General Surface Research Articles

A characterization of constant p-mean curvature surfaces in the Heisenberg group H1

In Euclidean 3-space, it is well known that the Sine-Gordon equation was considered in the nineteenth century in the course of investigations of surfaces of constant Gaussian curvature K=−1. Such a surface can be constructed from a solution to the Sine-Gordon equation, and vice versa. With this as motivation, employing the fundamental theorem of surfaces in the Heisenberg group H1, we show in this paper that the existence of a constant p-mean curvature surface (without singular points) is equivalent to the existence of a solution to a nonlinear second-order ODE (1.2), which is a kind of Liénard equations. Therefore, we turn to investigate this equation. It is a surprise that we give a complete set of solutions to (1.2) (or (1.5)), and hence use the types of the solution to divide constant p-mean curvature surfaces into several classes. As a result, after a kind of normalization, we obtain a representation of constant p-mean curvature surfaces and classify further all constant p-mean curvature surfaces. In Section 9, we provide an approach to construct p-minimal surfaces. It turns out that, in some sense, generic p-minimal surfaces can be constructed via this approach. Finally, as a derivation, we recover the Bernstein-type theorem which was first shown in [14] (or see [19,20]).

A Generic Mission-Level Flight Control Surface EMA Power Consumption Simulation Tool

The use of electromechanical actuators (EMAs) for aeronautical applications promises substantial benefits regarding efficiency and operability. To advance the design of power electronics and secondary power supply, there is a need for the ability to swiftly study the effects of aircraft mission and operational aspects on the actuator energy consumption. Pursuant to this, the aim of the work presented in this paper is twofold: (i) to build a generic mission-level flight control surface EMA power consumption simulation framework and (ii) to apply this framework to a case study involving a small all-electric aircraft, in which selected factors that impact energy consumption are investigated. The core of the framework comprises physics-based EMA power estimators, linked with a six-degree-of-freedom flight dynamics and control simulation module. The case study results show that the actuator power consumption correlates positively with the proportional gains in the flight control system but is inversely proportional to the trajectory radius and linearly dependent on turbulence intensity. The developed framework could aid in the selection of the actuator, as well as in the optimisation of airborne electronics and secondary power supply.

Locking effects in plowing-induced nanorippling of polystyrene surfaces

We have investigated the influence of the geometric shape bounding the scan pattern on the orientation of the nanoripples evolved from a polystyrene surface scraped by a silicon tip. Atomic force microscopy measurements at ambient conditions are compared with numeric simulations based on a continuum model describing the time evolution of a generic compliant surface, which is indented viscoplastically and sheared elastically at the location occupied by the scanning tip. The tendency of the ripples to align at a steady angle defined by the scan pattern can be significantly modified by the geometric confinement. The analysis of surface ripples resulting from square, circle, star, pentagon, hexagon, and heart-shaped areas allows us to recognize the distinct influence of straight and curved edges, and of sharp corners, on the nanostructuring process. Straight edges inclined in the direction defined by the locking angle result in a uniform ripple pattern covering the entire scanned region originated from them. These results pave the way to predict plowing-induced surface structures by computer simulations and define the most suitable scan path required to obtain a desired ripple pattern. Potential applications include the controlled fabrication of nanostructures for optical devices, antibacterial surfaces, microfluidic channeling, and flexible electronics.

Random hyperbolic surfaces of large genus have first eigenvalues greater than $$\frac{3}{16}-\epsilon $$

Let $M_g$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil-Petersson metric. In this paper, we show that for any $\epsilon>0$, as genus $g$ goes to infinity, a generic surface $X\in M_g$ satisfies that the first eigenvalue $\lambda_1(X)>\frac{3}{16}-\epsilon$. As an application, we also show that a generic surface $X\in M_g$ satisfies that the diameter $\mathrm{diam}(X)<(4+\epsilon)\ln(g)$ for large genus.

