Simple Normal Form Research Articles

Searching for nonlinear feedback shift registers with parallel computing

Nonlinear feedback shift registers (NLFSRs) are used to construct pseudorandom generators for stream ciphers. Their theory is not so complete as that of linear feedback shift registers (LFSRs). In general, it is not known how to construct all NLFSRs with maximum period. The direct method is to search for such registers with suitable properties. Advanced technology of parallel computing has been applied both in software and hardware to search for maximum period NLFSRs having a fairly simple algebraic normal form.

Static analysis and optimization of semantic web queries

Static analysis is a fundamental task in query optimization. In this article we study static analysis and optimization techniques for SPARQL, which is the standard language for querying Semantic Web data. Of particular interest for us is the optionality feature in SPARQL. It is crucial in Semantic Web data management, where data sources are inherently incomplete and the user is usually interested in partial answers to queries. This feature is one of the most complicated constructors in SPARQL and also the one that makes this language depart from classical query languages such as relational conjunctive queries. We focus on the class of well-designed SPARQL queries, which has been proposed in the literature as a fragment of the language with good properties regarding query evaluation. We first propose a tree representation for SPARQL queries, called pattern trees, which captures the class of well-designed SPARQL graph patterns. Among other results, we propose several rules that can be used to transform pattern trees into a simple normal form, and study equivalence and containment. We also study the evaluation and enumeration problems for this class of queries.

Hypernormal Form at Cubic of Honeycomb Sandwich Plate Dynamics Model

Honeycomb sandwich plate have been widly applied in industry design in recent years. In this paper, we study the cubic hypernormal form (the simplest normal form and the unique normal form) for honeycomb sandwich plate dynamics model with the help of Maple symbolic computation. Firstly, we get the average equation of four dimensional cartesian form by using the method of multiple scales perturbation analysis. Based on the method combined new grading function with multiple Lie brackets, we obtain the hypernormal form of cubic truncated. The results will further enrich the research for dynamics of honeycomb sandwich plate model, and is also the basis for higher order normal form research.

Improved Adjoint Operator Method and Normal Form of Nonlinear Dynamical System

An improved adjoint operator based on the adjoint operator concept of linear operator and S-N decomposition is proposed to calculate the normal forms of k order general nonlinear dynamic systems.Firstly, the whole polynomial solution space of homogeneous nilpotent partial differential equation are obtained.Secondly, the polynomial solution mentioned above is introduced into homogeneous semi-simple partial differential equation to find the whole polynomial solution space of a homogeneous linear partial differential equation Therefore, more polynomial first integrals need not be found and the simplest normal form of nonlinear dynamical system can be obtained easily. The example shows that the method is very effective.

Delayed feedback control of unstable steady states with high-frequency modulation of the delay

We analyze the stabilization of unstable steady states by delayed feedback control with a periodic time-varying delay in the regime of a high-frequency modulation of the delay. The average effect of the delayed feedback term in the control force is equivalent to a distributed delay in the interval of the modulation, and the obtained distribution depends on the type of the modulation. In our analysis we use a simple generic normal form of an unstable focus, and investigate the effects of phase-dependent coupling and the influence of the control loop latency on the controllability. In addition, we have explored the influence of the modulation of the delays in multiple delay feedback schemes consisting of two independent delay lines of Pyragas type. A main advantage of the variable delay is the considerably larger domain of stabilization in parameter space.

Implementation of Boolean functions with a bounded number of zeros by disjunctive normal forms

The problem of constructing simple disjunctive normal forms (DNFs) of Boolean functions with a small number of zeros is considered. The problem is of interest in the complexity analysis of Boolean functions and in its applications to data analysis. The method used is a further development of the reduction approach to the construction of DNFs of Boolean functions. A key idea of the reduction method is that a Boolean function is represented as a disjunction of Boolean functions with fewer zeros. In a number of practically important cases, this technique makes it possible to considerably reduce the complexity of DNF implementations of Boolean functions.

Constructing the Second Order Poincaré Map Based on the Hopf-Zero Unfolding Method

We investigate the Shilnikov sense homoclinicity in a 3D system and consider the dynamical behaviors in vicinity of the principal homoclinic orbit emerging from a third order simplified system. It depends on the application of the simplest normal form theory and further evolution of the Hopf-zero singularity unfolding. For the Shilnikov sense homoclinic orbit, the complex form analytic expression is accomplished by using the power series of the manifolds surrounding the saddle-focus equilibrium. Then, the second order Poincaré map in a generally analytical style helps to portrait the double pulse dynamics existing in the tubular neighborhood of the principal homoclinic orbit.

