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This article proposes a real-time method to control the contour error for industrial five-axis machine tools by combining the generalized predictive control (GPC) and the feedback correction (FBC) with dynamic con- tour error estimation (CEE). The CEE is developed by well considering the curves curvature and torsion based on Taylor series expansion and Frenet frame theory. By utilizing the spherical linear interpolation, the tool orientation and the tool tip position are synchronized with respect to the curve length. A novel dynamic foot point searching procedure is established to weaken the influences of the tracking errors magnitude on the CEE precision. To tackle the transmission effect, the GPC is adopted to predict the servo systems outputs, and then, the induced tool pose deviations, which are subsequently utilized to counteract the contour errors, are predicted by constructing Jacobian matrixes. Especially, the FBC loops are constructed to suppress the influences of disturbances and further to reduce the magnitudes of contour errors. Simulations and experiments are conducted to verify the effectiveness of the proposed methods.

Summary Recent developments in predicting microemulsion phase behavior for use in chemical flooding are based on the hydrophilic-lipophilic deviation (HLD) and net-average curvature (NAC) equation-of-state (EoS). The most advanced version of the HLD-NAC EoS assumes that the three-phase micelle characteristic length is constant as parameters like salinity and temperature vary. In this paper, we relax this assumption to improve the accuracy and thermodynamic consistency of these flash calculations. We introduce a variable characteristic length in the three-phase region based on experimental data that is monotonic with salinity or other formulation variables, such as temperature and pressure. The characteristic length at the boundary of the three-phase region is then used for flash calculations in the two-phase lobes for Winsor type II-/II+. The functional form of the characteristic length is made consistent with the Gibbs phase rule. The improved EoS can capture asymmetric phase behavior data around the optimum, whereas current HLD-NAC based models cannot. The variable characteristic length formulation also resolves the thermodynamic inconsistency of existing phase behavior models that give multiple solutions for the optimum. We show from experimental data and theory that the inverse of the characteristic length varies linearly with formulation variables. This important result means that it is easy to predict the characteristic length in the three-phase region, which also improves the estimation of surrounding two-phase lobes. This improved physical understanding of microemulsion phase behavior should greatly aid in the design of surfactant blends and improve recovery predictions in a chemical flooding simulator.

In this work we study a non-flat Friedmann-Robertson-Walker universe filled with a pressure-less dark matter (DM) and Barrow holographic dark energy (BHDE) whose IR cutoff is the apparent horizon. Among various DE models, (BHDE) model shows the dynamical enthusiasm to discuss the universe's transition phase. According to the new research, the universe transitioned smoothly from a decelerated to an accelerated period of expansion in the recent past. We exhibit that the development of $q$ relies upon the type of spatial curvature. Here we study the equation of state (EoS) parameter for the BHDE model to determine the cosmological evolution for the non-flat universe. The (EoS) parameter and the deceleration parameter (DP) shows a satisfactory behaviour, it does not cross the the phantom line. We also plot the statefinder diagram to characterize the properties of the BHDE model by taking distinct values of barrow exponent $\triangle$. Moreover, we likewise noticed the BHDE model in the $(\omega_{D}-\omega_{D}^{'})$ plane, which can furnish us with a valuable, powerful finding to the mathematical determination of the statefinder. In the statefinder trajectory, this model was found to be able to reach the $\Lambda CDM$ fixed point., Comment: 19 pages, 21 figures

The motion of blood conveying gold nanoparticles on a curved surface when suction, thermal radiation, and Lorentz force are significant is explored in this report with the aim to announce the increasing effects of Williamson fluid parameter, volume fraction, radius of curvature, thermal radiation, and Lorentz force on such a transport phenomenon. This report was designed to explore the upper and lower solutions of the model suitable to study the enhancement of the aforementioned variables. The similarity solution of the dimensional governing equation was sought for using the appropriate similarity variables. These dimensionless forms of ODEs are numerically solved using the 3-stage Lobatto formula, also known as bvp4c. The validation of the numerical scheme was considered. The drag force decelerated and then upsurges owing to the volume fraction of nanoparticles in the corresponding UBS and reduces in LBS, while the rate of heat transfer drastically decreases. The temperature and velocity gradient escalate and decelerate, respectively for both branches of results owing to the effect of higher curvature parameter. The temperature distribution decelerates in both outcomes due to the strengthened of mass suction while the velocity weakened in the lower branch solution (LBS) and augments in the upper branch solution (UBS).

