Spherical Topology Research Articles

Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach

We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.

Sphere theorems for RCD and stratified spaces

We prove topological sphere theorems for RCD(n-1, n) spaces which generalize Colding's results and Petersen's result to the RCD setting. We also get an improved sphere theorem in the case of Einstein stratified spaces.

Synthesis of atomically precise single-crystalline Ru2-based coordination polymers.

Methods to incorporate kinetically inert metal nodes and highly basic ligands into single-crystalline metal-organic frameworks (MOFs) are scarce, which prevents synthesis and systematic variation of many potential heterogeneous catalyst materials. Here we demonstrate that metallopolymerization of kinetically inert Ru2 metallomonomers via labile Ag-N bonds provides access to a family of atomically precise single-crystalline Ru2-based coordination polymers with varied network topology and primary coordination sphere.

Analysis of premixed flame kernel/turbulence interactions under engine conditions based on direct numerical simulation data

Although the evolution of premixed flames in turbulence has been frequently studied, it is not well understood how small flames interact with large-scale turbulent flow motion. Since this question is of practical importance for the occurrence of cycle-to-cycle variations in spark ignition engines, the objective of the present work is to fundamentally differentiate early flame kernel development from well-established turbulent flame configurations. For this purpose, a DNS database consisting of three flames propagating in homogeneous isotropic turbulence (Falkenstein et al., Combust. Flame, 2019) is considered. The flames feature different ratios of the initially laminar flame diameter to the integral length scale. To quantify flame kernel development, the time evolution of flame topology and flame front geometry are analysed in detail. It is shown that some realizations of the early flame kernel are substantially influenced by high compressive strain caused by large-scale turbulent flow motion with characteristic length scales greater than the flame kernel size. As a result, the initial spherical kernel topology may become highly distorted, which is reflected in the stochastic occurrence of excessive curvature variance. Two mechanisms of curvature production resulting from early flame kernel/turbulence interactions are identified by analysis of the mean curvature balance equation. Further, it is shown that the curvature distribution of small flame kernels becomes strongly skewed towards positive curvatures, which is contrary to developed turbulent flames. Hence, the transition of ignition kernels to self-sustaining turbulent flames is very different in nature compared with the development of a statistically planar flame brush.

Semi-separation axioms of the infinite Khalimsky topological sphere

In this paper we denote by ((Z2)⁎,(κ2)⁎) the Alexandroff one point compactification of the Khalimsky (K-, brevity) plane. We often call this space the infinite K-topological sphere or the infinite K-sphere for brevity in the present paper. After studying various properties of the infinite K-topological sphere such as a non-Alexandroff structure, we study low and semi-separation axioms of it. Finally, let (X,T) be a topological space which is semi-T2 and each point x(∈X) has an open neighborhood V(∋x) such that the closure of V (or ClX(V)) is compact, and (X⁎,T⁎) an Alexandroff one point compactification of (X,T). Then, we prove that (X⁎,T⁎) is a semi-T2-space. Thus it turns out that the infinite K-sphere is a semi-T2-space.

Stable currents and homology groups in a compact CR-warped product submanifold with negative constant sectional curvature

The main purpose of this paper is to highlight some striking features of the relationship between homology groups and compact warped product submanifolds geometry with negative constant sectional curvature that follows from Lawson and Simons (1973). Using the result of Fu and Xu (2008), we prove that there does not exist stable integral p-currents and their homology groups are zero in a compact CR-warped product submanifold Mn from a complex hyperbolic space ℂHm(−4) under extrinsic conditions involving the Laplacian of warped function and the squared norm of gradient of the warping function. Moreover, under the same extrinsic conditions and using the generalized Poincaré’s conjecture, we derive new topological sphere theorem on a compact CR-warped product submanifold Mn, we prove that Mn is homeomorphic to the sphere Sn if n=4. Also, if n=3 then M3 is homotopic to the sphere S3 and this result follows from the work of Sjerve (1973). Finally, the same types of theorems are constructed in terms of some mechanical tools such as Dirichlet energy and Hamiltonian as well.

Critical phenomena in causal dynamical triangulations

Four-dimensional CDT (causal dynamical triangulations) is a lattice theory of geometries which one might use in an attempt to define quantum gravity non-perturbatively, following the standard procedures of lattice field theory. Being a theory of geometries, the phase transitions which in usual lattice field theories are used to define the continuum limit of the lattice theory will in the CDT case be transitions between different types of geometries. This picture is interwoven with the topology of space which is kept fixed in the lattice theory, the reason being that ‘classical’ geometries around which the quantum fluctuations take place depend crucially on the imposed topology. Thus it is possible that the topology of space can influence the phase transitions and the corresponding critical phenomena used to extract continuum physics. In this article we perform a systematic comparison between a CDT phase transition where space has spherical topology and the ‘same’ transition where space has toroidal topology. The ‘classical’ geometries around which the systems fluctuate are very different it the two cases, but we find that the order of phase transition is not affected by this.

