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In this paper, explicit stable integrators based on symplectic and contact geometries are proposed for a family of non-autonomous ordinarily differential equations (ODEs) found in improving convergence rate of Nesterov's accelerated gradient method. Symplectic geometry is known to be suitable for describing Hamiltonian mechanics, and contact geometry is known as an odd-dimensional counterpart of symplectic geometry. Moreover, a procedure, called symplectization, is a known way to construct a symplectic manifold from a contact manifold, yielding autonomous Hamiltonian systems from contact ones. It is found in this paper that a previously investigated non-autonomous ODEs can be written as a contact Hamiltonian system family. Then, by developing and applying a symplectization of non-autonomous contact Hamiltonian vector fields expressing the non-autonomous ODEs, novel symplectic integrators are derived. Because the proposed symplectic integrators preserve hidden symplectic and contact structures in the ODEs, they are expected to be more stable than the Runge–Kutta method. Numerical experiments demonstrate that, as expected, the second-order symplectic integrator is stable and high convergence rates are achieved. [ABSTRACT FROM AUTHOR]

We study coisotropic deformations of a compact regular coisotropic submanifold C in a contact manifold (M , ξ) . Our main result states that C is rigid among nearby coisotropic submanifolds whose characteristic foliation is diffeomorphic to that of C. When combined with a classical rigidity result for foliations, this yields conditions under which C is rigid among all nearby coisotropic submanifolds. [ABSTRACT FROM AUTHOR]

We show that, apart from degeneracies, determining a plane curve whose curvature depends on its position essentially consists of determining a null curve in the Lorentzian 3-space when the null tangent direction depends on its position. We use this point of view to investigate the intrinsic equations for the n-elastic curves. We show how the problem of prescribed null tangent direction in terms of the position can be solved by quadratures when the prescription exhibits sufficient symmetries. This problem is formalized in terms of a convenient contact 3-form. [ABSTRACT FROM AUTHOR]

Suppressing unwanted vibrations has been a major challenge for more than a century. Semi-active dampers offer a compromise between the high energy costs of active solutions and their increased flexibility. A nonlinear system that utilizes dry friction and piecewise defined contact geometries is used as the basis for a semi-active damper. Furthermore, a slow frequency-based control that does not solely rely on dissipation is considered and compared to the conventional Skyhook Control. Simulations show that the strategy is an effective approach for vibration reduction. [ABSTRACT FROM AUTHOR]

In water-based experiments exploring ultralow or super low friction, the implementation of a running-in period before reaching such states is necessary and important. However, the effect of the change in contact geometry has not been fully realized. In this paper, a series of running-in tests on a Si3N4 ball and a WC plate have been performed in glycerol–water mixtures with different concentrations. The shape of the wear scars and the chemical compositions of the worn surfaces were characterized in detail. A numerical EHD and mixed lubrication model was established to comprehensively analyze the effects of geometric profiles, surface roughness, and working/lubrication conditions on ultralow or super low sliding friction. The experiment and simulation results of the study have provided an in-depth understanding of the mechanism of super low friction of liquid lubricated sliding point contacts. [ABSTRACT FROM AUTHOR]

In piled raft foundations, the bearing capacity is the result of the combination of the resistance of the pile group and the contribution of the contact between the pile cap and the soil. Although this contact contribution is still neglected, studies show that in certain cases, the load capacity directly absorbed by the pile cap can exceed the load absorbed by the pile group. Among the various factors that influence the behavior of piled raft foundations, the effect of contact geometry on the foundation has been inadequately researched. In this study, three-dimensional numerical analyses were conducted using the finite element method with the LCPC-CESAR software to investigate how pile cap–soil contact geometries affect the behavior of piled raft foundations. The net contact area, diameter, length, and number of piles were kept constant in all cases analyzed. Despite the complex interface of the LCPC-CESAR software, which hinders problem remodeling, validation securely demonstrated that this tool could replicate the behavior of piled raft foundations. The study results showed that, regardless of the spacing between the piles, the contact geometry does not affect the load–settlement curve of the system but significantly alters the distribution of loads among the foundation elements and the transfer of loads from the piles to the surrounding soil. When comparing pile cap geometries, it was observed that the square geometry supports a larger portion of the load compared to other geometries, likely due to better coverage of the pile inside the pile cap provided by this square configuration. [ABSTRACT FROM AUTHOR]