Micromechanical systems (MEMS) based on electrostatic actuators are commonly involved in driving and actuating high-speed micro-structures. For this to be efficient, these micro-actuators must produce sufficient actuating force in order to achieve the required large strokes for such operations (usually in the order of tens of microns). However, this larger amount of displacement will require high actuation electrostatic voltages (typically in the order of hundreds of volts) or larger actuator dimensions, resulting in a bulky and inefficient design. To overcome these challenges, this paper examines the possible use of the snap-through instability in an electrostatically actuated and initially curved (shallow arch) clamped-clamped microbeam actuator arrangement. An extended version of the variational Hamilton principle is applied to derive nonlinear equations governing the steady-state behavior of the structure under step DC load. These equations are solved by effectiveness meshless numerical Galerkin decomposition technique. A convergence study is performed to show the effectiveness and advantages of the applied numerical approach to the formulated nonlinear problem. The investigation inspects different initial curvature profiles (first three symmetric buckled modes: first, third, and fifth modes) for the shallow arch. It was reported that the presence of the snap-through instability, for all investigated initial profiles, in the structural behavior of the micro-actuator produces a non-trivial order of magnitude increase in its resultant displacement as compared to the classical parallel-plate actuator with identical geometrical dimensions and operating conditions. In addition, out of the assumed first three symmetric buckled modes to outline the initial curvature of the shallow arch microbeam, only the first and fifth modes showed a strong impact on the micro-actuator performances.

Given the Riemann, or the Weyl, or a generalized curvature tensor K, a symmetric tensor $b_{ij}$ is named `compatible' with the curvature tensor if $b_i{}^m K_{jklm} + b_j{}^m K_{kilm} + b_k{}^m K_{ijlm} = 0$. Amongst showing known and new properties, we prove that they form a special Jordan algebra, i.e. the symmetrized product of K-compatible tensors is K-compatible., 8 pages

This analysis reports an unsteady and incompressible flow of Williamson nanoliquid in presence of variable thermal characteristics are persuaded by a permeable stretching cylinder. The flow field investigation is established with the effect of mixed convection and non-uniform heat source/sink on flow and heat transfer. On the cylinder surface, the analysis is inspected with utilization of zero mass flux constraints. By using the appropriate similarity variables, the framed equations for the energy, momentum and mass is converted into non-linear ODEs. The numerical communication of the boundary value problem is successfully implemented using a computer algorithm programmed into the fifth Runge-Kutta scheme. Additionally, the wall shear factor and rate of heat transfer are calculated in two different cases namely, with curvature and without curvature. In addition, the results obtained are confirmed by making comparisons with previously published articles and we found an excellent match that guarantees the indemnity of current communication. A comprehensive change in velocity, temperature and concentration is examined for involved parameters like local Weissenberg number, space dependent heat source constant, magnetic number, curvature constant, thermophoretic parameter, buoyancy parameter, Brownian motion parameter, Prandtl number, Schmidt number, unsteadiness parameter, reaction rate parameter, activation energy parameter and temperature difference parameter. A reduction in velocity is observed for unsteady parameter and buoyancy constant. An enhanced nanofluid temperature is noted for space dependent heat source parameter, time dependent heat source parameter and unsteady parameter. Moreover, the nanofluid concentration is increases for temperature difference parameter while reverse observations are noticed for chemical reaction rate.

A nonlinear nonideal generalized magnetohydrodynamic (MHD) model is developed to explore the evolutionary dynamics of nonlinear wave-structural patterns naturalistically excitable in inhomogeneous protoplanetary disks (PPDs). A local nonlinear perturbation theory (multiscale analysis) over the magnetohydrostatic homogeneous equilibrium of the axisymmetric disk yields a second-order nonlinear partial differential equation in the perturbed gravitational potential curvature with a unique set of multiparametric coefficients. A numerical illustrative platform demonstrates the excitation of nonlinear coherent structures in the form of solitary lump waves (SLWs). The atypical SLW-characteristics are explored geometrically with the phase space trajectories. The SLW amplitude increases in magnitude with the Alfvenic Mach number, and vice-versa. It is interestingly inferred that the Mach number (reference frame velocity and density) acts as a destabilizing (stabilizing) agency to the PPD-dynamics against the nonlocal self-gravity effects, and so forth. It enlightens the long-standing gravitational Laplace 1796 conjecture having a real astronomic significance. At the last, the nontrivial applicability of our findings in the SLW-driven bounded planetary structure formation in varied astroenvirons is actualized.

Three-dimensional (3D) fabrics with negative Poisson’s ratio (NPR) are known as 3D auxetic fabrics. They expand (or contract) in the direction perpendicular to stretch (or compression) under uniaxial tensile (or compressing) conditions. Many enhanced properties come with this unusual auxetic behaviour, such as shear modulus, indentation resistance, fracture toughness, energy absorption, synclastic curvature, etc. Therefore, these 3D auxetic metamaterials have a broad prospect in applications like personal protective equipment as a replacement of polyurethane foams, fabric-reinforced composites with improved ballistic and impact resistance, and women's brassiere paddings with perfect shape-ability. This paper presents a review on 3D fabrics with NPR produced with different fabric technologies, including weaving, knitting, nonwoven, and other techniques.

