Warping Function Research Articles

Vanishing Homology of Warped Product Submanifolds in Complex Space Forms and Applications

In this paper, we prove the nonexistence of stable integral currents in compact oriented warped product pointwise semi-slant submanifold Mn of a complex space form M˜(4ϵ) under extrinsic conditions which involve the Laplacian, the squared norm gradient of the warped function, and pointwise slant functions. We show that i-the homology groups of Mn are vanished. As applications of homology groups, we derive new topological sphere theorems for warped product pointwise semi-slant submanifold Mn, in which Mn is homeomorphic to a sphere Sn if n≥4 and if n=3, then M3 is homotopic to a sphere S3 under the assumption of extrinsic conditions. Moreover, the same results are generalized for CR-warped product submanifolds.

Einstein Hypersurfaces of Warped Product Spaces

We consider Einstein hypersurfaces of warped products \(I\times _\omega {\mathbb {Q}}_\epsilon ^n,\) where \(I\subset {\mathbb {R}}\) is an open interval and \({\mathbb {Q}}_\epsilon ^n\) is the simply connected space form of dimension \(n\ge 2\) and constant sectional curvature \(\epsilon \in \{-1,0,1\}\). We show that, for all \(c\in {\mathbb {R}}\) (resp. \(c>0\)), there exist rotational hypersurfaces of constant sectional curvature c in \(I\times _\omega {\mathbb {H}}^n\) and \(I\times _\omega {\mathbb {R}}^n\) (resp. \(I\times _\omega {\mathbb {S}}^n\)), provided that \(\omega \) is nonconstant. We also show that the gradient T of the height function of any Einstein hypersurface of \(I\times _\omega \mathbb {Q} _\epsilon ^n\) (if nonzero) is one of its principal directions. Then, we consider a particular type of Einstein hypersurface of \(I\times _\omega {\mathbb {Q}}_\epsilon ^n\) with non vanishing T—which we call ideal—and prove that, for \(n>3,\) such a hypersurface \(\Sigma \) has either precisely two or precisely three distinct principal curvatures everywhere. We show that, in the latter case, there exist such a \(\Sigma \) for certain warping functions \(\omega ,\) whereas in the former case \(\Sigma \) is necessarily of constant sectional curvature and rotational, regardless the warping function \(\omega .\) We also characterize ideal Einstein hypersurfaces of \(I\times _\omega {\mathbb {Q}}_\epsilon ^n\) with no vanishing angle function as local graphs on families of isoparametric hypersurfaces of \({\mathbb {Q}}_\epsilon ^n.\)

Coupling 2D continuum and beam elements: a mixed formulation for avoiding spurious stresses

This paper presents a novel approach to coupling beam and solid elements. Connecting standard beam elements with solid elements results in a situation where the solid part of the model covers the cross-sectional deformation, while at the beam part assumptions only consider the rigid body movement of the cross-section. Therefore, kinematic assumptions do not include cross-section deformations such as warping and contraction. Due to this restriction, spurious stresses occur at the transition zone. To circumvent this problem, this contribution introduces a mixed hybrid transition element based on an extended Hu–Washizu functional. The extension allows the cross-section to contract and warp. Compared with other approaches that include these phenomena, precomputation of the warping function is unnecessary. The present approach considers the cross-sectional deformation using local variables. On the solid interface, there are only displacement degrees of freedom (depending on the solid discretization), whereas on the beam interface, there are two displacement degrees of freedom and one rotation. This allows beam-like parts of the solid model to be replaced with standard beam elements without affecting the overall accuracy of the model. Besides solid–beam coupling, the proposed formulation can also be used to apply beam-like boundary conditions as well as stress resultants to the solid interface. For the sake of simplicity and without restricting generality, the underlying element formulation is introduced for the 2D linear elastic case. Numerical examples demonstrate that the transition element causes no spurious stress distributions on the solid part. The results of full solid models are compared with mixed beam–solid models.

Characterization of Einstein Poisson warped product space

In this article, we study the problem of the existence and nonexistence of warping function associated with constant scalar curvature on pseudo-Riemannian Poisson warped product space under the assumption that fiber space has constant scalar curvature. We characterize the warping function on Einstein Poisson warped space by taking the various dimensions of base space B (i.e; (1). \(dim B=1,\) (2). \(dimB\ge 2\)).