We construct non-compact contact manifolds containing compact Legendrians which can be displaced from their Reeb flow with arbitrarily small oscillation energy. We use this to show the Shelukhin-Chekanov-Hofer pseudo-metric considered by Rosen and Zhang (Math. Res. Lett. 27:1165–1193, 2020) is degenerate on the isotopy class of the constructed Legendrians. Other aspects related to oscillation energy of Legendrian isotopies are explored. [ABSTRACT FROM AUTHOR]

In this short note, we prove that singular Reeb vector fields associated with generic -contact forms on three dimensional manifolds with compact embedded critical surfaces have either (at least) or an infinite number of escape orbits, where denotes the number of connected components of the critical set. In case where the first Betti number of a connected component of the critical surface is positive, there exist infinitely many escape orbits. A similar result holds in the case of -Beltrami vector fields that are not -Reeb. The proof is based on a more detailed analysis of the main result in [19]. [ABSTRACT FROM AUTHOR]

In this note, we formulate and study a Hamilton–Jacobi approach for describing thermodynamic transformations. The thermodynamic phase space assumes the structure of a contact manifold with the points representing equilibrium states being restricted to certain submanifolds of this phase space. We demonstrate that Hamilton–Jacobi theory consistently describes thermodynamic transformations on the space of externally controllable parameters or equivalently the space of equilibrium states. It turns out that in the Hamilton–Jacobi description, the choice of the principal function is not unique but, the resultant dynamical description for a given transformation remains the same irrespective of this choice. Some examples involving thermodynamic transformations of the ideal gas are discussed where the characteristic curves on the space of equilibrium states completely describe the dynamics. The geometric Hamilton–Jacobi formulation which has emerged recently is also discussed in the context of thermodynamics. [ABSTRACT FROM AUTHOR]

The thermodynamics of a conductive liquid is considered. The latter is described within the framework of contact geometry. The state and constitutive equations of the medium are obtained from the first principles and notion of symmetry under natural assumptions only. These thermodynamic relations are supplemented by the general laws of charge, mass, momentum, and energy conservation to form a complete system of partial differential equations that describe the motion of the media. Differential invariants of this system under the action of rotations and translations are studied. A complete set of such invariants is obtained. [ABSTRACT FROM AUTHOR]

This survey is based on lecture notes for an online mini-course taught for postgraduate students at UDELAR, Montevideo, Uruguay, in November 2020, remotely from the Mittag–Leffler Institute in Djursholm, Sweden. Lectures were recorded and are available online. [ABSTRACT FROM AUTHOR]

We improve the previous result "A complete Ricci soliton whose metric g is K-contact and the soliton vector field X is strictly contact, is compact Sasakian Einstein" and show that, if a complete Ricci soliton (M, g, X) whose metric g is a contact metric and the soliton vector field X is strictly contact, then X is an infinitesimal automorphism and g is Einstein. Finally, for a Ricci soliton with X as the Reeb vector field, we show that (M, g) is compact Einstein and and Sasakian. [ABSTRACT FROM AUTHOR]

Polyacrylamide (PAAm) hydrogels with brush-covered or crosslinked surfaces were produced and their tribological behavior was studied over a wide range of sliding speeds for two different contact geometries: sphere-on-flat and flat-pin-on-flat. Irrespective of the contact geometry, the brushy hydrogel surfaces displayed up to an order of magnitude lower coefficients of friction μ (COF) compared to the crosslinked surfaces, even achieving superlubricity (μ < 0.01). In general, a hydrogel sphere showed a lower coefficient of friction than a flat hydrogel pin at a similar contact pressure over the entire range of sliding speeds. However, after normalizing the friction force by the contact area, the shear stress of hydrogels with either crosslinked or brushy surfaces was found to be similar for both contact geometries at low speeds, indicating that hydrogel friction is unaffected by the contact geometry at these speeds. At high sliding speeds, the shear stress was found to be lower for a sphere-on-flat configuration compared to a flat-pin-on-flat configuration. This can be attributed to the larger equivalent hydrodynamic thickness due to the convergent inlet zone ahead of the sphere-on-flat contact, which presumably enhances the water supply in the contact, promotes rehydration, and thus reduces the friction at high sliding speeds compared to that measured for the flat-pin-on-flat contact. [ABSTRACT FROM AUTHOR]

