Sign Of Curvature Research Articles

Discrete Z4 Symmetry in Quantum Gravity

We consider the discrete Z4 symmetry i^, which takes place in the scenario of quantum gravity where the gravitational tetrads emerge as the order parameter—the vacuum expectation value of the bilinear combination of fermionic operators. Under this symmetry operation, i^, the emerging tetrads are multiplied by the imaginary unit, i^eμa=−ieμa. The existence of such symmetry and the spontaneous breaking of this symmetry are also supported by the consideration of the symmetry breaking scheme in the topological superfluid 3He-B. The order parameter in 3He-B is also the bilinear combination of the fermionic operators. This order parameter is the analog of the tetrad field, but it has complex values. The i^-symmetry operation changes the phase of the complex order parameter by π/2, which corresponds to the Z4 discrete symmetry in quantum gravity. We also considered the alternative scenario of the breaking of this Z4 symmetry, in which the i^-operation changes sign of the scalar curvature, i^R=−R, and thus the Einstein–Hilbert action violates the i^-symmetry. In the alternative scenario of symmetry breaking, the gravitational coupling K=1/16πG plays the role of the order parameter, which changes sign under i^-transformation.

Thermodynamic Topology of Topological Black Hole in F(R)-ModMax Gravity’s Rainbow

Abstract In order to include the effect of high energy and topological parameters on black holes in $\mathrm{ F}(R)$ gravity, we consider two corrections to this gravity: energy-dependent spacetime with different topological constants, and a nonlinear electrodynamics field. In other words, we combine $\mathrm{ F}(R)$ gravity’s rainbow with ModMax nonlinear electrodynamics theory to see the effects of high energy and topological parameters on the physics of black holes. For this purpose, we first extract topological black hole solutions in $\mathrm{ F}(R)$-ModMax gravity’s rainbow. Then, by considering black holes as thermodynamic systems, we obtain thermodynamic quantities and check the first law of thermodynamics. The effect of the topological parameter on the Hawking temperature and the total mass of black holes is obvious. We also discuss the thermodynamic topology of topological black holes in $\mathrm{ F}(R)$-ModMax gravity’s rainbow using the off-shell free energy method. In this formalism, black holes are assumed to be equivalent to defects in their thermodynamic spaces. For our analysis, we consider two different types of thermodynamic ensembles. These are: fixed q ensemble and fixed $\phi$ ensemble. We take into account all the different types of curvature hypersurfaces that can be constructed in these black holes. The local and global topology of these black holes are studied by computing the topological charges at the defects in their thermodynamic spaces. Finally, in accordance with their topological charges, we classify the black holes into three topological classes with total winding numbers corresponding to $-1, 0$, and 1. We observe that the topological classes of these black holes are dependent on the value of the rainbow function, the sign of the scalar curvature, and the choice of ensembles.

Semiclassical thermodynamic geometry.

In this work the thermodynamic geometry (TG) of semiclassical fluids is analyzed. We present results for two models. The first one is a semiclassical hard-sphere (SCHS) fluid whose Helmholtz free energy is obtained from path-integral Monte Carlo simulations. It is found that, due to quantum contributions in the thermodynamic potential, the anomaly found in TG for the classical hard-sphere fluid related to the sign of the scalar curvature is now avoided in a considerable region of the thermodynamic space. The second model is a semiclassical square-well fluid, described by a SCHS repulsive interaction coupled with a classical attractive square-well contribution. The behavior of the semiclassical curvature scalar as a function of the thermal de Broglie wavelength λ_{B} is analyzed for several attractive-potential ranges. A description of the semiclassical R Widom lines, defined by the maxima of the curvature scalar, is also obtained and results are compared with the corresponding classical systems for different square-well ranges.

