Unique Normal Forms Research Articles

Generic 2-parameter perturbations of parabolic singular points of vector fields in ℂ

We describe the equivalence classes of germs of generic 2 2 -parameter families of complex vector fields z ˙ = ω ϵ ( z ) \dot z = \omega _\epsilon (z) on C \mathbb {C} unfolding a singular parabolic point of multiplicity k + 1 k+1 : ω 0 = z k + 1 + o ( z k + 1 ) \omega _0= z^{k+1} +o(z^{k+1}) . The equivalence is under conjugacy by holomorphic change of coordinate and parameter. As a preparatory step, we present the bifurcation diagram of the family of vector fields z ˙ = z k + 1 + ϵ 1 z + ϵ 0 \dot z = z^{k+1}+\epsilon _1z+\epsilon _0 over C P 1 \mathbb {C}\mathbb {P}^1 . This presentation is done using the new tools of periodgon and star domain. We then provide a description of the modulus space and (almost) unique normal forms for the equivalence classes of germs.

Effective codescent morphisms in the varieties determined by convergent term rewriting systems

It is shown that the elements of amalgamated free products in a variety of universal algebras have unique normal forms if the variety is represented by a confluent term rewriting system satisfying some additional requirements for its signature and rules. Applying this fact it is proved that any codescent morphism is effective in such varieties. In particular, this is the case for the variety of Mal'tsev algebras, the varieties of magmas with unit and two-sided inverses, idempotent quasigroups, unipotent quasigroups, left Steiner loops, and right Steiner loops.

Clocked lambda calculus

One of the best-known methods for discriminating λ-terms with respect to β-convertibility is due to Corrado Böhm. The idea is to compute the infinitary normal form of a λ-term M, the Böhm Tree (BT) of M. If λ-terms M, N have distinct BTs, then M ≠βN, that is, M and N are not β-convertible. But what if their BTs coincide? For example, all fixed point combinators (FPCs) have the same BT, namely λx.x(x(x(. . .))).We introduce a clocked λ-calculus, an extension of the classical λ-calculus with a unary symbol τ used to witness the β-steps needed in the normalization to the BT. This extension is infinitary strongly normalizing, infinitary confluent and the unique infinitary normal forms constitute enriched BTs, which we call clocked BTs. These are suitable for discriminating a rich class of λ-terms having the same BTs, including the well-known sequence of Böhm's FPCs.We further increase the discrimination power in two directions. First, by a refinement of the calculus: the atomic clocked λ-calculus, where we employ symbols τp that also witness the (relative) positions p of the β-steps. Second, by employing a localized version of the (atomic) clocked BTs that has even more discriminating power.

Unique normal forms near a degenerate elliptic fixed point in two-parametric families of area-preserving maps

We derive simplified normal forms for an area-preserving map in a neighbourhood of a degenerate resonant elliptic fixed point. Such fixed points appear in generic families of area-preserving maps. We also derive a simplified normal form for a generic two-parametric family. The normal forms are used to analyse bifurcations of n-periodic orbits. In particular, for n ⩾ 6 we find regions of parameters where the normal form has ‘meandering’ invariant curves.

Generic Complexity of Finitely Presented Monoids and Semigroups

We study the generic properties of finitely presented monoids and semigroups, and the generic-case complexity of decision problems for them. We show that for a finite alphabet A of size at least 2 and positive integers k and m, the generic A-generated k-relation monoid and semigroup (defined using any of several stratifications) satisfies the small overlap condition C(m). It follows that the generic finitely presented monoid has a number of interesting semigroup-theoretic properties and, by a recent result of the author, admits a linear time solution to its word problem and a regular language of unique normal forms for its elements. Moreover, the uniform word problem for finitely presented monoids is generically solvable in polynomial time.

Unique normal forms for area preserving maps near a fixed point with neutral multipliers

We study normal forms for families of area-preserving maps which have a fixed point with neutral multipliers -1 or +1 at epsilon=0. Our study covers both the orientation-preserving and orientation-reversing cases. In these cases Birkhoff normal forms do not provide a substantial simplification of the system. In the paper we prove that the Takens normal form vector field can be substantially simplified. We also show that if certain non-degeneracy conditions are satisfied no further simplification is generically possible since the constructed normal forms are unique. In particular, we provide a full system of formal invariants with respect to formal coordinate changes.

Filtration-preserving mappings and centralizers within graded Lie algebras

We use spectral sequence techniques to compute centralizers of elements within graded Lie algebras, and the methods are then applied to the calculation of unique normal forms of elements within one-parameter matrix Lie algebras. A finiteness criterion for unique normal forms is presented.

GRÖBNER–SHIRSHOV BASES FOR FREE INVERSE SEMIGROUPS

A new construction for free inverse semigroups was obtained by Poliakova and Schein in 2005. Based on their result, we find Gröbner–Shirshov bases for free inverse semigroups with respect to the deg-lex order of words. In particular, we give the (unique and shortest) normal forms in the classes of equivalent words of a free inverse semigroup together with the Gröbner–Shirshov algorithm to transform any word to its normal form.

