Existence of Equilibrium on a Manifold

Existence of equilibrium of a continuous preference relation p or correspondence P on a compact topological space W can be proved either by assuming acyclicity or convexity (no point belongs to the convex hull of its preferred set). Since both properties may well be violated in both political and economic situations, this paper considers instead a "local" convexity property appropriate to a "local" preference relation or preference field. The local convexity property is equivalent to the nonexistence of "local" cycles. When the state space W is a convex set, or is a smooth manifold of a certain topological type, then the "local" convexity property is sufficient to guarantee the existence of a set of critical optima

Existence of Equilibrium on a Manifold

Existence of equilibrium of a continuous preference relation p or correspondence P on a compact topological space W can be proved either by assuming acyclicity or convexity (no point belongs to the convex hull of its preferred set). Since both properties may well be violated in both political and economic situations, this paper considers instead a "local" convexity property appropriate to a "local" preference relation or preference field. The local convexity property is equivalent to the nonexistence of "local" cycles. When the state space W is a convex set, or is a smooth manifold of a certain topological type, then the "local" convexity property is sufficient to guarantee the existence of a set of critical optima

Multiscale dynamics of open three-level quantum systems with two quasi-degenerate levels

We consider a three-level quantum system interacting with a bosonic thermal reservoir. Two energy levels of the system are nearly degenerate but well separated from the third one. The system-reservoir interaction constant is larger than the energy difference of the degenerate levels, but it is smaller than the separation between the latter and the remaining level. We show that the quasi-degeneracy of energy levels leads to the existence of a manifold of quasi-stationary states, and the dynamics exhibits two characteristic time scales. On the first, shorter one, initial states approach the quasi-stationary manifold. Then, on the much longer second time scale, the final unique equilibrium is reached.Comment: Final text as publishe

Separatrix splitting at a Hamiltonian 02iω0^2 i\omega bifurcation

We discuss the splitting of a separatrix in a generic unfolding of a degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We assume that the unperturbed fixed point has two purely imaginary eigenvalues and a double zero one. It is well known that an one-parametric unfolding of the corresponding Hamiltonian can be described by an integrable normal form. The normal form has a normally elliptic invariant manifold of dimension two. On this manifold, the truncated normal form has a separatrix loop. This loop shrinks to a point when the unfolding parameter vanishes. Unlike the normal form, in the original system the stable and unstable trajectories of the equilibrium do not coincide in general. The splitting of this loop is exponentially small compared to the small parameter. This phenomenon implies non-existence of single-round homoclinic orbits and divergence of series in the normal form theory. We derive an asymptotic expression for the separatrix splitting. We also discuss relations with behaviour of analytic continuation of the system in a complex neighbourhood of the equilibrium

Equilibrium measures for uniformly quasiregular dynamics

We establish the existence and fundamental properties of the equilibrium measure in uniformly quasiregular dynamics. We show that a uniformly quasiregular endomorphism $f$ of degree at least 2 on a closed Riemannian manifold admits an equilibrium measure $\mu_f$, which is balanced and invariant under $f$ and non-atomic, and whose support agrees with the Julia set of $f$. Furthermore we show that $f$ is strongly mixing with respect to the measure $\mu_f$. We also characterize the measure $\mu_f$ using an approximation property by iterated pullbacks of points under $f$ up to a set of exceptional initial points of Hausdorff dimension at most $n-1$. These dynamical mixing and approximation results are reminiscent of the Mattila-Rickman equidistribution theorem for quasiregular mappings. Our methods are based on the existence of an invariant measurable conformal structure due to Iwaniec and Martin and the \cA-harmonic potential theory.Comment: 17 page

Normal form for travelling kinks in discrete Klein-Gordon lattices

We study travelling kinks in the spatial discretizations of the nonlinear Klein--Gordon equation, which include the discrete $\phi^4$ lattice and the discrete sine--Gordon lattice. The differential advance-delay equation for travelling kinks is reduced to the normal form, a scalar fourth-order differential equation, near the quadruple zero eigenvalue. We show numerically non-existence of monotonic kinks (heteroclinic orbits between adjacent equilibrium points) in the fourth-order equation. Making generic assumptions on the reduced fourth-order equation, we prove the persistence of bounded solutions (heteroclinic connections between periodic solutions near adjacent equilibrium points) in the full differential advanced-delay equation with the technique of center manifold reduction. Existence and persistence of multiple kinks in the discrete sine--Gordon equation are discussed in connection to recent numerical results of \cite{ACR03} and results of our normal form analysis

Spin Foams and Noncommutative Geometry

We extend the formalism of embedded spin networks and spin foams to include topological data that encode the underlying three-manifold or four-manifold as a branched cover. These data are expressed as monodromies, in a way similar to the encoding of the gravitational field via holonomies. We then describe convolution algebras of spin networks and spin foams, based on the different ways in which the same topology can be realized as a branched covering via covering moves, and on possible composition operations on spin foams. We illustrate the case of the groupoid algebra of the equivalence relation determined by covering moves and a 2-semigroupoid algebra arising from a 2-category of spin foams with composition operations corresponding to a fibered product of the branched coverings and the gluing of cobordisms. The spin foam amplitudes then give rise to dynamical flows on these algebras, and the existence of low temperature equilibrium states of Gibbs form is related to questions on the existence of topological invariants of embedded graphs and embedded two-complexes with given properties. We end by sketching a possible approach to combining the spin network and spin foam formalism with matter within the framework of spectral triples in noncommutative geometry.Comment: 48 pages LaTeX, 30 PDF figure

An economic growth system based on cause-n-effect

In this paper an economic system is represented as a Causal Dynamical Network, (CDN). Each link of CDN exists due to cause-n-effect. each node is either a demand or supply node. The link between demand, and supply exists due to the existence of causality, here is taken as consumer/producer surplus which is a function of preference manifold. Growth is the existence of entropy in CDN. Entropy is measured as a metric probability, which measures change in local equilibrium. Localequilibrium is equilibrium on disordering locality links