Substantiating the expedient route parameters for the location of the site outgassing wells in the Western Donbas conditions

The paper deals with the issues of substantiating the parameters of location outgassing wells, determining their expedient routes in the mining-geological and mining-technical Western Donbas conditions. Based on the conducted multifactorial computer experiment, the patterns have been obtained of influence of the coal-overlaying formation texture on the parameters of its shear into the mined-out area. When determining the patterns of influence of mechanical properties of the coal-overlaying formation lithotypes, they are divided into three generalized groups: low, averaged, high. At the same time, the influence of the height and distance parameters the strongest and hardest lithotype occurrence, such as sandstone, has been studied. It is noted that with low mechanical characteristics, the gradient angle of the line of changing curvature sign of the rock layers bending in the direction of the coal seam dip is steadily decreasing. Conditions have been identified and criteria have been developed for the most effective routes for the location of site outgassing wells. The developed criteria, in combination with the dependences of the coal seam shear parameters, make it possible to create a methodology for determining the expedient coordinates for drilling outgassing wells based on the results of modeling by the finite element method.

Chirality effects in molecular chainmail.

Motivated by the observation of positive Gaussian curvature in kinetoplast DNA networks, we consider the effect of linking chirality in square lattice molecular chainmail networks using Langevin dynamics simulations and constrained gradient optimization. Linking chirality here refers to ordering of over-under versus under-over linkages between a loop and its neighbors. We consider fully alternating linking, maximally non-alternating, and partially non-alternating linking chiralities. We find that in simulations of polymer chainmail networks, the linking chirality dictates the sign of the Gaussian curvature of the final state of the chainmail membranes. Alternating networks have positive Gaussian curvature, similar to what is observed in kinetoplast DNA networks. Maximally non-alternating networks form isotropic membranes with negative Gaussian curvature. Partially non-alternating networks form flat diamond-shaped sheets which undergo a thermal folding transition when sufficiently large, similar to the crumpling transition in tethered membranes. We further investigate this topology-curvature relationship on geometric grounds by considering the tightest possible configurations and the constraints that must be satisfied to achieve them.

On Human-like Biases in Convolutional Neural Networks for the Perception of Slant from Texture

Depth estimation is fundamental to 3D perception, and humans are known to have biased estimates of depth. This study investigates whether convolutional neural networks (CNNs) can be biased when predicting the sign of curvature and depth of surfaces of textured surfaces under different viewing conditions (field of view) and surface parameters (slant and texture irregularity). This hypothesis is drawn from the idea that texture gradients described by local neighborhoods—a cue identified in human vision literature—are also representable within convolutional neural networks. To this end, we trained both unsupervised and supervised CNN models on the renderings of slanted surfaces with random Polka dot patterns and analyzed their internal latent representations. The results show that the unsupervised models have similar prediction biases as humans across all experiments, while supervised CNN models do not exhibit similar biases. The latent spaces of the unsupervised models can be linearly separated into axes representing field of view and optical slant. For supervised models, this ability varies substantially with model architecture and the kind of supervision (continuous slant vs. sign of slant). Even though this study says nothing of any shared mechanism, these findings suggest that unsupervised CNN models can share similar predictions to the human visual system. Code: github.com/brownvc/Slant-CNN-Biases.

Curvature effects on the structure of near-wall turbulence

The interaction between near-wall turbulence and wall curvature is described for the incompressible flow in a plane channel with a small concave–convex–concave bump on the bottom wall, with height comparable to the wall-normal location of the main turbulent structures. The analysis starts from a database generated by a direct numerical simulation and hinges upon the anisotropic generalised Kolmogorov equations, i.e. the exact budget equations for the second-order structure function tensor. The influence of the bump on the wall cycle and on the energy production, redistribution and transfers is described in the physical and scale spaces. Over the upstream side of the bump, the energy drained from the mean flow to sustain the streamwise fluctuations decreases, and the streaks of high and low streamwise velocity weaken and are stretched spanwise. After the bump tip, instead, the production of streamwise fluctuations grows and the streaks intensify, progressively recovering their characteristic spanwise scale. The wall-normal fluctuations, and thus the quasi-streamwise vortices, are sustained by the mean flow over the upstream side of the bump, while energy flows from the vertical fluctuations to the mean field over the downstream side. On the concave portion of the upstream side, the near-wall fluctuations form structures of spanwise velocity which are consistent with Taylor–Görtler vortices at an early stage of development. Their evolution is described by analysing the scale-space pressure–strain term. A schematic description of the bump flow is presented, in which various regions are identified according to the signs of curvature and streamwise pressure gradient.

