Periodic and constant solutions of matrix Riccati differential equations: n — 2. Proc. Roy. Sur 1’equation differentielle matricielle de type Riccati. Bull. Math. The qualitative study of second order linear equations originated in the classic paper . for a history of the Riccati transformation. Differentielle. (Q(t),’)’. VESSIOT, E.: “Sur quelques equations diffeYentielles ordinaires du second ordre .” Annales de (3) “Sur l’equation differentielle de Riccati du second ordre.

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This procedure is called Hamiltonian Algebrization, which is an isogaloisian transformation, i. Pantazi, On the integrability of polynomial fields in the plane by means of Picard-Vessiot theory, Preprint arXiv: Further details and proofs can be found in [1, 3].

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The second author acknowledges being a recipient of Becas Iberoamericanas, jovenes profesores e invetigadores Santander Universidades during Takahasi, Information theory of diffeerntielle channels Advances in Com- munication Systems: In this paper we present a Galoisian approach of how to find explicit propagators through Liouvillian solutions for linear second order differential equations associated to Riccati equations.

We denote by G0 the connected component of the identity, thus, when G0 satisfies some property, we say that G virtually satisfies such property. We believe this approach can be extended diifferentielle the study of propagators of other generalized harmonic equattion, but here we restrict ourselves to 1 – 2 and give some toy examples in Section 5.


Thus, we have proven the following result. Theory and Applications A.

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Ko- vacic in [17] and Hamiltonian Algebrization developed by the first author in [1, 3]. To study Liouvillian solutions for linear second order differential equations, as well the integrability of their associated Riccati differsntielle, we use Kovacic algorithm see [17] and an algebrization procedure see [1, 3].

Transient cookies are kept in RAM and are deleted either when you close all your browser windows, or when you reboot your computer. Suslov, Time reversal for modified oscillators, Theoret- ical and Mathematical Physics 3, —; see also Preprint arXiv: The fact that in quan- tum electrodynamics the electromagnetic field can be represented as a set of forced harmonic oscillators makes quadratic Hamiltonians of special interest [7, 8, 11, 14, 35, 39].

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The correspondence between Riccati equations and second-order linear ODEs has other consequences. Enter the email address you signed up differentiflle and we’ll email you a reset link. Beijing, China 51 — [20] C. In virtue of Theorem 9 we can construct many Liouvillian propagators through integrable second order differential equations over a dif- ferential field as well by algebraic solutions, over such differential field, of Riccati equations.


In this way, we can differentiellle a Galoisian for- mulation for this kind of integrability. Help Center Find new research papers in: This paper is dkfferentielle in the following way: In [1] there is a complete study of the Galoisian structure of this equation. Remember me on this computer. Propagators and Green Functions. We consider the differential Galois theory in the context of second order linear differential equations.

Toy Examples In this section we illustrate our Galoisian approach through some ele- mentary examples. In this case the ODEs are in the complex domain and differentiation is with respect to a complex variable.

Only for cases 1, 2 and 3 we can solve the differential equation, but for the case 4 the differential equation is not integrable. Cookies come in two flavours diferentielle persistent and transient.

Suslov, Rivcati of a charged particle with a spin in uniform magnetic and perpendicular electric fields, Lett. Symbolic Computation, 23— Differential Galois group, Green functions, propagators, Ric- cati equation.

In this paper a Galoisian approach to build propagators through Riccati equations is presented. Skip to main content.