We study 2-point correlation functions for scalar operators in position space through holography including bulk cubic couplings as well as higher curvature couplings to the square of the Weyl tensor. We focus on scalar operators with large conformal dimensions. This allows us to use the geodesic approximation for propagators. In addition to the leading order contribution, captured by geodesics anchored at the insertion points of the operators on the boundary and probing the bulk geometry thoroughly studied in the literature, the first correction is given by a Witten diagram involving both the bulk cubic coupling and the higher curvature couplings. As a result, this correction is proportional to the VEV of a neutral operator $O_k$ and thus probes the interior of the black hole exactly as in the case studied by Grinberg and Maldacena [13]. The form of the correction matches the general expectations in CFT and allows to identify the contributions of $T^nO_k$ (being $T^n$ the general contraction of n energy-momentum tensors) to the 2-point function. This correction is actually the leading term for off-diagonal correlators (i.e. correlators for operators of different conformal dimension), which can then be computed holographically in this way., Comment: 23 pages

Regulation of hot spots exhibits excellent potential in many applications including nanolasers, energy harvesting, sensing, and subwavelength imaging. Here, hat-shaped hierarchical nanostructures with different space curvatures have been proposed to enhance hot spots for facilitating surface-enhanced Raman scattering (SERS) and plasmon-driven catalysis applications. These novel nanostructures comprise two layers of metal nanoparticles separated by hat-shaped MoS 2 films. The fabrication of this hybrid structure is based on the thermal annealing and thermal evaporation of self-assembled polystyrene spheres, which are convenient to control the metal particle size and the curvature of hat-shaped nanostructures. Based on the narrow gaps produced by the MoS 2 films and the curvature of space, the constructed platform exhibits superior SERS capability and achieves ultrasensitive detection for toxic molecules. Furthermore, the surface catalytic conversion of p-nitrothiophenol (PNTP) to p, p′-dimercaptobenzene (DMAB) was in situ monitored by the SERS substrate. The mechanism governing this regulation of hot spots is also investigated via theoretical simulations.

The present paper numerically investigates viscoelastic fluid flow in the developing flow regime through both straight and 90-degree curve ducts. The aim is to investigate the effects of the first and second normal-stress differences (N1 and N2) as well as the curvature ratio ( $$\kappa = D/R$$ , where R and D are the curvature radius and channel side, respectively) and Reynolds number on the secondary flow patterns. Simulations were carried out in developing flow conditions in curved ducts using the finite volume method. The Reynolds numbers were 10, 20, 30, 40, 50, and 100, and the curvature ratios were 0.05, 0.066, 0.1, and 0.2. The Giesekus constitutive equation was utilized to model the non-linear rheological behavior of a 5.0 wt.% solution of polyisobutylene (PIB) in tetradecane (C14H30). The results reveal that a secondary flow with eight corner vortices is generated in a fully developed flow of viscoelastic fluid through a straight duct. This behavior is attributed to the difference in the second normal stress in the flow field. The results of the current study confirm that the second normal stress difference and change in the sign of N2 around the cross section’s corners trigger these corner vortices. Furthermore, the effects of the curvature ratio on the distributions of first and second normal-stress differences in the developing Dean flow were studied for the first time. The vorticity vector method was used for mixing-enhancement assessment. The results show that the intensity of secondary flows was significantly higher in viscoelastic fluid at all curvature ratios and at all considered Reynolds numbers than in Newtonian fluid flow.

Introduction. A kinematic model of group pursuit of a set of targets on a plane is considered. Pursuers use a technique similar to parallel approach method to achieve goals. Unlike the parallel approach method, the speed vectors of pursuers and targets are directed arbitrarily. In the parallel approach method, the instantaneous directions of movement of the pursuer and the target intersect at a point belonging to the circle of Apollonius. In the group model of pursuing multiple goals, the pursuers try to adhere to a network of predictable trajectories.Materials and Methods. The model sets the task of achieving goals by pursuers at designated points in time. This problem is solved by the methods of multidimensional descriptive geometry using the Radishchev diagram. The predicted trajectory is a composite line that moves parallel to itself when the target moves. On the projection plane “Radius of curvature — speed value”, the permissible speed range of the pursuer is displayed in the form of level lines (these are straight lines parallel to one of the projection planes). Images of speed level lines are displayed on the projection plane “Radius of curvature — time to reach the goal”. The search for points of intersection of the speed line images and the appointed time level line is being conducted. Along the communication lines, the values of the intersection points are lowered to the plane “Radius of curvature — speed value”. Using the obtained points, we construct an approximating curve and look for the intersection point with the line of the assigned speed. As a result, we get values of the radius of the circle at the predicted line of the trajectory of the pursuer.Results. Based on the results of the conducted research, test programs have been created, and animated images have been made in the computer mathematics system.Discussion and Conclusions. This method of constructing trajectories of pursuers to achieve a variety of goals at a given time values can be in demand by developers of autonomous unmanned aerial vehicles.