Microstructure design and degradation performance in vitro of three-dimensional printed bioscaffold for bone tissue engineering

In bone tissue engineering, three-dimensional printed biological scaffolds play an important role in the development of bone regeneration. The ideal scaffolds should have the ability to match the bone degradation rate and osteogenic ability. This article optimizes the unit cell model of the microstructure including spherical pore, gyroid, and topology to explore degradation performance of scaffolds. Boolean operation of array microstructure unit cells and selected part of a computer-aided design (CAD) femur model are adopted to create a reconstructed scaffold model. Polylactic acid/[Formula: see text]-tricalcium phosphate/hydroxyapatite scaffolds with spherical pore, gyroid, and topology-optimized structures are manufactured by three-dimensional printing utilizing the composition of bio-ink including polylactic acid, [Formula: see text]-tricalcium phosphate, and hydroxyapatite. After degradation of the scaffolds in vitro for several days, the mechanical properties are analyzed to study the effects of different microstructures on the degradation properties. The results show that the gyroid scaffolds with favorable degradability still maintain excellent mechanical properties after degradation. Mechanical properties of the scaffolds with topology-optimized structure and spherical pore microstructure scaffolds have a significant decrease after degradation.

Calcium phosphate-PLA scaffolds fabricated by fused deposition modeling technique for bone tissue applications: Fabrication, characterization and simulation

With the aid of fast development in manufacturing techniques, it is now possible to produce structures with considerable geometrical complexity. In the current investigation, 3D printing machine is utilized to produce beam-type bone implant made of porous polymers with three different periodic cellular topologies. The stress-strain curves and mechanical properties of the bone implants are extracted experimentally corresponding to each porosity shape with various pore sizes. Scanning electron microscope (SEM) is utilized to assess the biological capability in apatite formation on the free surface of the fabricated scaffolds after soaking in the simulated body fluid (SBF). Thereafter, based upon the obtained mechanical properties and using generalized differential quadrature (GDQ) method, the nonlinear primary resonance of bone implants made of the polymer scaffolds and subjected to the soft harmonic excitation is predicted corresponding. The frequency-response and amplitude-response of the bone implants are presented associated with the cubic, spherical and honeycomb periodic cellular topologies and different pore sizes. The analysis of soaked samples in SBF reveals that among different porosity shapes, the samples with cubic and hexagonal porosities have the maximum and minimum permeability, respectively. However, this pattern is vice versa for apatite formation. Moreover, it is seen that for the samples with cubic and hexagonal porosity shapes, the oscillation amplitude as well as the associated excitation frequency at the peak of jump phenomenon are maximum and minimum, respectively.

Topological effects in continuum two-dimensional U(N) gauge theories

We study the $\theta$ dependence of the continuum limit of 2d $U(N)$ gauge theories defined on compact manifolds, with special emphasis on spherical ($g=0$) and toroidal ($g=1$) topologies. We find that the coupling between $U(1)$ and $SU(N)$ degrees of freedom survives the continuum limit, leading to observable deviations of the continuum topological susceptibility from the $U(1)$ behavior, especially for $g=0$, in which case deviations remain even in the large $N$ limit.

Finding hexahedrizations for small quadrangulations of the sphere

This paper tackles the challenging problem of constrained hexahedral meshing. An algorithm is introduced to build combinatorial hexahedral meshes whose boundary facets exactly match a given quadrangulation of the topological sphere. This algorithm is the first practical solution to the problem. It is able to compute small hexahedral meshes of quadrangulations for which the previously known best solutions could only be built by hand or contained thousands of hexahedra. These challenging quadrangulations include the boundaries of transition templates that are critical for the success of general hexahedral meshing algorithms. The algorithm proposed in this paper is dedicated to building combinatorial hexahedral meshes of small quadrangulations and ignores the geometrical problem. The key idea of the method is to exploit the equivalence between quad flips in the boundary and the insertion of hexahedra glued to this boundary. The tree of all sequences of flipping operations is explored, searching for a path that transforms the input quadrangulation Q into a new quadrangulation for which a hexahedral mesh is known. When a small hexahedral mesh exists, a sequence transforming Q into the boundary of a cube is found; otherwise, a set of pre-computed hexahedral meshes is used. A novel approach to deal with the large number of problem symmetries is proposed. Combined with an efficient backtracking search, it allows small shellable hexahedral meshes to be found for all even quadrangulations with up to 20 quadrangles. All 54, 943 such quadrangulations were meshed using no more than 72 hexahedra. This algorithm is also used to find a construction to fill arbitrary domains, thereby proving that any ball-shaped domain bounded by n quadrangles can be meshed with no more than 78 n hexahedra. This very significantly lowers the previous upper bound of 5396 n.

