Finite Critical Value Research Articles

Structure of Julia sets of polynomial semigroups with bounded finite postcritical set

Let G be a semigroup of complex polynomials (under the operation of composition of functions) such that there exists a bounded set in the plane which contains any finite critical value of any map g ∈ G. We discuss the dynamics of such polynomial semigroups as well the structure of the Julia set of G. In general, the Julia set of such a semigroup G may be disconnected, and each Fatou component of such G is either simply connected or doubly connected. In this paper, we show that for any two distinct Fatou components of certain types (e.g., two doubly connected components of the Fatou set), the boundaries are separated by a Cantor set of quasicircles (with uniform dilatation) inside the Julia set of G. Furthermore, we provide results concerning the (semi-)hyperbolicity of such semigroups as well as discuss various topological consequences of the postcritically boundedness condition.

Flow instability and shear localization in a drilling mud

To have a better knowledge of problems occurring with drilling fluids in complex wells, we carried out a detailed rheological analysis of a typical drilling mud at low shear rates using both conventional rheometry and MRI velocimetry. We show the existence of a viscosity bifurcation effect: Below a critical stress value, the mud tends to completely stop flowing, whereas beyond this critical stress, it reaches an apparent shear rate larger than a finite (critical) value, and no stable flows can be obtained between this critical shear rate value and zero. These results are confirmed by MRI velocity profiles, which exhibit a slope break at the interface between the solid and the liquid phases inside the Couette geometry. Moreover, this viscosity bifurcation is a transient phenomenon, the progressive development of which can be observed by MRI. A further examination of MRI data shows that, in the transient regime, the shear rate does not vary monotonously in the rheometer gap and is particularly large along the outer (rough) cylinder, which might be at the origin of the development of a region of constant shear rate in the apparent flow curve.

Finite-Temperature Charge-Ordering Transition and Fluctuation Effects in Quasi One-Dimensional Electron Systems at Quarter Filling

Finite-temperature charge-ordering phase transition in quasi one-dimensional (1D) molecular conductors is investigated theoretically, based on a quasi 1D extended Hubbard model at quarter filling with interchain Coulomb repulsion $V_\perp$. The interchain term is treated within mean-field approximation whereas the 1D fluctuations in the chains are fully taken into account by the bosonization theory. Three regions are found depending on how the charge ordered state appears at finite temperature when $V_\perp$ is introduced: (i) weak-coupling region where the system transforms from a metal to a charge ordered insulator with finite transition temperature at a finite critical value of $V_\perp$, (ii) an intermediate region where this transition occurs by infinitesimal $V_\perp$ due to the stability of inherent 1D fluctuation, and (iii) strong-coupling region where the charge ordered state is realized already in the purely 1D case, of which the transition temperature becomes finite with infinitesimal $V_\perp$. Analytical formula for the $V_\perp$ dependence of the transition temperature is derived for each region.

Dynamic analysis of parametrically excited laminated composite curved panels under non-uniform edge loading with damping

The present work deals with the problem of the occurrence of combination resonances in parametrically excited simply supported laminated composite doubly curved panels under non-uniform edge loading. The first-order shear deformation theory is used to model the panels, considering the effects of transverse shear deformation and rotary inertia. Finite element technique is applied for obtaining the non-uniform stress distribution within the curved panels. The method of multiple scale is used to derive the analytical expressions for the instability regions. It is shown that, besides the principal instability region at Ω = 2 ω 1, other cases of Ω = ω m + ω n , related to other modes, can be of major importance and yield a significantly enlarged instability regions. The effects of non-uniform loading, damping, number of layers, orthotropy, thickness and the static load factor on the dynamic instability behavior of the simply supported laminated composite curved panels are studied. The results show that under localized edge loading, combination resonance zones are as important as simple resonance zones. The effects of damping show that there is finite critical value of dynamic load factor for each instability region below which the curved panels cannot become unstable.

Non-Mean-Field Behavior of the Contact Process on Scale-Free Networks

We present an analysis of the classical contact process on scale-free networks. A mean-field study, both for finite and infinite network sizes, yields an absorbing-state phase transition at a finite critical value of the control parameter, characterized by a set of exponents depending on the network structure. Since finite size effects are large and the infinite network limit cannot be reached in practice, a numerical study of the transition requires the application of finite size scaling theory. Contrary to other critical phenomena studied previously, the contact process in scale-free networks exhibits a nontrivial critical behavior that cannot be quantitatively accounted for by mean-field theory.

