Solving Parametric Ideal Membership Problems and Computing Integral Numbers in a Ring of Convergent Power Series Via Comprehensive Gröbner Systems

2018 ◽  
Vol 13 (1-2) ◽  
pp. 185-194 ◽  
Author(s):  
Katsusuke Nabeshima ◽  
Shinichi Tajima
Keyword(s):  
Power Series ◽  
Convergent Power Series ◽  
Ideal Membership ◽  
Comprehensive Gröbner Systems ◽  
Membership Problems

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.

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.

Caputo fractional reduced differential transform method for SEIR epidemic model with fractional order

2021 ◽  
Vol 8 (3) ◽  
pp. 537-548
Author(s):  
S. E. Fadugba ◽  
◽  
F. Ali ◽  
A. B. Abubakar ◽  
◽  
...  
Keyword(s):  
Power Series ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Epidemic Model ◽  
Differential Transform Method ◽  
Host Community ◽  
Convergent Power Series ◽  
Transform Method ◽  
Differential Transform ◽  
Reduced Differential Transform Method

This paper proposes the Caputo Fractional Reduced Differential Transform Method (CFRDTM) for Susceptible-Exposed-Infected-Recovered (SEIR) epidemic model with fractional order in a host community. CFRDTM is the combination of the Caputo Fractional Derivative (CFD) and the well-known Reduced Differential Transform Method (RDTM). CFRDTM demonstrates feasible progress and efficiency of operation. The properties of the model were analyzed and investigated. The fractional SEIR epidemic model has been solved via CFRDTM successfully. Hence, CFRDTM provides the solutions of the model in the form of a convergent power series with easily computable components without any restrictive assumptions.

scholarly journals Local zero estimates and effective division in rings of algebraic power series

2018 ◽  
Vol 2018 (737) ◽  
pp. 111-160 ◽  
Author(s):  
Guillaume Rond
Keyword(s):  
Power Series ◽  
Necessary Condition ◽  
Effective Solution ◽  
Gap Theorem ◽  
Division Theorem ◽  
Ideal Membership ◽  
Formal Power ◽  
Algebraic Power Series ◽  
Ideal Membership Problem ◽  
The Ideal

AbstractWe give a necessary condition for algebraicity of finite modules over the ring of formal power series. This condition is given in terms of local zero estimates. In fact, we show that this condition is also sufficient when the module is a ring with some additional properties. To prove this result we show an effective Weierstrass Division Theorem and an effective solution to the Ideal Membership Problem in rings of algebraic power series. Finally, we apply these results to prove a gap theorem for power series which are remainders of the Grauert–Hironaka–Galligo Division Theorem.

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