Fractals via iterated functions and multifunctions

Fractals have wide applications in biology, computer graphics, quantum physics and several other areas of applied sciences (see, for instance [Daya Sagar BS, Rangarajan Govindan, Veneziano Daniele. Preface – fractals in geophysics. Chaos, Solitons & Fractals 2004;19:237–39; El Naschie MS. Young double-split experiment Heisenberg uncertainty principles and cantorian space-time. Chaos, Solitons & Fractals 1994;4(3):403–09; El Naschie MS. Quantum measurement, information, diffusion and cantorian geodesics. In: El Naschie MS, Rossler OE, Prigogine I, editors. Quantum mechanics, diffusion and Chaotic fractals. Oxford: Elsevier Science Ltd; 1995. p. 191–205; El Naschie MS. Iterated function systems, information and the two-slit experiment of quantum mechanics. In: El Naschie MS, Rossler OE, Prigogine I, editors. Quantum mechanics, diffusion and Chaotic fractals. Oxford: Elsevier Science Ltd; 1995. p. 185–9; El Naschie MS, Rossler OE, Prigogine I. Forward. In: El Naschie MS, Rossler OE, Prigogine I, editors. Quantum mechanics, diffusion and Chaotic fractals. Oxford: Elsevier Science Ltd; 1995; El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19:209–36; El Naschie MS. Fractal black holes and information. Chaos, Solitons & Fractals 2006;29:23–35; El Naschie MS. Superstring theory: what it cannot do but E-infinity could. Chaos, Solitons & Fractals 2006;29:65–8). Especially, the study of iterated functions has been found very useful in the theory of black holes, two-slit experiment in quantum mechanics (cf. El Naschie, as mentioned above). The intent of this paper is to give a brief account of recent developments of fractals arising from IFS. We also discuss iterated multifunctions.     

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].     

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.     

A note on a fixed point theorem in Menger probabilistic quasi-metric spaces

The main result in Rezaiyan et al. [Rezaiyan R, Cho YJ, Saadati R. A common fixed point theorem in Menger probabilistic quasi-metric spaces, Chaos, Solitons & Fractals 2008;37:1153–7] is proved under two (necessary) additional conditions.     

Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system

To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. In this paper, we attempt to investigate the ultimate bound and positively invariant set for two specific systems, the Lorenz system and a unified chaotic system. We derive an ellipsoidal estimate of the ultimate bound and positively invariant set for the Lorenz system, for all the positive values of its parameters a, b and c, and obtain the minimum value of volume for the ellipsoid. Comparing with the best results in the current literature [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534; X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419], our new results fill up the gap of the estimate for the cases of 0<a<1 and 0<b<2 [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419]. Furthermore, the estimation derived here contains the results given in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419] as special cases. Along the same line, we also provide estimates of cylindrical and ellipsoidal bounds for a unified chaotic system, for its parameter range , and obtain the minimum value of volume for the ellipsoid. The estimate is more accurate than and also extends the result of [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529-534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404-1419].     

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].     

Multiplicatively badly approximable numbers and generalised Cantor sets

Let p be a prime number. The p-adic case of the Mixed Littlewood Conjecture states that for all α∈R. We show that with the additional factor of the statement is false. Indeed, our main result implies that the set of α for which is of full dimension. The result is obtained as an application of a general framework for Cantor sets developed in this paper.     

The SC property of an expected residual function arising from stochastic complementarity problems

The stochastic nonlinear complementarity problem has been recently reformulated as an expected residual minimization problem which minimizes an expected residual function defined by an NCP function. In this work, we show that the expected residual function defined by the Fischer–Burmeister function is an function.     

Asymptotic method for Dougall's bilateral hypergeometric sums

By means of the modified Abel lemma on summation by parts, a recurrence relation for Dougall's bilateral -series is established with an extra natural number parameter m. Then the steepest descent method allows us to compute the limit for m→∞, which leads us surprisingly to a completely new proof of the celebrated bilateral -series identity due to Dougall (1907). The same approach applies also to the bilateral very well-poised -series identity [J. Dougall, On Vandermonde's theorem and some more general expansions, Proc. Edinburgh Math. Soc. 25 (1907) 114-132].     

Some fixed point theorems in fuzzy reflexive Banach spaces

In this paper, we first show that there are some gaps in the fixed point theorems for fuzzy non-expansive mappings which are proved by Bag and Samanta, in [Bag T, Samanta SK. Fixed point theorems on fuzzy normed linear spaces. Inf Sci 2006;176:2910–31; Bag T, Samanta SK. Some fixed point theorems in fuzzy normed linear spaces. Inform Sci 2007;177(3):3271–89]. By introducing the notion of fuzzy and α- fuzzy reflexive Banach spaces, we obtain some results which help us to establish the correct version of fuzzy fixed point theorems. Second, by applying Theorem 3.3 of Sadeqi and Solati kia [Sadeqi I, Solati kia F. Fuzzy normed linear space and it’s topological structure. Chaos, Solitons & Fractals, in press] which says that any fuzzy normed linear space is also a topological vector space, we show that all topological version of fixed point theorems do hold in fuzzy normed linear spaces.     