Sign problems in elliptic integral solution of planar elastica theory

Elastica theory studies large deformation of slender beams and rods. The elliptic integral solution (EIS) is an accurate and efficient theoretical solution to elastica problem. However, the sign of elliptic integral (SEI) will change with the sign of curvature along an elastica, and also depends on parameters of the coordinate origin, such as elliptical modular angle and curvature. Due to the uncertainty of SEIs, it is quite difficult to obtain an EIS with a uniform final expression.In this study, the sign of elliptic integral (SEI) will be discussed in detail, by classifying the elastica problems into two loading cases: force dominant case (FDC) and bending moment dominant case (MDC). A complete set of EISs and the general solving procedure of EISs, which are applicable to various planar elastica without axial deformation, are presented. The accuracy of current elastica analysis is confirmed by corresponding FE simulations. Employing this EIS method, some metamaterials under large deformation are analyzed efficiently. It is found that the predictions of the equivalent modulus, stress-strain relations, and deformation profiles are both accurate.

Spatial curvature and thermodynamics

ABSTRACTReasonable parametrizations of the current Hubble data set of the expansion rate of our homogeneous and isotropic universe, after suitable smoothing of these data, strongly suggest that the area of the apparent horizon increases irrespective of whether the spatial curvature of the metric is open, flat, or closed. Put in another way, any sign of the spatial curvature appears consistent with the second law of thermodynamics.

Discontinuous Metric Programming in Liquid Crystalline Elastomers.

Liquid crystalline elastomers (LCEs) are shape-changing materials that exhibit large deformations in response to applied stimuli. Local control of the orientation of LCEs spatially directs the deformation of these materials to realize a spontaneous shape change in response to stimuli. Prior approaches to shape programming in LCEs utilize patterning techniques that involve the detailed inscription of spatially varying nematic fields to produce sheets. These patterned sheets deform into elaborate geometries with complex Gaussian curvatures. Here, we present an alternative approach to realize shape-morphing in LCEs where spatial patterning of the crosslink density locally regulates the material deformation magnitude on either side of a prescribed interface curve. We also present a simple mathematical model describing the behavior of these materials. Further experiments coupled with the mathematical model demonstrate the control of the sign of Gaussian curvature, which is used in combination with heat transfer effects to design LCEs that self-clean as a result of temperature-dependent actuation properties.

Gravitational lensing in Brill spacetimes

We consider the Brill metric which is an electrovacuum solution to Einstein's field equation. It depends on three parameters, a mass parameter $m$, a NUT parameter $l$ and a charge parameter $e$. If the charge parameter is small, the metric describes a black hole; if it is sufficiently big, it describes a wormhole. We determine the relevant lensing features both in the black-hole and in the wormhole case. In particular, we give formulas for the photon spheres, for the angular radius of the shadow and for the deflection angle. We illustrate the lensing features with the help of an effective potential and in terms of embedding diagrams. To that end we make use of the fact that each lightlike geodesic is contained in a (coordinate) cone and that it is a geodesic of a Riemannian optical metric on this cone. By the Gauss-Bonnet theorem, the sign of the Gaussian curvature of the optical metric determines the sign of the deflection angle. In the wormhole case the deflection angle may be negative which means that light rays are repelled from the center.

The Jacobi metric approach for dynamical wormholes

We present the Jacobi metric formalism for dynamical wormholes. We show that in isotropic dynamical spacetimes , a first integral of the geodesic equations can be found using the Jacobi metric, and without any use of geodesic equation. This enables us to reduce the geodesic motion in dynamical wormholes to a dynamics defined in a Riemannian manifold. Then, making use of the Jacobi formalism, we study the circular stable orbits in the Jacobi metric framework for the dynamical wormhole background. Finally, we also show that the Gaussian curvature of the family of Jacobi metrics is directly related, as in the static case, to the flare-out condition of the dynamical wormhole, giving a way to characterize a wormhole spacetime by the sign of the Gaussian curvature of its Jacobi metric only.

