scholarly journals Rings of convergent power series and Weierstrass Preparation Theorem

1981 ◽  
Vol 81 ◽  
pp. 73-78
Author(s):  
Takasi Sugatani
Keyword(s):  
Power Series ◽  
Integral Domain ◽  
Multiplicative Group ◽  
Convergent Power Series ◽  
Weierstrass Preparation ◽  
Preparation Theorem

Let B be a B-ring with a nonarchimedean valuation | |, i.e., B is an integral domain satisfying the following conditions: (i) B is bounded (| a | ≤ 1 for every a ∊ B), (ii) the boundary forms a multiplicative group.

Convex Directed Subgroups of a Group of Divisibility

1974 ◽  
Vol 26 (3) ◽  
pp. 532-542 ◽  
Author(s):  
Joe L. Mott
Keyword(s):  
Integral Domain ◽  
Multiplicative Group ◽  
Ordered Group ◽  
Quotient Field ◽  
Partially Ordered ◽  
Group Of Units ◽  
Partially Ordered Group ◽  
Group Of Divisibility

If D is an integral domain with quotient field K, the group of divisibility G(D) of D is the partially ordered group of non-zero principal fractional ideals with aD ≦ bD if and only if aD contains bD. If K* denotes the multiplicative group of K and U(D) the group of units of D, then G(D) is order isomorphic to K*/U(D), where aU(D) ≦ bU(D) if and only if b/a ∊ D.

Weak Normalization of Power Series Rings

1995 ◽  
Vol 38 (4) ◽  
pp. 429-433 ◽  
Author(s):  
David E. Dobbs ◽  
Moshe Roitman
Keyword(s):  
Power Series ◽  
Integral Domain ◽  
Power Series Ring ◽  
Series Ring ◽  
Power Series Rings

AbstractIt is proved that if r* is the weak normalization of an integral domain r, then the weak normalization of the power series ring r[[x1,....xn]] is contained in R*[[X1,....Xn]]. Consequently, if R is a weakly normal integral domain, then R[[X1,....Xn]] is also weakly normal.

scholarly journals Note on a special determinant

1937 ◽  
Vol 30 ◽  
pp. xxvii-xviii
Author(s):  
A. C. Aitken
Keyword(s):  
Power Series ◽  
Convergent Power Series ◽  
Image Position

Suppose a polynomial or convergent power seriesis raised to powers j = 0, 1, 2, 3, … The coefficients of xk in [f(x)]j, k = 0, 1, 2, …, may be entered as elements in positions (j, k) in an array or matrix F, thus:By construction all elements in column (k) have weight (sum of suffixes) equal to k.

scholarly journals An explicit solution of coupled viscous Burgers' equation by the decomposition method

2001 ◽  
Vol 27 (11) ◽  
pp. 675-680 ◽  
Author(s):  
Doğan Kaya
Keyword(s):  
Perturbation Theory ◽  
Power Series ◽  
Explicit Solution ◽  
Decomposition Method ◽  
Series Solution ◽  
Burgers Equation ◽  
Coupled System ◽  
The Self ◽  
Convergent Power Series ◽  
Weak Nonlinearity

We consider a coupled system of viscous Burgers' equations with appropriate initial values using the decomposition method. In this method, the solution is calculated in the form of a convergent power series with easily computable components. The method does not need linearization, weak nonlinearity assumptions or perturbation theory. The decomposition series solution of the problem is quickly obtained by observing the existence of the self-canceling “noise” terms where the sum of components vanishes in the limit.

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