We study a long-recognised but under-appreciated symmetry called dynamical similarity and illustrate its relevance to many important conceptual problems in fundamental physics. Dynamical similarities are general transformations of a system where the unit of Hamilton's principal function is rescaled, and therefore represent a kind of dynamical scaling symmetry with formal properties that differ from many standard symmetries. To study this symmetry, we develop a general framework for symmetries that distinguishes the observable and surplus structures of a theory by using the minimal freely specifiable initial data for the theory that is necessary to achieve empirical adequacy. This framework is then applied to well-studied examples including Galilean invariance and the symmetries of the Kepler problem. We find that our framework gives a precise dynamical criterion for identifying the observables of those systems, and that those observables agree with epistemic expectations. We then apply our framework to dynamical similarity. First we give a general definition of dynamical similarity. Then we show, with the help of some previous results, how the dynamics of our observables leads to singularity resolution and the emergence of an arrow of time in cosmology. [ABSTRACT FROM AUTHOR]

We give a "soft" proof of Alberti's Luzin-type theorem in Alberti (J Funct Anal 100:110–118, 1991), using elementary geometric measure theory and topology. Applications to the C 2 -rectifiability problem are also discussed. [ABSTRACT FROM AUTHOR]

This paper proposes a new way of considering wheel–rail contact in multibody systems simulation that goes beyond the traditional planar constraint and elastic approaches. In this approach, wheel–rail interaction is modelled as a force element with pressures and shear stresses distributed over a contact area that may be curved, supporting conformal contact situations. This by-passes the selection of the contact reference location and reference angle, which are delicate aspects of planar contact approaches. The idea is worked out introducing the curved reference surface as the new backbone for the computations, instead of the tangent plane used previously in planar contact approaches. The steps are described by which the curved reference is constructed in CONTACT, using generic facilities for markers, grids, and coordinate transformations, by which generic wheel/rail configurations can be analyzed in a fully automated way. Numerical results show the capabilities of the new method for measured, worn profiles, suppressing discontinuities in the forces when multiple contact patches split or merge. A further application concerns the evaluation of strategies used in planar contact approaches. There we find that the tangent plane's inclination is of the biggest importance. This should be defined in an averaged way to achieve maximum correspondence to the more detailed curved contact approach. [ABSTRACT FROM AUTHOR]

We show that a hypersurface of the regularized, spatial circular restricted three-body problem is of contact type whenever the energy level is below the first critical value (the energy level of the first Lagrange point) or if the energy level is slightly above it. A dynamical consequence is that there is no blue sky catastrophe in this energy range. [ABSTRACT FROM AUTHOR]

We construct a natural prequantization space over a monotone product of a toric manifold and an arbitrary number of complex Grassmannians of 2-planes in even-dimensional complex spaces, and prove that the universal cover of the identity component of the contactomorphism group of its total space carries a nonzero homogeneous quasi-morphism. The construction uses Givental's nonlinear Maslov index and a reduction theorem for quasi-morphisms on contactomorphism groups previously established together with M. S. Borman. We explore applications to metrics on this group and to symplectic and contact rigidity. In particular we obtain a new proof that the quaternionic projective space H P n - 1 , naturally embedded in the Grassmannian G 2 (C 2 n) as a Lagrangian, cannot be displaced from the real part G 2 (R 2 n) by a Hamiltonian isotopy. [ABSTRACT FROM AUTHOR]

We solve an optimal control problem for thermodynamic processes in an ideal gas. The thermodynamic state is given by a Legendrian manifold in a contact space. Pontryagin's maximum principle is used to find an optimal trajectory (thermodynamic process) on this manifold that maximizes the work of the gas. In the case of ideal gases, it is shown that the corresponding Hamiltonian system is completely integrable and its quadrature-based solution is given. Keywords: : contact geometry, thermodynamics, optimal control, Hamiltonian systems, integrability [ABSTRACT FROM AUTHOR]

