Inverse Curvature Research Articles

An inverse Gauss curvature flow to the Lp-Gauss Minkowski problem

In this paper, we confirm the existence of asymmetric smooth solutions to the Lp-Gauss Minkowski problem for p>0 by a class of inverse Gauss curvature flows. Furthermore, asymmetric weak solution for p>0 is obtained by a parabolic approximation method.

Reversible shape memory behavior of knitting-fabric reinforced polymer matrix composites

Reversible self-shaping behavior has been occurred with chemically crosslinked polymers and multi-layer structures while this behavior is difficultly realized for the fabric reinforced polymeric composites. In this work, a thin weft plain knitted fabric reinforced shape memory polymer composites (SMPCs) with reversible shape memory effect were developed. The effect of sizes, knitting densities and actuated temperatures on reversible shape memory behavior of SMPC was investigated. The results show that the maximum self-bending curvature of SMPCs is 35.8 m-1 and the shape recovery ratio reaches 82.2 %. A simplified finite element model was established to simulate and analyze the process of self-bending with heating and self-recovery with cooling of SMPC. Interestingly, the SMPC in wale direction (90°) has the inverse self-bending curvature of −18.4 m-1 as compared with that of SMPC in course direction (0°). The complex double curvature self-bending behavior of SMPC was also demonstrated. This work helps to broaden the development and research of reversible shape memory composites.

Stability of quermassintegral inequalities along inverse curvature flows

Stability of quermassintegral inequalities along inverse curvature flows

Two nonlocal inverse curvature flows of convex closed plane curves

In this paper, we introduce two [Formula: see text]-type ([Formula: see text]) curvature flows for closed convex planar curves. Along the flows the length of the curve is decreasing while the enclosed area is increasing. Finally, the evolving curves converge smoothly to a finite circle if they do not develop singularity during the evolution process.

A Study on Normal Motion of the Torus of Revolution in ℝ3

Abstract In the present research paper, we investigate the motion of surfaces in ℝ3 according to their curvatures. We study the motion of the torus of revolution via the normal velocity. We consider two cases: when the normal velocity is a function of both the time and the coordinates of the torus, and when it is a function of time only. We also study how the torus moves under different types of curvature flows, such as inverse mean curvature flow, inverse Gaussian curvature flow, and harmonic mean curvature flow. Moreover, we present some new applications of these flows.

Real-Time pH-Dependent Self-Assembly of Ionisable Lipids from COVID-19 Vaccines and In Situ Nucleic Acid Complexation.

Ionisable amino-lipid is a key component in lipid nanoparticles (LNPs), which plays a crucial role in the encapsulation of RNA molecules, allowing efficient cellular uptake and then releasing RNA from acidic endosomes. Herein, we present direct evidence for the remarkable structural transitions, with decreasing membrane curvature, including from inverse micellar, to inverse hexagonal, to two distinct inverse bicontinuous cubic, and finally to a lamellar phase for the two mainstream COVID-19 vaccine ionisable ALC-0315 and SM-102 lipids, occurring upon gradual acidification as encountered in endosomes. The millisecond kinetic growth of the inverse cubic and hexagonal structures and the evolution of the ordered structural formation upon ionisable lipid-RNA/DNA complexation are quantitatively revealed by in situ synchrotron radiation time-resolved small angle X-ray scattering coupled with rapid flow mixing. We found that the final self-assembled structural identity, and the formation kinetics, were controlled by the ionisable lipid molecular structure, acidic bulk environment, lipid compositions, and nucleic acid molecular structure/size. The implicated link between the inverse membrane curvature of LNP and LNP endosomal escape helps future optimisation of ionisable lipids and LNP engineering for RNA and gene delivery.

Real‐Time pH‐Dependent Self‐Assembly of Ionisable Lipids from COVID‐19 Vaccines and In Situ Nucleic Acid Complexation

AbstractIonisable amino‐lipid is a key component in lipid nanoparticles (LNPs), which plays a crucial role in the encapsulation of RNA molecules, allowing efficient cellular uptake and then releasing RNA from acidic endosomes. Herein, we present direct evidence for the remarkable structural transitions, with decreasing membrane curvature, including from inverse micellar, to inverse hexagonal, to two distinct inverse bicontinuous cubic, and finally to a lamellar phase for the two mainstream COVID‐19 vaccine ionisable ALC‐0315 and SM‐102 lipids, occurring upon gradual acidification as encountered in endosomes. The millisecond kinetic growth of the inverse cubic and hexagonal structures and the evolution of the ordered structural formation upon ionisable lipid‐RNA/DNA complexation are quantitatively revealed by in situ synchrotron radiation time‐resolved small angle X‐ray scattering coupled with rapid flow mixing. We found that the final self‐assembled structural identity, and the formation kinetics, were controlled by the ionisable lipid molecular structure, acidic bulk environment, lipid compositions, and nucleic acid molecular structure/size. The implicated link between the inverse membrane curvature of LNP and LNP endosomal escape helps future optimisation of ionisable lipids and LNP engineering for RNA and gene delivery.

