scholarly journals The exponent of certain finite p-groups

1990 ◽  
Vol 33 (3) ◽  
pp. 483-490 ◽  
Author(s):  
I. O. York
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Group Operation ◽  
Image Position ◽  
Formal Power ◽  
Quotient Groups

In this paper, for R a commutative ring, with identity, of characteristic p, we look at the group G(R) of formal power series with coefficients in R, of the formand the group operation being substitution. The results obtained give the exponent of the quotient groups Gn(R) of this group, n∈ℕ.

Substitution Groups of Formal Power Series

1954 ◽  
Vol 6 ◽  
pp. 325-340 ◽  
Author(s):  
S. A. Jennings
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Point Of View ◽  
Group Operation ◽  
Several Variables ◽  
Special Cases ◽  
Image Position ◽  
Formal Power ◽  
Complex Coefficients

In this paper we are concerned with the group of formal power series of the form,the coefficients being elements of a commutative ring R and the group operation being substitution. Little seems to be known of the properties of groups of this type, except in special cases, although groups of formal power series in several variables with complex coefficients have been investigated from a different point of view by Bochner and Martin (1, chap. I) and Gotô (2).

scholarly journals A non-embeddable composite of embeddable functions

1968 ◽  
Vol 8 (1) ◽  
pp. 109-113 ◽  
Author(s):  
A. Ran
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Complex Variable ◽  
Complex Parameter ◽  
Complex Numbers ◽  
Group Operation ◽  
Image Position ◽  
Formal Power

Let Ω be the group of the functions ƒ(z) of the complex variable z, analytic in some neighborhood of z = 0, with ƒ(0) = 0, ƒ′(0) = 1, where the group operation is the composition g[f(z)](g(z), f(z) ∈ Ω). For every function f(z) ∈ Ω there exists [4] a unique formal power series where the coefficients ƒq(s) are polynomials of the complex parameter s, with ƒ1(s) = 1, such that and, for any two complex numbers s and t, the formal law of composition is valid.

scholarly journals An algebraically closed field

1968 ◽  
Vol 9 (2) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Rayner
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Zero Characteristic ◽  
Image Position ◽  
Formal Power ◽  
Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

Chromatic Solutions

1982 ◽  
Vol 34 (3) ◽  
pp. 741-758 ◽  
Author(s):  
W. T. Tutte
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Chromatic Polynomial ◽  
Basic Theory ◽  
Root Vertex ◽  
Independent Variables ◽  
Image Position ◽  
Formal Power ◽  
Two Independent Variables

Early in the Seventies I sought the number of rooted λ-coloured triangulations of the sphere with 2p faces. In these triangulations double joins, but not loops, were permitted. The investigation soon took the form of a discussion of a certain formal power series l(y, z, λ) in two independent variables y and z.The basic theory of l is set out in [1]. There l is defined as the coefficient of x2 in a more complicated power series g(x, y, z, λ). But the definition is equivalent to the following formula.1Here T denotes a general rooted triangulation. n(T) is the valency of its root-vertex, and 2p(T) is the number of its faces. P(T, λ) is the chromatic polynomial of the graph of T.

scholarly journals Solutions of first level of meromorphic differential equations

1982 ◽  
Vol 25 (2) ◽  
pp. 183-207 ◽  
Author(s):  
W. Balser
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Differential Equations ◽  
Formal Power Series ◽  
Convergent Series ◽  
Constant Matrix ◽  
Image Position ◽  
Formal Power ◽  
Meromorphic Differential Equation ◽  
Meromorphic Differential

Let a meromorphic differential equationbe given, where r is an integer, and the series converges for |z| sufficiently large. Then it is well known that (0.1) is formally satisfied by an expressionwhere F( z) is a formal power series in z–1 times an integer power of z, and F( z) has an inverse of the same kind, L is a constant matrix, andis a diagonal matrix of polynomials qj( z) in a root of z, 1≦ j≦ n. If, for example, all the polynomials in Q( z) are equal, then F( z) can be seen to be a convergent series (see Section 1), whereas if not, then generally the coefficients in F( z) grow so rapidly that F( z) diverges for every (finite) z.

scholarly journals A p-adic analogue to a theorem by J. Popken

1973 ◽  
Vol 16 (2) ◽  
pp. 176-184 ◽  
Author(s):  
K. Mahler
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Formal Power Series ◽  
Algebraic Differential Equation ◽  
Image Position ◽  
Formal Power

AbstractIt is proved that if is a formal power series with algebraic p-adic coefficients which satisfies an algebraic differential equation, then a constant y4 > 0 and a constant integer h1 ≧ 0 exist such that .

