Singly generated planar algebras of small dimension, Part II

We classified in Bisch and Jones (Duke Math. J. 101 (2000) 41) all spherical C∗-planar algebras generated by a non-trivial 2-box subject to the condition that the dimension of N′∩M₂ is ?12. We showed that they are given by the Fuss-Catalan systems discovered in Bisch and Jones (Invent. Math. 128 (1997) 89) and one exceptional planar algebra. In the present paper, we extend these results and show that there is only one spherical C∗-planar algebra generated by a single non-trivial 2-box if the dimension of N′∩M₂ is 13. It is given by the standard invariant of the crossed product subfactor , where D₅ denotes the dihedral group with 10 elements.     

Hydrodynamics of dielectric fluid films

We introduce a general hydrodynamic model to study the stability of lipid films against thermal fluctuations. As one novel aspect the model accounts before all for a complete intrinsic surface rheology of the film interfaces. Thus the rheological behaviour of the surface adsorbed lipids is modelled which screen the hydrophobic film interior against the aqueous exterior. For coloured films we demonstrate first the influence of electrical forces on the dynamics and film stability. For that we perform a linear stability analysis on a simplified mechanically symmetric film with i) symmetric surface charge distribution and ii) linear electric potential drop across the film. Based on the complete film model we then categorize the complete set of solutions of the linearized equations of motion and we study the growth rates of unstable film modes. Finally we discuss the stability properties of a black film after introducing a repulsive mechanism due to the steric hindrance of the interfacial lipids.     

Horizontal Redistribution of Two Fluid Phases in a Porous Medium: Experimental Investigations

Classical models for flow and transport processes in porous media employ the so-called extended Darcy’s Law. Originally, it was proposed empirically for one-dimensional isothermal flow of an incompressible fluid in a rigid, homogeneous, and isotropic porous medium. Nowadays, the extended Darcy’s Law is used for highly complex situations like non-isothermal, multi-phase and multi-component flow and transport, without introducing any additional driving forces. In this work, an alternative approach by Hassanizadeh and Gray identifying additional driving forces were tested in an experimental setup for horizontal redistribution of two fluid phases with an initial saturation discontinuity. Analytical and numerical solutions based on traditional models predict that the saturation discontinuity will persist, but a uniform saturation distribution will be established in each subdomain after an infinite amount of time. The pressure field, however, is predicted to be continuous throughout the domain at all times and is expected to become uniform when there is no flow. In our experiments, we also find that the saturation discontinuity persists. But, gradients in both saturation and pressure remain in both subdomains even when the flow of fluids stops. This indicates that the identified additional driving forces present in the truly extended Darcy’s Law are potentially significant.     

Nonspecific interaction forces at water-membrane interface by forced molecular dynamics simulations

Nonspecific interactions are the main driving forces for the behavior of molecules with great affinity for biologic membranes. To investigate not only the molecular details of these interactions but to estimate their magnitude as well, the theoretical method of Forced Molecular Dynamics Simulations, based on the Atomic Force Spectroscopy experimental technique, was applied. In this approach, an additional one-dimensional elastic force, representing the cantilever probe, was incorporated to the force field of a Molecular Dynamics computational program. This force represents a spring fixed on one end to a selected atom of the molecule; the other end of the spring is displaced at constant velocity to pull the molecule out of the membrane. The force experimented by the molecule due to the spring, is proportional to the spring elongation relative to its equilibrium position. This value is registered during the entire simulation, and its maximum value will determine the molecule-membrane interaction force. Nonexplicit medium simulations were carried out. Polar and apolar media were considered according to their polarizability degree and a specific dielectric constant value was assigned. In this approach, the membrane was considered as the apolar region limited by two flat surfaces with a polar aqueous medium. The potential energy discontinuity at the interfaces was smoothed by considering the polarization-induced effects using the image method. The results of this methodology are presented using a small system, a single Alanine amino acid model, which enables extended simulations in a microsecond time scale. The confinement of this amino acid at the interface reduces its degrees of freedom and forces it to adopt one of the six defined conformations. A correlation between these stable structures at the water-membrane interface and the interaction force value was determined.     