Complete Stationary Spacelike Surfaces in an n-Dimensional Generalized Robertson–Walker Spacetime

Several uniqueness results for non-compact complete stationary spacelike surfaces in an \(n(\ge 3)\)-dimensional Generalized Robertson Walker spacetime are obtained. In order to do that, we assume a natural inequality involving the Gauss curvature of the surface, the restrictions of the warping function and the sectional curvature of the fiber to the surface. This inequality gives the parabolicity of the surface. Using this property, a distinguished non-negative superharmonic function on the surface is shown to be constant, which implies that the stationary spacelike surface must be totally geodesic. Moreover, non-trivial examples of stationary spacelike surfaces in the four dimensional Lorentz–Minkowski spacetime are exposed to show that each of our assumptions is needed.

A general optimization framework for dynamic time warping

The goal of dynamic time warping is to transform or warp time in order to approximately align two signals. We pose the choice of warping function as an optimization problem with several terms in the objective. The first term measures the misalignment of the time-warped signals. Two additional regularization terms penalize the cumulative warping and the instantaneous rate of time warping; constraints on the warping can be imposed by assigning the value $$+\infty$$ to the regularization terms. Different choices of the three objective terms yield different time warping functions that trade off signal fit or alignment and properties of the warping function. The optimization problem we formulate is a classical optimal control problem, with initial and terminal constraints, and a state dimension of one. We describe an effective general method that minimizes the objective by discretizing the values of the original and warped time, and using standard dynamic programming to compute the (globally) optimal warping function with the discretized values. Iterated refinement of this scheme yields a high accuracy warping function in just a few iterations. Our method is implemented as an open source Python package GDTW.

A Topological sphere theorem for contact CR-warped product submanifolds of an odd-dimensional unit sphere

AbstractIn this paper, we investigate topological sphere theorems for compact minimal contact CR-submanifolds of odd dimensional unit sphere. We show that if an inequality involving the warping function and the scalar curvature of the fibers is satisfied, a compact minimal contact CR-warped product submanifold of the odd dimensional unit sphere is homeomorphic to the sphere. In particular case, for 5-dimensional unit sphere, we show that a 4-dimensional compact minimal contact CR-warped product submanifold is homeomorphic to a sphere if ∣∇lnf∣2 < 1 is satisfied. By using Bonnet-Myers’s theorem we give a result about fundamental group and by Leung’s theorem we obtain a result about homology groups of a contact CR-warped submanifold.

Characterizations of PR-Pseudo-Slant Warped Product Submanifold of Para-Kenmotsu Manifold with Slant Base

In this article, we study the properties of PR-pseudo-slant submanifold of para-Kenmotsu manifold and obtain the integrability conditions for the slant distribution and anti-invariant distribution of such submanifold. We derived the necessary and sufficient conditions for a PR-pseudo-slant submanifold of para-Kenmotsu manifold to be a PR-pseudo-slant warped product which are in terms of warping functions and shape operator. Some examples of PR-pseudo-slant warped products of para-Kenmotsu manifold are also illustrated in the article.

A new structural model of warping torsion. II: Evaluation for a model problem

This paper presents an evaluation of the mixed formulation of warping torsion developed in the first part of this work. Specifically, we present a comparative study with two widely-used existing formulations, namely, the TWKV formulation of Timoshenko–Wagner–Kappus–Vlasov involving only the twist rotation (thus constraining the twist to the warping) and the RBV formulation of Reissner–Benscoter–Vlasov incorporating an additional warping field, as the newly proposed mixed formulation. However, in big contrast to those two formulations, the new formulation involves an alternative kinematic relation between the twist and warping motivated by a full mixed treatment of the axial strain–stress response. The evaluations are based on the exact closed-form solution for a model problem of a prismatic elastic shaft, with the torsional constants appearing in the different formulations computed with plane finite element approximations of the relevant warping functions on the cross section. The structural models are also compared with full 3D finite element simulations of the elastic solids. The new mixed formulation is found to improve upon the over-stiff constrained TWKV formulation as well as the RBV formulation with its physically incorrect stresses.

The nonexistence of gradient almost Ricci solitons warped product

In this paper, by slightly modifying Li-Yau's technique so that we can handle drifting Laplacians, we were able to find three different gradient estimates for the warping function, one for each sign of the Einstein constant of the fiber manifold. As an application, we exhibit a nonexistence theorem for gradient almost Ricci solitons possessing certain metric properties on the base of the warped product.

