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Multiple solutions of indefinite elliptic systems via a Galerkin-type Conley index theory

Marek Izydorek, Krzysztof P. Rybakowski (2003)

Fundamenta Mathematicae

Let Ω be a bounded domain in N with smooth boundary. Consider the following elliptic system: - Δ u = v H ( u , v , x ) in Ω, - Δ v = u H ( u , v , x ) in Ω, u = 0, v = 0 in ∂Ω. (ES) We assume that H is an even "-"-type Hamiltonian function whose first order partial derivatives satisfy appropriate growth conditions. We show that if (0,0) is a hyperbolic solution of (ES), then (ES) has at least 2|μ| nontrivial solutions, where μ = μ(0,0) is the renormalized Morse index of (0,0). This proves a conjecture by Angenent and van der Vorst.

Multiplicative integrable models from Poisson-Nijenhuis structures

Francesco Bonechi (2015)

Banach Center Publications

We discuss the role of Poisson-Nijenhuis (PN) geometry in the definition of multiplicative integrable models on symplectic groupoids. These are integrable models that are compatible with the groupoid structure in such a way that the set of contour levels of the hamiltonians in involution inherits a topological groupoid structure. We show that every maximal rank PN structure defines such a model. We consider the examples defined on compact hermitian symmetric spaces studied by F. Bonechi, J. Qiu...

Multivalued Lyapunov functions for homeomorphisms of the 2-torus

Patrice Le Calvez (2006)

Fundamenta Mathematicae

Let F be a homeomorphism of 𝕋² = ℝ²/ℤ² isotopic to the identity and f a lift to the universal covering space ℝ². We suppose that κ ∈ H¹(𝕋²,ℝ) is a cohomology class which is positive on the rotation set of f. We prove the existence of a smooth Lyapunov function of f whose derivative lifts a non-vanishing smooth closed form on 𝕋² whose cohomology class is κ.

Multivariate smooth transition AR model with aggregation operators and application to exchange rates

Tomáš Bacigál (2007)

Kybernetika

An overview of multivariate modelling based on logistic and exponential smooth transition models with transition variable generated by aggregation operators and orders of auto and exogenous regression selected by information criterion separately for each regime is given. Model specification procedure is demonstrated on trivariate exchange rates time series. The application results show satisfactory improvement in fit when particular aggregation operators are used. Source code in the form of Mathematica...

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