Simple Normal Form Research Articles

Equilibrium play and best response to (stated) beliefs in normal form games

We report experimental results on a series of ten one-shot two-person 3 × 3 normal form games with unique equilibrium in pure strategies played by non-economists. In contrast to previous experiments in which game theory predictions fail dramatically, a majority of actions taken coincided with the equilibrium prediction (70.2%) and were best-responses to subjects' stated beliefs (67.2%). In constant-sum games, 78% of actions taken were predicted by the equilibrium model, outperforming simple K-level reasoning models. We discuss how non-trivial game characteristics related to risk aversion, efficiency concerns and social preferences may affect the predictive value of different models in simple normal form games.

FURTHER REDUCTION OF NORMAL FORMS AND UNIQUE NORMAL FORMS OF SMOOTH MAPS

Further reduction for classical normal forms of smooth maps is considered in this paper. Firstly, based on the idea of computation of simplest normal forms for vector fields [Yu & Yuan, 2003], we compute the transformed map of a given smooth map under a near identity formal transformation, and give a recursive formula for the homogeneous terms of the transformed map, which is a powerful tool for further reduction of classical normal forms of smooth maps. Secondly, by using the recursive formula, the idea of [Chen & Della Dora, 1999] for further reduction of normal forms for maps and the method introduced by Kokubu et al. [1996] for further reduction of normal forms of vector fields, we develop the concepts of Nth order normal forms and infinite order normal forms of smooth maps, and give some sufficient conditions for uniqueness of normal forms of smooth maps. As an application, we show the occurrence of the flip–Neimark–Sacker bifurcation in a financial model.

The simplest parametrized normal forms of Hopf and generalized Hopf bifurcations

This paper considers the computation of the simplest parameterized normal forms (SPNF) of Hopf and generalized Hopf bifurcations. Although the notion of the simplest normal form has been studied for more than two decades, most of the efforts have been spent on the systems that do not involve perturbation parameters due to the restriction of the computational complexity. Very recently, two singularities – single zero and Hopf bifurcation – have been investigated, and the SPNFs for these two cases have been obtained. This paper extends a recently developed method for Hopf bifurcation to compute the SPNF of generalized Hopf bifurcations. The attention is focused on a codimension-2 generalized Hopf bifurcation. It is shown that the SPNF cannot be obtained by using only a near-identity transformation. Additional transformations such as time and parameter rescaling are further introduced. Moreover, an efficient recursive formula is presented for computing the SPNF. Examples are given to demonstrate the applicability of the new method.

Streamline patterns and their bifurcations near a wall with Navier slip boundary conditions

We consider the two-dimensional topology of streamlines near a surface where the Navier slip boundary condition applies. Using transformations to bring the streamfunction in a simple normal form, we obtain bifurcation diagrams of streamline patterns under variation of one or two external parameters. Topologically, these are identical with the ones previously found for no-slip surfaces. We use the theory to analyze the Stokes flow inside a circle, and show how it can be used to predict new bifurcation phenomena.

Normal forms of vector fields with perturbation parameters and their application

Abstract This paper is concerned with the normal forms of vector fields with perturbation parameters. Usually, such a normal form is obtained via two steps: first find the normal form of a “reduced” system of the original vector field without perturbation parameters, and then add an unfolding to the normal form. This way, however, it does not yield the relation (transformation) between the original system and the normal form. A study is given in this paper to consider the role of near-identity transformations in the computation of normal forms. It is shown that using only near-identity transformations cannot generate the normal form with unfolding as expected. Such normal forms, which contain many nonlinear terms involving perturbation parameters, are not very useful in bifurcation analysis. Therefore, additional transformations are needed, resulting in a further reduction of normal forms – the simplest normal form. Examples are presented to illustrate the theoretical results.

Research on Decomposition Problem of Temporal Elementary Key Normal form and Temporal Simple Normal Form in Temporal Database with Multiple Time Granularities

Research on Decomposition Problem of Temporal Elementary Key Normal form and Temporal Simple Normal Form in Temporal Database with Multiple Time Granularities

Hypernormal form theory: foundations and algorithms

Hypernormal forms (unique normal forms, simplest normal forms) are investigated both from the standpoint of foundational theory and algorithms suitable for use with computer algebra. The Baider theory of the Campbell–Hausdorff group is refined, by a study of its subgroups, to determine the smallest substages into which the hypernormalization process can be divided. This leads to a linear algebra algorithm to compute the generators needed for each substage with the least amount of work. A concrete interpretation of Jan Sanders’ spectral sequence for hypernormal forms is presented. Examples are given, and a proof is given for a little-known theorem of Belitskii expressing the hypernormal form space (in the inner product style) as the kernel of a higher-order differential operator.

