Convex Boundary Research Articles

The first Dirichlet eigenvalue and the width

For a geodesic ball with non-negative Ricci curvature and mean convex boundary, it is known that the first Dirichlet eigenvalue of this geodesic ball has a sharp lower bound in term of its radius. We show a quantitative explicit inequality, which bounds the width of geodesic ball in terms of the spectral gap between the first Dirichlet eigenvalue and the corresponding sharp lower bound.

The integration of En route flow optimization, complex network clustering, and rule-based approach to airspace sub-sectorization for enhanced air traffic monitoring

This study proposes the use of a novel integrated framework for 2D en route airspace sub-sectorization. The integrated framework combines the multi-commodity flow optimization approach, complex network clustering approach, and Minimum Bounding Geometry (MBG)-coupled Rule-based Approach for boundary design. A decomposition-based discrete particle swarm optimization (DPSO) is used to solve the clustering problem. The output of the flow optimization is used as a guiding standard for the DPSO. Experimentations were performed using the Indian airspace sector to validate the framework and DPSO was run for different maximum number of generations (maxgen). The findings reveal that the multi-commodity flow approach captures system-wide flow operations. Clustering results corresponding to maxgen=100 and maxgen=150 perform best in terms of equitable and balanced distribution of cluster size and traffic load. The MBG-coupled Rule-based Approach leads to compact and convex sub-sector boundary design. Major implications of this research include dynamic adaptability of the integrated framework, increased sensitivity of sector design to network evolution, and a computationally tractable framework. The higher controllability of the proposed framework also offers an increased acceptance among practitioners.

Impingement of vortex dipole on heated boundaries and related thermal plume dynamics

The profound influence of an externally induced vortex dipole on thermal plume dynamics is numerically studied for varying Rayleigh numbers (Ra) employing the Bhatnagar–Gross–Krook collision model-based lattice Boltzmann method with a double distribution function approach. This study is extended to vortex dipole impingement with different types of heated bottom boundaries of two-dimensional domain, such as flat, “V-shaped,” and “inverted-V-shaped.” The vortex dipole impingement with the heated boundaries generates secondary vortices, which in turn produce vortex-driven thermal plumes, thereby advancing plume generation. The subsequent merging of the plumes enhances heat transport and leads to a continuous plume ascent. The presence of convex corners facilitates flow separation and also gives rise to the formation of secondary vortex dipoles, thereby significantly impacting the continuous generation of jet-like plumes when compared to concave configurations. The lack of an external vortex in pure buoyancy-driven flows produces less pronounced jet-like plumes and a relatively low Nusselt number. The boundary types and Ra significantly influence the vorticity production, resulting in higher enstrophy and palinstrophy for convex boundaries compared to flat and concave ones. A lower Prandtl number increases secondary vortices and corner rolls, leading to larger velocity gradients, higher thermal diffusivity, resulting in increased kinetic energy and thermal dissipation rates. The increased cell height enhances heat transfer at the top boundary due to improved heat convection from the slanted boundary and influence of early dipole impingement. Furthermore, kinetic energy dissipates in the dipole-driven flows and increases in the buoyancy-dominated flows.

Double b-fibrations and desingularization of the X-ray transform on manifolds with strictly convex boundary

Double b-fibrations and desingularization of the X-ray transform on manifolds with strictly convex boundary

Vortex-dipole impingement with convex and concave boundaries

The vortex-dipole interactions with convex and concave boundaries in a two-dimensional domain are analyzed using the Bhatnagar–Gross–Krook (BGK) collision-based lattice Boltzmann method (LBM). The formation and detachment of the boundary layers at the flat wall creates new dipoles of non-identical vortices, following cyclic trajectories and exhibiting logarithmic variations in the production of maximum vorticity with Reynolds number (Re). The vortex-dipole interactions with concave boundaries produce a series of secondary dipoles, whose relative strength linearly decreases for the given Re in subsequent vortex-releasing events. Oblique interactions with cavity corners cause secondary dipoles to undergo head-on collisions at the domain center, influencing partner exchange and orthogonal propagation. Conversely, convex boundaries split the primary vortex-dipole upon impingement, intensifying the vorticity production and strain effects. Whereas, the larger ingestion of vorticity at the convex corners of the “T-shaped” cavity forms new dipoles, which deflect, rebound, and follow a parabolic trajectory for the vortex exchange. Normalized enstrophy, Ω(t), and palinstrophy, P(t), show distinct peaks during dipole interactions with boundary walls, influencing enhanced kinetic energy, E(t), decay. Moreover, the evolution of E(t) and Ω(t) satisfies the relation valid for no-slip boundaries. The collision behavior, vorticity production, and vortex rebound are functions of Re. The convex boundaries modify the scaling results of maximum Ω(t) and P(t) to higher exponent values than the dipole interactions with flat and concave boundaries.

