Curvature Conditions Research Articles

η-Ricci solitons in Kenmotsu manifolds

Abstract The object of the present paper is to study generalized weakly symmetric and generalized weakly Ricci symmetric Kenmotsu manifolds whose metric tensor is η-Ricci soliton. The paper also aims to bring out curvature conditions for which η-Ricci solitons in Kenmotsu manifolds are sometimes shrinking or expanding and some other time remain steady. The existence of each generalized weakly symmetric and generalized weakly Ricci symmetric Kenmotsu manifold is ensured by an example.

On Three Dimensional Pseudosymmetric Alpha-Kenmotsu Manifolds

The main purpose of this paper is to study pseudosymmetric conditions on alpha-Kenmotsu manifolds with dimension . In particular, we obtain some results satisfying some certain curvature conditions on such manifolds depending on.

Hypersurfaces in space forms satisfying some generalized Einstein metric condition

The difference tensor C⋅R−R⋅C of Einstein manifolds, some quasi-Einstein manifolds and Roter type manifolds, of dimension n≥4, satisfy the following curvature condition: (∗)C⋅R−R⋅C=Q(S,C)−(κ∕(n−1))Q(g,C). We investigate hypersurfaces M in space forms N satisfying (∗). The main result states that if the tensor C⋅R−R⋅C of a non-quasi-Einstein hypersurface M in N is a linear combination of the tensors Q(g,C) and Q(S,C) then (∗) holds on M. In the case when M is a quasi-Einstein hypersurface in N and some additional assumptions are satisfied then (∗) also holds on M.

Global Convergence of A Modified Dia-yuan Conjugate Gradient for Unconstrained Optimization

ABSTRACT A new modified formula of the parameter for the conjugate gradient method is proposed in this paper. The algorithm is a very effective method for solving unconstrained optimization, especially for large-scale problems. The suggested method based on famous Dia-Yuan method with approaches new which can give the new algorithm. Generally, the numerator of the plays a vital role in having strong convergent property. However, it has jamming issue in practice but the new algorithm has better numerical results in practice and demonstrates performance well. The new method has descent condition under curvature condition and sufficient descent condition. Furthemore, the global convergence of the proposed method is established as well. That’s why, preliminary numerical result compared with standard Dia-Yuan of method show that our method is more robust and effective.

A Note on f-biharmonic Legendre Curves in S-Space Forms

In this paper, we study f-biharmonic Legendre curves in S-space forms. Our aim is to find curvature conditions for these curves and determine their types, i.e., a geodesic, a circle, a helix or a Frenet curve of osculating order r with specific curvature equations. We also give a proper example of f-biharmonic Legendre curves in the S-space form R^(2m+s)(−3s), with m = 2 and s = 2.

Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions

We prove existence and uniqueness of solutions to a nonlinear stochastic evolution equation on the d-dimensional torus with singular p-Laplace-type or total variation flow-type drift with general sublinear doubling nonlinearities and Gaussian gradient Stratonovich noise with divergence-free coefficients. Assuming a weak defective commutator bound and a curvature-dimension condition, the well-posedness result is obtained in a stochastic variational inequality setup by using resolvent and Dirichlet form methods and an approximative Itô-formula.

Take-over requests during highly automated driving: How should they be presented and under what conditions?

Even with highly automated driving, there will exist the occasional need for a human driver to resume control. The design of take-over requests (TOR) and the driving scenario upon their presentation are factors that can be expected to determine the effective resumption of vehicle control. We conducted a driving simulator study in the laboratory that examined how either an abstract or skeuomorphic TOR design influenced driving resumption, given different conditions of road curvature and time headway. We found that drivers react faster and more accurately when experiencing a skeuomorphic visualisation of a TOR prompt compared to an abstract interface, especially when there is less time headway. Moreover, participants reported subjective preferences for the skeuomorphic TOR design and considered them to be more intuitively comprehensible. Furthermore, more accurate TOR responses were found when TORs of lateral control were presented before longitudinal control. Based on these results, we formulated five conclusions relating to the design of take-over requests. These conclusions should be validated in more realistic settings. However, in the future, they could be the basis for more research developing design guidelines that could create safe and reliable communications between drivers and vehicles.