The total Steenrod operation is induced by an A∞ ring homomorphism

1987 ◽  
Vol 101 (3) ◽  
pp. 469-476 ◽  
Author(s):  
A. Kozlowski
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Homotopy Type ◽  
Cohomology Ring ◽  
Ring Homomorphism ◽  
Group Operation ◽  
Steenrod Operation ◽  
Steenrod Square ◽  
Formal Power

Let X be a (based) space of the homotopy type of a CW-complex. Let H(X) denote the classical (ungraded) cohomology ring Πi≥0Hi (X;Z/2). In [1] Atiyah and Hirzebruch described the group of natural ring automorphisms of H(X) (‘cohomology automorphisms’) with group operation given by composition. They showed that is isomorphic to the group of formal power series of the form with group operation given by ‘substitution’ of one power series into another. In particular the most famous ‘cohomology automorphism’, the total Steenrod Square, corresponds to x + x2.

Algebras of Acyclic Type

1981 ◽  
Vol 33 (1) ◽  
pp. 129-141 ◽  
Author(s):  
Phil Hanlon
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Homomorphic Image ◽  
Incidence Algebra ◽  
Section 8 ◽  
Slight Generalization ◽  
Background Material ◽  
Binomial Type ◽  
Formal Power

In this paper we consider the problem of determining when an algebra of formal power series over a commutative ring R is the homomorphic image of a reduced incidence algebra P(R, ∽). The question of when two such algebras are isomorphic is answered in Section 8 of [1]. A slight generalization of their notion of full binomial type is introduced here.Section 1 contains background material together writh a summary of the results of [1]. In Section 2 we present the desired characterization, and to conclude an application appears in Section 3. In Section 3 the tools of Section 2 are used to derive an equation of R. W. Robinson and R. P. Stanley which counts labelled, acyclic digraphs.

Commutative subsemigroups of the composition semigroup of formal power series over an integral domain

1979 ◽  
Vol 27 (3) ◽  
pp. 313-318
Author(s):  
Hermann Kautschitsch
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Integral Domain ◽  
Positive Order ◽  
Formal Power

AbstractLet R be a commutative ring with identity. R[[x]] denotes the ring of formal power series, in which we consider the composition ○, defined by f(x)○g(x)=f(g(x)). This operation is well defined in the subring R+[[x]] of formal power series of positive order. The algebra= 〈R+[[x]], ○〉 is learly a semigroup, which is not commutative for ∣R∣>1. In this paper we consider all those commutative subsemigroups of , which consist of power series of all positive orders, which are called ‘permutable chains’.

On transfer of annihilator conditions of rings

2018 ◽  
Vol 17 (10) ◽  
pp. 1850199
Author(s):  
Abdollah Alhevaz ◽  
Ebrahim Hashemi ◽  
Rasul Mohammadi
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Noetherian Rings ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Base Ring ◽  
Polynomial Formula ◽  
Formal Power

It is well known that a polynomial [Formula: see text] over a commutative ring [Formula: see text] with identity is a zero-divisor in [Formula: see text] if and only if [Formula: see text] has a non-zero annihilator in the base ring, where [Formula: see text] is the polynomial ring with indeterminate [Formula: see text] over [Formula: see text]. But this result fails in non-commutative rings and in the case of formal power series ring. In this paper, we consider the problem of determining some annihilator properties of the formal power series ring [Formula: see text] over an associative non-commutative ring [Formula: see text]. We investigate relations between power series-wise McCoy property and other standard ring-theoretic properties. In this context, we consider right zip rings, right strongly [Formula: see text] rings and rings with right Property [Formula: see text]. We give a generalization (in the case of non-commutative ring) of a classical results related to the annihilator of formal power series rings over the commutative Noetherian rings. We also give a partial answer, in the case of formal power series ring, to the question posed in [1 Question, p. 16].

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