Equal Curvature Research Articles

Phase-Field Modeling and Simulations of Lipid Membranes Coupling Composition with Membrane Mechanical Properties

The plasma membrane, a lipid bilayer membrane surrounding all mammalian cells, is not homogeneous, but rather contains domains termed ‘rafts,’ defined as regions enriched with cholesterol and saturated lipids. Understanding how and why these rafts form is of great importance to cell biologists and immunologists, since they are involved in many important cell functions and processes including endocytosis, cell adhesion, signaling, protein organization, lipid regulation, and infection by pathogens. These raft structures also show great potential for technological applications, especially in connection with biosensors and drug delivery systems. We examine the formation and evolution of lipid raft-like domains in multicomponent lipid membrane vesicles using a continuum-level simulation method. Our objective is to investigate how various physical parameters input into the model, such as spontaneous curvature, bending rigidity, and phase fraction, affect the dynamics and equilibrium morphological phases formed in two-phase lipid membrane systems. This model is applied to membranes with spherical background geometries, simulating the compositional and shape evolution of lipid vesicles, coupled using a modified Helfrich free energy. The compositional evolution is modeled using a phase-field method and is described by a Cahn-Hilliard-type equation, while the shape changes are described by relaxation dynamics in which the vesicle surface area is conserved. We find that the compositional and morphological evolution are significantly altered when the mechanical coupling is present by comparing the results with those of systems where this coupling is absent. More specifically, we find that the evolution is significantly slowed when the phases have equal and opposite spontaneous curvatures and are present in roughly equal amounts. We also investigate equilibrium shapes formed by completely phase-separated vesicles as a function of spontaneous curvature, bending rigidity, and phase fraction.

Gluing constructions amongst constant mean curvature hypersurfaces in $${\mathbb {S}^{n+1}}$$

Four constructions of constant mean curvature (CMC) hypersurfaces in \({\mathbb {S}^{n+1}}\) are given, which should be considered analogues of ‘classical’ constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an equator, are shown to exist for all the values of the mean curvature. Second, a hypersurface is constructed which consists of two chains of spheres winding around a pair of orthogonal equators, showing that Delaunay-like hypersurfaces can be fused together in a symmetric manner. Third, a Delaunay-like handle can be attached to a generalized Clifford torus of the same mean curvature. Finally, two generalized Clifford tori of equal but opposite mean curvature of any magnitude can be attached to each other by symmetrically positioned Delaunay-like ‘arms’. This last result extends Butscher and Pacard’s doubling construction for generalized Clifford tori of small mean curvature.

Obtaining the Varshni potential function using the 2-body Kaxiras-Pandey parameters

A generalized version of the Varshni potential function was adopted by Kaxiras and Pandey for describing the 2-body energy portion of multi-body condensed matter. The former's simplicity and resemblance to a Morse potential allows faster computation while the latter's greater number of parameters allows better curve-fitting of spectroscopic data. This paper shows one set of parameter conversion from the Varshni function to the 2-body portion of the Kaxiras-Pandey function, and vice versa two sets of parameter conversion. The latter two sets reveal good correlation between plotted curves, and were verified by the imposition of equal energy curvatures at equilibrium and equal energy integral from equilibrium to dissociation. These parameter conversions can also be attained more easily by equating the product of indices (for short range) and the summation of index reciprocals (for long range).

Alternative scaling factor between Lennard-Jones and Exponential-6 potential energy functions

In spite of the ease of computing the Lennard-Jones function for describing van der Waals energy of interaction, the Exponential-6 function is normally adopted due to its better description of short range interaction. Hence scaling factors of 13.772 and 12 have been used to give equal curvature at the minimum well-depth and equal long-range energy description, respectively, when compared to the Lennard-Jones (12-6) potential. This paper proposes an intermediate scaling factor of 12.6533 on the basis of zero integral error from the well-depth to infinite separation. It is herein shown that the use of this intermediate scaling factor divides the entire Exponential-6 computation into three stages instead of two, thereby smoothening the energy curve by reducing the computed energy drop at the scaling factor switching point. In cases where a single scaling factor is desired as a results highly complex dynamical molecular system, the proposed intermediate scaling factor gives significantly smaller overall error.

