The nondegenerate center problem and the inverse integrating factor

In this paper we study some aspects of the nondegenerate center problem for analytic and, in particular, for polynomial vector fields. The relation between the existence of an inverse integrating factor and the center problem is studied. The relationship between the conditions for a center using the Poincaré formal series and the inverse integrating factor formal series for systems with a linear center perturbed by homogeneous polynomials is proved.     

On formal Laurent series

Several kinds of formal Laurent series have been introduced with some restrictions so far. This paper systematically sets up a natural definition and structure of formal Laurent series without those restrictions, including introducing a multiplication between formal Laurent series. This paper also provides some results on the algebraic structure of the space of formal Laurent series, denoted by \mathbbL\mathbb{L}. By means of the results of the generalized composition of formal power series, we define a composition of a Laurent series with a formal power series and provide a necessary and sufficient condition for the existence of such compositions. The calculus about formal Laurent series is also introduced.     

Associative formal power series in two indeterminates

Investigating the associativity equation for formal power series in two variables we show that the transcendental associative formal power series are of order one or two and that they can be represented by an invertible formal power series in one variable. We also discuss the convergence of associative formal power series.     

On Polynomials Over Rickart Rings and Their Generalizations I

This paper continues the investigation of polynomials and formal power series over a ring with various annihilator conditions which were originally used by Rickart and Kaplansky to abstract the algebraic properties of von Neumann algebras. Results of Armendariz on polynomial rings over a PP ring are extended to analogous annihilator conditions in nearrings of polynomials and nearrings of formal power series. These results are somewhat striking since, in contrast to the polynomial ring case, the nearring of polynomials or formal power series has substitution for its “multiplication” operation. These investigations provide an alternative viewpoint in illustrating the structure of polynomials and formal power series. Extensions of Rickart rings to formal power series rings are also discussed. The author was partially supported by the National Science Council, Taiwan under the grant number NSC 93-2115-M-143-001.     

Lifting endomorphisms of formal A-modules over finite fields

Lubin conjectures that for an invertible series to commute with a noninvertible series with only simple roots of iterates, two such commuting power series must be endomorphisms of a single formal group. In this paper, we show that if the reduction of these two commuting power series are endomorphisms of a formal group, then themselves are endomorphisms of a formal group.     

CYCLIC CODES OVER FORMAL POWER SERIES RINGS

In this article, cyclic codes and negacyclic codes over formal power series rings are studied. The structure of cyclic codes over this class of rings is given, and the relationship between these codes and cyclic codes over finite chain rings is obtained. Using an isomorphism between cyclic and negacyclic codes over formal power series rings, the structure of negacyclic codes over the formal power series rings is obtained.     

Iterations and logarithms of formal automorphisms

Using the decomposition of an automorphism of the ring of formal power series in several variables, in a semisimple and a unipotent automorphism, I prove in this paper that an automorphism allows a continuous iteration if and only if it is the exponential of a derivation. This result implies a number of results recently obtained by Reich, Schwaiger, and Bucher.     

Formal power series with cyclically ordered exponents

We define and study a notion of ring of formal power series with exponents in a cyclically ordered group. Such a ring is a quotient of various subrings of classical formal power series rings. It carries a two variable valuation function. In the particular case where the cyclically ordered group is actually totally ordered, our notion of formal power series is equivalent to the classical one in a language enriched with a predicate interpreted by the set of all monomials.Received: 24 February 2003     

SELF-DUAL PERMUTATION CODES OVER FORMAL POWER SERIES RINGS AND FINITE PRINCIPAL IDEAL RINGS

In this paper, we study self-dual permutation codes over formal power series rings and finite principal ideal rings. We first give some results on the torsion codes associated with the linear codes over formal power series rings. These results allow for obtaining some conditions for non-existence of self-dual permutation codes over formal power series rings. Finally, we describe self-dual permutation codes over finite principal ideal rings by examining permutation codes over their component chain rings.     

Non-commutative Hopf algebra of formal diffeomorphisms

This paper deals with two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second is the set of formal diffeomorphisms with the group law being a composition of series. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with non-commutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual non-commutative algebra there exists a natural co-associative co-product defining a non-commutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a non-commutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semi-direct co-product of the previous Hopf algebras, and to series in several variables. Finally, we show how the non-commutative Hopf algebras of formal series are related to some renormalization Hopf algebras, which are combinatorial Hopf algebras motivated by the renormalization procedure in quantum field theory, and to the renormalization functor given by the double-tensor algebra on a bi-algebra.     

On universal formal power series

The point source of this work is Seleznev's theorem which asserts the existence of a power series which satisfies universal approximation properties in C^∗. The paper deals with a strengthened version of this result. We establish a double approximation theorem on formal power series using a weighted backward shift operator. Moreover we give strong conditions that guarantee the existence of common universal series of an uncountable family of weighted backward shift with respect to the simultaneous approximation. Finally we obtain results on admissible growth of universal formal power series. We especially prove that you cannot control the defect of analyticity of such a series even if there exist universal series in the well-known intersection of formal Gevrey classes.     