Null surface thermodynamics

We establish that boundary degrees of freedom associated with a generic codimension one null surface in $D$-dimensional pure Einstein gravity naturally admit a thermodynamical description. We expect the null surface thermodynamics to universally follow as a result of the diffeomorphism invariance of the theory, not relying on other special features of the null surface or the gravity theory. Using standard surface charge analysis and covariant phase-space method, we formulate laws of null surface thermodynamics which are local equations over an arbitrary null surface paralleling local versions of the zeroth- and first laws and the Gibbs-Duhem equation. This thermodynamical system is generally an open system and can be closed only when there is no flux of gravitons through the null surface. Our analysis extends the usual black hole thermodynamics to a universal feature of any area element on a generic null surface. We discuss the relevance of our study for the membrane paradigm and black hole microstates.

A refined derived Torelli theorem for enriques surfaces, II: the non-generic case

We prove that two Enriques surfaces defined over an algebraically closed field of characteristic different from 2 are isomorphic if their Kuznetsov components are equivalent. This improves and completes our previous result joint with Nuer where the same statement is proved for generic Enriques surfaces.

Fourier transform of surface–carried measures of two-dimensional generic surfaces and applications

&lt;p style='text-indent:20px;'&gt;We give a simple proof of the sharp decay of the Fourier-transform of surface-carried measures of two-dimensional generic surfaces. The estimates are applied to prove Strichartz and resolvent estimates for elliptic operators whose characteristic surfaces satisfy the generic assumptions. We also obtain new results on the spectral and scattering theory of discrete Schrödinger operators on the cubic lattice.&lt;/p&gt;

Continuation methods for principal foliations of embedded surfaces

&lt;p style='text-indent:20px;'&gt;Continuation methods are a well established tool for following equilibria and periodic orbits in dynamical systems as a parameter is varied. Properly formulated, they locate and classify bifurcations of these key components of phase portraits. Principal foliations of surfaces embedded in &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathbb{R}^3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; resemble phase portraits of two dimensional vector fields, but they are not orientable. In the spirit of dynamical systems theory, Gutierrez and Sotomayor investigated qualitative geometric features that characterize structurally stable principal foliations and their bifurcations in one parameter families. This paper computes return maps and applies continuation methods to obtain new insight into bifurcations of principal foliations.&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;Umbilics are the singularities of principal foliations and lines of curvature connecting umbilics are analogous to homoclinic and heteroclinic bifurcations of vector fields. Here, a continuation method tracks a periodic line of curvature in a family of surfaces that deforms an ellipsoid. One of the bifurcations of these periodic lines of curvature are connections between lemon umbilics. Differences between these bifurcations and analogous saddle connections in two dimensional vector fields are emphasized. A second case study tracks umbilics in a one parameter family of surfaces with continuation methods and locates their bifurcations using Taylor expansions in "Monge coordinates." Return maps that are generalized interval exchange maps of a circle are constructed for generic surfaces with no monstar umbilics.&lt;/p&gt;

Thermal nature of a generic null surface

Dynamical properties of a generic null surface are known to have a thermodynamic interpretation. Such an interpretation is completely based on an analogy between the usual law of thermodynamics and structure of gravitational field equation on the surface. Here we materialize this analogy and show that assigning a temperature on the null surface for a local observer is indeed physically relevant. We find that for a local frame, chosen as outgoing massless chargeless particle (or field mode), perceives a ``local unstable Hamiltonian'' very near to the surface. Due to this it has finite quantum probability to escape through acausal null path which is given by Maxwell-Boltzmann like distribution, thereby providing a temperature on the surface.

Thermodynamic structure of a generic null surface and the zeroth law in scalar-tensor theory

We show that the equation of motion of scalar-tensor theory acquires thermodynamic identity when projected on a generic null surface. The relevant projection is given by $E_{ab}l^ak^b$, where $E_{ab} =8\pi T_{ab}^{(m)}$ represents the equation motion for gravitational field in presence of external matter, $l^a$ is the generator of the null surface and $k^a$ is the corresponding auxiliary null vector. Our analysis is done completely in a covariant way. Therefore all the thermodynamic quantities are in covariant form and hence can be used for any specific form of metric adapted to a null surface. We show this both in Einstein and Jordan frames and find that these two frames provide equivalent thermodynamic quantities. This is consistent with the previous findings for a Killing horizon. Also, a concrete proof of the zeroth law in scalar-tensor theory is provided when the null surface is defined by a Killing vector.