Several dynamical systems of interest in Celestial Mechanics can be written in the form of a Newton equation with time-dependent damping, linear in the velocities. For instance, the modified Kepler problem, the spin–orbit model and the Lane–Emden equation all belong to such class. In this work, we start an investigation of these models from the point of view of contact geometry. In particular, we focus on the (contact) Hamiltonisation of these models and on the construction of the corresponding geometric integrators. [ABSTRACT FROM AUTHOR]

To investigate the effects of the contact geometry, interfacial friction, and substrate recovery on the behavior of polymer scratching using a conical tip, an analytical model is proposed. The normal stress acting on the contact surface between the tip and the substrate is described as a function of the included angle θ, representing the angle between two planes across the axis of the conical tip, and the attack angle β, representing the angle between the conical surface and the substrate material surface. The effects of the rear contact geometry on the scratch friction between the tip and substrate, represented by recovery angle φ, owing to the instantaneous elastic recovery of the polymer substrate, are also introduced. Validated by the experimental and numerical results from the literature, the proposed analytical model can describe well the scratch coefficient of friction (SCOF), which is defined as the ratio of the tangential force to the normal force. Meaningful guidance is provided to understand the scratch friction behavior. [ABSTRACT FROM AUTHOR]

In this article, we give a brief survey of major historical developments in the field of Contact and Symplectic Geometry. This field has grown into an area in its own right due to rapid progress seen in the last five decades. The community of Indian mathematicians working on this field is small but steadily growing. The contribution from Indian mathematicians to this field is noted in the article. [ABSTRACT FROM AUTHOR]

A Jacobi structure J on a line bundle L → M is weakly regular, if the sharp map J ♯ : J 1 L → D L has constant rank. A generalized contact bundle with a weakly regular Jacobi structure possesses a transverse complex structure. Similarly to conditions obtained by Bailey on generalized complex structures, we find condition on a pair consisting of a weakly regular Jacobi structure and a transverse complex structure to come from a generalized contact structure. In this way, we are able to construct interesting examples of generalized contact bundles. As applications: (1) we prove that every five-dimensional nilmanifold is equipped with an invariant generalized contact structure; (2) we show that unlike the generalized complex case, all contact bundles over a complex manifold possess a compatible generalized contact structure. Finally, we provide a counterexample presenting a locally conformal symplectic bundle over a generalized contact manifold of complex type that does not possess a compatible generalized contact structure. [ABSTRACT FROM AUTHOR]

In the present research work, the modular parametric design plug-in Grasshopper, available in Rhinoceros 5, is utilized as a pre-processor for the estimation of the projection of the contact length between ring and tools in the radial-axial ring rolling process. The estimated lengths, for each round of the process, are then used in a slip line based force model for the precise estimation of the radial forming force. The proposed method allows reducing the inaccuracies of the traditional approaches since it supersedes the concept of common thickness draft on both mandrel and main roll side, allowing a more precise estimation of the projection of the contact arc between ring and tools, considered to have a unique value on both mandrel side and main roll side. The fulfillment of this last assumption ensures the forming force to have the same value regardless it is calculated on the mandrel side or on the main roll side. The model has been validated by cross-comparing the analytical results with those of laboratory experiment and finite element simulation. The developed analytical model has been also applied to three different study cases where the previous literature models for the calculation of the projection of the contact arc have shown inaccuracies, demonstrating that the proposed approach can overcome these limitations. The positive cross comparisons among laboratory experiment, FEM simulations, and analytical estimations prove the reliability of the proposed approach, as well as its good integration with authors’ previous analytical algorithms. [ABSTRACT FROM AUTHOR]

Let Pfaffian system ω define an intrinsically nonlinear control system on manifold M that is invariant under the free, regular action of a Lie group G. The problem of identifying and constructing static feedback linearizable G-quotients of ω was solved in De Doná et al. (2016). Building on these results, the present paper proves that the trajectories of ω can often be expressed as the composition of the trajectories of a static feedback linearizable quotient control system, ω/G, on quotient manifold M/G, and those of a separate control system, γG, evolving on a principal G-bundle over a jet space. Furthermore, we point out that ω may not only have a static feedback linearizable quotient, ω/G but additionally, γG itself may possess a static feedback linearizable reduction as well. This enables one to express the trajectories of an intrinsically nonlinear control system as the composition of the trajectories of static feedback linearizable control systems, thereby providing a geometric criterion for the explicit integrability of intrinsically nonlinear systems. Moreover, special integrability properties arise when G is solvable. Examples are presented in which the above phenomena are explicitly demonstrated. An important aspect of the examples is that they gather evidence for the conjecture that our sufficient conditions for explicit integrability are also necessary. [ABSTRACT FROM AUTHOR]

