scholarly journals Formal Weierstrass Nonintegrability Criterion for Some Classes of Polynomial Differential Systems in ℂ2

2020 ◽  
Vol 30 (04) ◽  
pp. 2050064
Author(s):  
Jaume Giné ◽  
Jaume Llibre
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Differential System ◽  
Integrable Systems ◽  
Necessary Conditions ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Formal Power

In this paper, we present a criterion for determining the formal Weierstrass nonintegrability of some polynomial differential systems in the plane [Formula: see text]. The criterion uses solutions of the form [Formula: see text] of the differential system in the plane and their associated cofactors, where [Formula: see text] is a formal power series. In particular, the criterion provides the necessary conditions in order that some polynomial differential systems in [Formula: see text] would be formal Weierstrass integrable. Inside this class there exist non-Liouvillian integrable systems. Finally we extend the theory of formal Weierstrass integrability to Puiseux Weierstrass integrability.

scholarly journals On the equivalence of three-dimensional differential systems

2020 ◽  
Vol 18 (1) ◽  
pp. 1164-1172
Author(s):  
Jian Zhou ◽  
Shiyin Zhao
Keyword(s):  
Periodic Solutions ◽  
Differential System ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Systems ◽  
Structural Form ◽  
Differential Systems

Abstract In this paper, firstly, we study the structural form of reflective integral for a given system. Then the sufficient conditions are obtained to ensure there exists the reflective integral with these structured form for such system. Secondly, we discuss the necessary conditions for the equivalence of such systems and a general three-dimensional differential system. And then, we apply the obtained results to the study of the behavior of their periodic solutions when such systems are periodic systems in t.

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.

Cyclic codes over formal power series rings

2011 ◽  
Vol 31 (1) ◽  
pp. 331-343 ◽  
Author(s):  
Steven T. Dougherty ◽  
Liu Hongwei
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Cyclic Codes ◽  
Power Series Rings ◽  
Formal Power
Sign in / Sign up

Export Citation Format

Share Document