Corneal Asphericity and Spherical Aberration

To summarize the various values of asphericity in different notations and present how corneal asphericity, corneal curvature, and entrance pupil diameter influence the longitudinal spherical aberration of the anterior corneal surface. After the conversion factors between the different asphericity notations were described, finite ray tracing through a conic section that models the anterior cornea profile was performed. The anterior cornea was given a range of curvatures and asphericities and a range of entrance pupil diameters. If the value of asphericity remains constant, longitudinal spherical aberration increases with the square of the entrance pupil diameter. If the pupil diameter remains fixed, the spherical aberration becomes a function of the value of asphericity, the refractive index, and the radius of curvature. If the refractive index, pupil diameter and asphericity are considered constant, the spherical aberration will decrease if the corneal surface flattens and increase as the cornea becomes steeper. In this way, with the same shape factor and with the same starting apical radius, longitudinal spherical aberration became a function of the surgically induced refractive change. With equal curvature, the longitudinal spherical aberration becomes negative if the surface is more prolate than perfect Cartesian oval; it will become positive if it is less prolate, spherical, or oblate. A conversion chart for corneal asphericity notations with the corresponding spherical aberration and a diagram reporting values of asphericity necessary to maintain the physiological value of the corneal spherical aberration after refractive procedures may be useful tools in corneal surgery.

Long range relationship between Morse and Lennard–Jones potential energy functions

Arising from the use of the Morse function–which is well-known for its applicability for describing bonded interaction energy–in van der Waals systems, an attempt is made herein to express parameters of the Lennard–Jones potential function in terms of the Morse function to enable normalized comparison. In a departure from previous work where the parameter relationships enforce equal curvature at the minimum well-depth, the present approach replaces this rule with equal area above the curves for 1 ≤ (r/R) ≤ ∞. Results show good approximations of the Morse function to the Lennard–Jones curve and vice versa. Comparison with the previous relation for short range interaction shows that the present relations offer superior agreement with the Lennard–Jones function over a longer range. The conversion relations provide a cost-effective, less time-consuming and reasonably reliable method for obtaining Morse parameters from those of the Lennard–Jones function and vice versa.

Relationship and discrepancies between the Extended-Rydberg and the Generalized Buckingham potential energy functions

This paper relates the interatomic energy according to the Extended-Rydberg and the Generalized Buckingham potential functions by applying a Maclaurin series expansion on the latter and thereafter comparing it with the former. In so doing, the plotted curves of these two functions not only show equal curvature at the equilibrium state, but also reveal a discrepancy for the finite distortion. It is shown that, when equated at equilibrium, the Generalized Buckingham gives a lower energy in comparison to the Extended-Rydberg at finite bond compression and stretching. However, the energy difference diminishes when the interatomic distance exceeds twice that at equilibrium. Due to such discrepancies upon comparative normalization, it would be beneficial for computational chemists to select the appropriate potential function for the purpose of conservative molecular modeling. .

Comparative study of six rotary nickel–titanium systems and hand instrumentation for root canal preparation

To compare ex vivo various parameters of root canal preparation using a manual technique and six different rotary nickel-titanium (Ni-Ti) instruments (FlexMaster, System GT, HERO 642, K3, ProTaper, and RaCe). A total of 147 extracted mandibular molars were divided into seven groups (n = 21) with equal mean mesio-buccal root canal curvatures (up to 70 degrees), and embedded in a muffle system. All root canals were prepared to size 30 using a crown-down preparation technique for the rotary nickel-titanium instruments and a standardized preparation (using reamers and Hedströem files) for the manual technique. Length modifications and straightening were determined by standardized radiography and a computer-aided difference measurement for every instrument system. Post-operative cross-sections were evaluated by light-microscopic investigation and photographic documentation. Procedural errors, working time and time for instrumentation were recorded. The data were analysed statistically using the Kruskal-Wallis test and the Mann-Whitney U-test. No significant differences were detected between the rotary Ni-Ti instruments for alteration of working length. All Ni-Ti systems maintained the original curvature well, with minor mean degrees of straightening ranging from 0.45 degrees (System GT) to 1.17 degrees (ProTaper). ProTaper had the lowest numbers of irregular post-operative root canal diameters; the results were comparable between the other systems. Instrument fractures occurred with ProTaper in three root canals, whilst preparation with System GT, HERO 642, K3 and the manual technique resulted in one fracture each. Ni-Ti instruments prepared canals more rapidly than the manual technique. The shortest time for instrumentation was achieved with System GT (11.7 s). Under the conditions of this ex vivo study all Ni-Ti systems maintained the canal curvature, were associated with few instrument fractures and were more rapid than a standardized manual technique. ProTaper instruments created more regular canal diameters.

Helicoidal surfaces under the cubic screw motion in Minkowski 3-space

In this paper, we construct helicoidal surfaces under the cubic screw motion with prescribed mean or Gauss curvature in Minkowski 3-space E 1 3 . We also find explicitly the relation between the mean curvature and Gauss curvature of them. Furthermore, we discuss helicoidal surfaces under the cubic screw motion with H 2 = K and prove that these surfaces have equal constant principal curvatures.