Based on the approximation of the bending curvature by Bernstein polynomials, a novel parametric model suitable for large deflection analysis is proposed for studying planar compliant mechanisms with uniform slender beam members. The numerical formulation is also presented based on Gaussian quadrature and Newton-Raphson iterations. By considering the bending curvature and coefficients of the Bernstein polynomial as the deformation field and unknown parameters, respectively, the equilibrium equations of the compliant mechanisms with geometric nonlinearity were established based on the principle of the minimum total potential energy. Further, the proposed method effectively avoided the numerical error introduced by using the deformation displacement to derive and calculate strain energy and greatly reduced the smoothness requirement of the trial function. Hence, the novel model possesses the benefits of high modeling accuracy, fast convergence speed in the numerical calculation, and versatility for application to straight and initially curved beams. Meanwhile, the parameterized curvature solutions have great geometric and mechanical significance that allows not only to realize the unified kinematic modeling of rigid and flexible bodies but also to efficiently present the complex and rich deformation information of slender beams under large deflections. The improved performance of the developed models was demonstrated through some classic numerical examples.

We analyze the stability of elastic systems composed of the shells of revolution with variable curvature and complex structures in the field of conservative axisymmetric loads of different nature. Within the framework of classical and refined theories of shells, we determine the limit and bifurcation critical values of the acting loads based on the geometrically nonlinear statement of the problem and a criterion of dynamic stability. To solve the corresponding nonlinear and eigenvalue problems, we propose to use a numerical-analytic approach based on their rational reduction to one-dimensional linear boundaryvalue problems in the meridional coordinate and their numerical solution by the discrete-orthogonalization method. We present test examples that confirm the applicability of the proposed procedure to the analyzed class of problems. The limit and bifurcation values of the critical loads in the shell system are analyzed depending on its geometric parameters.

Three-dimensional (3D) direct numerical simulations (DNS) of the viscous incompressible fluid flow through a helical pipe with circular cross section were performed. The flow is governed by three parameters: the Dean number (or the Reynolds number), curvature, and torsion. First, we obtained steady solutions by steady 3D calculations, where dual solutions were found, one was uniform in the pipe axial direction and the other varied very slowly, if torsion exceeded a critical value. Then, the instability of the steady solutions obtained was studied by unsteady 3D calculations. We obtained critical Reynolds numbers of steady to unsteady transition by observing the behaviors of the unsteady solutions. The present results of the critical Reynolds number nearly agreed with those by the 2D linear stability analysis (Yamamoto et al. [9] ) except for the lowest critical Reynolds number region, where the present study gave the critical Reynolds number much less than that obtained by the 2D linear stability analysis. We found the vortical structures in the form of a standing wave slightly above the marginal instability state, which is a trigger of explosive 3D instability.

We prove that a probability solution of the stationary Kolmogorov equation generated by a first order perturbation v of the Ornstein–Uhlenbeck operator L possesses a highly integrable density with respect to the Gaussian measure satisfying the non-perturbed equation provided that v is sufficiently integrable. More generally, a similar estimate is proved for solutions to inequalities connected with Markov semigroup generators under the curvature condition $CD(\theta ,\infty )$ . For perturbations from Lp an analog of the Log-Sobolev inequality is obtained. It is also proved in the Gaussian case that the gradient of the density is integrable to all powers. We obtain dimension-free bounds on the density and its gradient, which also covers the infinite-dimensional case.

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.

Over a century has passed since Einstein formulated general relativity, a remarkable theory of gravitation about which he grappled for about a decade following his theory of special relativity. The...

In this paper, the linear stability of sand waves sheared by a turbulent flow is analyzed. The velocity distribution in the streamwise direction is considered to follow the logarithmic law. The modified pressure distribution owing to the streamline curvature induced by the sand waves is employed in the formulation. The pressure distribution is derived using the Boussinesq approximation for the variation of streamline curvature over the flow depth. The flow model is coupled with the sediment transport model to study the stability of sand waves. Both the modes of sediment transport as bedload and suspended load are considered. The linear stability analysis reveals the favorable region for the formation of dunes and antidunes on the plane formed by the Froude number and the dimensionless wavenumber. The region of instability increases with an increase in particle parameter, while it decreases with an increase in relative roughness. The analysis reveals that the sand waves form beyond a limiting Froude number corresponding to the threshold Shields number. The theoretical results compare well with the available experimental data.

Considering a non-symmetric linear connection and its dual one appear 6 independent curvature tensors [15]. In the present paper we study properties of curvature tensors of semi-symmetric metric co...