In the present note, we start by observing that in the classical JKR theory of adhesion, using the usual Hertzian approximations, the pull-off load grows unbounded when the clearance goes to zero in a conformal “ball-in-socket” geometry. To consider the case of the conforming geometry, we use a recent rigorous general extension of the original JKR energetic derivation, which requires only adhesionless solutions, and an approximate adhesionless solution given in the literature. We find that depending on a single governing parameter of the problem, θ=ΔR/2πwR/E∗ where E∗ is the plane strain elastic modulus of the material couple, w the surface energy, ΔR the clearance and R the radius of the sphere, the system shows the classical bi-stable behaviour for a single sinusoid or a dimpled surface: pull-off is approximately that of the JKR theory for θ>0.82 only if the system is not “pushed” strong enough, otherwise a “strong adhesion” regime is found. Below this value, θ<0.82, a strong spontaneous adhesion regime is found similar to “full contact”. From the strong regime, pull-off will require a separate investigation depending on the actual system at hand. [ABSTRACT FROM AUTHOR]

Most of the previous studies on fretting fatigue have been accomplished under constant contact load condition, and a less attention has been paid to cyclic contact load. That is the case in many engineering applications such as bearing cap bolted joints of the V-type engines and dovetail joint of turbines. In this study, the effect of contact geometry on fretting fatigue behavior of Al7075-T6 under cyclic normal contact loading is investigated by experiment, numerical simulation and analysis. Two contact types including flat-on-flat and cylinder-on-flat are considered in this study. The crack initiated at a lower cycle for cylinder-on-flat contacts. It was found that in comparison with the constant contact loading condition, the cyclic contact loadings have more damaging effect on fretting fatigue life, particularly for cylinder-on-flat contact. With the increase in the pad width, fretting fatigue life increased for flat and cylindrical pads, particularly for high cycle fatigue (HCF) conditions. Examination of the fretting scars was performed using optical microscopy. Crack propagation life was determined by numerical simulation using the commercial FE codes ABAQUS and FRANC2D/L. Crack initiation life was calculated by a combination of numerical simulation and experiment. [ABSTRACT FROM AUTHOR]

We usually think of 2-dimensional manifolds as surfaces embedded in Euclidean 3-space. Since humans cannot visualise Euclidean spaces of higher dimensions, it appears to be impossible to give pictorial representations of higher-dimensional manifolds. However, one can in fact encode the topology of a surface in a 1-dimensional picture. By analogy, one can draw 2-dimensional pictures of 3-manifolds (Heegaard diagrams), and 3-dimensional pictures of 4-manifolds (Kirby diagrams). With the help of open books one can likewise represent at least some 5-manifolds by 3-dimensional diagrams, and contact geometry can be used to reduce these to drawings in the 2-plane. In this paper, I shall explain how to draw such pictures and how to use them for answering topological and geometric questions. The work on 5-manifolds is joint with Fan Ding and Otto van Koert. [ABSTRACT FROM AUTHOR]

We define a conformal reference frame, i.e., a special projection of the six-dimensional sky bundle of a Lorentzian manifold ( or the five-dimensional twistor space) to a three-dimensional manifold. We construct an example, a conformal compactification, for Minkowski space. Based on the complex structure on the skies, we define the celestial transformation of Lorentzian vectors, a kind of spinor correspondence. We express a 1- form generating the contact structure in the twistor space ( when it is smooth) explicitly as a form taking line-bundle values. We prove a theorem on the projection of this 1- form to the fiberwise normal bundle of a reference frame; its corollary is an equation for the flow of time. [ABSTRACT FROM AUTHOR]