A smooth tool path generation method for ultraprecision turning discontinuous multi-freeform-surfaces with high machining efficiency

The typical tool path of turning freeform surfaces is spiral. Usually, this kind of tool path only can deal with continuous surfaces within a circular area. If the designed surfaces are not continuous, the spiral tool path will break, which will lead to a significant increase in servo following error of ultraprecision lathe during machining. In order to reduce the servo following error, the processing speed needs to be significantly reduced, which will greatly decrease the machining efficiency. To solve this problem, a smooth tool path interpolation algorithm in the non-cutting area for ultraprecision turning discontinuous multi-freeform-surfaces is proposed. In addition, more than circular area, this method can generate smooth tool paths for discontinuous generic surfaces of quasi-revolution based on the optimized space Archimedean spiral. Finally, the effectiveness and adaptability of the method are proved through two examples.

Individualized and controlled laser beam pre-treatment process for adhesive bonding of fiber-reinforced plastics. III. Effects of contaminants

Adhesive bonding of plastic components is a well-established process for various applications. Specifically for thermoplastic parts, surface pre-treatment is required to achieve optimum bonding quality. However, automated surface pre-treatment is not established yet. Reasons are missing process monitoring and significant challenges arising from conventional monitoring techniques that hardly identify surface contaminants. In a recent research project, the authors investigated how process monitoring by spectrometry during a laser-based surface pre-treatment might solve this challenge for fiber-reinforced plastics and generic surface contaminants. This article describes experiments conducted on two composite materials with different contaminants. A nanosecond-pulsed UV-laser was used to pretreat a glass-fiber reinforced 2C-polyurethane (GF-PUR) and a glass-fiber reinforced polyamide 6 (GF-PA6). Each composite had coupons contaminated with an industry standard mould release agent or artificial silicate-based dust as well as uncontaminated samples. The samples were laser pretreated and compared to reference coupons that did not receive laser treatment. The samples were analyzed after laser pre-treatment for their free surface energies by the detection of the contact angle and measured for the surface roughness. The bonding improvement due to the laser process was determined by shear strength and peel resistance tests. The results show that UV-laser pre-treatment has an advantageous impact on the bonding strength of GF-PUR and GF-PA6. Combined with a fast spectrometric process monitoring that robustly identifies contaminants during the laser process, an inline process control is feasible.

Adsorption of Strontium onto Synthetic Iron(III) Oxide up to High Ionic Strength Systems

In this work, the adsorption behavior of Sr onto a synthetic iron(III) oxide (hematite with traces of goethite) has been studied. This solid, which might be considered a representative of Fe3+ solid phases (iron corrosion products), was characterized by X-Ray Diffraction (XRD) and X-Ray Photoelectron Spectroscopy (XPS), and its specific surface area was determined. Both XRD and XPS data are consistent with a mixed solid containing more than 90% hematite and 10% goethite. The solid was further characterized by fast acid-base titrations at different NaCl concentrations (from 0.1 to 5 M). Subsequently, for each background NaCl concentration used for the acid-base titrations, Sr-uptake experiments were carried out involving two different levels of Sr concentration (1 × 10−5 and 5 × 10−5 M, respectively) at constant solid concentration (7.3 g/L) as a function of −log([H+]/M). A Surface Complexation Model (SCM) was fitted to the experimental data, following a coupled Pitzer/surface complexation approach. The Pitzer model was applied to aqueous species. A Basic Stern Model was used for interfacial electrostatics of the system, which includes ion-specific effects via ion-specific pair-formation constants, whereas the Pitzer-approach involves ion-interaction parameters that enter the model through activity coefficients for aqueous species. A simple 1-pK model was applied (generic surface species, denoted as &gt;XOH−1/2). Parameter fitting was carried out using the general parameter estimation software UCODE, coupled to a modified version of FITEQL2. The combined approach describes the full set of data reasonably well and involves two Sr-surface complexes, one of them including chloride. Monodentate and bidentate models were tested and were found to perform equally well. The SCM is particularly able to account for the incomplete uptake of Sr at higher salt levels, supporting the idea that adsorption models conventionally used in salt concentrations below 1 M are applicable to high salt concentrations if the correct activity corrections for the aqueous species are applied. This generates a self-consistent model framework involving a practical approach for semi-mechanistic SCMs. The model framework of coupling conventional electrostatic double layer models for the surface with a Pitzer approach for the bulk solution earlier tested with strongly adsorbing solutes is here shown to be successful for more weakly adsorbing solutes.