Dynamics of one-resonant biholomorphisms

Our first main result is a construction of a simple formal normal form for holomorphic diffeomorphisms in \mathbb C^n whose differentials have one-dimensional family of resonances in the first m eigenvalues, m\leq n (but more resonances are allowed for other eigenvalues). Next, we provide invariants and give conditions for the existence of basins of attraction. Finally, we give applications and examples demonstrating the sharpness of our conditions.

Normal forms for Hopf-Zero singularities with nonconservative nonlinear part

In this paper we are concerned with the simplest normal form computation of the systems(0.1)x˙=2xf(x,y2+z2),y˙=z+yf(x,y2+z2),z˙=−y+zf(x,y2+z2), where f is a formal function with real coefficients and without any constant term. These are the classical normal forms of a larger family of systems with Hopf-Zero singularity. Indeed, these are defined such that this family would be a Lie subalgebra for the space of all classical normal form vector fields with Hopf-Zero singularity. The simplest normal forms and simplest orbital normal forms of this family with nonzero quadratic part are computed. We also obtain the simplest parametric normal form of any non-degenerate perturbation of this family within the Lie subalgebra. The symmetry group of the simplest normal forms is also discussed. This is a part of our results in decomposing the normal forms of Hopf-Zero singular systems into systems with a first integral and nonconservative systems.

Hypernormal Form at Cubic of a Rectangular Symmetric Cross-Ply Laminated Composite Plate

Based on wings flutter on flying aircraft in this paper, the authors study the mechanical model of the rectangular symmetric cross-ply composite laminated plates. Frist, the method of multiple scales is employed to obtain the four-dimensional averaged equations of the model. Then, the method of new grading function and multiple Lie brackets is utilized to obtain the hypernormal form (simplest normal form, unique normal form) at cubic of above averaged equations.

Simple disjunctive normal forms of Boolean functions with a restricted number of zeros

This paper studies the complexity of disjunctive normal forms (DNFs) of Boolean functions defined by their matrices of zeros. The best available bounds are obtained for the minimum number of conjunc� tions making up a DNF of a Boolean function with a restricted number of zeros. The results can be used to implement characteristic functions of classes in pat� tern recognition with binary data [1]. 1. The standard method for constructing DNFs of characteristic classes of functions can be described as follows. Let , , …, and , , …, be given binary descriptions of reference objects of classes K1 and K2, respectively, where , i ∈ {0, 1}, j ∈ {1, 2, …, ki}, is a binary vector of dimension n.

Internet Service Pricing: Flat or Volume?

In this paper, we study the pricing of Internet services under monopoly and duopoly environments using an analytic model in which a service provider and users try to maximize their respective payoffs. We compare a few popular pricing schemes, including flat, volume-based, two-part, and nonlinear tariffs, with respect to revenue, social welfare, and user surplus. We perform a study of the sensitivity of these schemes to the estimation errors. In the duopoly situation, we formulate a simple normal form game between two service providers and study their equilibrium behaviors. Our main findings include: (1) the flat pricing generates higher revenue than the pure volume pricing when the elasticity of demand is low; (2) the volume-based pricing is better for society and users than the flat pricing regardless of the elasticity; (3) the market is segmented into two when one provider provides flat pricing and another provides volume based pricing.

Global dynamics of two coupled parametrically excited van der Pol oscillators

Using a combination of analytical and numerical methods, the global bifurcations and chaotic dynamics of two non-linearly coupled parametrically excited van der Pol oscillators are investigated in detail. With the aid of the method of multiple scales, the slow flow equations are obtained. Based on the slow flow equations, normal form theory and the techniques of choosing complementary space are applied to find the explicit expressions of the simpler normal form associated with a double zero and a pair of pure imaginary eigenvalues. By the simpler normal form, using the global perturbation method developed by Kovacic and Wiggins, the analysis of global bifurcation and chaotic dynamics of two non-linearly coupled parametrically excited van der Pol oscillators is given. The results indicate that there exists a Silnikov type single-pulse homoclinic orbit for this class of system which implies the chaotic motions can occur. Numerical simulations are also given and verify the analytical predictions.