Combinatorial and metric properties of Thompson’s group 𝑇

We discuss metric and combinatorial properties of Thompson’s group T T , including normal forms for elements and unique tree pair diagram representatives. We relate these properties to those of Thompson’s group F F when possible, and highlight combinatorial differences between the two groups. We define a set of unique normal forms for elements of T T arising from minimal factorizations of elements into natural pieces. We show that the number of carets in a reduced representative of an element of T T estimates the word length, that F F is undistorted in T T , and we describe how to recognize torsion elements in T T .

Normal Form in Filtered Lie Algebra Representations

This paper gives a setup for normal form theory and the computation of normal forms with emphasis on the dual character of the transformation generators and the objects to be transformed into normal form. Spectral sequence techniques will be used to define unique normal forms. Theoretical estimates are given for the rate of convergence in terms of the spectral sequence.

Unique Orbital Normal Form for Vector Fields of Hopf-Zero Singularity

Normal forms for vector fields of Hopf-zero singularity in R3 are studied. Multiple Lie bracket method is used to give unique normal forms under both conjugacy and orbital equivalence for such vector fields with a generic quadratic part.

A SUFFICIENT CONDITION FOR THE UNIQUENESS OF NORMAL FORMS AND UNIQUE NORMAL FORMS OF GENERALIZED HOPF SINGULARITIES

A sufficient condition for the uniqueness of the Nth order normal form is provided. A new grading function is proposed and used to prove the uniqueness of the first-order normal forms of generalized Hopf singularities. Recursive formulas for computation of coefficients of unique normal forms of generalized Hopf singularities are also presented.

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.

Unique normal forms for Hopf-zero vector fields

We consider normal forms of Hopf-zero vector fields in R 3 . Unique normal forms under conjugacy and orbital equivalence for the generic case are given. To cite this article: G. Chen et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Remarks concerning finitely generated semigroups having regular sets of unique normal forms

AbstractProperties such as automaticity, growth and decidability are investigated for the class of finitely generated semigroups which have regular sets of unique normal forms. Knowledge obtained is then applied to the task of demonstrating that a class of semigroups derived from free inverse semigroups under certain closure operations is not automatic.

Objects and their lambda calculus

A kind of parallel typed lambda calculus is presented based on the language and structure of objects. The term “object” is used here in a sense different from that related to the expression “object-oriented language (programming)”. By “objects” here we mean any class of entities which (a) are resource dependent and (b) combine to each other (via some fitness relation) to form more complex ones. Two operators, λ and its dual λ ̄ , are used, and two operations, a binary one, ⊙, for juxtaposition, and an n-ary one, |, for every n, for branching. The construct λv.x represents, roughly, a receiving scheme producing copies of x when fed with proper objects y to fill the empty place v of x, while the dual construct λ ̄ y.z represents a sending scheme that throws y out z in proper surroundings. The interaction of these two constructs takes place when they are matched together by ⊙ and yields an exchange of resources in a way that preserves the total amount of them. The calculus captures such notions as concurrency, interaction and branching in a way analogous to that of (Berrey and Boudol, Theoret. Comput. Sci. 96 (1992) 217–248; Boudol, Lecture Notes in Computer Science, Vol. 351, Springer, Berlin, pp. 149–161) [3,4], but with a quite different meaning of the operations. What is described here is “situations” of coexisting entities rather than computations, and resource-preserving transformations between them. The terms are shown to have unique normal forms. Object structures that model the simple theory of objects are extended here in suitable graph structures that provide sound and complete semantics of the calculus.

Unique normal forms for the Takens–Bogdanov singularity in a special case

We consider further reduction of normal forms for nilpotent planar vector fields. We give a unique normal form for a special case of an open problem for the Takens–Bogdanov singularity.

Normal forms for differentiable maps near a fixed point

We propose in this paper a significant refinement of normal forms for differentiable maps near a fixed point. We give a method to obtain further reduction of classical normal forms with concrete and interesting applications. Our method leads to unique normal forms either with respect to general diffeomorphisms in certain cases or with respect to near identity diffeomorphisms in other cases. Our approach is rational in the sense that if the coefficients of a map are in a field K, so is its normal form.

Unique normal forms: The nilpotent Hamiltonian case

The problem we consider in this paper, namely the characterization of unique normal forms, derives from bifurcation theory. In a typical situation, one has an unperturbed vector field near a degenerate equilibrium. In order to study the effect of generic perturbations, suitable coordinates are introduced and the simplified system is said to be in normal form. A well known case is the Hopf bifurcation where a pair of conjugate eigenvalues cross the imaginary axis. Using averaging, one can in this case compute the periodic orbit that splits off from the equilib~um, A second familiar example is the Takens-Bogdanov bifurcation, where two eigenvalues are zero. In all these situations it is necessary to make certain choices, both in the definition of what constitutes a “normal form” and in the computation of the transformation. The idea here is to use this second kind of choice to sharpen the very definition of normal form. While in standard treatments only the linear part of the field plays a role, we show that one can effectively use its non-linear part to remove additional terms, and thus simplify the problem further. The underlying philosophy in this type of game is that

Completion for rewriting modulo a congruence

Completion modulo a congruence is a method for constructing a presentation of an equational theory as a rewrite system that defines unique normal forms with respect to the congruence. We formulate this completion method as an equational inference system and present techniques for proving the correctness of procedures based on the inference system. Our correctness results cover generalized and improved versions of the Peterson-Stickel and the Jouannaud-Kirchner procedure.