Locally constrained inverse curvature flow and an Alexandrov-Fenchel inequality in de Sitter space

The long-time existence and convergence are shown for the locally constrained inverse curvature flow of compact spacelike convex hypersurfaces in de Sitter space. As an application, the Alexandrov-Fenchel inequality between the quermassintegrals A0 and An−1 is established in view of their opposite monotonicity along the considered flow. With the help of the dual relationship, a geometric inequality concerning quermassintegrals of convex hypersurfaces in a hyperbolic space of even dimension follows.

Abstract 275: A New, Novel Shape Feature Space Classifier For Acute Type B Aortic Dissection

Background: Early detection and treatment of acute type B aortic dissection (TBAD) is associated with improved morbidity and mortality. Artificial intelligence and machine-learning (ML) based approaches for expedited detection of TBAD have made strides in recent years with improving predictive power and accuracy. These deep-learning neural networks are trained with iterative exposure to patient imaging and clinical information of patients with aortic pathology to identify consistent patterns. AI-decision support tools can be improved with more focused feature selection, variables of particular interest. Here, we identify a novel shape variable to describe the highly complex topography of the aorta. Methods: 46 patients with acute, complicated TBAD were identified from a now closed multi-center clinical trial of a conformable thoracic stent graft. An independent cohort of 87 patients without aortic pathology was obtained from a single center, retrospective database. Two-dimensional surface segmentations were generated from pre-operative CT angiography. Variance in total curvature, a size-independent shape measure, and inverse Casorati curvature, a size scalar, were calculated from the segmentations. Additional size measurements, including maximum diameter, were calculated using centerline methods. Results: 44 pre-operative scans of acute TBAD were analyzed. Mean maximum diameter in the acute TBAD cohort was 39.20 ± 6.75 mm compared to 21.44 ± 3.28 mm in the non-pathologic aorta cohort (p = 2.50E-17). Mean fluctuation in total curvature, normalized, was 7.09 ± 3.58 versus 4.18 ± 2.32 in the non-pathologic aorta cohort (p = 4.36E-13). Mean inverse Casorati curvature was 0.043 ± 0.005 in acute TBAD compared to 0.06 ± 0.01 in non-pathologic aortas (p = 4.67E-21). Logistic regression classification accuracy was 91.6 ± 2.4%. Conclusions: Acute TBAD demonstrates a unique shape signature defined by fluctuation in total curvature, in addition to traditional size measures. This easily quantifiable and reproducible dimension of aortic geometry classification has significant potential as a new, anatomically derived feature space to improve AI/ML-based aortic dissection detection.

Ell -convex Legendre curves and geometric inequalities

In this paper we consider ell -convex Legendre curves, which are natural generalizations of strictly convex curves. We generalize various optimal geometric inequalities, isoperimetric inequality, Bonnesen’s inequality and Green–Osher inequality, for strictly convex curves to ones for ell -convex Legendre curves. Moreover we generalize the inverse curvature curve flow for this class of Legendre curves and prove that it always converges to a compact soliton after rescaling. Unlike in the class of regular curves, there are infinitely many compact solitons, which include circles and astroids.

Bayesian analysis for a class of α-attractor inflationary models

We perform a Bayesian study of a generalization of the basic α-attractor T model given by the potential V(ϕ) = V 0[1-sech p (ϕ/√(6α)M pl)] where ϕ is the inflaton field and the parameter α corresponds to the inverse curvature of the scalar manifold in the conformal or superconformal realizations of the attractor models. Such generalization is characterized by the power p which includes the basic or base model for p = 2. Once the priors for the parameters of the α-attractor potential are set by numerical exploration, we perform the corresponding statistical analysis for the cases p = 1, 2, 3, 4, and derive posteriors. Considering the original α-attractor potential as the base model, we calculate the evidence for our generalization, and conclude that the p = 4 model is preferred by the CMB data. We also present constraints for the parameter α. Interestingly, all the cases studied prefer a specific value for the tensor-to-scalar ratio given by r ≃ 0.0025.

On an area-preserving locally constrained inverse curvature flow of convex curves

This paper will deal with a locally constrained inverse curvature flow which keeps the convexity and preserves the enclosed area of the evolving curve, and the limiting curve converges to a circle in the C∞ sense. As applications of this flow, an inequality for the integral of the power of support function for convex curves and the classical isoperimetric inequality are obtained.

Shifted inverse curvature flows in hyperbolic space

We introduce the shifted inverse curvature flow in hyperbolic space. This is a family of hypersurfaces in hyperbolic space expanding by $F^{-p}$ with positive power $p$ for a smooth, symmetric, strictly increasing and $1$-homogeneous curvature function $f$ of the shifted principal curvatures with some concavity properties. We study the maximal existence and asymptotical behavior of the flow for horo-convex hypersurfaces. In particular, for $0<p\leq 1$ we show that the limiting shape of the solution is always round as the maximal existence time is approached. This is in contrast to the asymptotical behavior of the (non-shifted) inverse curvature flow, as Hung and Wang [18] constructed a counterexample to show that the limiting shape of inverse curvature flow in hyperbolic space is not necessarily round.