Uniform Torsion Analysis Of Composite Shafts Using Point Collocation Method Based On Pascal Polynomials

Torsion bars are important elements in the marine structure applications. The usage of the composite torsion bars has increased recently because it takes advantage of the different material property of each constituent. The current research focuses mainly on the uniform warping analysis of composite torsion shafts by using the point collocation method (PCM). The PCM is based on Pascal's triangle polynomial expansion and coupled with subdomain technique that is indispensable for the analysis of composite bars. In addition to satisfying the field equation (Laplace equation for warping function) and traction-free boundary condition for each component, the continuity conditions in both warping functions and tractions need to be imposed along the interfaces of all components. In this paper, multiple-scale factors are introduced into the pol-ynomial series to improve the numerical stability. The current scheme is easy to be implemented, and the algebraic equations for warping functions can be simply established. This nu-merical method can effectively calculate the structural re-sponses of the composite shafts. The numerical examples of this study include the composite shafts having elliptical and rectangular cross sections. The structural responses such as warping functions, shear stresses and torsional rigidity are reported. The PCM solutions are compared with the exact solutions and with the finite element solutions obtained from COMSOL. The results show that this simple numerical method can effectively achieve the solutions with a high de-gree of accuracy.

Ricci curvature for biwarped product submanifolds of the type N𝜃1 ×f2N𝜃2 ×f3N𝜃3 in Kenmotsu space forms

The major goal of this work is to express the Ricci curvature inequality for biwarped product submanifold of the type [Formula: see text] isometrically immersed in a Kenmotsu space form via terms of the squared norm of mean curvature vector and warping functions, where [Formula: see text], [Formula: see text] and [Formula: see text] are the slant submanifolds with slant angles [Formula: see text], [Formula: see text] and [Formula: see text]. The interpretations of equality are also explored. In addition, we provide several applications of these inequalities.

Several cracks in a rectangular bar reinforced by a piezoelectric layer subjected to torsion

This paper provides the torsional solution of multiple arbitrarily shaped cracks in an isotropic rectangular cross-section coated by a piezoelectric layer. A uniform electric displacement is applied on the left and right edges of the piezoelectric layer. First, the stress components and the warping function of the rectangular cross-section bar with the piezoelectric coating containing a Volterra-type screw dislocation are found in terms of the screw dislocation density function using Fourier integral transform. The dislocation solution is then reduced to a set of integral equations with a singularity of the Cauchy-type to examine multiple arbitrarily shaped cracks situated in the coated rectangular bar subjected to combining torsion and the forced electric displacement by making use of the distributed dislocation technique (DDT). The numerical solution of the integral equations is presented to calculate stress intensity factors (SIFs) and torsional stiffness. Next, several examples are presented to evaluate the effects of significant parameters on the fracture behavior of the problem. Furthermore, a novel idea is presented to reduce stress intensity factors to a minimum through suitable values of the uniform electric displacement on the left and right edges of the piezoelectric coating.

Finite element formulation for the dynamic analysis of shear deformable thin-walled beams

The present study develops a finite element formulation for the natural vibration and dynamic analysis of thin-walled members with asymmetric cross-sections. The formulation captures shear deformation within the middle surface, global and local warping, viscous damping, translational and rotatory inertial effects, and inertial coupling. The formulation starts with the extended variational form of Hamilton’s principle to account for non-conservative damping effects, in conjunction with a special interpolation scheme aimed at circumventing shear locking. The resulting finite element has two nodes with seven degrees of freedom per node. The discretized equations of motion are integrated using the unconditionally stable Newmark integration scheme with constant average acceleration. For members with doubly symmetric sections, comparisons with shell solutions show that the formulation reliably predicts the natural frequencies and vibration modes associated with flexure/shear and torsion/warping responses. Further comparisons with other non-shear deformable thin-walled beam solutions show that shear deformation effects gain significance in shorter spans, higher vibration modes, and in flexural vibration about the major axis. For monosymmetric and asymmetric sections, comparisons with benchmark solutions showcase the ability of the formulation to capture the inertial coupling arising between the flexure/shear and torsion/warping responses, while keeping the computational cost to a minimum. Comparisons to shell-based solutions demonstrate the ability of the solution to accurately capture the damped time history response of members with doubly symmetric, monosymmetric, and asymmetric sections subjected to harmonic, ramped, and blast-like forces. A separate treatment is provided for degenerate cross-sections with a vanishing global warping function.

Oligoclonal Band Straightening Based on Optimized Hierarchical Warping for Multiple Sclerosis Diagnosis.