Motivated by mathematical structures which arise in string vacua and gauge theories with $${{\mathcal{N}=2}}$$ supersymmetry, we study the properties of certain generalized theta series which appear as Fourier coefficients of functions on a twisted torus. In Calabi-Yau string vacua, such theta series encode instanton corrections from k Neveu-Schwarz five-branes. The theta series are determined by vector-valued wave-functions, and in this work we obtain the transformation of these wave-functions induced by Kontsevich-Soibelman symplectomorphisms. This effectively provides a quantum version of these transformations, where the quantization parameter is inversely proportional to the five-brane charge k. Consistency with wall-crossing implies a new five-term relation for Faddeev's quantum dilogarithm $${\Phi_b}$$ at b = 1, which we prove. By allowing the torus to be non-commutative, we obtain a more general five-term relation valid for arbitrary b and k, which may be relevant for the physics of five-branes at finite chemical potential for angular momentum. [ABSTRACT FROM AUTHOR]

On a Riemannian manifold, any parallel form is preserved by the flow of any Killing vector field with constant magnitude. As a consequence, on a 2n+1-dimensional K-contact manifold, there are no nontrivial parallel forms except of degrees 0 and 2n+1. Flat contact metrics on 3-manifolds are characterized by reducible holonomy. [ABSTRACT FROM AUTHOR]

On a five-dimensional simply connected Sasaki-Einstein manifold, one can construct Yang-Mills theories coupled to matter with at least two supersymmetries. The partition function of these theories localises on the contact instantons, however, the contact instanton equations are not elliptic. It turns out that these equations can be embedded into the Haydys-Witten equations (which are elliptic) in the same way the 4D anti-self-dual instanton equations are embedded in the Vafa-Witten equations. We show that under some favourable circumstances, the latter equations will reduce to the former by proving some vanishing theorems. It was also known that the Haydys-Witten equations on product manifolds $${M_5 = M_4 \times \mathbb{R}}$$ arise in the context of twisting the 5D maximally supersymmetric Yang-Mills theory. In this paper, we present the construction of twisted N = 2 Yang-Mills theory on Sasaki-Einstein manifolds, and more generally on K-contact manifolds. The localisation locus of this new theory thus provides a covariant version of the Haydys-Witten equation. [ABSTRACT FROM AUTHOR]

We consider an integrable Hamiltonian system weakly coupled with a pendulum-type system. For each energy level within some range, the uncoupled system is assumed to possess a normally hyperbolic invariant manifold diffeomorphic to a three-sphere, which bounds a strictly convex domain, and whose stable and unstable invariant manifolds coincide. The Hamiltonian flow on the three-sphere is equivalent to the Reeb flow for the induced contact form. The strict convexity condition implies that the contact structure on the three-sphere is tight. When a small, generic coupling is added to the system, the normally hyperbolic invariant manifold is preserved as a three-sphere, and the stable and unstable manifolds split, yielding transverse intersections. We show that there exist trajectories that follow any prescribed collection of invariant tori and Aubry-Mather sets within some global section of the flow restricted to the three-sphere. In this sense, we say that the perturbed system exhibits global diffusion on the tight three-sphere. [ABSTRACT FROM AUTHOR]

We provide some examples of harmonic and minimal unit vector fields as normalized gradients of isoparametric functions coming from a K-contact geometric setting. [ABSTRACT FROM AUTHOR]

In this paper we develop a geometrical model of functional architecture for the processing of spatio-temporal visual stimuli. The model arises from the properties of the receptive field linear dynamics of orientation and speed-selective cells in the visual cortex, that can be embedded in the definition of a geometry where the connectivity between points is driven by the contact structure of a 5D manifold. Then, we compute the stochastic kernels that are the approximations of two Fokker Planck operators associated to the geometry, and implement them as facilitation patterns within a neural population activity model, in order to reproduce some psychophysiological findings about the perception of contours in motion and trajectories of points found in the literature. [ABSTRACT FROM AUTHOR]

A long-standing open problem asks whether a Riemannian metric on the real projective space with the same volume as the canonical metric carries a periodic geodesic whose length is at most π. A contact-geometric reformulation of systolic geometry and the use of canonical perturbation theory allow us to solve a parametric version of this problem: if g is a smooth, constant-volume deformation of the canonical metric that is not formally trivial, the length of the shortest periodic geodesic of the metric g attains π as a strict local maximum at s = 0. This result still holds for complex and quaternionic projective spaces as well as for the Cayley plane. Moreover, the same techniques can be applied to show that Zoll Finsler manifolds are the unique smooth critical points of the systolic volume. [ABSTRACT FROM AUTHOR]