Cosmic strings, deformed lattices and spontaneous symmetry breaking

External conditions have a dramatic impact on the way dynamical symmetry breaking occurs. In a curved background, the natural expectation is that curvature works toward the restoration of the internal symmetry. We show instead that, for topological defects, the competing action of the locally induced curvature and boundary conditions generated by the non-trivial topology allows configurations where symmetries can be spontaneously broken close to the core. Inspired by the effect of geometrical deformations on 2D lattices, we then propose a novel mechanism to induce a superconducting phase by triggering condensation along cosmic strings.

Cosmological singularities from high matter density without global topological assumptions

Cosmological singularity theorems such as that of Hawking and Penrose assume local curvature conditions as well as global ones like the existence of a compact (achronal) slice. Here, we prove a new singularity theorem for chronological spacetimes that satisfy what we call a ‘past null focusing’ condition. Such a condition forces all null geodesics gamma :[0,a)rightarrow M with future endpoint gamma (0) to develop a pair of conjugate points if past complete. By the Einstein field equations, such a condition will be satisfied if the density of matter fields remains sufficiently high towards the past of the spacetime, as may be expected in certain cosmological scenarios. The theorem obtained doesn’t make starting assumptions about the spacetime’s topology, such as the existence of a compact achronal slice, and if in addition to a ‘past null focusing’ condition we assume the timelike convergence condition, then further consequences pertaining to the existence of CMC foliations and the character of the singularity are obtained. With the addition of the timelike convergence condition, we obtain the conclusion that all timelike geodesics are past incomplete, rather than the existence of a single incomplete non-spacelike geodesic.

On the Oberlin affine curvature condition

We generalize the well-known notions of affine arclength and affine hypersurface measure to submanifolds of any dimension d in Rn, 1≤d≤n−1. We show that a canonical equiaffine-invariant measure exists and that, modulo sufficient regularity assumptions on the submanifold, the measure satisfies the affine curvature condition of Oberlin with an exponent which is best possible. The proof combines aspects of geometric invariant theory, convex geometry, and frame theory. A significant new element of the proof is a generalization to higher dimensions of an earlier result concerning inequalities of reverse Sobolev type for polynomials on arbitrary measurable subsets of the real line.

Investigation on adaptive preview distance path tracking control with directional error compensation

Path tracking control cannot effectively satisfy the stability requirements of intelligent vehicles under large curvature conditions. To solve this problem, an adaptive preview distance path tracking controller with a hierarchical structure is proposed in this study. The vehicle centroid lateral acceleration and lateral error of the preview point are taken as the inputs of the upper controller, and the optimal preview distance is obtained based on fuzzy inference. To eliminate the subjective influence of the membership function and fuzzy rule selection in the fuzzy controller design, a genetic algorithm is used for optimization. The lower controller is a sliding mode controller that aims to achieve intelligent vehicle self-tracking. Moreover, a radial basis function neural network is adopted to combine with the sliding mode controller to eliminate output chattering. However, adaptive adjustment of the preview distance deteriorates the vehicle directional tracking error, which makes controlling the vehicle at the road curvature switching point difficult. Thus, a directional error compensation controller is designed based on the iterative learning theory to compensate the front wheel steering angle. Simulations under two standard conditions are carried out to verify the control effect. The results show that, in a double lane change test, the peak centroid acceleration and coaxial load transfer rates decreased by 26.91% and 19.83% at low velocity, respectively, and the improvements at high velocity were 42.71% and 39.22%, respectively. In the pylon course slalom test, all three performance indicators decreased by more than 30%, which indicates the modified adaptive preview distance path tracking controller with a hierarchical structure can effectively improve the vehicle handling performance and roll stability and can ensure the tracking accuracy.