Tube formation and spontaneous budding in a fluid charged membrane

We determine the equations governing the equilibrium shape of an axially symmetric charged fluid membrane under the action of an external field. We consider that the charges are free to diffuse along the membrane, and we neglect the effect of screening counterions. By numerically integrating these equations for a membrane spread across a hole by a constant tension, we show that there exists a threshold electric field above which an infinitely long tube can be pulled. The threshold field for pulling a tube decreases as the surface charge density of the membrane increases, reaching zero at a finite critical value. Above this critical density of charge, the membrane spontaneously buds, in the absence of externally applied fields, because of the electrostatic repulsion of its charges.

Continuous-time diffusion Monte Carlo method applied to the quantum dimer model

A continuous-time formulation of the Diffusion Monte Carlo method for lattice models is presented. In its simplest version, without the explicit use of trial wavefunctions for importance sampling, the method is an excellent tool for investigating quantum lattice models in parameter regions close to generalized Rokhsar-Kivelson points. This is illustrated by showing results for the quantum dimer model on both triangular and square lattices. The potential energy of two test monomers as a function of their separation is computed at zero temperature. The existence of deconfined monomers in the triangular lattice is confirmed. The method allows also the study of dynamic monomers. A finite fraction of dynamic monomers is found to destroy the confined phase on the square lattice when the hopping parameter increases beyond a finite critical value. The phase boundary between the monomer confined and deconfined phases is obtained.

Bosons in Optical Lattices – from the Mott Transition to the Tonks–Girardeau Gas

We present results from quantum Monte Carlo simulations of trapped bosons in optical lattices, focusing on the crossover from a gas of softcore bosons to a Tonks-Girardeau gas in a one-dimensional optical lattice. We find that depending on the quantity being measured, the behavior found in the Tonks-Girardeau regime is observed already at relatively small values of the interaction strength. A finite critical value for entering the Tonks-Girardeau regime does not exist. Furthermore, we discuss the computational efficiency of two quantum Monte Carlo methods to simulate large scale trapped bosonic systems: directed loops in stochastic series expansions and the worm algorithm.

Nonadiabatical approach to the Dzyaloshinsky-Moriya interaction in theXYspin chain coupled with quantum phonons

Using the model of the DM interaction in $XY$ antiferromagnetic spin chain with spin-phonon coupling, the nonadiabatic effects on the DM interaction have been studied by developing a nonadiabatic analytical approach. The results show that in the nonadiabatic case, to a certain finite phonon frequency, there exists a finite critical value of spin-phonon coupling. As the spin-phonon coupling decreases to the critical value, the system becomes gapless and the spin dimerization is destroyed. The DM interaction leads the system to have a finite critical value of spin-phonon coupling even in the adiabatic limit. The increase in staggered DM interaction will decrease the critical value of spin-phonon coupling, but increase that of phonon frequency, therefore, favors the dimerization. The nonadiabaticity plays an important role in suppressing the enhancement effects of the DM interaction to the dimerization. The dimerized ground state when DM interaction is present can be destroyed by the quantum lattice fluctuations. The threshold value of $\ensuremath{\beta}$ changing the effect of the DM interaction on the dimerization from suppression to promotion is not simply a constant but a crossover. For appropriate fixed values of spin-phonon coupling, phonon frequency and $\ensuremath{\beta}$, as $D$ increases, the system undergoes a reentry of phase between spin-dimer state and gapless state.

Puncturing a thin elastic sheet

We use membrane theory to analyze the puncturing of a thin solid circular isotropic elastic sheet by a rigid axisymmetric indenter. A solution is obtained in which a hole is formed at the center of the sheet with an interior annulus in frictionless contact with the cylindrical surface. The contacting part is in a state of pure hoop stress with the corresponding hoop stretch exhibiting a strong singularity at the origin. Conditions are given ensuring that the solution has finite total energy and it is shown to be energetically favored over unpunctured states for transverse indenter displacements exceeding a finite critical value.

