Search results for "Nonlinear"
showing 10 items of 3684 documents
$O^\star$-algebras and quantum dynamics: some existence results
2008
We discuss the possibility of defining an algebraic dynamics within the settings of O -algebras. Compared to our previous results on this subject, the main improvement here is that we are not assuming the existence of some Hamiltonian for the full physical system. We will show that, under suitable conditions, the dynamics can still be defined via some limiting procedure starting from a given regularized sequence. © 2008 American Institute of Physics.
An isoperimetric type problem for primitive Pythagorean hodograph curves
2012
An isoperimetric type problem for primitive Pythagorean hodograph curves is studied. We show how to compute, for each possible degree, the Pythagorean hodograph curve of a given perimeter enclosing the greatest area. We also discuss the existence and construction of smooth solutions, obtaining a relationship with an interesting sequence of Appell polynomials.
Sign-indefinite second order differential operators on finite metric graphs
2012
The question of self-adjoint realizations of sign-indefinite second order differential operators is discussed in terms of a model problem. Operators of the type $-\frac{d}{dx} \sgn (x) \frac{d}{dx}$ are generalized to finite, not necessarily compact, metric graphs. All self-adjoint realizations are parametrized using methods from extension theory. The spectral and scattering theory of the self-adjoint realizations are studied in detail.
Quantum moment maps and invariants for G-invariant star products
2002
We study a quantum moment map and propose an invariant for $G$-invariant star products on a $G$-transitive symplectic manifold. We start by describing a new method to construct a quantum moment map for $G$-invariant star products of Fedosov type. We use it to obtain an invariant that is invariant under $G$-equivalence. In the last section we give two simple examples of such invariants, which involve non-classical terms and provide new insights into the classification of $G$-invariant star products.
Closed star products and cyclic cohomology
1992
We define the notion of a closed star product. A (generalized) star product (deformation of the associative product of functions on a symplectic manifold W) is closed iff integration over W is a trace on the deformed algebra. We show that for these products the cyclic cohomology replaces the Hochschild cohomology in usual star products. We then define the character of a closed star product as the cohomology class (in the cyclic bicomplex) of a well-defined cocycle, and show that, in the case of pseudodifferential operators (standard ordering on the cotangent bundle to a compact Riemannian manifold), the character is defined and given by the Todd class, while in general it fails to satisfy t…
Perturbed eigenvalue problems for the Robin p-Laplacian plus an indefinite potential
2020
AbstractWe consider a parametric nonlinear Robin problem driven by the negativep-Laplacian plus an indefinite potential. The equation can be thought as a perturbation of the usual eigenvalue problem. We consider the case where the perturbation$$f(z,\cdot )$$f(z,·)is$$(p-1)$$(p-1)-sublinear and then the case where it is$$(p-1)$$(p-1)-superlinear but without satisfying the Ambrosetti–Rabinowitz condition. We establish existence and uniqueness or multiplicity of positive solutions for certain admissible range for the parameter$$\lambda \in {\mathbb {R}}$$λ∈Rwhich we specify exactly in terms of principal eigenvalue of the differential operator.
Superstable cycles for antiferromagnetic Q-state Potts and three-site interaction Ising models on recursive lattices
2013
We consider the superstable cycles of the Q-state Potts (QSP) and the three-site interaction antiferromagnetic Ising (TSAI) models on recursive lattices. The rational mappings describing the models' statistical properties are obtained via the recurrence relation technique. We provide analytical solutions for the superstable cycles of the second order for both models. A particular attention is devoted to the period three window. Here we present an exact result for the third order superstable orbit for the QSP and a numerical solution for the TSAI model. Additionally, we point out a non-trivial connection between bifurcations and superstability: in some regions of parameters a superstable cyc…
Multiple solutions for nonlinear nonhomogeneous resonant coercive problems
2018
We consider a nonlinear, nonhomogeneous Dirichlet problem driven by the sum of a \begin{document}$p$\end{document} -Laplacian ( \begin{document}$2 ) and a Laplacian. The reaction term is a Caratheodory function \begin{document}$f(z,x)$\end{document} which is resonant with respect to the principal eigenvalue of ( \begin{document}$-\Delta_p,\, W^{1,p}_0(\Omega)$\end{document} ). Using variational methods combined with truncation and comparison techniques and Morse theory (critical groups) we prove the existence of three nontrivial smooth solutions all with sign information and under three different conditions concerning the behavior of \begin{document}$f(z,\cdot)$\end{document} near zero. By …
Unitary Representations of U q (𝔰𝔩}(2,ℝ)),¶the Modular Double and the Multiparticle q -Deformed¶Toda Chain
2002
The paper deals with the analytic theory of the quantum q-deformed Toda chains; the technique used combines the methods of representation theory and the Quantum Inverse Scattering Method. The key phenomenon which is under scrutiny is the role of the modular duality concept (first discovered by L. Faddeev) in the representation theory of noncompact semisimple quantum groups. Explicit formulae for the Whittaker vectors are presented in terms of the double sine functions and the wave functions of the N-particle q-deformed open Toda chain are given as a multiple integral of the Mellin–Barnes type. For the periodic chain the two dual Baxter equations are derived.
The graded Lie algebra structure of Lie superalgebra deformation theory
1989
We show how Lie superalgebra deformation theory can be treated by graded Lie algebras formalism. Rigidity and integrability theorems are obtained.