Hysteresis in the static aerodynamic characteristics of a curved-profile wing

The effect of profile asymmetry, Reynolds number, and angles of deflection of high-lift devices on the hysteresis loop shape in the dependences of the static aerodynamic forces and moments on angles of attack is studied. These dependences are measured in wind tunnels during testing of rectangular wings of equal thickness but different curvature. The wings have deflectable flaps spanning the wing trailing edge.

Poincaré’s closed geodesic on a convex surface

We present a new proof for the existence of a simple closed geodesic on a convex surface $M$. This result is due originally to Poincaré. The proof uses the ${2k}$-dimensional Riemannian manifold ${_{k}\Lambda M} = \operatorname {(briefly)} \Lambda$ of piecewise geodesic closed curves on $M$ with a fixed number $k$ of corners, $k$ chosen sufficiently large. In $\Lambda$ we consider a submanifold $\overset {\approx }{\Lambda }_{0}$ formed by those elements of $\Lambda$ which are simple regular and divide $M$ into two parts of equal total curvature $2\pi$. The main burden of the proof is to show that the energy integral $E$, restricted to $\overset {\approx }{\Lambda }_{0}$, assumes its infimum. At the end we give some indications of how our methods yield a new proof also for the existence of three simple closed geodesics on $M$.

4-Webs in the Plane and Their Linearizability

We investigate the linearizability problem for different classes of 4-webs in the plane. In particular, we apply the linearizability conditions, recently found by Akivis, Goldberg and Lychagin, to confirm that a 4-web MW (Mayrhofer's web) with equal curvature forms of its 3-subwebs and a nonconstant basic invariant is always linearizable (this result was first obtained by Mayrhofer in 1928; it also follows from the papers of Nakai). Using the same conditions, we further prove that such a 4-web with a constant basic invariant (Nakai's web) is linearizable if and only if it is parallelizable. Next we study four classes of the so-called almost parallelizable 4-webs APW a ,a=1,2,3,4 (for them the curvature K=0 and the basic invariant is constant on the leaves of the web foliation X a ), and prove that a 4-web APW a is linearizable if and only if it coincides with a 4-web MW a of the corresponding special class of 4-webs MW. The existence theorems are proved for all the classes of 4-webs considered in the paper.

Angle theorems for the Lagrangian mean curvature flow

We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle $\alpha$ for the corresponding Lagrangian submanifold in the cross product space $L\times M$ satisfies $\text{osc}(\alpha)\le \pi$ . If one considers a 4-dimensional Kahler-Einstein manifold $\overline{M}$ of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that $L\subset\overline{M}$ is a compact oriented Lagrangian submanifold w.r.t. J such that the Kahler form $\overline{\kappa}$ w.r.t.K restricted to L is positive and $\text{osc}(\alpha)\le \pi$ , then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. $\overline{\kappa}$ .

Reliability-Based Risk Assessment of Rubble Mound Breakwaters Under Tsunami Attack

In the reliability-based risk assessment, the second order reliability index (βII) method and the Conditional Expectation Monte Carlo (CEMC) simulation were utilised in order to analyse the safety levels of Haydarpasa Port main breakwater, Sea of Marmara, Turkey. The failure probability was forecasted in the βII method by approximating the Hudson performance surface with a second-degree polynomial having an equal curvature at the design point where the design and structural parameters were taken into consideration. In the past, applications of reliability-based risk assessment methodology in Turkey for design conditions, only wave characteristics, tidal range, storm surge, wave set-up and the structural system parameters were included. Tsunami risk was not considered a major design parameter and was not included in the computations.In this study, a structural stability criterion in coastal engineering is suggested to achieve a common definition of reliability including the tsunami risk. The model introduced in this study is a practical technique for reliability-based risk assessment of breakwaters subject to tsunami risk. In order to determine the occurrence probability under design conditions, which is a function of storm waves, tidal range, storm surge and tsunami height, the CEMC simulation was interrelated with the βII method. From the reliability-based risk assessment model applied to Haydarpasa port as a pilot study in Turkey it was found that, inclusion of the tsunami risk increases the failure risk of the structure, and as lifetime of the structure increases, the impact of tsunami risk on the failure mechanism is more important. For Haydarpasa port main breakwater, tsunami was not the key design parameter when compared to storm wave, however, in places with great seismic activity, tsunami risk may be very significant depending on the occurrence probability and the magnitude of the tsunami.