Effect of the Dzyaloshinsky–Moriya interaction on the spin-Peierls dimerization in the XY spin chain with spin–phonon coupling

Using the model of the DM interaction in XY antiferromagnetic spin chain with spin–phonon coupling, the influences of the DM interaction and the effects of the quantum lattice fluctuations on the spin-Peierls instability have been studied by developing a nonadiabatic analytical approach. The results show that in the nonadiabatic case, to a certain finite phonon frequency, there exists a finite critical value of spin–phonon coupling. As the spin–phonon coupling decreases to the critical value, the system becomes gapless and the spin dimerization is destroyed. The increase in staggered DM interaction will decrease the critical value of spin–phonon coupling, but increase that of phonon frequency, therefore, favours the dimerization. Whether the effect of the DM interaction on the dimerization is suppression or promotion depends on the value of the staggered DM interaction parameter β. There exists a finite threshold value of β and this value is not simply a constant but a crossover. For appropriate fixed values of the spin–phonon coupling, the phonon frequency and β, the system undergoes a reentry into spin-dimer state as D/ J increases. The spin–phonon coupling has a finite divide-line value determining the change of the spin-dimer order parameter to be increase or decrease as β increases. The nonadiabaticity plays an important role in suppressing the enhancement effects of the DM interaction to the dimerization.

Transition from ciliary to flapping mode in a swimming mollusc: flapping flight as a bifurcation in ${\hbox{\it Re}}_\omega$

From observations of swimming of the shell-less pteropod mollusc Clione antarctica we compare swimming velocities achieved by the organism using ciliated surfaces alone with velocities achieved by the same organism using a pair of flapping wings. Flapping dominates locomotion above a swimming Reynolds number Re in the range 5-20. We test the hypothesis that Re 5-20 marks the onset of 'flapping flight' in these organisms. We consider the proposition that forward, reciprocal flapping flight is impossible for locomoting organisms whose motion is fully determined by a body length L and a frequency w below some finite critical value of the Reynolds number Re ω = ωL 2 /ν. For a self-similar family of body shapes, the critical Reynolds number should depend only upon the geometry of the body and the cyclic movement used to locomote. We give evidence of such a critical Reynolds number in our data, and study the bifurcation in several simplified theoretical models. We argue further that this bifurcation marks the departure of natural locomotion from the low Reynolds number or Stokesian realm and its entry into the high Reynolds number or Eulerian realm

Flow and Thermal Convection in Full-Zone Liquid Bridges of Wide-Ranging Aspect Ratio

Flow and thermal effects concerned with liquid crystal growth are studied nonlinearly for low Prandtl numbers, in an axisymmetric steady configuration with end walls present. Full solutions are obtained by finite-element simulation to examine the influences of aspect ratio, Marangoni number and Rayleigh number. In particular, a “lemonhead” phenomenon is found in which the velocity profiles acquire a very localised bell-shaped form as jet-like flow starts to emerge, when the relative size of the Marangoni number increases above a finite critical value which is identified. Wide-domain and narrow-domain analyses are then presented, showing favourable agreement with the full solutions and explaining the impact of the end walls on the motion, especially through slender flow modelling in the narrow-domain context. Sensitive scalings for the relative effects of the Marangoni and Rayleigh numbers emerge, and the lemonhead phenomenon is also accounted for as a substantial change in the convective flow structure.

Effect of gauge boson mass on chiral symmetry breaking in three-dimensional QED

In three-dimensional quantum electrodynamics (QED$_{3}$) with massive gauge boson, we investigate the Dyson-Schwinger equation for the fermion self-energy in the Landau gauge and find that chiral symmetry breaking (CSB) occurs when the gauge boson mass $\xi$ is smaller than a finite critical value $\xi_{cv}$ but is suppressed when $\xi > \xi_{cv}$. We further show that the critical value $\xi_{cv}$ does not qualitatively change after considering higher order corrections from the wave function renormalization and vertex function. Based on the relation between CSB and the gauge boson mass $\xi$, we give a field theoretical description of the competing antiferromagnetic and superconducting orders and, in particular, the coexistence of these two orders in high temperature superconductors. When the gauge boson mass $\xi$ is generated via instanton effect in a compact QED$_{3}$ of massless fermions, our result shows that CSB coexists with instanton effect in a wide region of $\xi$, which can be used to study the confinement-deconfinement phase transition.