Principal Foliations of Surfaces near Ellipsoids

The lines of curvature of a surface embedded in $\R^3$ comprise its principal foliations. Principal foliations of surfaces embedded in $\R^3$ resemble phase portraits of two dimensional vector fields, but there are significant differences in their geometry because principal foliations are not orientable. The Poincar\'e-Bendixson Theorem precludes flows on the two sphere $S^2$ with recurrent trajectories larger than a periodic orbit, but there are convex surfaces whose principal foliations are closely related to non-vanishing vector fields on the torus $T^2$. This paper investigates families of such surfaces that have dense lines of curvature at a Cantor set $C$ of parameters. It introduces discrete one dimensional return maps of a cross-section whose trajectories are the intersections of a line of curvature with the cross-section. The main result proved here is that the return map of a generic surface has \emph{breaks}; i.e., jump discontinuities of its derivative. Khanin and Vul discovered a qualitative difference between one parameter families of smooth diffeomorphisms of the circle and those with breaks: smooth families have positive Lebesgue measure sets of parameters with irrational rotation number and dense trajectories while families of diffeomorphisms with a single break do not. This paper discusses whether Lebesgue almost all parameters yield closed lines of curvature in families of embedded surfaces.

Hydrodynamic drag reduction of shear-thinning liquids in superhydrophobic textured microchannels

Super-hydrophobic textured surfaces reduce hydrodynamic drag in pressure-driven laminar flows in micro-channels. However, despite the wide usage of non-Newtonian liquids in microfluidic devices, the flow behaviour of such liquids was rarely examined so far in the context of friction reduction in textured super-hydrophobic micro-channels. Thus, we have investigated the influence of topologically different rough surfaces on friction reduction of shear-thinning liquids in micro-channels. First, the friction factor ratio (a ratio of friction factor on a textured surface to a plain surface) on generic surface textures, such as posts, holes, longitudinal and transverse ribs, was estimated numerically over a range of Carreau number as a function of microchannel constriction ratio, gas fraction and power-law exponent. Resembling the flow behaviour of Newtonian liquids, the longitudinal ribs and posts have exhibited significantly less flow friction than the transverse ribs and holes while the friction factor ratios of all textures has exhibited non-monotonic variation with the Carreau number. While the minima of the friction factor ratio were noticed at a constant Carreau number irrespective of the microchannel constriction ratio, the minima have shifted to a higher Carreau number with an increase in the power-law index and gas fraction. Experiments were also conducted with aqueous Xanthan Gum liquids in micro-channels. The flow enhancement (the flow rate with super-hydrophobic textures with respect to a smooth surface) exhibited a non-monotonic behaviour and attenuated with an increase in power-law index tantamount to simulations. The results will serve as a guide to design frictionless micro-channels when employing non-Newtonian liquids.