This study investigates the dynamic pressure at the flip buckets of chute spillways, which is one of the most important design parameters of these structures. First, the dimensionless parameters affecting pressure were determined by dimensional analysis. Following that, according to the characteristics of selected dams with chute spillways leading to flip buckets, certain Froude number intervals of inflow to the flip bucket, as well as the chute slope angle, radius, and flip bucket curvature angle were selected for analysis. The combination of these parameters resulted in a total of 137 models simulated in FLOW-3D to obtain bottom pressure and maximum pressure values in the flip bucket. Next, based on the dimensionless parameters considered, equations were proposed to determine the bottom pressure and maximum pressure in the flip bucket downstream of the chute, using multiple regression analysis. Using the numerical modeling run results, along with multiple regression analyses, the unknown coefficients of the dimensionless pressure relationship were determined, and final equations for the bottom pressure and maximum pressure were presented. The correlation coefficient and Mean Absolute Percentage Error (MAPE) values of the proposed equations for determining the bottom pressure and maximum pressure were 0.94 and 0.96, and, 6.75% and 8.49%, respectively. These values indicate the appropriate accuracy of the proposed equations. In the proposed equations, the Froude number, relative curvature, chute slope angle, takeoff angle, and flip bucket’s curvature angle, respectively, had the highest impacts on the bottom pressure and maximum pressure.

Motion-trajectory (MT) curves are used to introduce Frenet oscillations. Time-varying orientation of the motion plane that contains the absolute velocity and acceleration vectors is defined in terms of three Frenet–Euler angles; the curvature, vertical-development, and bank angles, referred to as the Frenet angles for brevity. The Frenet bank angle and the associated Frenet super-elevation of the motion plane, which measure deviation of the centrifugal inertia force from the horizontal plane, can be used to shed light on definition of the balance speed used in practice. The concept of the pre-super-elevated osculating (PSEO) plane is introduced and Rodrigues’ formula is employed to develop an orthogonal rotation matrix that provides a geometric interpretation of the PSEO plane. A new inverse-dynamics problem that utilizes experimentally or simulation recorded motion trajectories (RMT) is used to define the Frenet inertia forces and demonstrate their equivalence to the Cartesian form of the inertia forces. New expressions for the curvature vector in terms of the velocity and acceleration, limit on the magnitude of the tangential acceleration for a given forward velocity, condition required for the centrifugal force to remain horizontal, and condition of curvature singular points are derived. The Frenet bank angle can be used to prove existence of the normal vectors at the curvature singular points. It is shown that the inertia force can assume different forms, depending on the curve parameter used. The results of a simple analytical curve demonstrate Frenet oscillations and importance of distinguishing between the highway-ramp and railroad track bank angles and super-elevations, which are time-invariant, and the Frenet bank angle and super-elevation, which are motion-dependent.

The objective of this article is to investigate the effects of electromagnetic field on the generalization of Lemaître–Tolman–Bondi (LTB) spacetime by keeping in view the Palatini f(R) gravity and dissipative dust fluid. For performing this analysis, we followed the strategy deployed by Herrera et al. (Phys Rev D 82(2):024021, 2010). We have explored the modified field equations along with kinematical quantities and mass function and constructed the evolution equations to study the dynamics of inhomogeneous universe along with Raychauduary and Ellis equations. We have developed the relation for Palatini f(R) scalar functions by splitting the Riemann curvature tensor orthogonally and associated them with metric coefficients using modified field equations. We have formulated these scalar functions for LTB and its generalized version, i.e., GLTB under the influence of charge. The properties of GLTB spacetime are consistent with those of the LTB geometry and the scalar functions found in both cases are comparable in the presence of charge and Palatini f(R) curvature terms. The symmetric properties of generalized LTB spacetime are also studied using streaming out and diffusion approximations.

PurposeIn this paper, we consider the Heisenberg groups which play a crucial role in both geometry and theoretical physics.Design/methodology/approachIn the first part, we retrieve the geometry of left-invariant Randers metrics on the Heisenberg group H2n+1, of dimension 2n + 1. Considering a left-invariant Randers metric, we give the Levi-Civita connection, curvature tensor, Ricci tensor and scalar curvature and show the Heisenberg groups H2n+1 have constant negative scalar curvature.FindingsIn the second part, we present our main results. We show that the Heisenberg group H2n+1 cannot admit Randers metric of Berwald and Ricci-quadratic Douglas types. Finally, the flag curvature of Z-Randers metrics in some special directions is obtained which shows that there exist flags of strictly negative and strictly positive curvatures.Originality/valueIn this work, we present complete Reimannian geometry of left invarint-metrics on Heisenberg groups. Also, some geometric properties of left-invarainat Randers metrics will be studied.

The Weil–Petersson metric for the moduli space of Riemann surfaces has negative sectional curvature. Surfaces represented in the complement of a compact set in the moduli space have short geodesics. At such surfaces the Weil–Petersson metric is approximately a product metric. An almost product metric has sections with almost vanishing curvature. We bound the sectional curvature away from zero in terms of the product of lengths of short geodesics on Riemann surfaces. We give examples and an expectation for the actual vanishing rate.