Miniversal deformations of matrices of bilinear forms

Arnold [V.I. Arnold, On matrices depending on parameters, Russian Math. Surveys 26 (2) (1971) 29–43] constructed miniversal deformations of square complex matrices under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We construct miniversal deformations of matrices under congruence.

Bifurcations of multiple homoclinics in general dynamical systems

By using the local active coordinates consisting of tangent vectors of the invariant subspaces, as well as the Silnikov coordinates, the simple normal form is established in the neighborhood of the double homoclinic loops with bellows configuration in a general system, then the dynamics near the homoclinic bellows is investigated, and the existence, uniqueness of the homoclinic orbits and periodic orbits with various patterns bifurcated from the primary orbits are demonstrated, and the corresponding bifurcation curves (or surfaces) and existence regions are located.

RULES FOR SUBATOMIC DERIVATION

In proof-theoretic semantics the meaning of an atomic sentence is usually determined by a set of derivations in an atomic system which contain that sentence as a conclusion (see, in particular, Prawitz, 1971, 1973). The paper critically discusses this standard approach and suggests an alternative account which proceeds in terms of subatomic introduction and elimination rules for atomic sentences. A simple subatomic normal form theorem by which this account of the semantics of atomic sentences and the terms from which they are composed is underpinned, shows moreover that the proof-theoretic analysis of first-order logic can be pursued also beneath the atomic level.

Simplest normal forms of generalized Neimark-Sacker bifurcation

The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal forms. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcation can be further simplified. The simplest normal forms of generalized Neimark-Sacker bifurcation are calculated. Based on the conventional normal form, using appropriate nonlinear transformations, it is found that the generalized Neimark-Sacker bifurcation has at most two nonlinear terms remaining in the amplitude equations of the simplest normal forms up to any order. There are two kinds of simplest normal forms. Their algebraic expression formulas of the simplest normal forms in terms of the coefficients of the generalized Neimark-Sacker bifurcation systems are given.

SIMPLEST NORMAL FORMS FOR PLANAR SYSTEMS ON EQUILIBRIUM MANIFOLDS

Equilibrium manifold is a manifold that consists of equilibrium points. Planar systems with one-dimensional equilibrium manifolds are considered in this paper. First, for such planar systems, a unified equation with the horizontal axis as the equilibrium curve is formulated. Then, according to the corresponding linearized systems, different cases are discussed: For the nondegenerate case, the simplest normal form of a system with simplified Bogdanov–Takens singularities is obtained; for the general first-order degenerative case, the simplest normal forms are completely characterized; finally, for the general higher-order degenerative case, deduction of the simplest normal form is illustrated.

On the hyperbolicity of 3D plasticity equations in isostatic coordinates

We consider the problem of classifying partial differential equations of the three-dimensional problem of ideal plasticity (for stress states corresponding to an edge of the Tresca prism) and the problem of finding a change of independent variables reducing these equations to the simplest normal Cauchy form. The original system of equations is represented in an isostatic coordinate system and is substantially nonlinear. We state a criterion for the simplest normal Cauchy form and find a coordinate system reducing the original system to the simplest normal Cauchy form. We show that the condition obtained in the present paper for a system to take the simplest normal form is stronger than the Petrovskii t-hyperbolicity condition if t is understood as the canonical isostatic coordinate whose level surfaces in space form fibers normal to the principal direction field corresponding to the maximum (minimum) principal stress.

Computation of the simplest normal form of a resonant double Hopf bifurcation system with the complex normal form method

The complex normal form method is presented to find the simplest normal form (SNF) of an odd ratio double Hopf bifurcation system. New nonlinear transformations and rules of getting the form of the SNF coefficients are presented to find an efficient approach to get the final explicit expressions of the SNF. During the course of obtaining the key equations, based on the algorithm of complex operations, all the matrix deduction is substituted by complex operation and no former matrix operation is involved compared with the matrix representation method. Explicit iterative formulas have been derived to determine the coefficients of the SNF and associated nonlinear transformations up to an arbitrary order. Meanwhile, a rule of analyzing the general expression of the SNF is also summarized, which enables to predict the components of the final explicit equations before any substantive analyses. As the result of these innovations is deduced, the computation process is basically simplified. An ordinary differential equation (ODE) system shows the feasibility and convenience of the new method.