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Quadratic functionals: positivity, oscillation, Rayleigh's principle

Werner Kratz (1998)

Archivum Mathematicum

In this paper we give a survey on the theory of quadratic functionals. Particularly the relationships between positive definiteness and the asymptotic behaviour of Riccati matrix differential equations, and between the oscillation properties of linear Hamiltonian systems and Rayleigh’s principle are demonstrated. Moreover, the main tools form control theory (as e.g. characterization of strong observability), from the calculus of variations (as e.g. field theory and Picone’s identity), and from matrix...

Quadratic functionals with a variable singular end point

Zuzana Došlá, PierLuigi Zezza (1992)

Commentationes Mathematicae Universitatis Carolinae

In this paper we introduce the definition of coupled point with respect to a (scalar) quadratic functional on a noncompact interval. In terms of coupled points we prove necessary (and sufficient) conditions for the nonnegativity of these functionals.

Quadratic Isochronous centers commute

M. Sabatini (1999)

Applicationes Mathematicae

We prove that every quadratic plane differential system having an isochronous center commutes with a polynomial differential system.

Quadratic systems with a unique finite rest point.

Bartomeu Coll, Armengol Gasull, Jaume Llibre (1988)

Publicacions Matemàtiques

We study phase portraits of quadratic systems with a unique finite singularity. We prove that there are 111 different phase portraits without limit cycles and that 13 of them are realizable with exactly one limit cycle. In order to finish completely our study two problems remain open: the realization of one topologically possible phase portrait, and to determine the exact number of limit cycles for a subclass of these systems.

Quadratic vector fields with a weak focus of third order.

Joan C. Artés, Jaume Llibre (1997)

Publicacions Matemàtiques

We study phase portraits of quadratic vector fields with a weak focus of third order at the origin. We show numerically the existence of at least 20 different global phase portraits for such vector fields coming from exactly 16 different local phase portraits available for these vector fields. Among these 20 phase portraits, 17 have no limit cycles and three have at least one limit cycle.

Qualitative investigation of nonlinear differential equations describing infiltration of water

Xingbao Wu (1995)

Annales Polonici Mathematici

A nonlinear differential equation of the form (q(x)k(x)u')' = F(x,u,u') arising in models of infiltration of water is considered, together with the corresponding differential equation with a positive parameter λ, (q(x)k(x)u')' = λF(x,u,u'). The theorems about existence, uniqueness, boundedness of solution and its dependence on the parameter are established.

Qualitative theory of half-linear second order differential equations

Ondřej Došlý (2002)

Mathematica Bohemica

Some recent results concerning properties of solutions of the half-linear second order differential equation ( r ( t ) Φ ( x ' ) ) ' + c ( t ) Φ ( x ) = 0 , Φ ( x ) : = | x | p - 2 x , p > 1 , ( * ) are presented. A particular attention is paid to the oscillation theory of ( * ) . Related problems are also discussed.

Quasialgebraic functions

G. Binyamini, D. Novikov, S. Yakovenko (2011)

Banach Center Publications

We introduce and discuss a new class of (multivalued analytic) transcendental functions which still share with algebraic functions the property that the number of their isolated zeros can be explicitly counted. On the other hand, this class is sufficiently rich to include all periods (integral of rational forms over algebraic cycles).

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