We study contact structures on nonnegatively graded manifolds equipped with homological contact vector fields. In the degree 1 case, we show that there is a one-to-one correspondence between such structures (with fixed contact form) and Jacobi manifolds. This correspondence allows us to reinterpret the Poissonization procedure, taking Jacobi manifolds to Poisson manifolds, as a supergeometric version of symplectization. [ABSTRACT FROM AUTHOR]

On a 2n+1-dimensional K-contact manifold, there are no nontrivial parallel forms except of degrees 0 and 2n+1. [ABSTRACT FROM AUTHOR]

In railroad turnouts, geometries of tongue and crossing rails are very complex and their shapes are changing along the track. Therefore, wheels are subjected to not only tread and flange contacts, but also the back-of-flange and top-of-flange contacts in the case of spring switches of tram vehicles. For this reason, one needs to deal with significant jumps in contact points for solving wheel/rail contact problems in turnout, and an accurate prediction of jumps in contact points is one of the most important issues that need to be carefully handled in the dynamic simulation of vehicle/turnout interactions. In this investigation, a numerical procedure that can be used for solving such a complex wheel/rail contact problem in turnout is proposed. In particular, a combined nodal and non-conformal contact approach is developed such that significant jumps in contact points are detected using the nodal search, while the exact location of contact point is then determined with continuous surface parameterizations using non-conformal contact equations. With this combined nodal and non-conformal contact approach for the contact geometry analysis of vehicle/turnout interactions, multiple look-up contact tables can be generated in an efficient way without losing accuracy. Since detailed contact search is performed off-line to obtain look-up contact tables, significant changes in contact points in turnout can be efficiently predicted on-line with tabular data to be interpolated in a standard way. Several numerical examples are presented in order to demonstrate the use of the numerical procedure developed in this investigation. [ABSTRACT FROM AUTHOR]

We construct a self-dual geometry of quasi-Sasakian 5-manifolds. Namely, we intrinsically define the notion of contact conformally semiflat (i.e., contact self-dual or contact antiself-dual) almost contact metric manifolds and also obtain a number of results concerning contact conformally semiflat quasi-Sasakian 5-manifolds. Themost important results concerning Sasakian and cosymplectic manifolds reveal interesting relationships between the characteristics of these manifolds such as contact self-duality and constancy of the Φ-holomorphic sectional curvature, contact anti-self-duality and Ricci flatness, etc. [ABSTRACT FROM AUTHOR]

Nano-scale adhesive contact mediated by intermolecular van der Waals forces has become a typical fundamental problem in many areas. Interpretation and control of the strength and efficiency of the nano-scale adhesive contacts require a proper modeling considering the actual interfacial forces, the varying contact area, and clearance. In this article, the finite-element (FE) method is developed to model the nano-scale adhesive contact of elastic bodies with an adhesive pressure derived from the interatomic interaction Lennard-Jones potential, which permits numerical solutions for a variety of interface geometries. Compared with the analytical results from conventional Hertz, JKR, and DMT models, the validity of the FE model is verified. For nano-scale contact, the assumption of equivalent radius adopted in the Hertz model is initially investigated and proved to be improper for nano-scale adhesive contact due to the distribution variations of interfacial force caused by local contact geometry. Then adhesive contact behaviors of four typical nano-scale contact geometries inspired by tip shapes of bio-adhesive pads are investigated in detail, which are flat punch tip, sphere tip, mushroom tip, and empty cup tip. The simulation results indicate that the nano-scale tip geometry plays a dominant role on the pull-off strength. Within the investigated geometries, cup tip results in a highest adhesion efficiency followed by flat punch tip, sphere tip, and mushroom tip, respectively, which are highly geometry dependent and verified by former experimental results. The dominant effect is found coming from the contact area ratio of the adhesive area to the sticking area or the whole contact area. The FE modeling can serve a useful purpose in revealing the nano-scale geometry-based adhesion contact for surface topography design in MEMS to avoid stiction failure and for the artificial sticky feet in bionics to increase adhesion strength. [ABSTRACT FROM AUTHOR]