Asymmetric kinks: stabilization by entropic forces.

Asymmetric kinks bridging two adjacent potential valleys of equal depth but different curvature are unstable against phonon modes. When coupled to a heat bath, a kink-bearing string tends to cross over into the shallower valley; kinks are thus predicted to drift in the appropriate direction with velocity proportional to the temperature, in close agreement with numerical simulation. When contrasted by a mechanical bias, these entropic forces give rise to a rich phenomenology that includes configurational phase transitions, double-kink dissociation, and noise-directed signal transmission.

NOISE RECTIFICATION AS AN ENTROPIC EFFECT

Asymmetric kinks bridging two adjacent potential valleys of equal depth but different curvature are predicted to drift with velocity proportional to their temperature, in close agreement with numerical simulation. Such a (white) noise rectification phenomenon, not to be mistaken for a ratchet-like effect, is an entropic mechanism with potential impact on noise-sustained signal propagation in linear bistable arrays, of both biological and microelectronic nature.

On the infinitesimal affine rigidity of ellipsoids in the n-space

Due to R. Schneider 1967 an ellipsoid E in the affine space $\Bbb A^n$ is affinely rigid, i.e. every other ovaloid F in $\Bbb A^n$ with the same affine Blaschke metric as for E equals E up to an equiaffine motion of E. Due to M. Kozlowski 1985 resp. W. Blaschke 1922 for n = 3 ellipsoids are moreover S-rigid resp. infinitesimally S-rigid in the sense of equal resp. infinitesimally equal affine scalar curvature S (unknown until now for n >3). – In this article it is proved that ellipsoids in $\Bbb A^n$ are also infinitesimally S-rigid for any n.

Curvature of curvilinear 4-webs and pencils of one forms: Variation on a theorem of Poincar�, Mayrhofer and Reidemeister

A curvilinear d-web W = (F1 , . . . , Fd) is a configuration of d curvilinear foliations Fi on a surface. When d = 3, Bott connections of the normal bundles of Fi extend naturally to equal affine connection, which is called Chern connection. For 3 < d, this is the case if and only if the modulus of tangents to the leaves of Fi at a point is constant. A d-web is associative if the modulus is constant and weakly associative if Chern connections of all 3-subwebs have equal curvature form. We give a geometric interpretation of the curvature form in terms of fake billiard in §2, and prove that a weakly associative d-web is associative if Chern connections of triples of the members are non flat, and then the foliations are defined by members of a pencil (projective linear family of dim 1) of 1-forms. This result completes the classification of weakly associative 4-webs initiated by Poincaré, Mayrhofer and Reidemeister for the flat case. In §4, we generalize the result for n + 2-webs of n-spaces.

Effect of Director Pretilt on Field Induced Periodic Distortions in Nematics

This work reports a theoretical study on the linear threshold of static director distortions in a rigidly anchored nematic sample subjected to the simultaneous actions of electric (E) and magnetic (B) fields applied along certain symmetry directions with respect to the initial uniform director alignment n 0. The results mainly concern a material having susceptibility anisotropics of opposite signs (εa > 0, χa < 0). The deformation above the electric threshold may be homogeneous (HD) or periodic (PD) depending upon material and geometric parameters. PD is quenched when n 0 is tilted sufficiently away from the homeotropic or when B is impressed away from E. For a given director pretilt, the direction of post threshold periodicity changes continuously with the magnetic tilt. At a given magnetic tilt, the PD threshold parameters exhibit a discontinuous change when the director pretilt is varied. Phase diagrams in different planes of the parameter space demarcate the regions of existence of PD and HD. The importance of elastic anisotropy in determining the nature of PD emerges from a study of a hypothetical material having two equal curvature elastic constants.

Wide angular aperture holograms in photorefractive crystals by the use of orthogonally polarized write and read beams

We demonstrate a method of simultaneous holographic recording and readout in photorefractive crystals that provides high write-read beam isolation and wide angular bandwidth. The method uses orthogonally polarized read and write beams and parallel tangent diffraction geometry near the equal curvature condition to provide spatially separable, orthogonally polarized diffracted output beams with high isolation and wide Bragg-matched angular bandwidth. The available angular bandwidth of this read-write technique is analyzed, simulated, and experimentally investigated. The measured angular bandwidth internal to the crystal is approximately 18° × 6° for our 45°-cut BaTiO(3) crystal, yet the entire hologram still demonstrates high Bragg selectivity. In contrast, traditional nonparallel-tangent geometries yield angular apertures of the order of 1° × 4°.