Dynamics of collapse of free-surface bubbles: effects of gravity and viscosity

The rupture of the thin film at the top of a bubble at a liquid–gas interface leads to an axisymmetric collapse of the bubble cavity. We present scaling laws for such a cavity collapse, established from experiments conducted with bubbles spanning a wide range of Bond ( ${10^{-3}< Bo\leq 1}$ ) and Ohnesorge numbers ( ${10^{-3}< Oh<10^{-1}}$ ), defined with the bubble radius $R$ . The cavity collapse is a capillary-driven process, with a dependency on viscosity and gravity, affecting respectively, precursory capillary waves on the cavity boundary and the static bubble shape. The collapse is characterised by the normal interface velocity ( $U_n$ ) and by the tangential wave propagation velocity of the kink ( $U_t$ ), defined by the intersection of the concave cavity boundary formed after the rupture of the thin film with the convex boundary of the bubble cavity. During the collapse, $U_t$ remains constant and is shown to be $U_t=4.5U_c{\mathcal {W}}_R$ , where $U_c$ is the capillary velocity and ${\mathcal {W}}_R(Oh,Bo)={(1-\sqrt {Oh {\mathscr {L}}} )^{-1/2}}$ is the wave resistance factor due to the precursory capillary waves, with $\mathscr {L}(Bo)$ being the path correction of the kink motion. The movement of the kink in the normal direction is part of the inward shrinkage of the whole cavity due to the sudden reduction of gas pressure inside the bubble cavity after the thin film rupture. This normal velocity is shown to scale as $U_c$ in the equatorial plane, while at the bottom of the cavity $\bar {U}_{nb}=U_c(Z_c/R)({\mathcal {W}_R}/ {\mathscr {L}})$ , where $Z_c(Bo)$ is the static cavity depth. The filling rate of the cavity, which remains a constant throughout the collapse, is shown to be entirely determined by the shrinking velocity and scales as ${Q_T\simeq 2{\rm \pi} R Z_c U_c}$ . From $Q_T$ we recover the jet velocity scaling, thereby relating the cavity collapse with the jet velocity scaling.

The First Width of Non-negatively Curved Surfaces with Convex Boundary

The First Width of Non-negatively Curved Surfaces with Convex Boundary

Comprehensive study on the through-process Goss texture evolution in Fe-3.78 wt.%Si grain oriented electrical steel

Cold rolled grain oriented steel is used in transformer cores due to its superior magnetic properties. Large Goss oriented grains develop after the final secondary recrystallization provide low hysteresis loss. The present study represents the development of Goss orientation after each processing stage of Fe-3.78 wt% Si steel with the help of electron backscatter diffraction and x-ray diffraction techniques. The Goss oriented grains developed during hot rolling and hot band annealing did not remain stable in cold rolling. On the other hand, shear bands developed inside {111}<112> component of γ-fiber texture during cold rolling. Partial recrystallization of the cold rolled sheet revealed that Goss oriented grains started evolving inside the shear bands of {111}<112> oriented grains. Intermediate annealing between cold rolling passes was found to be not essential to achieve Goss orientation in primary recrystallization. The optimum primary recrystallization temperature was in between 923 and 948 K where a significant amount of Goss oriented grains were present. Further, secondary recrystallization was attempted at different temperatures between 1173 and 1473 K. Grain boundary curvature of Goss oriented grains and all other grains after both primary and secondary recrystallization was studied. Despite the percentage of convex boundaries surrounding Goss oriented grains was found slightly higher than that around other grains, it was insufficient to cause abnormal grain growth.