Let ( M, θ) be a pseudo-Hermitian space of real dimension 2 n + 1, that is M is a CR-manifold of dimension 2 n + 1 and θ is a contact form on M giving the Levi distribution $${HT(M) \subset TM}$$. Let $${M^\theta \subset T^* M}$$ be the canonical symplectization of ( M, θ) and let M be identified with the zero section of M. Then M is a manifold of real dimension 2( n + 1) which admits a canonical foliation by surfaces parametrized by $${\mathbb{C} \ni t+i\sigma\mapsto \phi^{\theta}_{p}(t+i\sigma)=\sigma\theta_{g_t(p)}}$$, where $${p \in M}$$ is arbitrary and g is the flow generated by the Reeb vector field associated to the contact form θ. Let J be an (integrable) complex structure defined in a neighbourhood U of M in M. We say that the pair ( U, J) is an adapted complex tube on M if all the parametrizations $${\phi^{\theta}_{p}(t+i\sigma)}$$ defined above are holomorphic on $${(\phi^{\theta}_{p})^{-1}(U)}$$. In this paper we prove that if ( U, J) is an adapted complex tube on M, then the real function E on $${M^\theta\subset T^*M}$$ defined by the condition $${\alpha=E (\alpha)\theta_{\pi(\alpha)}}$$, for each $${\alpha \in M^\theta}$$, is a canonical defining function for M which satisfies the homogeneous Monge-Ampère equation (dd E) = 0. We also prove that if M and θ are real analytic then the symplectization M admits an unique maximal adapted complex tube. [ABSTRACT FROM AUTHOR]

We solve a problem of local contact equivalence of hyperbolic and elliptic Monge-Ampère equations to linear equations with constant coefficients. We find normal forms for such equations: the telegraph equation and the Helmholtz equation. [ABSTRACT FROM AUTHOR]

Feedback force is very important for novices to simulate tooth preparation by using the haptic interaction system (dental training system) in a virtual environment. In the process of haptic simulation, the fidelity of generated forces by a haptic device decides whether the simulation is successful. A force model computes feedback force, and we present an analytical force model to compute the force between a tooth and a dental pin during tooth preparation. The force between a tooth and a dental pin is modeled in two parts: (1) force to resist human’s operation and (2) friction to resist the rotation of the dental engine. The force to resist the human’s operation is divided into three parts in the coordinates that are constructed on the bottom center of the dental pin. In addition, we also consider the effects of dental-pin type, tooth stiffness, and contact geometry in the force model. To determine the parameters of the force model, we construct a measuring system by using machine vision and a force/torque sensor to track the human’s operations and measure the forces between the dental pins and teeth. Based on the measuring results, we construct the relation between the force and the human’s operation. The force model is implemented in the prototype of a dental training system that uses the Phantom as the haptic interface. Dentists performing virtual operations have confirmed the fidelity of feedback force. [ABSTRACT FROM AUTHOR]

A simple classification of triples of Lie cycles is given. The class of each triad determines the number of solutions to the associated oriented Apollonius contact problem. The classification is derived via 2-dimensional Lie contact geometry in the form of two of its subgeometries—Laguerre geometry and oriented Möbius geometry. The method of proof illustrates interactions between the two subgeometries of Lie geometry. Two models of Laguerre geometry are used: the classic model and the 3-dimensional affine Minkowski space model. [ABSTRACT FROM AUTHOR]

The theory of Arnold's invariants of plane curves and wave fronts is applied to the study of the geometry of wave fronts in the standard 2-sphere, in the Euclidean plane and in the hyperbolic plane. Some enumerative formulae similar to the Plücker formulae in algebraic geometry are given in order to compute the generalized Bennequin invariant J+ in terms of the geometry of the front. It is shown that in fact every coefficient of the polynomial invariant of Aicardi can be computed in this way. In the case of affine wave fronts, some formulae previously announced by S.L. Tabachnikov are proved. This geometric point of view leads to a generalization to generic wave fronts of a result shown by Viro for smooth plane curves. As another application, the Fabricius-Bjerre and Weiner formulae for smooth plane and spherical curves are generalized to wave fronts. [ABSTRACT FROM AUTHOR]