A Stratification Given by Artin-Rees Estimates

1992 ◽  
Vol 44 (1) ◽  
pp. 194-205 ◽  
Author(s):  
Ti Wang
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Open Subset ◽  
Meromorphic Functions ◽  
Finite Sequence ◽  
Analytic Subset ◽  
Formal Power

Let = ℝ or ℂ. Let U be an open subset of n. Let X be a closed analytic subset of U and let Z be a proper closed analytic subset of X. Let M(X;Z) denote the ring of meromorphic functions on Xwhose poles lie in Z. Let M be the families of formal power series generated by a finite sequence ƒi,… ,ƒq ∈ M(X; Z) ⟦ y⟧p (For details, see § 2).

Fréchet-valued meromorphic extension along a pencil of complex lines

2021 ◽  
pp. 2250086
Author(s):  
Lien Vuong Lam ◽  
Nguyen Van Dai
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Meromorphic Functions ◽  
Extension Theorem ◽  
Homogeneous Polynomials ◽  
Complete Space ◽  
Complex Lines ◽  
Formal Power

The aim of paper is to find the condition under which a Fréchet-valued function [Formula: see text] admitting meromorphic extension along some pencil of complex lines can be meromorphically extended to a neighborhood of [Formula: see text] Some auxiliary results concerning the domains of existence for Fréchet-valued meromorphic functions, Rothstein’s theorem, Levi extension theorem for meromorphic functions with values in a locally complete space, convergence of formal power series of Fréchet-valued homogeneous polynomials are also proved in this work.

scholarly journals Non-integrability of the 1:1:2–resonance

1984 ◽  
Vol 4 (4) ◽  
pp. 553-568 ◽  
Author(s):  
J. J. Duistermaat
Keyword(s):  
Hamiltonian System ◽  
Power Series ◽  
Equilibrium Point ◽  
Formal Power Series ◽  
Open Subset ◽  
Taylor Expansion ◽  
Degrees Of Freedom ◽  
Third Order ◽  
Formal Power ◽  
F System

AbstractA Hamiltonian system of n degrees of freedom, defined by the function F, with an equilibrium point at the origin, is called formally integrable if there exist A A formal power series , functionally independent, in involution, and such that the Taylor expansion of F is a formal power series in the .Take n = 3, , F(k) homogeneous of degree k, F(2) > 0 and the eigenfrequencies in ratio 1:1:2. If F(3) avoids a certain hypersurface of ‘symmetric’ third order terms, then the F system is not formally integrable. If F(3) is symmetric but F(4) is in a non-void open subset, then homoclinic intersection with Devaney spiralling occurs; the angle decays of order 1 when approaching the origin.

Criteria for Algebraicity of Analytic Functions

2019 ◽  
Author(s):  
Jacek Bochnak ◽  
Janusz Gwoździewicz ◽  
Wojciech Kucharz
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Formal Power Series ◽  
Open Subset ◽  
Analytic Functions ◽  
Algebraic Curves ◽  
Polynomial Function ◽  
Algebraic Set ◽  
Formal Power

Abstract We consider functions defined on an open subset of a nonsingular, either real or complex, algebraic set. We give criteria for an analytic function to be a Nash (resp. regular, resp. polynomial) function. Our criteria depend only on the behavior of such a function along irreducible nonsingular algebraic curves passing through a given point. In the proofs we use results on algebraicity of formal power series, which are also established in this paper.

scholarly journals On transformations of formal power series

2003 ◽  
Vol 184 (2) ◽  
pp. 369-383 ◽  
Author(s):  
Manfred Droste ◽  
Guo-Qiang Zhang
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Formal Power

scholarly journals On the Square Root of a Bell Matrix

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Formal Power Series ◽  
Computational Method ◽  
Square Root ◽  
Riordan Arrays ◽  
Formal Power

AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = R ( f ( t ) / t , f ( t ) ) defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ f ( t ) = ∑ k ≥ 0 f k t k with $$f(0)=f_0=0$$ f ( 0 ) = f 0 = 0 is presented. It is proved that if $$f^\prime (0)=1$$ f ′ ( 0 ) = 1 and $$f^{\prime \prime }(0)\ne 0$$ f ″ ( 0 ) ≠ 0 then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H = R ( h ( t ) / t , h ( t ) ) such that $$H*H=R;$$ H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices.

scholarly journals Two Interacting Coordinate Hopf Algebras of Affine Groups of Formal Series on a Category

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Laurent Poinsot
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Semidirect Product ◽  
Hopf Algebras ◽  
Formal Series ◽  
Product Structure ◽  
Locally Finite ◽  
Commutative Algebras ◽  
Affine Groups ◽  
Formal Power

A locally finite category is defined as a category in which every arrow admits only finitely many different ways to be factorized by composable arrows. The large algebra of such categories over some fields may be defined, and with it a group of invertible series (under multiplication). For certain particular locally finite categories, a substitution operation, generalizing the usual substitution of formal power series, may be defined, and with it a group of reversible series (invertible under substitution). Moreover, both groups are actually affine groups. In this contribution, we introduce their coordinate Hopf algebras which are both free as commutative algebras. The semidirect product structure obtained from the action of reversible series on invertible series by anti-automorphisms gives rise to an interaction at the level of their coordinate Hopf algebras under the form of a smash coproduct.

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