Monadic datalog and the expressive power of languages for Web information extraction

Research on information extraction from Web pages (wrapping) has seen much activity recently (particularly systems implementations), but little work has been done on formally studying the expressiveness of the formalisms proposed or on the theoretical foundations of wrapping. In this paper, we first study monadic datalog over trees as a wrapping language. We show that this simple language is equivalent to monadic second order logic (MSO) in its ability to specify wrappers. We believe that MSO has the right expressiveness required for Web information extraction and propose MSO as a yardstick for evaluating and comparing wrappers. Along the way, several other results on the complexity of query evaluation and query containment for monadic datalog over trees are established, and a simple normal form for this language is presented. Using the above results, we subsequently study the kernel fragment Elog − of the Elog wrapping language used in the Lixto system (a visual wrapper generator). Curiously, Elog − exactly captures MSO, yet is easier to use. Indeed, programs in this language can be entirely visually specified.

The simplest normal form of Hopf bifurcation

Recently, further reduction on normal forms of differential equations leading to the simplest normal forms (SNFs) has received considerable attention. However, the computation of the SNF has been mainly restricted to systems which do not contain perturbation parameters (unfolding), since the computation of the SNF with unfolding is much more complicated than that of the SNF without unfolding. From the practical point of view, only the SNF with perturbation (bifurcation) parameters is useful in analysing physical or engineering problems. It is shown that the SNF with unfolding cannot be obtained using only near-identity transformation. Additional transformations such as time and parameter rescaling need to be introduced. An efficient computational method is presented for computing the algebraic equations that can be used to find the SNF. A physical example is given to show the applicability of the new method.

Computation of the simplest normal forms with perturbation parameters based on Lie transform and rescaling

Normal form theory is one of the most power tools for the study of nonlinear differential equations, in particular, for stability and bifurcation analysis. Recently, many researchers have paid attention to further reduction of conventional normal forms (CNF) to so called the simplest normal form (SNF). However, the computation of normal forms has been restricted to systems which do not contain perturbation parameters (unfolding). The computation of the SNF is more involved than that of CNFs, and the computation of the SNF with unfolding is even more complicated than the SNF without unfolding. Although some author mentioned further reduction of the SNF, no results have been reported on the exact computation of the SNF of systems with perturbation parameters. This paper presents an efficient method for computing the SNF of differential equations with perturbation parameters. Unlike CNF theory which uses an independent nonlinear transformation at each order, this approach uses a consistent nonlinear transformation through all order computations. The particular advantage of the method is able to provide an efficient recursive formula which can be used to obtain the nth-order equations containing the nth-order terms only. This greatly saves computational time and computer memory. The recursive formulations have been implemented on computer systems using Maple. As an illustrative example, the SNF for single zero singularity is considered using the new approach.

Polynomial normal forms for vector fields on ℝ3

The present paper is devoted to studying a class of smoothly (C∞) finitely determined vector fields on ℝ3. Given any such generic local system of the form $\dot{x}$=Ax+⋯, where A is a 3×3 matrix, we find the minimal possible number i(A) such that the vector field is i(A)-jet determined, and we find the number μ(A) of moduli in the C∞ classification. We also give a list of the simplest normal forms, that is, polynomials of degree i(A) containing exactly μ(A) parameters.

Algorithmic Reduction of Poincaré–Dulac Normal Forms and Lie Algebraic Structure

The Poincare–Dulac normal form of a given resonant system is in general nonunique; given a specific normal form, one would like to further reduce it to a simplest normal form. In this Letter we give an algorithm, based on the Lie algebraic structure of the set of normal forms, to obtain this. The algorithm can be applied under some condition, nongeneric but often met in applications. When applicable, it is only necessary to solve linear equations, and is more powerful than the one proposed in previous work by the same author [Lett. Math. Phys. 42 (1999), 103–114; and Ann. Inst. H. PoincarePhys. Theor. 70 (1999), 461–514].

Isochronicity via normal form

We analyze the isochronicity of centres, by means of a procedure to obtain hypernormal forms (simplest normal forms) for the Hopf bifurcation, that uses the theory of transformations based on the Lie transforms. We establish the relation between the period constants and the normal form coefficients, and prove that an equilibrium point is an isochronous centre if and only if a property of commutation holds. Also, we give necessary and sufficient conditions, expressed in terms of the Lie product, to determine if an equilibrium point is a centre. Several examples are also included in order to show the usefulness of the method. In particular, the isochronicity of the origin for the Lienard equation is analyzed in some cases.