We examine the geometry of the subspace of homogeneous probability measures in terms of the 2-Wasserstein metric on the space of all probability measures on Hilbert spaces of functions. We show that, on appropriate Hilbert spaces, the geodesics joining homogeneous measures stay in the space of homogeneous measures and, as a result, the homogeneous measures themselves form a space of nonpositive curvature in the sense of A. D. Alexandrov.

We study the Knieper measures of the geodesic flows on non-compact rank 1 manifolds of non-positive curvature. We construct the Busemann density on the ideal boundary, and prove that if there is a Knieper measure on T1M with finite total mass, then the Knieper measure is unique, up to a scalar multiple. Our result partially extends Paulin-Pollicott-Shapira’s work on the uniqueness of finite Gibbs measure of geodesic flows on negatively curved non-compact manifolds to non-compact manifolds of non-positive curvature.

By the classical Li–Yau inequality, an immersion of a closed surface in ℝ n {\mathbb{R}^{n}} with Willmore energy below 8 ⁢ π {8\pi} has to be embedded. We discuss analogous results for curves in ℝ 2 {\mathbb{R}^{2}} , involving Euler’s elastic energy and other possible curvature functionals. Additionally, we provide applications to associated gradient flows.

Continuous‐curvature paths play an important role in the area of driving robots: as vehicles usually cannot changethe steering angle in zero‐time, real trajectories must not have discontinuities in the curvature profile. Typical continuous‐curvature paths are thus built of straight lines, arcs and clothoids. Due to the geometric nature of clothoids, some questions in the area of trajectory planning are difficult the answer – usually we need approximations here. In this paper we describe a full approach for continuous‐curvature trajectory planning for mobile robots – it covers a maneuver‐based planning with Viterbi optimization and geometric approximations required to construct the respective clothoid trajectories.

We solve the task of representing free forms by an arrangement of panels that are manufacturable by precise isometric bending of surfaces made from a small number of molds. In fact we manage to solve the paneling task with surfaces of constant Gaussian curvature alone. This includes the case of developable surfaces which exhibit zero curvature. Our computations are based on an existing discrete model of isometric mappings between surfaces which for this occasion has been refined to obtain higher numerical accuracy. Further topics are interesting connections of the paneling problem with the geometry of Killing vector fields, designing and actuating isometries, curved folding in the double-curved case, and quad meshes with rigid faces that are nevertheless flexible.

This article presents and illustrates several important subset design approaches for Gaussian nonlinear regression models and for linear models where interest lies in a nonlinear function of the model parameters. These design strategies are particularly useful in situations where currently- used subset design procedures fail to provide designs which can be used to fit the model function. Our original design technique is illustrated in conjuction with D-optimality, Bayesian D-optimality and Kiefer'sk-optimality, and is extended to yield subset designs which take account of curvature.

We examine the Kerr/CFT correspondence in Einstein gravity extended with quadratic curvature invariants. We consider two explicit examples in four and five dimensions and compute the central charges of the asymptotic symmetry algebras of the near horizon geometries, using the improved version of the BBC formalism that encompasses the information of the Lagrangian. We find that the resulting Cardy entropy differs from the Wald entropy, caused by the Riemann-squared $R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}$ term., Comment: Latex, 17 pages

When the Froude number F of a free-surface flow ranges between 0.3 and 3, the flow is unstable and frequently characterised by free surface undulations, with the undular hydraulic jump being a seminal flow in hydro-environmental mechanics. The presence of the free surface undulations significantly affects the flow field, with major velocity and pressure field redistributions between successive crests and troughs. All the current theoretical models to simulate undular hydraulic jumps are limited to two-dimensional flow conditions, ignoring all the relevant three-dimensional flow effects, namely shock-wave drag, turbulent breaking and turbulent stresses. These aspects are critically accounted for in this review article, where a depth-averaged Boussinesq model which approximately accounted for 3D flow effects was presented, constituting the first attempt in this line. The model predictions were compared with experimental results from different sources for F1

In this paper, our purpose is to establish new Nomizu–Smyth type results concerning complete Riemannian immersions in the hyperbolic and de Sitter spaces. Assuming that the image of the Gauss map of such an immersion lies in a totally umbilical hypersurface and supposing the existence of a point of the immersion where the Gauss–Kronecker curvature does not vanish, we prove that the immersion must be a totally umbilical hypersurface of the ambient space.