Local structure theorems for smooth maps of formal schemes

We continue our study on infinitesimal lifting properties of maps between locally noetherian formal schemes started in [L. Alonso Tarrío, A. Jeremías López, M. Pérez Rodríguez, Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes, Comm. Alg. 35 (2007) 1341-1367]. In this paper, we focus on some properties which arise specifically in the formal context. In this vein, we make a detailed study of the relationship between the infinitesimal lifting properties of a morphism of formal schemes and those of the corresponding maps of usual schemes associated to the directed systems that define the corresponding formal schemes. Among our main results, we obtain the characterization of completion morphisms as pseudo-closed immersions that are flat. Also, the local structure of smooth and étale morphisms between locally noetherian formal schemes is described: the former factors locally as a completion morphism followed by a smooth adic morphism and the latter as a completion morphism followed by an étale adic morphism.     

Gevrey character of formal solutions of differential—difference equations with polynomials coefficients

In this paper we prove that formal factorial series as well as formal power series in 1/x solutions of a differential—difference equation with polynomial coefficients are Gevrey of some order which can be determined from a suitable Newton polygon     

Formal power series, operator calculus, and duality on Lie algebras

This paper presents an operator calculus approach to computing with non-commutative variables. First, we recall the product formulation of formal exponential series. Then we show how to formulate canonical boson calculus on formal series. This calculus is used to represent the action of a Lie algebra on its universal enveloping algebra. As applications, Hamilton's equations for a general Hamiltonian, given as a formal series, are found using a double-dual representation, and a formulation of the exponential of the adjoint representation is given. With these techniques one can represent the Volterra product acting on the enveloping algebra. We illustrate with a three-step nilpotent Lie algebra.     

Convergence and formal manipulation in the theory of series from 1730 to 1815

In the early calculus mathematicians used convergent series to represent geometrical quantities and solve geometrical problems. However, series were also manipulated formally using procedures that were the infinitary extension of finite procedures. By the 1720s results were being published that could not be reduced to the original conceptions of convergence and geometrical representation. This situation led Euler to develop explicitly a more formal approach which generalized the early theory. Formal analysis, which was predominant during the second half of the 18th century despite criticisms of it by some researchers, contributed to the enlargement of mathematics and even led to a new branch of analysis: the calculus of operations. However, formal methods could not give an adequate treatment of trigonometric series and series that were not the expansions of elementary functions. The need to use trigonometric series and introduce nonelementary functions led Fourier and Gauss to reject the formal concept of series and adopt a different, purely quantitative notion of series.     

A representation of solution of stochastic differential equations

We prove that the logarithm of the formal power series, obtained from a stochastic differential equation, is an element in the closure of the Lie algebra generated by vector fields being coefficients of equations. By using this result, we obtain a representation of the solution of stochastic differential equations in terms of Lie brackets and iterated Stratonovich integrals in the algebra of formal power series.     

Analyticity without differentiability

In this article we derive all salient properties of analytic functions, including the analytic version of the inverse function theorem, using only the most elementary convergence properties of series. Not even the notion of differentiability is required to do so. Instead, analytical arguments are replaced by combinatorial arguments exhibiting properties of formal power series. Along the way, we show how formal power series can be used to solve combinatorial problems and also derive some results in calculus with a minimum of analytical machinery.     

Hahn-Ternärkörper mit speziellen Faktorsystemen

In a Hahn ternary field, we consider the properties of linearity and distributivity and of associativity and commutativity with respect to addition and multiplication on the set of all formal power series with finite support and investigate their consequences for the generalized factor system and for the whole Hahn ternary field.     

Some geometric-differential models in the class of formal operator power series

We consider an example of a formal construction of local differential geometry in which smooth functions regarded as morphisms are replaced by formal operator power series.     

Cosheaves and connectedness in formal topology

The localic definitions of cosheaves, connectedness and local connectedness are transferred from impredicative topos theory to predicative formal topology. A formal topology is locally connected (has base of connected opens) iff it has a cosheaf π₀ together with certain additional structure and properties that constrain π₀ to be the connected components cosheaf. In the inductively generated case, complete spreads (in the sense of Bunge and Funk) corresponding to cosheaves are defined as formal topologies. Maps between the complete spreads are equivalent to homomorphisms between the cosheaves. A cosheaf is the connected components cosheaf for a locally connected formal topology iff its complete spread is a homeomorphism, and in this case it is a terminal cosheaf.A new, geometric proof is given of the topos-theoretic result that a cosheaf is a connected components cosheaf iff it is a “strongly terminal” point of the symmetric topos, in the sense that it is terminal amongst all the generalized points of the symmetric topos. It is conjectured that a study of sites as “formal toposes” would allow such geometric proofs to be incorporated into predicative mathematics.