Convergence of nonstationary cascade algorithms

Summary. A nonstationary multiresolution of is generated by a sequence of scaling functions We consider that is the solution of the nonstationary refinement equations where is finitely supported for each k and M is a dilation matrix. We study various forms of convergence in of the corresponding nonstationary cascade algorithm as k or n tends to It is assumed that there is a stationary refinement equation at with filter sequence h and that The results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of the transition operator associated with h. Received September 19, 1997 / Revised version received May 22, 1998 / Published online August 19, 1999     

Convergence of multidimensional cascade algorithm

A necessary and sufficient condition on the spectrum of the restricted transition operator is given for the convergence in of the multidimensional cascade algorithm for any starting function whose shifts form a partition of unity. Received September 12, 1995 / Revised version received August 2, 1996     

Eddy cascade of instabilities and transition to turbulence

Numerical simulation is used to investigate a shear layer influenced by a constant external forcing in the theory of turbulence (Kolmogorov’s problem). The dynamics of flows developing in the case of various initial streamwise velocity profiles are studied. The transition from a two-dimensional laminar flow to a three-dimensional turbulent flow is considered. It is shown that developing hydrodynamic instabilities give rise to an eddy cascade, which, in the transition of the flow to a turbulent stage, corresponds to an eddy cascade developing in the energy and, then, inertial ranges.     

Regenerative cascade homotopies for solving polynomial systems

A key step in the numerical computation of the irreducible decomposition of a polynomial system is the computation of a witness superset of the solution set. In many problems involving a solution set of a polynomial system, the witness superset contains all the needed information. Sommese and Wampler gave the first numerical method to compute witness supersets, based on dimension-by-dimension slicing of the solution set by generic linear spaces, followed later by the cascade homotopy of Sommese and Verschelde. Recently, the authors of this article introduced a new method, regeneration, to compute solution sets of polynomial systems. Tests showed that combining regeneration with the dimension-by-dimension algorithm was significantly faster than naively combining it with the cascade homotopy. However, in this article, we combine an appropriate randomization of the polynomial system with the regeneration technique to construct a new cascade of homotopies for computing witness supersets. Computational tests give strong evidence that regenerative cascade is superior in practice to previous methods.     

Convergence of cascade algorithms in Sobolev spaces and integrals of wavelets

Summary. The cascade algorithm with mask a and dilation M generates a sequence by the iterative process from a starting function where M is a dilation matrix. A complete characterization is given for the strong convergence of cascade algorithms in Sobolev spaces for the case in which M is isotropic. The results on the convergence of cascade algorithms are used to deduce simple conditions for the computation of integrals of products of derivatives of refinable functions and wavelets. Received May 5, 1999 / Revised version received June 24, 1999 / Published online June 20, 2001     

A comparison of mathematical models for a recycling cascade in glycogenolysis

Mathematical models are derived which describe a cyclic-AMP initiated recycling enzyme cascade for the activation of glycogen phosphorylase. The components of the cascade are protein kinase, phosphorylase kinase, phosphorylase kinase phosphatase, phosphorylase, and phosphorylase phosphatase. Two classes of models are formulated and compared. The models differ only in the molecular hypothesis for activation of protein kinase by cyclic-AMP; in one case the assumption leads to second order kinetics, in the other third order kinetics are indicated. Amplification and a fast response time are the prime features of both classes of models for the activation of phosphorylase. The role of free calcium in optimizing the activation and deactivation of the cascade is discussed.     

Building confidence in models for multiple audiences: The modelling cascade

This paper reports on a model building process developed to enable multiple audiences, particularly non-experts, to appreciate the validity of the models being built and their outcomes. The process is a four stage reversible cascade. This cascade provides a structured, auditable/transparent, formalized process from “real world” interviews generating a rich qualitative model through two intermediate steps before arriving at a quantitative simulation model. There are a number of advantages of the cascade process including; achieving comprehensiveness, developing organizational learning, testing the veracity of multiple perspectives, modeling transparency, achieving common understanding across many audiences and promoting confidence building in the models. The paper, based on extensive work with organizations, discusses both the cascade process and its inherent benefits.     

A study of bounded cascade models of random fields on a plane

Cascade models of random processes and fields are extensively used in mathematical modeling of various objects and phenomena. This paper treats of properties of bounded cascade models on a plane as well as their one-dimensional distributions and correlation structure. The results obtained are helpful in analyzing the suitability of cascade models for solving various practical problems.     

Convergence on cooperative cascade systems with length one

Motivated by the open problems on the generic convergence of cooperative systems without the assumption of irreducibility independently proposed by Smith and Sontag, this paper investigates the generic convergence for the solutions of cooperative cascade systems with length one. First, by fixing a solution of a base system converging to an equilibrium, we establish both the Nonordering of Limit Sets and the Limit Set Dichotomy for the solutions of the cascade system. Combining these tools with the idea of limiting equation, we then prove the Sequential Limit Set Trichotomy and hence the quasiconvergence in generic meaning. The generic convergent result is finally obtained by improving the Limit Set Dichotomy.     