Failure mechanism of a (Ni, Pt)Al/EB-PVD 8YSZ deposited on substrate with different curvature signs in thermal shock test

A thermal shock test was conducted on an 8 wt% Y2O3 stabilized ZrO2 electron beam-physical vapor deposited (EB-PVD 8YSZ) thermal barrier coating with a (Ni, Pt)Al bond coating on the substrate with different curvature signs. The microstructural evolution and durability have been characterized. The microstructure of the top ceramic layer is strongly dependent on the substrate geometry. The results of the thermal shock test indicated that the sample with a positive curvature exhibits mixed mode spalling by the linking of cracks at TBC/TGO interface and TGO/BC interface. Spallation occurs primarily at the TGO/BC interface and inside the bond coats near to the surface of bond coats in planar samples. The spalling occurs principally at the TGO/BC interface in specimens with a negative curvature. The failure mechanism is elucidated integrate with stress analysis.

Deposition path-dependent lightweight support design and its implication to self-support topology optimization

Abstract This paper presents a lightweight support design method for material extrusion-type three-dimensional printed panel structures that innovatively involves the deposition path curvature information for support point determination. Specifically, this support design method provides a robust segmentation algorithm to divide the filament deposition paths into segments based on the curvature sign alternating condition, and then searches for the fewest support points for the filaments counting on the experimentally calibrated relationship between the maximum allowable self-support distance and the local mean curvature. The proposed method features in generating thin-walled skeleton-ray styled support structures that are lightweight while providing firm support for the panels. More importantly, the support design method provides a new type of self-support criterion for structural topology optimization involving non-designable planar panels, i.e., only a sparse point set would be sufficient to support the panel. Consequently, more materials could be spent on enhancing the load-bearing capacity instead of being wasted on oversupporting. The achievable structural performances from self-support topology optimization with this new self-support criterion can improve significantly. Support design and printing tests were conducted on a few panel structures that validated the improved support effect compared with equal-volume supports generated by commercial software. Equidistant and gap-free deposited filaments, no filament collapse due to insufficient support, and no isolated voids reflect the improved support effect. The improved self-support topological design was also validated through a comparative numerical case study, and a compliance reduction of 7.76% was achieved.

Active membrane recycling induced morphology changes in vesicles

Membranes of organelles in the intracellular trafficking pathway continuously undergo recycling through fission and fusion processes. The effect of these recycling processes on the large-scale morphology of organelles is not well understood. Using a dynamically triangulated surface model, we developed a membrane morphology simulator that allows for membrane trafficking, and analyzed the steady state shape of vesicles subjected to such active remodeling. We study a two-component vesicle composed of 1) active species which can have nonzero spontaneous curvature and participate in the recycling and 2) inactive species which do not participate in the recycling. We obtain a plethora of steady state morphologies as a function of the activity rate, spontaneous curvature, and the strength of interaction between species. We observe that morphology changes, as a function of rate of activity, are diametrically opposite for the two signs of the spontaneous curvature, but only have a weak effect on its magnitude. The interplay between the in-plane diffusion, the activity rate, and the spontaneous curvature are shown to determine the vesicle morphology at the steady state. It is shown that the spontaneous curvature and activity inhibits the formation of clusters of active species on the surface. We carry out linear stability analysis of a continuum model and show that the spherical shape of a vesicle is indeed unstable when subjected to active membrane recycling above a certain activity rate.

The influence of the Gaussian curvature sign of the compound shell structure’s middle surface on local and overall buckling under combined loading

The buckling problem of an elastic compound shell structure with a variable Gaussian curvature of the middle surface, especially the middle surface meridian curvature sign, under the action of external pressure and axial loading is considered. In continuation of the previous research of the authors, this paper is devoted, in particular, to examining the influence of the negative Gaussian curvature sign of one of its compartments on stability. The solution is based on using the method of finite differences for basic stability equations of each compartment in the case when one of them can have a negative curvature of the meridian, taking into account the discreteness of the intermediate rib location and their rigidity from the initial curvature plane as well. The obtained solution allows visualizing the buckling modes under various combinations of external loading and identifying rational, according to overall buckling modes, geometric and rigidity parameters of the system being investigated.