The geometry description plays a central role in many engineering applications and directly influences the quality of the computer simulation results. The geometry of a space curve can be completely defined in terms of two parameters: the horizontal and vertical curvatures, or equivalently, the curve curvature and torsion. In this paper, distinction is made between the track angle and space-curve bank angle, referred to in this paper as the Frenet bank angle. In railroad vehicle systems, the track bank angle measures the track super-elevation required to define a balance speed and achieve a safe vehicle operation. The formulation of the track space-curve differential equations in terms of Euler angles, however, shows the dependence of the Frenet bank angle on two independent parameters, often used as inputs in the definition of the track geometry. This paper develops the general differential equations that govern the track geometry using the Euler angle sequence adopted in practice. It is shown by an example that a curve can be twisted and vertically elevated but not super-elevated while maintaining a constant vertical-development angle. The continuity conditions at the track segment transitions are also examined. As discussed in the paper, imposing curvature continuity does not ensure continuity of the tangent vectors at the curve/spiral intersection. Several curve geometries that include planar and helix curves are used to explain some of the fundamental issues addressed in this study.

In this paper, we have investigated some aspects of gradient $$\rho$$ -Einstein Ricci soliton in a complete Riemannian manifold. First, we have proved that the compact gradient $$\rho$$ -Einstein soliton satisfying some curvature conditions is isometric to the Euclidean sphere by showing that the scalar curvature becomes constant. Second, we have shown that in a non-compact gradient $$\rho$$ -Einstein soliton satisfying an integral condition, the scalar curvature vanishes.

In this paper, we study locally projectively flat Finsler metrics of constant flag curvature. We find equations that characterize these metrics by warped product. Using the obtained equations, we manufacture new locally projectively flat Finsler warped product metrics of vanishing flag curvature. These metrics contain the metric introduced by Berwald and the spherically symmetric metric given by Mo-Zhu.

To uncover the chewing mechanism of Cyrtotrachelus buqueti Guer, a mathematical model was created and a kinematic analysis of its rostrum mouthparts was conducted for, to our knowledge, the first time. To reduce noise and improve the quality of scanning electron micrographs of the weevil's mouthparts, nonlocal means and integral nonlocal means algorithms were proposed. Additionally, based on a comparison and analysis of five classical edge detection algorithms, a multiscale edge detection algorithm based on the B-spline wavelet was used to obtain the boundaries of structural features. The least squares method was used to analyze the data of the mouthparts to fit the mathematical model and fitted curves were obtained using Gaussian equations. The results show that curvature and concave–convex variations of the weevil's mouthparts can highlight fluctuations in friction effects when it chews bamboo shoots, which is helpful in preventing debris from bamboo shoots or other debris from sticking to the mouthpart surfaces. Moreover, this paper highlights the utility of micro-computed tomography (microCT) for three-dimensional (3D) reconstruction and a flowchart is suggested. The reconstructed slices were 9.0 μm thick and an accurate 3D rendered model was obtained from a series of microCT slices. Finally, a real model of the rostrum mouthparts was analyzed using finite-element analysis. The results provide a biological template for the design of a novel bionic drilling mechanism.

This paper aims to study spacelike surfaces from a given spacelike curve in Minkowski 3–space. Also, we investigate the necessary and sufficient conditions for the given space-like curve to be the line of curvature on the space-like surface. Depending on the causal character of the curve, the necessary and sufficient conditions for the given space-like curve to satisfy the line of curvature and the geodesic (resp. asymptotic) requirements are also analyzed. Furthermore, we give with illustration some computational examples in support of our main results.

Covariant Lyapunov vectors or CLVs span the expanding and contracting directions of perturbations along trajectories in a chaotic dynamical system. Due to efficient algorithms to compute them that only utilize trajectory information, they have been widely applied across scientific disciplines, principally for sensitivity analysis and predictions under uncertainty. In this paper, we develop a numerical method to compute the directional derivatives of the first CLV along its own direction; the norm of this derivative is also the curvature of one-dimensional unstable manifolds. Similar to the computation of CLVs, the present method for their derivatives is iterative and analogously uses the second-order derivative of the chaotic map along trajectories, in addition to the Jacobian. We validate the new method on a super-contracting Smale–Williams Solenoid attractor. We also demonstrate the algorithm on several other examples including smoothly perturbed Arnold Cat maps, and the Lorenz’63 attractor, obtaining visualizations of the curvature of each attractor. Furthermore, we reveal a fundamental connection of the derivation of the CLV self-derivative computation with an efficient computation of linear response of chaotic systems.

In this paper, we give some vanishing theorems for harmonic p-forms on complete noncompact Riemannian manifolds satisfying a weighted p-Poincare inequality with nonnegative scalar curvature and under pointwise curvature pinching conditions which are bounded from above by the weight function.

PurposeLet (M, g) be a n-dimensional smooth Riemannian manifold. In the present paper, the authors introduce a new class of natural metrics denoted by gf and called gradient Sasaki metric on the tangent bundle TM. The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature, scalar and sectional curvatures.Design/methodology/approachIn this paper the authors introduce a new class of natural metrics called gradient Sasaki metric on tangent bundle.FindingsThe authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.Originality/valueThe authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.