Towards an UV fixed point in CDT gravity

CDT is an attempt to formulate a non-perturbative lattice theory of quantum gravity. We describe the phase diagram and analyse the phase transition between phase B and phase C (which is the analogue of the de Sitter phase observed for the spherical spatial topology). This transition is accessible to ordinary Monte Carlo simulations when the topology of space is toroidal. We find that the transition is most likely first order, but with unusual properties. The end points of the transition line are candidates for second order phase transition points where an UV continuum limit might exist.

Completing characterization of photon orbits in Kerr and Kerr-Newman metrics

Recently, several new characteristics have been introduced to describe null geodesic structure of stationary spacetimes, such as photon regions (PR) and transversely trapping surfaces (TTS). The former are three-dimensional domains confining the spherical photon orbits, while the latter are closed two-surface of spherical topology which fill other regions called TTR. It is argued that in generic stationary axisymmetric spacetime it is natural to consider also the non-closed TTSs of the geometry of spherical cups, satisfying the same conditions ("partial" TTS or PTTS), which fill the three-dimensional regions, PTTR. We then show that PR, TTR and PTTR together with the corresponding anti-trapping regions constitute the complete set of regions filling the entire three-space (where timelike surfaces are defined) of Kerr-like spacetimes. This construction provides a novel optical description of such spacetimes without recurring to explicit solution of the geodesic equations. Applying this analysis to Kerr-Newman metrics (including the overspinning ones) we reveal four different optical types for different sets of the rotation and charge parameters. To illustrate their properties we extend Synge analysis of photon escape in the Schwarzschild metric to stationary spacetimes and construct density graphs describing escape of photons from all the above regions.

Four-dimensional traversable wormholes and bouncing cosmologies in vacuum

In this letter we point out the existence of solutions to General Relativity with a negative cosmological constant in four dimensions, which contain solitons as well as traversable wormholes. The latter connect two asymptotically locally AdS4 spacetimes. At every constant value of the radial coordinate the spacetime is a spacelike warped AdS3. We compute the dual energy momentum tensor at each boundary showing that it yields different results. We also show that these vacuum wormholes can have more than one throat and that they are indeed traversable by computing the time it takes for a light signal to go from one boundary to the other, as seen by a geodesic observer. We generalize the wormholes to include rotation and charge. When the cosmological constant is positive we find a cosmology that is everywhere regular, has either one or two bounces and that for late and early times matches the Friedmann-Lemaître-Robertson-Walker metric with spherical topology and an exponential scale factor.

Stability of constant mean curvature surfaces in three-dimensional warped product manifolds

In this paper we prove that stable, compact without boundary, oriented, nonzero constant mean curvature surfaces in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds are the slices, provided its mean curvature satisfies some positive lower bound. More generally, we prove that stable, compact without boundary, oriented nonzero constant mean curvature surfaces in a large class of three dimensional warped product manifolds are embedded topological spheres, provided the mean curvature satisfies a positive lower bound depending only on the ambient curvatures. We conclude the paper proving that a stable, compact without boundary, nonzero constant mean curvature surface in a general Riemannian is a topological sphere provided its mean curvature has a lower bound depending only on the scalar curvature of the ambient space and the squared norm of the mean curvature vector field of the immersion of the ambient space in some Euclidean space.

A 6-Letter ‘DNA’ for Baskets with Handles

Fabric surfaces, made using techniques such as crochet and net-making, are typically worked in a linear order that meanders, without crossing itself, to ultimately visit and build the entire surface. For a closed basket, whose surface is a topological sphere, it is known that the construction can be described by a codeword on a 4-letter alphabet via Mullin’s encoding of plane graphs. Mullin’s code exemplifies the formal language known as the Shuffled Dyck Language with 2 Types of Parenthesis ( S D L 2 ). Besides its 4-letter alphabet, S D L 2 has some other similarities to DNA: Any word can be ‘evolved’ via a sequence of local mutations (rewriting rules), and ‘gene-splicing’ two S D L 2 words, by an insertion or concatenation, produces another S D L 2 word. However, S D L 2 comes up short when we attempt to make a basket with handles. I show that extending the language to S D L 3 , by addition of a third type of parenthesis, succeeds for orientable surfaces with handles—provided an appropriate choice of cut graph is made.