Nano-spheroid formation on YAG surfaces induced by single ultrafast Bessel laser pulses

We report on single pulse ultrafast Bessel laser beam processing of YAG ceramic surfaces as a method for producing contamination-free submicron particles and size adjustable surface hemisphere structures of different curvature signs. The micro-machining process was performed in both air and liquid environments, leading to a variety of surface structures depending on the processing parameters. Particularly, a transformation of the surface structure morphology from micro-hole profiles to hemisphere extrusions was observed. The size of the hemisphere structure is found to be highly sensitive to laser parameters, such as pulse energy, pulse duration and beam focusing position. Through careful analyses of the influence of the laser pulse parameters, a precise regulation of the lateral diameter and height of the hemisphere structure was achieved. Large area hemisphere arrays with low standard deviation in size were fabricated. The detachment of the emerging structures and subsequent particle deposition can be observed in liquid environments when the height-diameter aspect ratio of the hemisphere exceeds a factor of 0.65. The Mechanisms for the formation and detaching of the hemisphere structure are discussed with cross-sectional and morphology imaging via Scanning Electron Microscopy and Atomic Force Microscopy. As a preliminary step towards submicron particle generation in liquid environments, the observation of surface hemispheres has interest in exploring the initial mechanisms of particles formation under laser ablation in liquids. The presented method allows for the fabrication of contamination-free and size adjustable YAG submicron convex structures which have potential applications in integrated optics, biotechnology and other advanced processing techniques.

Smooth Interpolating Curves with Local Control and Monotone Alternating Curvature

We propose a method for the construction of a planar curve based on piecewise clothoids and straight lines that intuitively interpolates a given sequence of control points. Our method has several desirable properties that are not simultaneously fulfilled by previous approaches: Our interpolating curves are C2 continuous, their computation does not rely on global optimization and has local support, enabling fast evaluation for interactive modeling. Further, the sign of the curvature at control points is consistent with the control polygon; the curvature attains its extrema at control points and is monotone between consecutive control points of opposite curvature signs. In addition, we can ensure that the curve has self-intersections only when the control polygon also self-intersects between the same control points. For more fine-grained control, the user can specify the desired curvature and tangent values at certain control points, though it is not required by our method. Our local optimization can lead to discontinuity w.r.t. the locations of control points, although the problem is limited by its locality. We demonstrate the utility of our approach in generating various curves and provide a comparison with the state of the art.

Effect of peculiar velocities of inhomogeneities on the shape of gravitational potential in spatially curved universe

We investigate the effect of peculiar velocities of inhomogeneities and the spatial curvature of the universe on the shape of the gravitating potential. To this end, we consider scalar perturbations of the FLRW metric. The gravitational potential satisfies a Helmholtz-type equation which follows from the system of linearized Einstein equations. We obtain analytical solutions of this equation in the cases of open and closed universes, filled with cold dark matter in presence of the cosmological constant. We demonstrate that, first, peculiar velocities significantly affect the screening length of the gravitational interaction and, second, the form of the gravitational potential depends on the sign of the spatial curvature.

Compactness of Dirac–Einstein Spin Manifolds and Horizontal Deformations

In this paper, we consider the Hilbert–Einstein–Dirac functional, whose critical points are pairs, metrics-spinors, that satisfy a system coupling the Riemannian and the spinorial part. Under some assumptions, on the sign of the scalar curvature and the diameter, we prove a compactness result for this class of pairs, in dimension three and four. This can be seen as the equivalent of the study of compactness of sequences of Einstein manifolds as in references (Anderson in J Am Math Soc 2:455–490, 1989; Nakajima in J Fac Sci Univ Tokyo Sect IA Math 35:411–424, 1988). Indeed, we study the compactness of sequences of critical points of the Hilbert–Einstein–Dirac functional which is an extension of the Hilbert–Einstein functional having Einstein manifolds as critical points. Moreover we will study the second variation of the energy, characterizing the horizontal deformations for which the second variation vanishes. Finally we will exhibit some explicit examples.