Narrow Results

In this paper, we are concerned with spectral-theoretic features of general iterated function systems (IFS). Such systems arise from the study of iteration limits of a finite family of maps τ_i, i=1,…,N, in some Hausdorff space Y. There is a standard construction which generally allows us to reduce to the case of a compact invariant subset X⊂Y. Typically, some kind of contractivity property for the maps τ_i is assumed, but our present considerations relax this restriction. This means that there is then not a natural equilibrium measure μ available which allows us to pass the point-maps τ_i to operators on the Hilbert space L²(μ). Instead, we show that it is possible to realize the maps τ_i quite generally in Hilbert spaces ℋ(X) of square-densities on X. The elements in ℋ(X) are equivalence classes of pairs (φ,μ), where φ is a Borel function on X, μ is a positive Borel measure on X, and ∫_X|φ|² dμ<∞. We say that (φ,μ)~(ψ,ν) if there is a positive Borel measure λ such that μ≪λ, ν≪λ, and

We further prove that this representation in the Hilbert space ℋ(X) has several universal properties.

Bond-percolation processes are studied for random lattices on the surface of a sphere, and for their duals. The estimated threshold is 0.3326 ± 0.0005 for spherical random lattices and 0.6680 ± 0.0005 for the duals of spherical random lattices, and the exact threshold is conjectured as 1/3 for two-dimensional random lattices and 2/3 for their duals. A suitably defined spanning probability at the threshold, E_p(p_c), for both spherical random lattices and their duals is 0.980±0.005, which may be universal for a 2-d lattice with this spanning definition. The shift-to-width ratio of the distribution function of the threshold concentration and the universal values of the critical value of the effective coordination number can be extended from regular lattices to spherical random lattices and their duals. The results of critical exponents are consistent with the assertion from the universality hypothesis. Finite-size scaling is also examined.

This study incorporates bulk dissipation described by a losing probability f into a modified Manna model on an L × L square lattice. The crossover behavior between bulk and boundary dissipation is investigated using the characteristic lengths produced by bulk dissipation. The toppling number N_n and area N_a are studied. For a probability distribution of N_s, where s = n or a, the scaling form including the finite-size scaling (f = 0) and the critical scaling (L → ∞) are determined. Subsequently, this paper investigates the joint probability distribution and provides the scaling relation between the toppling number and area.

This paper presents the general method for the adaptive function Q-S synchronization of different chaotic (hyper-chaotic) systems. Based upon the Lyapunov stability theory, the dynamical evolution can be achieved by the Q-S synchronization with a desired scaling function between the different chaotic (hyper-chaotic) systems. This approach is successfully applied to two examples: Chen hyper-chaotic system drives the Lorenz hyper-chaotic system; Lorenz system drives Lü hyper-chaotic system. Numerical simulations are used to validate and demonstrate the effectiveness of the proposed scheme.

This work is concerned with the general methods for the function projective synchronization (FPS) of chaotic (or hyperchaotic) systems. The aim is to investigate the FPS of different chaotic (hyper-chaotic) systems with unknown parameters. The adaptive control law and the parameter update law are derived to make the states of two different chaotic systems asymptotically synchronized up to a desired scaling function by Lyapunov stability theory. The general approach for FPS of Chen hyperchaotic system and Lü system is provided. Numerical simulations are also presented to verify the effectiveness of the proposed scheme.

The time evolution of the roughness is investigated for one-dimensional systems described by the Kargar–Parisi–Zhang equation. Scaling behavior of the roughness is studied, and the scaling function is obtained.

The oscillating multifractal formalism is a formula conjectured by Jaffard expected to yield the spectrum d(h, β) of oscillating singularity exponents from a scaling function ζ(p, s'), for p > 0 and s' ∈ ℝ, based on wavelet leaders of fractional primitives f_-s' of f. In this paper, using some results from Jaffard et al., we first show that ζ(p, s') can be extended on p ∈ ℝ to a function that is concave with respect to p ∈ ℝ and independent on orthonormal wavelet bases in the Schwartz class. We also establish its concavity with respect to s' when p > 0. Then, we prove that, under some assumptions, the extended scaling function ζ(p, s') is the Legendre transform of the wavelet leaders density of f_-s'. Finally, as an application, we study the validity of the extended oscillating multifractal formalism for random wavelet series (under the assumption of independence and laws depending only on the scale).

In this paper, under a mild condition, the construction of compactly supported -wavelets is obtained. Wavelets inherit the symmetry of the corresponding scaling function and satisfy the vanishing moment condition originating in the symbols of the scaling function. An example is also given to demonstrate the general theory.

It is well-known that the different kinds of multiresolution analysis (MRA) structures generate different wavelets. In this paper, we give two uniform formulas on the number of mother functions for various wavelets associated with MRA structures. These formulas show that the number of mother functions of wavelets is determined by the support of the Fourier transform of the scaling function in MRA structure.