Classification of 2-dimensional contracting rigid germs and Kato surfaces: I

A holomorphic germ f: ( C n,0)⥀ is rigid if the increasing union of the critical set of all its iterate is a divisor with normal crossings. The aim of this article is to classify completely contracting rigid germs of ( C 2,0). There are seven different main classes of them. For each of them, we give simple (polynomial) normal forms, and we study their main features, their unicity and their group of automorphisms. We give an application to the study of the geometry of the basin of attraction of the superattracting point of a Hénon mapping: this reproves old results of Hubbard and Obersta-Vorth and precise them.

Spinal-Formed Context-Free Tree Grammars

In this paper we introduce a restricted model of context-free tree grammars called spine grammars, and study their formal properties including considerably simple normal forms. Recent research on natural languages has suggested that formalisms for natural languages need to generate a slightly larger class of languages than context-free grammars, and for that reason tree adjoining grammars have been widely studied relating them to natural languages. It is shown that the class of string languages generated by spine grammars coincides with that of tree adjoining grammars. We also introduce acceptors called linear pushdown tree automata, and show that linear pushdown tree automata accept exactly the class of tree languages generated by spine grammars. Linear pushdown tree automata are obtained from pushdown tree automata with a restriction on duplicability for the pushdown stacks.

SIMPLEST NORMAL FORMS OF HOPF AND GENERALIZED HOPF BIFURCATIONS

The normal forms of Hopf and generalized Hopf bifurcations have been extensively studied, and obtained using the method of normal form theory and many other different approaches. It is well known that if the normal forms of Hopf and generalized Hopf bifurcations are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal form. In this paper, three theorems are presented to show that the conventional normal forms of Hopf and generalized Hopf bifurcations can be further simplified. The forms obtained in this paper for Hopf and generalized Hopf bifurcations are shown indeed to be the "simplest", and at most only two terms remain in the amplitude equation of the "simplest normal form" up to any order. An example is given to illustrate the applicability of the theory. A computer algebra system using Maple is used to derive all the formulas and verify the results presented in this paper.

Input-output decoupling of linear time-varying singular systems

The problem of input-output decoupling by state feedback is considered for linear time-varying singular systems. Assumptions are introduced, and an algorithm is developed such that the system can be expressed in a simple normal form. A time-varying feedback law is then constructed which ensures, under appropriate conditions, that the closed-loop system is regular, impulse-free, and noninteractive.

Hypernormal Form for the Hopf-Zero Bifurcation

In this paper we present hypernormal forms up to an arbitrary order for equilibria of tridimensional systems having a linear degneracy corresponding to a pair of pure imaginary eigenvalues and a third one zero. These simplest normal forms are obtained assuming some generic conditions on the quadratic terms, and using C∞-conjugacy as well as C∞-equivalence. Also, the case of ℤ2-symmetric systems is considered. In this situation, the hypernormal forms are characterized under generic conditions on the cubic terms. In all the cases, we provide recursive algorithms that compute explicitly the hypernormal form coefficients, in terms of the normal form coefficients.

Periodic orbit theory for resonant tunneling diodes: Comparison with quantum and experimental results

We investigate whether the quantal and experimental amplitudes of current oscillations of resonant tunneling diodes in tilted fields are obtainable from periodic orbit (PO) theories by considering recently proposed PO approaches. We show that accurate amplitude and frequency shifts for the current oscillations (typically to within a few %) can be obtained from a simple analytical formula both in the stable (torus-quantization) limit and the unstable regimes of the experiments that are dominated by isolated PO's. But we find that the PO approach does not describe quantitatively the dynamically interesting intermediate experimental regimes that appear to be dominated by contributions from complex orbits and multiple nonisolated PO's. We conclude that these regimes will not easily be described by the usual PO approach, even with simple normal forms.

Strong equilibria in claim games corresponding to convex games

This paper deals with a specific aspect of the problem of coalition formation in a situation described by a TU-game. First, we define a very simple normal form game which models the process of coalition formation. To define the payoff functions of the players we use an allocation rule for TU-games. The main objective of this paper is ascertain what conditions of the allocation rule lead to the grand coalition being a strong equilibrium of the normal form game, when the original TU-game is convex.