Compact Sub‐Semicircular Axial Gradient Index Lens Synthesized by Metasurfaces: A Promising Solution for Miniaturized Antenna and Subwavelength Focusing Systems

AbstractWe present the design, fabrication, and experimental measurements of an Axial Gradient‐index (AGRIN) flat lens synthesized by 2D periodic metallic patches on a grounded dielectric slab, known as metasurfaces. Metasurfaces can be considered as artificial materials that possess unique characteristics, which mainly revolve around the two‐dimensional distribution of the structure. By manipulating the phase and amplitude of the near‐fields over the surface of the metasurface, specific values of the effective permittivity and permeability can be achieved, thereby regulating the overall field behavior by gaining control of the refractive index that is related to the aforementioned two material parameters. By configuring the refractive index, metasurfaces enable the manipulation of the direction in which electromagnetic waves propagate, as dictated by Snell's law. By varying the geometric dimensions of unit cells in each segment that compose the lens, the gradient refractive index profile required for our AGRIN lens can be achieved. A primary feature of this lens lies with its semicircular structure, in contrast to conventional circular ones. Its perimeter comprises a convex boundary and a straight one. In comparison to traditional circular lenses like the Luneburg lens and Maxwell fisheye lens, our proposed AGRIN lens offers the advantage of being more compact. Despite its smaller area, the AGRIN lens maintains its focusing ability. The AGRIN lens is capable of focusing over 71.75% of the energy from the incident wave onto the focal point. This efficient focusing ability, combined with its reduced size, makes the AGRIN lens highly advantageous for various applications.

Rigidity of nonpositively curved manifolds with convex boundary

We show that a compact Riemannian 3 3 -manifold M M with strictly convex simply connected boundary and sectional curvature K ≤ a ≤ 0 K\leq a\leq 0 is isometric to a convex domain in a complete simply connected space of constant curvature a a , provided that K ≡ a K\equiv a on planes tangent to the boundary of M M . This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard 3 3 -manifolds, and extends some rigidity results of Greene-Wu, Gromov, and Schroeder-Strake. Our proof is based on a recent comparison formula for total curvature of Riemannian hypersurfaces, which also yields some dual results for K ≥ a ≥ 0 K\geq a\geq 0 .

Uryson width of three dimensional mean convex domain with non-negative Ricci curvature

We prove that for every three dimensional manifold with nonnegative Ricci curvature and strictly mean convex boundary (non-empty), there exists a Morse function so that each connected component of its level sets has a uniform diameter bound, which depends only on the lower bound of mean curvature. This gives an upper bound of Uryson 1-width for those three manifolds with boundary.

The wave trace and Birkhoff billiards

The purpose of this article is to develop a Hadamard–Riesz-type parametrix for the wave propagator in bounded planar domains with smooth, strictly convex boundary. This parametrix then allows us to rederive an oscillatory integral representation for the wave trace appearing in the work by Marvizi and Melrose (1982) and compute its principal symbol explicitly in terms of geometric data associated to the billiard map. This results in new formulas for the wave invariants. The order of the principal symbol, which appears to be inconsistent in the works by Marvizi and Melrose (1982) and Popov (1994), is also corrected. In those papers, the principal symbol was never actually computed and to our knowledge, this paper contains the first explicit formulas for the principal symbol of the wave trace. The wave trace formulas we provide are localized near both simple lengths corresponding to nondegenerate periodic orbits and degenerate lengths associated to one parameter families of periodic orbits tangent to a single rational caustic. Existence of a Hadamard–Riesz-type parametrix with explicit symbol and phase calculations in the interior appears to be new in the literature, with the exception of the author’s previous work [J. Geom. Anal. 31 (2021), 2238–2295] in the special case of elliptical domains. This allows us to circumvent the symbol calculus used by Duistermaat and Guillemin (1975) and Hezari and Zelditch (2012) when computing trace formulas, which are instead derived from integrating our explicit parametrix over the diagonal.