We present convective instability analyses of the boundary-layer flows originated over cones (with half-angle ψ ≤ 9 0 ∘ ) rotating in an otherwise still conducting fluid. A uniform magnetic field (with magnetic strength parameter, m ≥ 0 ) is acting normally on the surface of each cone. In the non-magnetic case of cones, comparison of present results with existing experimental and theoretical studies lead *to propose that onset of instabilities may be attributed to crossflow (type I) modes and streamline curvature (type II) modes for broad cones (with ψ ≥ 4 0 ∘ ) and more slender cones (with ψ ≤ 1 5 ∘ ), respectively. For slender cones with half-angle ψ in the vicinity of 25° the presented results validate the hypothesis of centrifugal (type III) modes ( discovered by Garrett et al. (2014)). In the magnetic case, increasing magnetic strength m ∈ ( 0 , 11 ] considerably delays the onset of instabilities (due to both type I and type II modes) over each rotating cone with fixed ψ . The stabilising influence of magnetism is in agreement with the meagrely available theoretical studies for magnetic rotating disk case ψ = 9 0 ∘ . In addition, our results lead to suggest that streamline curvature mode over each cone with fixed half-angle ψ become sensitive with increasing m. A minimum m 0 ∈ [ 0 , 11 ] exists for each fixed half-angle ψ 0 ≤ 9 0 ∘ such that whenever m ≥ m 0 the onset of instabilities (in the sense of minimum critical Reynolds numbers) over every cone with half-angle ( ψ ≤ ψ 0 ) are commenced by type II mode instead of type I mode whilst the aforementioned conjecture of stabilising effect of magnetism is not violated in the considered range of parameters. Under non-stationary vortices assumption with magnetism, we show that crossflow modes travelling at around 75% of the cone surface are likely to be selected in applications where smooth and frictionless surfaces are used. This finding is in complete agreement with the non-magnetic case of rotating cone in existing studies.

For a hypersurface in ${\mathbb R}^3$, Willmore flow is defined as the $L^2$--gradient flow of the classical Willmore energy: the integral of the squared mean curvature. This geometric evolution law is of interest in differential geometry, image reconstruction and mathematical biology. In this paper, we propose novel numerical approximations for the Willmore flow of axisymmetric hypersurfaces. For the semidiscrete continuous-in-time variants we prove a stability result. We consider both closed surfaces, and surfaces with a boundary. In the latter case, we carefully derive weak formulations of suitable boundary conditions. Furthermore, we consider many generalizations of the classical Willmore energy, particularly those that play a role in the study of biomembranes. In the generalized models we include spontaneous curvature and area difference elasticity (ADE) effects, Gaussian curvature and line energy contributions. Several numerical experiments demonstrate the efficiency and robustness of our developed numerical methods., 52 pages, 19 figures

The authors propose an adaptive, general and data‐driven curvature penalty for signal denoising via the Schrödinge operator. The term is derived by assuming noise to be generally Gaussian distributed, a widely applied assumption in most 1D signal denoising applications. The proposed penalty term is simple and in closed‐form, and it can be adapted to different types of signals as it depends on data‐driven estimation of the smoothness term. Combined with semi‐classical signal analysis, we refer this method as C‐SCSA in the context. Comparison with existing methods is done on pulse shaped signals. It exhibits higher signal‐to‐noise ratio and also preserves peaks without much distortion, especially when noise levels are high. ECG signal is also considered, in scenarios with real and non‐stationary noise. Experiments validate that the proposed denoising method does indeed remove noise accurately and consistently from pulse shaped signals compared to some of the state‐of‐the‐art methods.

In this paper, we give a new lower bound for the eigenvalues of the Dirac operator defined on the Spin Riemannian hypersurface manifold endowed with 2-tensor, in terms of the Energy-Momentum tensor, scalar curvature and extrinsic curvature. Then this estimate is improved in two different ways by considering the conformal invariance of the Dirac operator. The first is given in term of the first eigenvalue of the Yamabe operator. The latter, is given in terms of the the area of a topological 2-sphere., TUBITAK The Scientific and Technological Research Council of Turkey [120F109]; TUBITAK (The Scientific and Technological Research Council of Turkey), This study was supported by TUBITAK The Scientific and Technological Research Council of Turkey (Project Number: 120F109). Also, The authors would like to appreciate TUBITAK (The Scientific and Technological Research Council of Turkey) for supporting this research.

In this paper, we give a full classification of pseudo-Riemannian Lie groups of dimension four with non-trivial recurrent curvature tensor (i.e., non-locally symmetric). Then we investigate Ricci solitons on recurrent curvature Lie groups. Locally conformally flat examples of our classification have also been presented.

This paper is devoted to examine a special case of Walker metric on a 4-dimensional manifold, and some of its curvature properties are studied, e.g., conditions for a Walker metric to be Einstein or locally conformally flat. Finally, a necessary and sufficient condition for a function c(x, y, z, t) of Walker metric of 4-dimensional manifold to have vector field X on M to be a Killing vector field, is constructed.