The detection of immunoglobulin G (IgG) oligoclonal bands (OCB) in cerebrospinal fluid (CSF) by isoelectric focusing (IEF) is a valuable tool for the diagnosis of multiple sclerosis. Over the last decade, the results of our clinical research have suggested that tears are a non-invasive alternative to CSF. However, since tear samples have a lower IgG concentration than CSF, a sensitive OCB detection is therefore required. We are developing the first automatic tool for IEF analysis, with a view to speeding up the current visual inspection method, removing user variability, reducing misinterpretation, and facilitating OCB quantification and follow-up studies. The removal of band distortion is a key image enhancement step in increasing the reliability of automatic OCB detection. Here, we describe a novel, fully automatic band-straightening algorithm. The algorithm is based on a correlation directional warping function, estimated using an energy minimization procedure. The approach was optimized via an innovative coupling of a hierarchy of image resolutions to a hierarchy of transformation, in which band misalignment is corrected at successively finer scales. The algorithm’s performance was assessed in terms of the bands’ standard deviation before and after straightening, using a synthetic dataset and a set of 200 lanes of CSF, tear, serum and control samples on which experts had manually delineated the bands. The number of distorted bands was divided by almost 16 for the synthetic lanes and by 7 for the test dataset of real lanes. This method can be applied effectively to different sample types. It can realign minimal contrast bands and is robust for non-uniform deformations.

Characterizing Base in Warped Product Submanifolds of Complex Projective Spaces by Differential Equations

In this study, a link between the squared norm of the second fundamental form and the Laplacian of the warping function for a warped product pointwise semi-slant submanifold Mn in a complex projective space is presented. Some characterizations of the base NT of Mn are offered as applications. We also look at whether the base NT is isometric to the Euclidean space Rp or the Euclidean sphere Sp, subject to some constraints on the second fundamental form and warping function.

Model-based joint curve registration and classification

In this paper, we consider the problem of classification of misaligned multivariate functional data. We propose to use a model-based approach for the joint registration and classification of such data. The observed functional inputs are modeled as a functional nonlinear mixed effects model containing a nonlinear functional fixed effect constructed upon warping functions to account for curve alignment, and a nonlinear functional random effects component to address the variability among subjects. The warping functions are also modeled to accommodate common effect within groups and the variability between subjects. Then, a functional logistic regression model defined upon the representation of the aligned curves and scalar inputs is used to account for curve classification. EM-based algorithms are developed to perform maximum likelihood inference of the proposed models. The identifiability of the registration model and the asymptotical properties of the proposed method are established. The performance of the proposed procedure is illustrated via simulation studies and an analysis of a hyoid bone movement data application. The statistical developments proposed in this paper were motivated by the hyoid bone movement study, the methodology is designed and presented generality and can be applied to numerous areas of scientific research.

Ricci curvature of semi-slant warped product submanifolds in generalized complex space forms

<abstract><p>The objective of this paper is to achieve the inequality for Ricci curvature of a semi-slant warped product submanifold isometrically immersed in a generalized complex space form admitting a nearly Kaehler structure in the expressions of the squared norm of mean curvature vector and warping function. In addition, the equality case is likewise discussed. We provide numerous physical applications of the derived inequalities. Later, we proved that under a certain condition the base manifold $ N_T^{n_1} $ is isometric to a $ n_1 $-dimensional sphere $ S^{n_1}(\frac{\lambda_1}{n_1}) $ with constant sectional curvature $ \frac{\lambda_1}{n_1}. $</p></abstract>

Closed-form expressions of shear deformability tensors associated with a new beam model

The application of a recent beam model requires the evaluation of a suitably defined shear deformability tensor in order to ensure both energetic and geometric consistency with the 3D Saint Venant model subjected to non-uniform flexure and torsion. Actually, whatever is the position of the shearing force, the beam model provides the same elastic energy and the same tip displacement of the barycentric axis of the 3D Saint Venant model. The expression of the shear deformability tensor is rather involved since it depends upon three harmonic functions, namely the warping and the warpage functions associated, respectively, with torsion and shear. For this reason, in order to promote its use and shed some light on its properties, we derive explicit expressions of the shear deformability tensor for circular, elliptical and rectangular sections exploiting the relevant analytical formulas of the warping and warpage functions.

Bayesian semiparametric modelling of phase-varying point processes

We propose a Bayesian semiparametric approach for registration of multiple point processes. Our approach entails modelling the mean measures of the phase-varying point processes with a Bernstein–Dirichlet prior, which induces a prior on the space of all warp functions. Theoretical results on the support of the induced priors are derived, and posterior consistency is obtained under mild conditions. Numerical experiments suggest a good performance of the proposed methods, and a climatology real-data example is used to showcase how the method can be employed in practice.