Cup product in bounded cohomology of negatively curved manifolds

Let M M be a negatively curved compact Riemannian manifold with (possibly empty) convex boundary. Every closed differential 2 2 -form ξ ∈ Ω 2 ( M ) \xi \in \Omega ^2(M) defines a bounded cocycle c ξ ∈ C b 2 ( M ) c_\xi \in C_b^2(M) by integrating ξ \xi over straightened 2 2 -simplices. In particular Barge and Ghys [Invent. Math. 92 (1988), pp. 509–526] proved that, when M M is a closed hyperbolic surface, Ω 2 ( M ) \Omega ^2(M) injects this way in H b 2 ( M ) H_b^2(M) as an infinite dimensional subspace. We show that the cup product of any class of the form [ c ξ ] [c_\xi ] , where ξ \xi is an exact differential 2-form, and any other bounded cohomology class is trivial in H b ∙ ( M ) H_b^{\bullet }(M) .

Abstract P4-02-29: A 3D Visualization Method for Breast Cancer Surgeons and Patients

Abstract Background: Surgical oncologists currently rely on 2D image slices, such as those from mammograms, to assess the locations and extents of their patients’ breast cancer tumors. In order to accurately evaluate the potential success of various surgical options, a surgeon must mentally translate these 2D images into more realistic, 3D image to visualize breast and tumor morphologies. For a patient without the same experience, it is even more difficult to imagine the true, 3D shape, size and location of their tumor and to participate fully in decision-making and surgical planning. Since the type of surgery chosen (nipple-sparing versus skin-sparing, mastectomy versus lumpectomy), and its ultimate clinical and cosmetic consequences, depends on not only a surgeon’s accurate pre-operative assessment, but also on the patient’s understanding; a method of clear, precise, 3D visualization in the clinic would represent a significant advancement. Methods: Using DCE-MRI as input, we developed TumorSight (TSi) to provide surgical oncologists with an accurate, 3D representation of a patient’s breast tissue and tumor, and to allow computation of useful metrics, such as the tumor-to-breast volume ratio and distance to surgically useful anatomical landmarks such as the nipple. TSi was used to create 3D models of breast cancer patient tumor using MRIs from the SimBioSys Virtual TumorBank which currently houses thousands of patients. Results: After creating the models, labels for tumor, as well as skin, chest wall, nipple, adipose tissue, glandular tissue, and vasculature were created. After tumor segmentation validation studies (of 49 patients) were completed, the median absolute volumetric error was 0.32 and the median maximum Hausdorff distance was 17.72mm. From these representations, we determine whether the tumor is monocentric, multifocal or diffuse, compute the tumor volume-to-breast volume ratio, and measure the distance of closest approach between the tumor and the nipple. In addition, by adding a margin to the tumor and computing the volume within a convex boundary containing the tumor+margin, TSi provides a pre-operative estimate of the extirpative volume and its fraction of the total breast volume. Conclusion: The ability to visualize, in 3D, a tumor’s location and distribution in the breast, and to accurately measure important metrics such as tumor-to-nipple distance, multifocality, and tumor-to-breast volume, are critical to surgical planning in breast cancer. To assist in this planning, TSi provides a surgeon and their patient with a three-dimensional breast and tumor representation and reports measurements of key features of interest. Citation Format: Amanda Parker, Tyler M. Earnest, Arda Pekis, Vignesh Kannan, Joseph Peterson, John A. Cole, Anuja K. Antony, Daniel Cook. A 3D Visualization Method for Breast Cancer Surgeons and Patients [abstract]. In: Proceedings of the 2022 San Antonio Breast Cancer Symposium; 2022 Dec 6-10; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2023;83(5 Suppl):Abstract nr P4-02-29.