Observer-based Geometric Impedance Control of a Fully-Actuated Hexarotor for Physical Sliding Interaction with Unknown Generic Surfaces

Aerial physical interaction is a promising field for unmanned aerial vehicles in future applications. This paper presents a novel paradigm for automatic aerial contact-based sliding interaction (inspection/cleaning) tasks in aerial robotics allowing a 3D force with a constant norm to be applied on generic surfaces with unknown geometry. The interaction task is achieved by a fully-actuated hexarotor equipped with a rigidly attached end-effector under a passivity-based geometric impedance controller and a new sliding-mode extended state observer to estimate the interaction wrench. In order to increase the observer performance and reduce the estimation chattering phenomenon, the observer is innovatively incorporated with a super-twisting algorithm and a sigmoid function with a switching gain being adaptively updated by a fuzzy logic system. A detailed stability analysis for the observer is presented based on the Lyapunov stability theory. The proposed control approach is validated in several simulations in which we try to accomplish the aerial physical sliding interaction task with different types of objects under various sliding speeds.

The monodromy conjecture for a space monomial curve with a plane semigroup

This article investigates the monodromy conjecture for a space monomial curve that appears as the special fiber of an equisingular family of curves with a plane branch as generic fiber. Roughly speaking, the monodromy conjecture states that every pole of the motivic, or related, Igusa zeta function induces an eigenvalue of monodromy. As the poles of the motivic zeta function associated with such a space monomial curve have been determined in earlier work, it remains to study the eigenvalues of monodromy. After reducing the problem to the curve seen as a Cartier divisor on a generic embedding surface, we construct an embedded Q-resolution of this pair and use an A’Campo formula in terms of this resolution to compute the zeta function of monodromy. Combining all results, we prove the monodromy conjecture for this class of monomial curves.

Genus 0 Gopakumar–vafa Invariants of the Banana Manifold

Abstract The Banana manifold $X_{Ban}$ is a compact Calabi–Yau threefold constructed as the conifold resolution of the fiber product of a generic rational elliptic surface with itself, which was first studied by Bryan. We compute Katz’s genus 0 Gopakumar–Vafa invariants of fiber curve classes on the Banana manifold $X_{Ban}\to \mathbf{P} ^1$. The weak Jacobi form of weight −2 and index 1 is the associated generating function for these genus 0 Gopakumar–Vafa invariants. The invariants are shown to be an actual count of structure sheaves of certain possibly non-reduced genus 0 curves on the universal cover of the singular fibers of $X_{Ban}\to\mathbf{P}^1$.

Computer geometric modeling of quasi-rotation surfaces

The paper presents an analytic description of quasi-rotation correspondence. The formulas for calculation of values characterizing geometric objects are provided on the basis of constructive description of the quasi-rotation correspondence. The values characterizing circles generated through quasi-rotation of the initial point are calculated. A system of parametric equations describing a generic quasi-rotation surface is provided. Every type of curvilinear second-degree axis – circular, elliptic, parabolic, and hyperbolic – is considered. An algorithm for surface modeling through computer algebra systems has been developed on the basis of the analytic description and tested by means of Maple software. The algorithm allows one to acquire plots of separate sheets of the surface as well as all four sheets simultaneously. The capability to acquire 3D models of quasi-rotation surfaces given the desired shape and mutual location of the axis and the generatrix is demonstrated. The developed algorithm is applicable to any flat generating curve belonging to the axis plane and defined by a single-parameter system of equations. The resultant 3D model polygon count is user-defined and only limited by computer’s processing capability assuring high surface quality. The images of several four-sheet quasi-rotation surfaces are provided. The developed algorithm allows one to design surfaces of pre-defined properties.

Extracting vibrational anharmonicities from short driven molecular dynamics trajectories

Anharmonicities provide a wealth of information about the vibrational dynamics, mode coupling and energy transfer within a polyatomic system. In this contribution, we show how driven molecular dynamics trajectories can be used to extract anharmonicity properties under very short times of a few hundreds of vibrational periods, using two exciting fields at- and slightly off-resonance. Detailed analyses on generic quartic potential energy surfaces and applications to various model systems are presented, giving good agreement with perturbation theory. Application to a realistic molecule, cubane (C $$_8$$ H $$_8$$ ), modelled with a tight-binding quantum force field, further indicates how the method can be applied in practical cases.