Interest rate option pricing and volatility forecasting: An application to Brazil

Assessing the markets perception of future interest and inflation rate volatility is of crucial importance to assess the evolution of expectations in an inflation targeting framework. This article aims to evaluate the information content of implied volatilities extracted from a Brazilian interest-rate call option. We compared the predictive performance of three different approaches: one using the traditional [Black F. The pricing of commodity contracts. J Financ Econ 1976;3:167–79] method, another one using the extended-Vasicek model, and in the third approach, we use a GARCH(2, 1) model. The empirical evidence was more favorable to the extended-Vasicek method. Moreover, extended-Vasicek’s implied volatilities could predict around 33% (adjusted R²) of the variations in realized volatility. Further research could test for the predictive content of long memory options such as those suggested in Wang et al. [Wang X-T, Qiu W-Y, Ren F-Y. Option pricing of fractional version of the Black–Scholes model with Hurst exponent H being in . Chaos, Solitons & Fractals 2001;12:599–608; Wang X-T, Ren F-Y, Liang X-Q. A fractional version of the Merton model. Chaos, Solitons & Fractals 2003;15:455–63].     

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].     

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.     

A modified variable-coefficient projective Riccati equation method and its application to (2 + 1)-dimensional simplified generalized Broer–Kaup system

A modified variable-coefficient projective Riccati equation method is proposed and applied to a (2 + 1)-dimensional simplified and generalized Broer–Kaup system. It is shown that the method presented by Huang and Zhang [Huang DJ, Zhang HQ. Chaos, Solitons & Fractals 2005; 23:601] is a special case of our method. The results obtained in the paper include many new formal solutions besides the all solutions found by Huang and Zhang.     

On perturbations of strongly admissible prior distributions

Consider a parametric statistical model, P(dx|θ), and an improper prior distribution, ν(dθ), that together yield a (proper) formal posterior distribution, Q(dθ|x). The prior is called strongly admissible if the generalized Bayes estimator of every bounded function of θ is admissible under squared error loss. Eaton [M.L. Eaton, A statistical diptych: Admissible inferences-recurrence of symmetric Markov chains, Annals of Statistics 20 (1992) 1147–1179] used the Blyth–Stein Lemma to develop a sufficient condition, call it , for strong admissibility of ν. Our main result says that, under mild regularity conditions, if ν satisfies and g(θ) is a bounded, non-negative function, then the perturbed prior distribution g(θ)ν(dθ) also satisfies and is therefore strongly admissible. Our proof has three basic components: (i) Eaton's [M.L. Eaton, A statistical diptych: Admissible inferences-recurrence of symmetric Markov chains, Annals of Statistics 20 (1992) 1147–1179] result that the condition is equivalent to the local recurrence of the Markov chain whose transition function is R(dθ|η)=∫Q(dθ|x)P(dx|η); (ii) a new result for general state space Markov chains giving conditions under which local recurrence is equivalent to recurrence; and (iii) a new generalization of Hobert and Robert's [J.P. Hobert, C.P. Robert, Eaton's Markov chain, its conjugate partner and -admissibility, Annals of Statistics 27 (1999) 361–373] result that says Eaton's Markov chain is recurrent if and only if the chain with transition function is recurrent. One important application of our results involves the construction of strongly admissible prior distributions for estimation problems with restricted parameter spaces.     

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].     

Hereditary properties of partitions, ordered graphs and ordered hypergraphs

In this paper we use the Klazar–Marcus–Tardos method (see [A. Marcus, G. Tardos, Excluded permutation matrices and the Stanley–Wilf conjecture. J. Combin. Theory Ser. A 107 (2004) 153–160]) to prove that, if a hereditary property of partitions has super-exponential speed, then, for every k-permutation π, contains the partition of [2k] with parts {{i,π(i)+k}:i[k]}. We also prove a similar jump, from exponential to factorial, in the possible speeds of monotone properties of ordered graphs, and of hereditary properties of ordered graphs not containing large complete, or complete bipartite ordered graphs.Our results generalize the Stanley–Wilf conjecture on the number of n-permutations avoiding a fixed permutation, which was recently proved by the combined results of Klazar [M. Klazar, The Füredi–Hajnal conjecture implies the Stanley–Wilf conjecture, in: D. Krob, A.A. Mikhalev, A.V. Mikhalev (Eds.), Formal Power Series and Algebraic Combinatorics, Springer, Berlin, 2000, pp. 250–255] and Marcus and Tardos [A. Marcus, G. Tardos, Excluded permutation matrices and the Stanley–Wilf conjecture, J. Combin. Theory Ser. A 107 (2004) 153–160]. Our main results follow from a generalization to ordered hypergraphs of the theorem of Marcus and Tardos.     

An upper bound on Jacobi polynomials

Let be an orthonormal Jacobi polynomial of degree k. We will establish the following inequality:

where δ_-1<δ₁ are appropriate approximations to the extreme zeros of . As a corollary we confirm, even in a stronger form, T. Erdélyi, A.P. Magnus and P. Nevai conjecture [T. Erdélyi, A.P. Magnus, P. Nevai, Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994) 602–614] by proving that

in the region .