Abstract P4-02-24: A 3D Visualization and Prediction Device for Breast Cancer Surgeons and their Patients

Abstract Background: Despite chemotherapeutic advances, surgery remains a putative treatment modality for breast cancer. The type of surgery chosen, and its ultimate clinical and cosmetic consequences, depends on a surgeon’s ability to accurately assess a tumor’s size, distribution, and position in the breast relative to anatomical landmarks. Moreover, in cases in which the tumor-to-breast volume ratio is substantial, neoadjuvant chemotherapy (NAC) offers the potential to shrink a tumor such that enhanced cosmesis or breast-conserving surgery becomes possible. Therefore, doctors must not only build an accurate picture of the current tumor shape, size and location, but anticipate how they might change over time and during treatment. SimBioSys’ TumorSight (TSi) offers doctors the ability to simulate the way a patient’s tumor might respond to various drug therapies, volumetrically and morphologically. These predictions are critical for surgical planning, in which a patient’s eligibility for breast-conserving surgery depends on the spatial distribution of the tumor. Primarily, this means consideration of (1) tumor multicentricity, (2) the extirpative tumor-to-breast volume ratio, and (3) the distance from the tumor to nipple. TSi provides physicians with estimates of each of these three key metrics, in 3D, over time, and as a function of regimen, to facilitate accurate evaluation, visualization and forecasting of NAC response with respect to surgical planning. Methods: TSi was used to create 3D models across multiple patient cohorts using DCE-MRI. These included segmentations for skin, chest wall, nipple, adipose tissue, glandular tissue, vasculature, and tumor. Volumetric and morphological changes in the tumor and adjacent tissues were simulated throughout a chemotherapy treatment regimen, as described previously. The resulting tumor distributions were analyzed and compared to distributions in post-treatment MRI segmentations. Metrics of interest in this study were the distance of closest approach between the tumor and nipple, tumor volume-to-breast volume ratio, tumor convex hull-to-breast volume ratio, and whether the tumor was monocentric. Results: In a set of N=292 patients, both the predicted tumor volume and tumor volume from a post-treatment MRI segmentation were computed. We took the difference in these volumes, normalized by the total breast volume (also computed from an MRI segmentation), to assess how well TSi predicts the final tumor volume-to-breast volume ratio after treatment. We found that the median difference in this metric was 3.26e-07, and 5th and 95th percentiles were 0.0035 and 0.0023, respectively. Similar performance was seen when using the volume of a convex boundary enclosing the tumor and a 1 cm margin (to simulate operative conditions). In a set of N=164 patients, the accuracy of our prediction of whether the tumor will be monocentric after treatment was assessed. We found that our simulations were 78.7% accurate in predicting monocentricity. Finally, in a set of N=134 patients, TSi predicted a distribution of changes in the closest distance between the nipple and tumor, with a median of 1.06 cm and standard deviation of 1.64 cm. In all studies, there was no difference in median values seen between cancer subtypes. Conclusion: The ability to accurately measure, visualize, and predict post-NAC metrics such as tumor-to-nipple distance, monocentricity, and tumor-to-breast volume ratio is critical to surgical planning in breast cancer. To assist in this planning, TSi provides a surgeon and their patient with a realistic three-dimensional representation of tumor volume, shape, and location in the breast, and reliably predicts how these metrics change during treatment. Citation Format: Amanda Parker, Joseph Peterson, John A. Cole, John Whitman, Nicole Hobbs, Daniel Cook, Anuja K. Antony. A 3D Visualization and Prediction Device for Breast Cancer Surgeons and their Patients [abstract]. In: Proceedings of the 2022 San Antonio Breast Cancer Symposium; 2022 Dec 6-10; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2023;83(5 Suppl):Abstract nr P4-02-24.

Neighbor Reweighted Local Centroid for Geometric Feature Identification.

Identifying geometric features from sampled surfaces is a significant and fundamental task. The existing curvature-based methods that can identify ridge and valley features are generally sensitive to noise. Without requiring high-order differential operators, most statistics-based methods sacrifice certain extents of the feature descriptive powers in exchange for robustness. However, neither of these types of methods can treat the surface boundary features simultaneously. In this paper, we propose a novel neighbor reweighted local centroid (NRLC) computational algorithm to identify geometric features for point cloud models. It constructs a feature descriptor for the considered point via decomposing each of its neighboring vectors into two orthogonal directions. A neighboring vector starts from the considered point and ends with the corresponding neighbor. The decomposed neighboring vectors are then accumulated with different weights to generate the NRLC. With the defined NRLC, we design a probability set for each candidate feature point so that the convex, concave and surface boundary points can be recognized concurrently. In addition, we introduce a pair of feature operators, including assimilation and dissimilation, to further strengthen the identified geometric features. Finally, we test NRLC on a large body of point cloud models derived from different data sources. Several groups of the comparison experiments are conducted, and the results verify the validity and efficiency of our NRLC method.