Gabardo and Nashed have studied nonuniform multiresolution analysis based on the theory of spectral pairs in a series of papers, see Refs. 4 and 5. Farkov,³ has extended the notion of multiresolution analysis on locally compact Abelian groups and constructed the compactly supported orthogonal p-wavelets on L²(ℝ₊). We have considered the nonuniform multiresolution analysis on positive half-line. The associated subspace V₀ of L²(ℝ₊) has an orthonormal basis, a collection of translates of the scaling function φ of the form {φ(x ⊖ λ)}_λ∈Λ+ where Λ₊ = {0, r/N} + ℤ₊, N > 1 (an integer) and r is an odd integer with 1 ≤ r ≤ 2N - 1 such that r and N are relatively prime and ℤ₊ is the set of non-negative integers. We find the necessary and sufficient condition for the existence of associated wavelets and derive the analogue of Cohen's condition for the nonuniform multiresolution analysis on the positive half-line.

We construct smooth nonseparable compactly supported refinable functions that generate multiresolution analyses on L²(ℝ^d), d > 1. Using these refinable functions we construct smooth nonseparable compactly supported orthonormal wavelet systems. These systems are nonseparable, in the sense that none of its constituent functions can be expressed as the product of two functions defined on lower dimensions. Both the refinable functions and the wavelets can be made as smooth as desired. Estimates for the supports of these scaling functions and wavelets, are given.

This paper presents a construction method of bivariate N-band wavelet tight frames. First, we discuss the properties of N-band scaling functions, and the construction of the corresponding wavelet tight frame as well as the explication of the formula of the wavelet tight frame. Thereafter, the decomposition and reconstruction formulas of the bivariate N-band wavelet tight frames are provided. Finally, the numerical example is given.

A multiresolution analysis (MRA) on local fields of positive characteristic was defined by Shah and Abdullah for which the translation set is a discrete set which is not a group. In this paper, we continue the study based on this nonstandard setting and introduce vector-valued nonuniform multiresolution analysis (VNUMRA) where the associated subspace V₀ of L²(K, ℂ^M) has an orthonormal basis of the form {Φ (x - λ)}_λ∈Λ where Λ = {0, r/N} + 𝒵, N ≥ 1 is an integer and r is an odd integer such that r and N are relatively prime and 𝒵 = {u(n) : n ∈ ℕ₀}. We establish a necessary and sufficient condition for the existence of associated wavelets and derive an algorithm for the construction of VNUMRA on local fields starting from a vector refinement mask G(ξ) with appropriate conditions. Further, these results also hold for Cantor and Vilenkin groups.

An explicit description of all Walsh polynomials generating tight wavelet frames is given. An algorithm for finding the corresponding wavelet functions is suggested, and a general form for all wavelet frames generated by an appropriate Walsh polynomial is described. Approximation properties of tight wavelet frames are also studied. In contrast to the real setting, it appeared that a wavelet tight frame decomposition has an arbitrary large approximation order whenever all wavelet functions are compactly supported.

The main purpose of this paper is to provide a characterization of scaling functions for non-uniform multiresolution analysis (NUMRA, in short). Some necessary and sufficient conditions for scaling functions of wavelet NUMRA in the frequency domain are also obtained.

The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.

In signal processing, rational $\frac{p}{q}$-wavelets are preferable than the wavelets corresponding to dyadic MRA because it allows more variations in scale factors of signal components. In this paper, for a rational number $\frac{p}{q},p,q\in \mathbb{N},p>q\ge 1,(p,q)=1$ and $\alpha \in (0,\frac{q}{p-1})$, we consider a collection ${{𝒜}}_{\alpha }^{p/q}$, the space of all continuous functions in $L^2(ℝ)$ that are linear on $[pqk+pl\alpha ,pqk+p(l+1)\alpha ]$ and $[pqk+(p-1)p\alpha ,pqk+pq]$ for all $l=0,1,\dots ,p-2,k\in \mathbb{Z}$. For $\frac{p}{q}=\frac{3}{2},\frac{4}{3}$, under certain conditions, we prove that, if $\frac{1}{\sqrt{q}}\varphi (\frac{\cdot }{q})\in {{𝒜}}_{\alpha }^{p/q}$ generates a $\frac{p}{q}$-MRA, then $\alpha =\frac{q}{p}$. Also, we show that if $\alpha =\frac{q}{p}$, there exists a function $\frac{1}{\sqrt{q}}\varphi (\frac{\cdot }{q})\in L^2(ℝ)$, satisfying the above conditions, that generates $\frac{p}{q}$-MRA. In addition, we construct orthonormal $\frac{p}{q}$-wavelets corresponding to $\frac{p}{q}$-MRA.

To overcome the disadvantages of dyadic wavelet and extend the scope of wavelet applications, the multiresolution analysis of the rank M wavelet and the orthogonal condition of the rank M wavelet are studied. A new method for the construction of the filter coefficients corresponding to the rank 3 wavelet scaling function is presented. And the conclusion is drawn that the number of the FIR coefficients corresponding to the rank M wavelet scaling function is at least MN, where N is the vanishing moment of the wavelet function. Rank M wavelet has potential advantages over the traditional wavelets in image compression.

To provide a criterion for existence of bivariate compactly supported tight wavelet frames in terms of some conditions on the symbol of the scaling function. In particular, if a scaling function is symmetric, then there exists symmetric or antisymmetric tight wavelet frame.

In this paper we investigate a framework for bivariate non-uniform Haar wavelets. We use tensor product to define the bivariate Haar scaling function, the bivariate non-uniform Haar wavelets, and the bivariate non-uniform multiresolution analysis associated to these functions. The direct and inverse bivariate non-uniform Haar wavelet transforms are given.