Critical Hypersurface Research Articles

Codimension one symplectic foliations and regular Poisson structures

In this short note we give a complete characterization of a certain class of compact corank one Poisson manifolds, those equipped with a closed one-form defining the symplectic foliation and a closed two-form extending the symplectic form on each leaf. If such a manifold has a compact leaf, then all the leaves are compact, and furthermore the manifold is a mapping torus of a compact leaf. These manifolds and their regular Poisson structures admit an extension as the critical hypersurface of a b-Poisson manifold as we will see in [9].

Quark recombination and heavy quark diffusion in hot nuclear matter

We discuss resonance recombination for quarks and show that it is compatible with quark and hadron distributions in local thermal equilibrium. We then calculate realistic heavy quark phase space distributions in heavy ion collisions using Langevin simulations with non-perturbative T-matrix interactions in hydrodynamic backgrounds. We hadronize the heavy quarks on the critical hypersurface given by hydrodynamics after constructing a criterion for the relative recombination and fragmentation contributions. We discuss the influence of recombination and flow on the resulting heavy meson and single electron RAA and elliptic flow. We will also comment on the effect of diffusion of open heavy flavor mesons in the hadronic phase.

On the stability of Mañé critical hypersurfaces

We construct examples of Tonelli Hamiltonians on \({\mathbb{T}^n}\) (for any n ≥ 2) such that the hypersurfaces corresponding to the Mane critical value are stable (i.e. geodesible). We also provide a criterion for instability in terms of closed orbits in free homotopy classes and we show that any stable energy level of a Tonelli Hamiltonian must contain a closed orbit.

Topological origin of the phase transition in a model of DNA denaturation.

The hypothesis that phase transitions originate from some topological change of the critical level hypersurface of the potential energy receives direct evident support by our study of the Bishop-Peyrard model of DNA thermal denaturation.

Stability balloon for the double-lid-driven cavity flow

The linear stability of the two-dimensional, steady, incompressible flow in a rectangular cavity is investigated numerically. The flow is driven by the parallel or antiparallel motion of two facing walls. When the cavity is infinitely extended in the spanwise direction a variety of different, three-dimensional flow instabilities can arise. We provide an overview on the global structure of the critical hypersurface in the three-dimensional parameter space spanned by the cross-sectional aspect ratio and the two side-wall Reynolds numbers. It turns out that the critical hypersurface is made up of six major segments belonging to different modes of instability. The interrelation and connectivity of these critical modes is elucidated.

On consistent symbolic representations of general dynamic systems

This paper deals with the issue of consistent symbolic (qualitative) representation of continuous dynamic systems. Consistency means here that the results of reasoning with the qualitative representation hold in the underlying (quantitative) dynamic system. In the formalization proposed in this paper, the quantitative structure is represented using the notion of a general dynamic system (GDS). The qualitative counterpart (QDS), is represented by a finite-state automaton structure. The two representational substructures are related through functions, called qualitative abstractions of dynamic systems. Qualitative abstractions associate inputs, states and outputs of the QDS, with partitions of appropriate GDS spaces. The paper shows how to establish such consistent partitions, given a partitioning of the system's output. To represent borders of these partitions, the notion of critical hypersurfaces is introduced. One of the main ideas that provides consistency is the interpretation of qualitative input events as elements of the partition of the Cartesian product of input, initial state and time sets. An example of a consistent qualitative/quantitative representation of a simple dynamic system, and of reasoning using such a representation, is provided.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

D.c. critical currents in granular sintered superconducting ceramics

Intergrain weak link coupling limits the superconducting critical current at zero voltage and thus electric power transportation. A classical model has been developed independently by Rhyner and Blatter and the present author which neglects phase fluctuations of the superconducting order parameter between grains and models the weak links as Josephson junctions, each possessing a limiting critical current. Rhyner and Blatter solved this problem on two-dimensional lattices using a transfer matrix method or simplex algorithm, and applied it to texturing. The present author has shown that the problem can be efficiently solved using an adaptation of the maximal flow theorem of Ford and Fulkerson, well known in economics. The minimal cut, i.e. limitation of the current by a critical hypersurface, can be derived directly from this. New results on two- and three-dimensional networks are presented in this paper.

Corrections to scaling at two-dimensional Ising transitions.

We investigate the completeness of the set of irrelevant critical exponents predicted by conformal invariance, particularly the question of whether the set of operators in the conformal block is complete in general, or only in a subspace of the critical hypersurface. The leading correction-to-scaling exponent is determined numerically at the critical points of some versions of the two-dimensional Ising model that enhance vacancy excitations: the antiferromagnetic Ising model in a magnetic field and the Blume-Capel model. The presence of corrections to scaling with the vacancy exponent ${y}_{\mathrm{vac}=\mathrm{\ensuremath{-}}(4/3}$ would be consistent with conformal invariance, but would contradict completeness of the closed operator algebra. Only corrections to scaling with an irrelevant critical exponent ${y}_{\mathrm{ir}=\mathrm{\ensuremath{-}}2.0}$ are found, consistent with completeness. However, an investigation of the random-cluster-model representation of the q=2+\ensuremath{\epsilon} state Potts model shows that a vacancy correction to scaling with exponent ${y}_{\mathrm{vac}=\mathrm{\ensuremath{-}}(4/3}$ appears. The correction is of order \ensuremath{\epsilon}: its amplitude vanishes in the Ising case q=2.

The dynamics of triple convection

Abstract In the parameter space of a fluid subject to triple convection, there is a critical hypersurface on which three growth rates of linear theory vanish and all the rest are distinctly negative. When parameter values are chosen to place the system very near to this polycritical condition, the temporal behavior of the system may be complicated and even chaotic. This remark, based on rather general considerations (Arneodo et al., 1984), is here illustrated by an example from GFD (Arneodo et al., 1982): two-dimensional Boussinesq thermohaline convection (or semi-convection) in a planeparallel layer rotating about a vertical axis and subject to mathematically convenient boundary conditions. The treatment is made in terms that show why the results may apply to many fluid dynamical systems or indeed to other kinds of triply unstable systems and, using both amplitude equations and mappings, we discuss the chaos that can arise.