The planar algebra of group-type subfactors

If G is a countable, discrete group generated by two finite subgroups H and K and P is a II₁ factor with an outer G-action, one can construct the group-type subfactor PH⊂P?K introduced by Haagerup and the first author to obtain numerous examples of infinite depth subfactors whose standard invariant has exotic growth properties. We compute the planar algebra of this subfactor and prove that any subfactor with an abstract planar algebra of “group type” arises from such a subfactor. The action of Jones' planar operad is determined explicitly.     

On the rational approximation of Markov functions,with applications to the computation of Markov functions of Toeplitz matrices

Numerical Algorithms - We investigate the problem of approximating the matrix function f(A) by r(A), with f a Markov function, r a rational interpolant of f, and A a symmetric Toeplitz matrix. In a...     

Polarization effects on peptide conformations at water–membrane interface by molecular dynamics simulation

The electrostatic image method was applied to investigate the conformation of peptides characterized by different hydrophobicities in a water–membrane interface model. The interface was represented by a surface of discontinuity between two media with different dielectric constants, taking into account the difference between the polarizabilities of the aqueous medium and the hydrocarbon one. The method consists of a substitution of the real problem, which involves the charges and the induced polarization at the surface of discontinuity, by a simpler problem formed with charges and their images. The electric field due to the polarization induced at the surface by charge q was calculated using a hypothetical charge q′ (image of q), symmetrically located on the opposite side of the surface. The value of q′ was determined using the appropriate electrostatic boundary conditions at the surface. By means of this procedure, the effect of the interface can be introduced easily in the usual force field. We included this extension in the computational package that we are developing for molecular dynamics simulations (Thor ). The peptides studied included hydrophilic tetraaspartic acid (Asp–Asp–Asp–Asp), tetralysine (Lys–Lys–Lys–Lys), hydrophobic tetrapeptide (His–Phe–Arg–Trp), an amphiphilic fragment of β-endorphin, and the signal sequence of the E. coli λ-receptor. The simulation results are in agreement with known experimental data regarding the behavior of peptides at the water–membrane interface. An analysis of the conformational dynamics of the signal sequence peptide at the interface was performed over the course of a few nanoseconds. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 971–982, 1999     

Algebras associated to intermediate subfactors

The Temperley-Lieb algebras are the fundamental symmetry associated to any inclusion of factors with finite index. We analyze in this paper the situation when there is an intermediate subfactor of . The additional symmetry is captured by a tower of certain algebras associated to . These algebras form a Popa system (or standard lattice) and thus, by a theorem of Popa, arise as higher relative commutants of a subfactor. This subfactor gives a free composition (or minimal product) of an and an subfactor. We determine the Bratteli diagram describing their inclusions. This is done by studying a hierarchy of colored generalizations of the Temperley-Lieb algebras, using a diagrammatic approach, à la Kauffman, that is independent of the subfactor context. The Fuss-Catalan numbers appear as the dimensions of our algebras. We give a presentation of the and calculate their structure in the semisimple case employing a diagrammatic method. The principal part of the Bratteli diagram describing the inclusions of the algebras is the Fibonacci graph. Our algebras have a natural trace and we compute the trace weights explicitly as products of Temperley-Lieb traces. If all indices are , we prove that the algebras and coincide. If one of the indices is , is a quotient of and we compute the Bratteli diagram of the tower . Our results generalize to a chain of intermediate subfactors. Oblatum 1-XII-1995 & 1-VII-1996     

Complete set of Hopf bifurcation in an autocatalytic ring network

We investigate the Hopf bifurcation for a five species chemical ring network with an autocatalytic reaction. We show that the bifurcation hypersurface in the rate constants space is the boundary of a simply connected set. We use a numerical method to calculate this hypersurface.