A note on singularities in finite time for the L^{2} gradient flow of the Helfrich functional

This work investigates the formation of singularities under the steepest descent L^2-gradient flow of {{,mathrm{{ W}},}}_{lambda _1, lambda _2} with zero spontaneous curvature, i.e., the sum of the Willmore energy, lambda _1 times the area, and lambda _2 times the signed volume of an immersed closed surface without boundary in mathbb {R}^3. We show that in the case that lambda _1>1 and lambda _2=0, any immersion develops singularities in finite time under this flow. If lambda _1 >0 and lambda _2 > 0, embedded closed surfaces with energy less than 8π+min16πλ13/3λ22,8π\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} 8\\pi +\\min \\left\\{ \\left( 16 \\pi \\lambda _1^3\\right) \\bigg /\\left( 3\\lambda _2^2\\right) , 8\\pi \\right\\} \\end{aligned}$$\\end{document}and positive volume evolve singularities in finite time. If in this case the initial surface is a topological sphere and the initial energy is less than 8 pi , the flow shrinks to a round point in finite time. We furthermore discuss similar results for the case that lambda _2 is negative. These results strengthen the ones of McCoy and Wheeler (Commun Anal Geom 24(4):843–886, 2016). For lambda _1 >0 and lambda _2 ge 0, they showed that embedded closed spheres with positive volume and energy close to 4pi , i.e., close to the Willmore energy of a round sphere, converge to round points in finite time.

Geometrothermodynamic analysis and P–V criticality of higher dimensional charged Gauss–Bonnet black holes with first order entropy correction

We consider a charged Gauss-Bonnet black hole in $d$-dimensional spacetime and examine the effect of thermal fluctuations on the thermodynamics of the concerned black hole. At first we take the first order logarithmic correction term in entropy and compute the thermodynamic potentials like Helmholtz free energy $F$, enthalpy $H$ and Gibbs free energy $G$ in the spherical, Ricci flat and hyperbolic topology of the black hole horizon, respectively. We also investigate the $P$-$V$ criticality and calculate the critical volume $V_c$, critical pressure $P_c$ and critical temperature $T_c$ using different equations when $P$-$V$ criticality appears. We show that there is no critical point without thermal fluctuations for this type of black hole. We find that the presence of logarithmic correction in it is necessary to have critical points and stable phases. Moreover, we study the stability of the black holes by employing the specific heat. Finally, we study the geometrothermodynamics and analyse the Ricci scalar of the Ruppeiner metric graphically for the same.

Holographic complexity growth rate in Horndeski theory

Based on the context of complexity = action (CA) conjecture, we calculate the holographic complexity of AdS black holes with planar and spherical topologies in Horndeski theory. We find that the rate of change of holographic complexity for neutral AdS black holes saturates the Lloyd’s bound. For charged black holes, we find that there exists only one horizon and thus the corresponding holographic complexity can’t be expressed as the difference of some thermodynamical potential between two horizons as that of Reissner–Nordstrom AdS black hole in Einstein–Maxwell theory. However, the Lloyd’s bound is not violated for charged AdS black hole in Horndeski theory.

Spherical U-Net on Cortical Surfaces: Methods and Applications.

Convolutional Neural Networks (CNNs) have been providing the state-of-the-art performance for learning-related problems involving 2D/3D images in Euclidean space. However, unlike in the Euclidean space, the shapes of many structures in medical imaging have a spherical topology in a manifold space, e.g., brain cortical or subcortical surfaces represented by triangular meshes, with large inter-subject and intra-subject variations in vertex number and local connectivity. Hence, there is no consistent neighborhood definition and thus no straightforward convolution/transposed convolution operations for cortical/subcortical surface data. In this paper, by leveraging the regular and consistent geometric structure of the resampled cortical surface mapped onto the spherical space, we propose a novel convolution filter analogous to the standard convolution on the image grid. Accordingly, we develop corresponding operations for convolution, pooling, and transposed convolution for spherical surface data and thus construct spherical CNNs. Specifically, we propose the Spherical U-Net architecture by replacing all operations in the standard U-Net with their spherical operation counterparts. We then apply the Spherical U-Net to two challenging and neuroscientifically important tasks in infant brains: cortical surface parcellation and cortical attribute map development prediction. Both applications demonstrate the competitive performance in the accuracy, computational efficiency, and effectiveness of our proposed Spherical U-Net, in comparison with the state-of-the-art methods.