Rigidity of capillary surfaces in compact 3-manifolds with strictly convex boundary

AbstractIn this paper, we obtain one sharp estimate for the length $L(\partial\Sigma)$ of the boundary $\partial\Sigma$ of a capillary minimal surface Σ2 in M3, where M is a compact three-manifolds with strictly convex boundary, assuming Σ has index one. The estimate is in term of the genus of Σ, the number of connected components of $\partial\Sigma$ and the constant contact angle θ. Making an extra assumption on the geometry of M along $\partial M$, we characterize the global geometry of M, which is saturated only by the Euclidean three-balls. For capillary stable CMC surfaces, we also obtain similar results.

Bordered Floer homology and contact structures

Abstract We introduce a contact invariant in the bordered sutured Heegaard Floer homology of a three-manifold with boundary. The input for the invariant is a contact manifold $(M, \xi , \mathcal {F})$ whose convex boundary is equipped with a signed singular foliation $\mathcal {F}$ closely related to the characteristic foliation. Such a manifold admits a family of foliated open book decompositions classified by a Giroux correspondence, as described in [LV20]. We use a special class of foliated open books to construct admissible bordered sutured Heegaard diagrams and identify well-defined classes $c_D$ and $c_A$ in the corresponding bordered sutured modules. Foliated open books exhibit user-friendly gluing behavior, and we show that the pairing on invariants induced by gluing compatible foliated open books recovers the Heegaard Floer contact invariant for closed contact manifolds. We also consider a natural map associated to forgetting the foliation $\mathcal {F}$ in favor of the dividing set and show that it maps the bordered sutured invariant to the contact invariant of a sutured manifold defined by Honda–Kazez–Matić.

Turbulent flow out of a convex curve in a channel using the SST turbulence model

The paper presents the results of the SST turbulence model in the Comsol Multiphysics software package for problems of convex boundary layer curvature, which are presented in the NASA database. In this work, the finite element method is used for the numerical implementation of the turbulence equations. To stabilize the discretized equations, Galerkin least squares (GLS) stabilization and crosswind propagation stabilization were used. The results obtained are compared with the results of experimental data.

Spontaneous multi-scale void formation and closure in densifying epithelial and fibroblast monolayers from the sub-confluent state.

Using time-lapse phase contrast microscopy, the formation and closure of spontaneously generated voids in the densifying monolayers of isotropic epithelial cells (ECs) and elongated fibroblast cells (FCs) through proliferation from the sub-confluent state are investigated. It is found that, in both types of monolayers after forming a connected network composed of nematic patches with different orientations, numerous multi-scale voids can be spontaneously formed and gradually close with increasing time. The isotropic fluctuations of deformation and crawling of ECs and the anisotropic axial motion/alignment polarizations of FCs are the two keys leading to the following different generic dynamical behaviors. In EC monolayers, voids exhibit irregular boundary fluctuations and easier cell re-orientation of front layer cells (FLCs) surrounding void boundaries. Void closures are mainly through pinching the gap between the opposite fluctuating void boundaries, and the inward crawling of FLCs to reduce void area associated with topological rearrangement to reduce FLC number. In FC monolayers, large voids have piecewise smooth convex boundaries, and cusp-shaped concave boundaries with cells orienting toward the void at cusp tips. The extension of a thin cell bridge from the cusp tip can bisect a large void into smaller voids. For smaller FC voids dominated by convex boundaries, along which cell alignment prohibits inward crawling, the reduction of FLC number through successive outward squeezing of single FLCs by neighboring FLCs sliding along the void boundary plays an important role for topological rearrangement and void closure. Unlike those surrounding artificial wounds in dense EC monolayers, the absence of ring-like purse-strings surrounding EC and FC voids allows topological rearrangements for reducing void perimeter and void area.