POWER SERIES PROOFS FOR LOCAL STABILITIES OF KÄHLER AND BALANCED STRUCTURES WITH MILD -LEMMA

2021 ◽  
pp. 1-50
Author(s):  
SHENG RAO ◽  
XUEYUAN WAN ◽  
QUANTING ZHAO
Keyword(s):  
Power Series ◽  
Local Stability ◽  
Differential Forms ◽  
Stability Theorem ◽  
Power Series Method ◽  
Stability Theorems ◽  
Type Complex ◽  
Chern Numbers ◽  
Balanced Manifold ◽  
Kähler Structures

Abstract By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira–Spencer’s local stability theorem of Kähler structures. We also obtain two new local stability theorems, one of balanced structures on an n-dimensional balanced manifold with the $(n-1,n)$ th mild $\partial \overline {\partial }$ -lemma by power series method and the other one on p-Kähler structures with the deformation invariance of $(p,p)$ -Bott–Chern numbers.

scholarly journals On local stabilities of -Kähler structures

2019 ◽  
Vol 155 (3) ◽  
pp. 455-483 ◽  
Author(s):  
Sheng Rao ◽  
Xueyuan Wan ◽  
Quanting Zhao
Keyword(s):  
Power Series ◽  
Local Stability ◽  
Natural Extension ◽  
Stability Theorem ◽  
Series Method ◽  
Power Series Method ◽  
Kähler Structures

By use of a natural extension map and a power series method, we obtain a local stability theorem for $p$-Kähler structures with the $(p,p+1)$th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$-lemma under small differentiable deformations.

A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation

2021 ◽  
Vol 477 (2255) ◽  
Author(s):  
Shou-Fu Tian ◽  
Mei-Juan Xu ◽  
Tian-Tian Zhang
Keyword(s):  
Power Series ◽  
Vector Fields ◽  
Higher Order ◽  
Beam Equation ◽  
Series Solutions ◽  
Local Conservation ◽  
Power Series Method ◽  
Potential System ◽  
Power Series Solutions ◽  
The Difference

Under investigation in this work is a generalized higher-order beam equation, which is an important physical model and describes the vibrations of a rod. By considering Lie symmetry analysis, and using the power series method, we derive the geometric vector fields, symmetry reductions, group invariant solutions and power series solutions of the equation, respectively. The convergence analysis of the power series solutions are also provided with detailed proof. Furthermore, by virtue of the multipliers, the local conservation laws of the equation are derived as well. Furthermore, an effective and direct approach is proposed to study the symmetry-preserving discretization for the equation via its potential system. Finally, the invariant difference models of the generalized beam equation are successfully constructed. Our results show that it is very useful to construct the difference models of the potential system instead of the original equation.

The Scope of the Power Series Method

2013 ◽  
Vol 86 (1) ◽  
pp. 56-62
Author(s):  
Richard Beals
Keyword(s):  
Power Series ◽  
Series Method ◽  
Power Series Method

scholarly journals Generalized Hyers–Ulam Stability of the Additive Functional Equation

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 76 ◽  
Author(s):  
Yang-Hi Lee ◽  
Gwang Kim
Keyword(s):  
Functional Equation ◽  
Additive Function ◽  
Additive Functional ◽  
Stability Theorem ◽  
Ulam Stability ◽  
Additive Functional Equation ◽  
Stability Theorems ◽  
The Stability

We will prove the generalized Hyers–Ulam stability and the hyperstability of the additive functional equation f(x1 + y1, x2 + y2, …, xn + yn) = f(x1, x2, … xn) + f(y1, y2, …, yn). By restricting the domain of a mapping f that satisfies the inequality condition used in the assumption part of the stability theorem, we partially generalize the results of the stability theorems of the additive function equations.

Stability theorems

1977 ◽  
Vol 9 (02) ◽  
pp. 336-361 ◽  
Author(s):  
Eugene Lukacs
Keyword(s):  
Stability Theorem ◽  
Stability Theorems ◽  
Primary Object ◽  
The Stability ◽  
Decomposition Theorems

A stability theorem determines the extent to which the conclusions of a given theorem are affected if the assumptions of the theorem are not exactly but only approximately satisfied. The meaning of the word ‘approximately’ has to be defined exactly. The stability of decomposition theorems, of characterizations by independence and by regression properties are the primary object of the paper.

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