Two beam toy model for dislocation contrast in ECCI

Dislocation contrast in the SEM, as observed though electron channelling contrast imaging (ECCI), is commonly treated analogously to the contrast in the TEM. This perception is based on early studies done for dislocations parallel with the surface where the surface relaxation is negligible. However, for threading dislocations (TD) that interact with the surface (normal or inclined), as is the case for nitrides materials, g b type invisibility criteria are no longer fully applicable to ECCI, especially in forward geometry [1]. Dislocations change locally the lattice curvature and Bragg diffraction conditions in the crystal, affecting the form and diffracting behaviour of the electron wavefunction in that region. More explicitly, Howie and Whelan [2] had shown that dislocation contrast is the result of interband transitions between Bloch waves states which, in turn, are caused by the change in the displacement field, u(r), around the dislocation or local “strain”. Dynamical models have been used successfully to both predict and characterise dislocations in ECCI [3]. Nevertheless, the behaviour of dislocation contrast in ECCI in particular and diffraction contrast in the SEM in general remains somewhat opaque. In the work we investigate the behaviour of contrast causing strain as a means of insight into this problem.

Integrable nonlocal Hirota equations

We construct several new integrable systems corresponding to nonlocal versions of the Hirota equation, which is a particular example of higher order nonlinear Schrödinger equations. The integrability of the new models is established by providing their explicit forms of Lax pairs or zero curvature conditions. The two compatibility equations arising in this construction are found to be related to each other either by a parity transformation P, by a time reversal T or a P T-transformation possibly combined with a conjugation. We construct explicit multisoliton solutions for these models by employing Hirota’s direct method as well as Darboux-Crum transformations. The nonlocal nature of these models allows for a modification of these solution procedures as the new systems also possess new types of solutions with different parameter dependence and different qualitative behavior. The multisoliton solutions are of varied type, being, for instance, nonlocal in space, nonlocal in time of time-crystal type, regular with local structures either in time/space or of rogue wave type.

Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical Laplacian comparison theorems for the sub-Riemannian distance may be obtained as a limit of horizontal and vertical Laplacian comparison theorems for the Riemannian distances approximations. As a corollary we prove that, under suitable curvature conditions, sub-Riemannian Sasakian spaces are actually limits of Riemannian spaces satisfying a uniform measure contraction property.

The Cartan–Hadamard conjecture and the Little Prince

The generalized Cartan–Hadamard conjecture says that if \Omega is a domain with fixed volume in a complete, simply connected Riemannian n -manifold M with sectional curvature K \le \kappa \le 0 , then \partial\Omega has the least possible boundary volume when \Omega is a round n -ball with constant curvature K=\kappa . The case n=2 and \kappa=0 is an old result of Weil. We give a unified proof of this conjecture in dimensions n=2 and n=4 when \kappa=0 , and a special case of the conjecture for \kappa < 0 and a version for \kappa > 0 . Our argument uses a new interpretation, based on optical transport, optimal transport, and linear programming, of Croke's proof for n=4 and \kappa=0 . The generalization to n=4 and \kappa \ne 0 is a new result. As Croke implicitly did, we relax the curvature condition K \le \kappa to a weaker candle condition Candle} (\kappa) or LCD} (\kappa) . We also find counterexamples to a naïve version of the Cartan–Hadamard conjecture: For every \epsilon > 0 , there is a Riemannian \Omega \cong B^3 with (1-\epsilon) -pinched negative curvature, and with |\partial\Omega| bounded by a function of \epsilon and |\Omega| arbitrarily large. We begin with a pointwise isoperimetric problem called "the problem of the Little Prince". Its proof becomes part of the more general method.

Design of a simple fisheye lens.

We present a simple design of fisheye lens that consists of a pre-group optics of two negative meniscus lenses and a post-group optics of three singlet lenses. The pre-group optics is first designed according to the requirements for the compression ratio of field angle and the transverse dimension of the lens. The balance conditions of the field curvature, longitudinal and transverse chromatic aberrations between the pre-group and post-group optics, and the requirement of the total optical power of the system are used to determine the optical power and the spacing of the lenses of the post-group optics in the thin-lens model; the initial curvature radii of three lenses are then determined according to the resultant optical powers of lenses, and their thicknesses are determined according to the exact balance of field curvature and longitudinal and transverse chromatic aberrations. Finally, we apply a self-adaptive and normalized real-coded genetic algorithm to optimize the starting design of the fisheye lens obtained in the above process.

