taking into consideration a great role played by the hyperbolic functions in geometry and physics, (“Lobatchevski's hyperbolic geometry”, “Four-dimensional Minkowski's world”, etc.), it is possible to expect that the new theory of the hyperbolic functions will bring to new results and interpretations on mathematics, biology, physics, and cosmology. In particular, the result is vital for understanding the relation between transfinitness i.e. fractal geometry and the hyperbolic symmetrical character of the disintegration of the neural vacuum, as pointed out by El Naschie [Chaos Solitons & Fractals 17 (2003) 631].     

A series of new soliton-like solutions and double-like periodic solutions of a (2 + 1)-dimensional dispersive long wave equation

In this paper, we extend the algebraic method proposed by Fan (Chaos, Solitons & Fractals 20 (2004) 609) and the improved extended tanh method by Yomba (Chaos, Solitons and Fractals 20 (2004) 1135) to uniformly construct a series of soliton-like solutions and double-like periodic solutions for nonlinear partial differential equations (NPDE). Some new soliton-like solutions and double-like periodic solutions of a (2 + 1)-dimensional dispersive long wave equation are obtained.     

Exact solution of a thin film flow of an Oldroyd 6-constant fluid over a moving belt

We obtain exact solutions for thin film flow of an Oldroyd 6-constant fluid on a vertically moving belt. These are compared with the homotopy perturbation results of Siddiqui et al. [Siddiqui AM, Ahmed M, Ghori QK. Thin film flow of non-Newtonian fluids on a moving belt. Chaos, Solitons & Fractals 2007;33(3):1006–1016.].     

A generalized poincaré-invariant action with possible application in strings and E-infinity theory

Using the semi-inverse method [He. Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos Solitons & Fractals 2004;19(4):847–851], a generalized Poincaré-invariant action using the tetrad in any number of dimensions is obtained. The so-obtained generalized action with a free parameter might find potential applications in bosonic strings and E-infinity theory.     

Symbolic computation and new families of exact non-travelling wave solutions of (2 + 1)-dimensional Konopelchenko–Dubrovsky equations

The improved tanh function method [Chaos, Solitons & Fractals 2005;24:257] is further improved by constructing new ansatz solution of the considered equation. As its application, the (2 + 1)-dimensional Konopelchenko–Dubrovsky equations are considered and abundant new exact non-travelling wave solutions are obtained.     

Families of non-travelling wave solutions to a generalized variable coefficient two-dimensional KdV equation using symbolic computation

Making use of symbolic computation and the generalized Riccati equation expansion method, some exact non-travelling wave solutions for a generalized variable coefficients two-dimensional KdV equation are obtained. By means of some suitable selections of the arbitrary functions including in the obtained solutions, the results obtained by Elwakil et al. [see: Chaos, Solitons & Fractals 19 (2004) 1083] can be recovered. From our results, some exact solutions for the cylindrical Kadomatsev–Petviashvilli equation can be also derived.     

Complex-plane approach for the analysis of an externally excited system with autoparametric resonance

An autoparametric system with external excitation was examined by Nabergoj R, Tondl A, and Virag Z. (Chaos, Solitons & Fractals 1994;4:263–273). For investigating the semi-trivial and the nontrivial solution, a different approach in terms of complex-plane variables is here presented. It is proved that autoparametric resonance is initiated when semi-trivial solution becomes unstable. The trajectory of the system transforms from a straight line to an elliptic path. Within a certain interval of the excitation frequency, chaotic motion becomes possible.     

Distribution of the units digit of primes

A sequence is formed by the units digit of consecutive prime numbers. The sequence is not random. To visualize the non-randomness of the sequence, we utilize a method put forward by Hao et al. [Chaos, Solitons & Fractals 11 (2000) 825]. A fractal-like structure is observed.     

Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems

This paper investigates the synchronization of two linearly coupled unified chaotic systems and two linearly coupled Lorenz systems. Some sufficient conditions for synchronization are attained through rigorous mathematical theory. Compared with the results in the reference [Chaos, Solitons & Fractals 2002;14:529], the sufficient condition for the synchronization of two linearly coupled Lorenz systems is simpler and less conservative. Numerical simulations are provided for illustration and verification.     

Direction and stability of bifurcating periodic solutions of a chemostat model with two distributed delays

We have considered a chemostat model with two distributed delays in a recent paper [Chaos, Solitons & Fractals 2004;20:995–1004], where, using the average time delay corresponding to the growth response as a bifurcation parameter, it is proven that the model undergoes Hopf bifurcations for two weak kernels. This article is a sequel to the previous work. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. The results are consistent with the numerical results in [Chaos, Solitons & Fractals 2004;20:995–1004].     

New exact soliton-like solutions and special soliton-like structures of the (2+1) dimensional Burgers equation

In this paper, new explicit exact soliton-like solutions and multi-sliton solutions to the (2+1) dimensional Burgers equation are obtained by using the further extended tanh method [Phys Lett A 307 (2003) 269, Chaos, Solitons & Fractals 17 (2003) 669]. Based on the derived exact solutions which contain arbitrary functions, special soliton-like structures are revealed.     

Statistical convergence of double sequences in intuitionistic fuzzy normed spaces

Recently, the concept of intuitionistic fuzzy normed spaces was introduced by Saadati and Park [Saadati R, Park JH. Chaos, Solitons & Fractals 2006;27:331–44]. Karakus et al. [Karakus S, Demirci K, Duman O. Chaos, Solitons & Fractals 2008;35:763–69] have quite recently studied the notion of statistical convergence for single sequences in intuitionistic fuzzy normed spaces. In this paper, we study the concept of statistically convergent and statistically Cauchy double sequences in intuitionistic fuzzy normed spaces. Furthermore, we construct an example of a double sequence to show that in IFNS statistical convergence does not imply convergence and our method of convergence even for double sequences is stronger than the usual convergence in intuitionistic fuzzy normed space.     

Nonlinear dynamics of regenerative cutting processes—Comparison of two models

Understanding the nonlinear dynamics of cutting processes is essential for the improvement of machining technology. We study machine cutting processes by two different models, one has been recently introduced by Litak [Litak G. Chaotic vibrations in a regenerative cutting process. Chaos, Solitons & Fractals 2002;13:1531–5] and the other is the classic delay differential equation model. Although chaotic solutions have been found in both models, well known routes to chaos, such as period-doubling or quasi-periodic motion to chaos are not observed in either model. Careful analysis shows that the chaotic motion from the Litak’s model has sharper spectral peaks, a smaller correlation dimension and a smaller value for the largest positive Lyapunov exponent. Implications to the control of chaos in cutting processes are discussed.