A ring-like cascade connection and a class of NFSRs with the same cycle structures

Nonlinear feedback shift registers (NFSRs) are widely used in stream cipher designs. In this paper, we propose a variant of cascade connections of NFSRs, called ring-like cascade connections. It is shown that given an initial state of a ring-like cascade connection, each register outputs the sequence of the same period. Based on this configuration, a class of NFSRs with the same cycle structure can be derived. Moreover, inspired by this result, two more general types of NFSRs with the same cycle structures are also studied.     

Stability of flow past cylinder cascade

The global stability analysis of the two–dimensional incompressible unsteady flow past a circular clinder cascade is studied using linear stability analysis. A new numerical technique is used to find the value of critical Reynolds number that causes the flow to undergo Hopf bifurcation. An attempt has also been made to study the blockage effect on critical Reynolds number and associated Strouhal number. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence of cascade algorithms by frequency approach

Starting with an initial function ?₀, the cascade algorithm generates a sequence by cascade operator Qa defined by

     

L2-convergence of vector cascade algorithm

In this paper, the L²-convergence of vector cascade algorithm (a very important concept in the wavelet theory) is characterized in terms of the behavior of the given filter matrix H(ξ) and its associated transition operator T_H. More precisely, it is shown that the cascade algorithm defined by H(ξ) converges in L²(R^d) if and only if H(ξ) satisfies Fundamental Condition (Condition F, a restriction on its values on the vertices of [0, π]^d), and satisfies either Condition E(1 is a simple eigenvalue and the others have modulees less than one) or Condition [(E)\tilde]\tilde E . Also, we describe a class of functions which can be chosen as initial functions.     

L2-convergence of vector cascade algorithm

In this paper, the L²-convergence of vector cascade algorithm (a very important concept in the wavelet theory) is characterized in terms of the behavior of the given filter matrix H(ξ) and its associated transition operator T_H. More precisely, it is shown that the cascade algorithm defined by H(ξ) converges in L²(R^d) if and only if H(ξ) satisfies Fundamental Condition (Condition F, a restriction on its values on the vertices of [0, π]^d), and satisfies either Condition E(1 is a simple eigenvalue and the others have modulees less than one) or Condition . Also, we describe a class of functions which can be chosen as initial functions. In memory of Professor M. T. Cheng Supported by the National Natural Sciences Foundation, of China.     

The function cascade synchronization scheme for discrete-time hyperchaotic systems

In this paper, the function cascade synchronization scheme is proposed to investigate the discrete-time hyperchaotic systems. By choosing some different error functions and with the aid of symbolic–numeric computation, the proposed scheme is applied to achieve the function cascade synchronization for two discrete-time hyperchaotic systems: the generalized Hénon map and the discrete-time Rössler system, respectively. Numerical simulations are used to verify the effectiveness and feasibility of the proposed technique.     

A new approach to the cascade theory

The cascade theory of cosmic rays is investigated from a new point of view; we deal with the number of particles produced in a certain thickness of matter, the energy of each particle being greater than E at the point of its production. This new approach leads to an elegant asymptotic theory for large thicknesses and it is better adapted to the interpretation of experiments in nuclear emulsions.     

A new result on irreducible NFSRs with respect to cascade connection

Nonlinear feedback shift registers (NFSRs) are widely used in stream cipher design as building blocks. The cascade connection of NFSRs, known as an important architecture, has been adopted in Grain family of stream ciphers. In this paper, a new sufficient condition under which an NFSR cannot be decomposed into the cascade connection of two smaller NFSRs is presented, which is easy to be verified from the algebraic normal form (ANF) of the characteristic function. In fact, our results are also applicable to nonsingular Boolean functions, which actually improve a previous research of Rhodes [6] where the characteristic functions of NFSRs cannot be contained.     

Convergence of cascade algorithms introduced by I. Daubechies

Orthonormal wavelets are often constructed by an iteration procedure commonly referred to as thecascade algorithm. We obtain an asymptotic expansion and deduce necessary and sufficient conditions for convergence in Sobolev norms.     

Structural cascade decomposition of stationary linear systems that is connected with the Kalman decomposition

The structure of a cascade decomposition of stationary linear systems is considered. It is shown that each system with Hilbert spaces of states of input and output data can be represented in the form of a cascade composition of a controlled and a completely uncontrolled (or of a completely unobservable and an observable) system. Also, the connection between the various decompositions, corresponding to the same invariant subspace, is considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 250–252, February, 1990.