The evolution of area-preserving and length-preserving inverse curvature flows for immersed locally convex closed plane curves

In this paper, we study an area-preserving inverse curvature flow and a length-preserving inverse curvature flow for immersed locally convex closed plane curves with rotation number m∈N+. The global-in-time flows are shown to converge smoothly to m-fold round circles as time goes to infinity. The sufficient conditions on initial curve are also found to guarantee the occurrence of the flow's singularity at finite time.

New weighted geometric inequalities for hypersurfaces in space forms

We prove a family of new sharp geometric inequalities involving weighted curvature integrals and quermassintegrals for smooth closed hypersurfaces in space forms. The tools we shall use are the inverse curvature flow by Gerhardt and Urbas and the locally constrained curvature flows introduced recently by Brendle, Guan and Li.

The emergence of spontaneous coordinated epithelial rotation on cylindrical curved surfaces.

Three-dimensional collective epithelial rotation around a given axis represents a coordinated cellular movement driving tissue morphogenesis and transformation. Questions regarding these behaviors and their relationship with substrate curvatures are intimately linked to spontaneous active matter processes and to vital morphogenetic and embryonic processes. Here, using interdisciplinary approaches, we study the dynamics of epithelial layers lining different cylindrical surfaces. We observe large-scale, persistent, and circumferential rotation in both concavely and convexly curved cylindrical tissues. While epithelia of inverse curvature show an orthogonal switch in actomyosin network orientation and opposite apicobasal polarities, their rotational movements emerge and vary similarly within a common curvature window. We further reveal that this persisting rotation requires stable cell-cell adhesion and Rac-1–dependent cell polarity. Using an active polar gel model, we unveil the different relationships of collective cell polarity and actin alignment with curvatures, which lead to coordinated rotational behavior despite the inverted curvature and cytoskeleton order.

Self-similar solutions to the curvature flow and its inverse on the 2-dimensional light cone

We consider a novel curvature flow for curves on the 2-dimensional light cone, contained in the 3-dimensional Minkowski space. We show that the solutions to the curvature flow (CF) for such curves are in correspondence with the solutions to the inverse curvature flow (ICF). We prove that the ellipses and the hyperbolas are the only curves that evolve under homotheties. The ellipses are the only closed ones and they are ancient solutions. We show that a spacelike curve on the cone is a self-similar solution to the CF (resp. (ICF)) if, only if, its curvature (resp. inverse of its curvature) differs by a constant c from being the inner product between its tangent vector field and a fixed vector v of the 3-dimensional Minkowski space. The curve is a soliton solution when \(c=0\). We prove that for each vector v there exists a 2-parameter family of self-similar solutions to the CF and to the ICF, on the light cone. Considering non-trivial solutions to the CF, we prove that the corresponding solutions to the ICF may have at most three connected components. Moreover, at each end of such a curve the curvature is either unbounded or it tends to 0 or to the constant c. Explicitly given soliton solutions are included and some self-similar solutions on the light cone are visualized.

A two-point function, non-collapsing property and pinching estimates for inverse curvature flow in space forms

A two-point function, non-collapsing property and pinching estimates for inverse curvature flow in space forms

Inverse Gauss Curvature Flows and Orlicz Minkowski Problem

Abstract Liu and Lu [27] investigated a generalized Gauss curvature flow and obtained an even solution to the dual Orlicz-Minkowski problem under some appropriate assumptions. The present paper investigates a inverse Gauss curvature flow, and achieves the long-time existence and convergence of this flow via a different C 0-estimate technique under weaker conditions. As an application of this inverse Gauss curvature flow, the present paper first arrives at a non-even smooth solution to the Orlicz Minkowski problem.

Joint-Space Kinematic Control of a Bionic Continuum Manipulator in Real-Time by Using Hybrid Approach

Continuum manipulators are a type of robot used for delicate applications, including safe human-robot interactions. Controlling these manipulators for an accurate trajectory, especially in the case of pneumatic actuation, is extremely challenging. Thus, this article proposes a real-time kinematic trajectory control of a pneumatically actuated multi-segment bionic continuum manipulator with a mobile base by combining a neural network and analytical model with a cascaded controller to overcome this challenge. The inverse kinematics solution of the multi-segment manipulator is developed by using a neural network and an inverse piecewise constant curvature approach. The neural network is trained by using a separate learning algorithm. Although hybrid inverse modeling gives better solutions than existing techniques, significant residual positional error of the manipulator tip remains due to inherent material hysteresis. Thus, a cascaded PI-controller is utilized to compensate for the residual positional error. The controller gains are updated in each step by predictions of the actuator length, where the Jacobian entries are computed from the neural network model. The proposed procedure is validated on Festo Didactics’ elephant trunk-like two-segment continuum manipulator, Robotino-XT. Three different cases are considered for real-time trajectory tracking, where the OptiTrack vision system is used for validation by tracking the manipulator tip pose. For the trajectory points outside the manipulator workspace, simultaneous trunk and base movements are used. In experimental validation, the proposed scheme is shown to give much reduced manipulator tip trajectory error as compared to the existing methods.