Using Naturalistic Vehicle-Based Data to Predict Distraction and Environmental Demand

This research utilized vehicle-based measures from a naturalistic driving dataset to detect distraction as indicated by long off-path glances (≥ 2 s) and whether the driver was engaged in a secondary (non-driving) task or not, as well as to estimate motor control difficulty associated with the driving environment (i.e. curvature and poor surface conditions). Advanced driver assistance systems can exploit such driver behavior models to better support the driver and improve safety. Given the temporal nature of vehicle-based measures, Hidden Markov Models (HMMs) were utilized; GPS speed and steering wheel position were used to classify the existence of off-path glances (yes vs. no) and secondary task engagement (yes vs. no); lateral (x-axis) and longitudinal (y-axis) acceleration were used to classify motor control difficulty (lower vs. higher). Best classification accuracies were achieved for identifying cases of long off-path glances and secondary task engagement with both accuracies of 77%.

Conformally Embedded Spacetimes and the Space of Null Geodesics

It is shown that the space of null geodesics of a causally simple Lorentzian manifold is Hausdorff if it admits an open conformal embedding into a globally hyperbolic spacetime. This provides an obstruction to conformal embeddings of causally simple spacetimes into globally hyperbolic ones irrespective of curvature conditions. Examples of causally simple spacetimes are given not conformally embeddable into globally hyperbolic ones.

New volume comparison results and applications to degeneration of Riemannian metrics

We consider a condition on the Ricci curvature involving vector fields, which is broader than the Bakry-Émery Ricci condition. Under this condition volume comparison, Laplacian comparison, isoperimetric inequality and gradient bounds are proven on the manifold.Specializing to the Bakry-Émery Ricci curvature condition, we initiate a short cut approach to work on the original manifold, which yields, under a weaker than usual assumption, the results mentioned above for the original manifold. These results are different from most well known ones in the literature where the conclusions are made on the weighted manifold instead.Applications on convergence and degeneration of Riemannian metrics under this curvature condition are given. To this effect, in particular for the Bakry-Émery Ricci curvature condition, the gradient of the potential function is allowed to have singularity of order close to 1, which is broader than those allowed by the traditional weighted method in general. Coupling with some new techniques in constructing model functions, this approach enables us to extend some of the results in the papers [14], [11], [36], [29] and [31]. The condition also covers general Ricci solitons instead of just gradient Ricci solitons.

Chaotic dynamics and primary resonance analysis of a curved carbon nanotube considering influence of thermal and magnetic fields

In this paper, the nonlocal Euler–Bernoulli beam theory is presented to study the primary resonance and chaotic vibration of a curved single-walled carbon nanotube (CSWCNT). The CSWCNT is exposed to axial thermomagnetic and transverse harmonic forces resting on a viscoelastic foundation. A single-mode Galerkin approximation is implemented to transform the nonlinear governing partial differential equation into an ordinary one. The multiple scales method is employed to determine the primary resonance response of a clamped–clamped CSWCNT under harmonic external force. The effects of magnetic field strength, temperature change and amplitude of sinusoidal curvature, mode number and different boundary conditions including clamped–free and free–free are examined to study the jumping phenomenon and instability states in primary resonance frequency response. Moreover, by applying the Runge–Kutta numerical method, the bifurcation diagram and the largest Lyapunov exponent curve are generated to detect the chaotic values of external amplitude of excitation. The phase plane trajectories along with Poincare map are presented to show the chaotic and periodic vibration of a CSWCNT. The results show that the axial thermomagnetic forces have significant effects on the frequency response of a CSWCNT and the nonlinear chaotic vibration has a strong dependency on the amplitude of excitation.