Phase diagram and correlation function of a spin-Peierls system in the ground state: Effects of the quantum lattice fluctuations

Using a one-dimensional antiferromagnetic Heisenberg chain model with spin-phonon interaction, the effects of quantum lattice fluctuations on the phase diagram and correlation functions in the ground state of the spin-Peierls system are investigated by developing an analytical approach based on the unitary transformation method. To study the nonadiabatic case, the Green's function method is used to implement the perturbation treatment. We show that when the spin-phonon coupling constant decreases or phonon frequency increases, the lattice dimerization and the spin dimerization gap decrease gradually. At some finite critical value, the Peierls dimerization is destroyed by the quantum lattice fluctuations and the system becomes gapless. The inverse-square-root singularity of the density of states at the gap edge in the adiabatic case disappears because of the nonadiabatic effect, which is consistent with the measurement of the optical absorption spectrum in quasi-one-dimensional spin-Peierls systems. The dimerization gap is much more reduced by the quantum lattice fluctuations than the spin dimer order parameter is. This could result in a true spin dimerization gap that is smaller than the activation energy of the optical absorption spectrum.

Flow of an Inviscid Rotating Liquid into an Elevated Sink

Long's solution (Q. Jl Mech. Appl. Math. 9 (1956) 385-393) for steady flow of a rotating liquid into a hydrodynamic sink at the base of a cylinder is extended to the case where the sink is elevated above the cylinder base. The solution satisfies the nonlinear Euler equations exactly for all sink heights and for swirl ratios (inverse Rossby numbers) less than a finite critical value. Flow above the elevated sink is qualitatively similar to that in Long's original solution, while flow beneath the sink is comprised of a very weak meridional circulation and an intense nearly-potential vortex.

Phase diagram and optical conductivity of the one-dimensional spinless Holstein model

The effects of quantum lattice fluctuations on the Peierls transition and the optical conductivity in the one-dimensional Holstein model of spinless fermions have been studied by developing an analytical approach, based on the unitary transformation method. We show that when the electron-phonon coupling constant decreases to a finite critical value the Peierls dimerization is destroyed by the quantum lattice fluctuations. The dimerization gap is much more reduced by the quantum lattice fluctuations than the phonon order parameter. The calculated optical conductivity does not have the inverse-square-root singularity but have a peak above the gap edge and there exists a significant tail below the peak. The peak of optical-conductivity spectrum is not directly corresponding to the dimerized gap. Our results of the phase diagram and the spectral-weight function agree with those of the density matrix renormalization group and the exact diagonalization methods.

Antiferromagnetic Phase Transition and Crossover to Fermi Liquid Phase in a Weakly Coupled Half-Filled Chain System

Effects of the intrachain electron-electron umklapp process and the interchain one-particle hopping, t ⊥ , in a weakly coupled half-filled chain system have been studied, based on the perturbative renormalization-group approach. By solving the scaling equations, we found that the intrachain umklapp process strongly suppresses an interchain one-particle process and causes a finite critical value for t ⊥ , t ⊥ c . For t ⊥ t ⊥ c , the system undergoes a crossover to the Fermi liquid phase where the interchain coherent propagation of quasi-particles occurs.

Simple and combination resonances of rectangular plates subjected to non-uniform edge loading with damping

The effects of damping on the behaviour of simple and combination resonances of simply-supported rectangular plates subjected to non-uniform in-plane periodic loading are studied. Finite element formulation is applied to obtain the equilibrium equation for the plate. The relations for the boundaries of parametric instability regions are obtained by using the method of multiple scales. The results show that under localized edge loading, combination resonance zones are as important as simple resonance zones. For nearly uniform loading, the combination resonance zones are very small in width and may disappear in the presence of slight damping. The effects of damping show that there is a finite critical value of the dynamic load factor for each zone below which the plate cannot become dynamically unstable. It is also shown that the effects of damping on the combination resonances may be destabilizing under certain conditions.

Free quantum fields on the Poincaré group

A class of free quantum fields defined on the Poincaré group is described by means of their two-point vacuum expectation values. They are not equivalent to fields defined on the Minkowski space–time and they are ‘‘elementary’’ in the sense that they describe particles that transform according to irreducible unitary representations of the symmetry group, given by the product of the Poincaré group and of the group SL(2,C) considered as an internal symmetry group. Some of these fields describe particles with positive mass and arbitrary spin and particles with zero mass and arbitrary helicity or with an infinite helicity spectrum. In each case the allowed SL(2,C) internal quantum numbers are specified. The properties of local commutativity and the limit in which one recovers the usual field theories in Minkowski space–time are discussed. By means of a superposition of elementary fields, one obtains an example of a field that presents a broken symmetry with respect to the group Sp(4,R) that survives in the short-distance limit. Finally, the interaction with an accelerated external source is studied and it is shown that, in some theories, the average number of particles emitted per unit of proper time diverges when the acceleration exceeds a finite critical value.