Geometric Setting Research Articles

Sharp Boundary Trace Theory and Schrödinger Operators on Bounded Lipschitz Domains

We develop a sharp boundary trace theory in arbitrary bounded Lipschitz domains which, in contrast to classical results, allows “forbidden” endpoints and permits the consideration of functions exhibiting very limited regularity. This is done at the (necessary) expense of stipulating an additional regularity condition involving the action of the Laplacian on the functions in question which, nonetheless, works perfectly with the Dirichlet and Neumann realizations of the Schrödinger differential expression − Δ + V -\Delta +V . In turn, this boundary trace theory serves as a platform for developing a spectral theory for Schrödinger operators on bounded Lipschitz domains, along with their associated Weyl–Titchmarsh operators. Overall, this pushes the present state of knowledge a significant step further. For example, we succeed in extending the Dirichlet and Neumann trace operators in such a way that all self-adjoint extensions of a Schrödinger operator on a bounded Lipschitz domain may be described with explicit boundary conditions, thus providing a final answer to a problem that has been investigated for more than 60 years in the mathematical literature. Along the way, a number of other open problems are solved. The most general geometric and analytic setting in which the theory developed here yields satisfactory results is that of Lipschitz subdomains of Riemannian manifolds and for the corresponding Laplace–Beltrami operator (in place of the standard flat-space Laplacian). In particular, such an extension yields results for variable coefficient Schrödinger operators on bounded Lipschitz domains.

Solvation, geometry, and assembly of the tobacco mosaic virus.

Biological self-assembly is a fundamental aspect in the development of complex structures in nature. A paradigm for such a process is the assembly of tobacco mosaic virus (TMV) capsid proteins into helical rods around the viral genome. The self-assembly process of the virus is typically modelled through attractive interactions between protein subunits, however capsid proteins also interact with their aqueous environment through solvation free energy. An open question is what role solvation plays in virus self-assembly. Here, we show that a purely geometric model of nonpolar solvation free energy, the morphometric approach, is sufficient to simulate the assembly of up to three protein subunits of TMV. The lowest solvation free energy states we find in a geometric simulation setting are remarkably close to the correctly assembled states of various experimentally determined structures. This demonstrates that van der Waals forces and entropic considerations are sufficient to guide assembly in the absence of attractive interactions between protein subunits. It further illustrates the impact of the morphometric approach as a computationally efficient model for nonpolar solvation free energy of solutes, in particular those with complicated geometry. This demonstration of the role of solvation raises important questions about the driving forces behind biological self-assembly and the paramount role of geometry.

Morse Index Stability of Biharmonic Maps in Critical Dimension

In December 2022, Da Lio, Gianocca, and Rivière developed a new theory to prove the upper semi-continuity of the sum of the Morse index and the nullity in geometric analysis (Da Lio, Gianocca, and Rivière in Morse index stability for critical points to conformally invariant Lagrangians, 2023, arXiv:2212.03124), and applied it to conformally invariant problems in dimension 2—which include harmonic maps. Their method, quickly extended to Willmore immersions and later to other geometric settings, is applied in this article to the Morse stability of biharmonic maps in critical dimension 4. A key step in the proof is to obtain an energy quantization in L2,1 (the pre-dual of the Marcinkiewicz space L2,∞ of weakly squared-integrable functions), and we prove a general result to establish this strong energy quantization, which allows us to recover previous results in a unified fashion.

On completeness of foliated structures, and null Killing fields

Abstract We consider a compact manifold $$(M,\mathfrak {F})$$ ( M , F ) with a foliation $$\mathfrak {F}$$ F , and a smooth affine connection $$\nabla $$ ∇ on the tangent bundle of the foliation $$T\mathfrak {F}$$ T F . We introduce and study a foliated completeness problem. Namely, under which conditions on $$\nabla $$ ∇ the leaves are complete? We consider different natural geometric settings: the first one is the case of a totally geodesic lightlike foliation of a compact Lorentzian manifold, and the second one is the case where the leaves have particular affine structures. In the first case, we characterize the completeness, and obtain in particular that if a compact Lorentzian manifold admits a null Killing field V such that the distribution orthogonal to V is integrable, then it defines a (totally geodesic) foliation with complete leaves. In the second case, we give a completeness result for a specific affine structure called “the unimodular affine lightlike geometry”, and characterize the completeness for a natural relaxation of the geometry. On the other hand, we study the global completeness of a compact Lorentzian manifold in the presence of a null Killing field. We give two non-complete examples, starting from dimension 3: one is a locally homogeneous manifold, and the other is a 3D example where the Killing field dynamics is equicontinuous.

Подход к твёрдотельному моделированию геометрических объектов в точечном исчислении

The development of computer-aided design systems involves a combination of fundamental and applied research. The conceptual foundation of the mathematical framework for such systems lies in the notion of a complete geometric body—a geometric set of points where the number of active parameters matches the dimensionality of the space, with the geometric body represented as a distinct part of that space. The analytical representation of these point sets is achieved through the mathematical apparatus of point calculus, which can be generalized to multidimensional spaces. The article compares this proposed approach to solid modeling of geometric objects with existing methods. Examples demonstrating the modeling of geometric bodies using the new approach are provided. The advantages of this ap-proach are emphasized, including the compactness of the analytical description, the elimination of transformation ma-trices, the facilitation of parallel computations within the mathematical framework, and more. Additionally, the article explores the capabilities of modeling geometric bodies in point calculus, such as the representation of isotropic and ani-sotropic bodies as solid geometric objects with a functionally controlled linear or nonlinear spatial structure.

On the geometric red-blue set cover problem

On the geometric red-blue set cover problem

Optimal Shape Design of 1D Fractional Order Heat Equation in Large Time

In this paper, we consider an optimal design problem for the 1D nonlocal heat equations involving the fractional Laplacian . The control here is the shape of the interval on which heat diffuses. We work in the geometric setting introduced by Šverák in [1] where the intervals under consideration are assumed to have a limited number of holes. Based on a -convergence approach, we can prove when \\frac{1}{2}$$\\end{document}]]>, the nonlocal parabolic optimal designs converge, in the complementary Hausdorff topology, to an optimal design for the corresponding stationary nonlocal elliptic equation when time tends to infinity.

Species scale, worldsheet CFTs and emergent geometry

We study infinite-distance limits in the moduli space of perturbative string vacua. The remarkable interplay of string dualities seems to determine a highly non-trivial dichotomy, summarized by the emergent string conjecture, by which in some duality frame either internal dimensions decompactify or a unique critical string becomes tensionless. We investigate whether this pattern persists in potentially non-geometric settings, showing that (a proxy for) the cutoff of the gravitational effective field theory in perturbative type II vacua extracted from a graviton scattering amplitude vanishes if and only if a light tower of states appears. Moreover, under some technical assumptions on the spectrum of conformal weights, the cutoff scales with the spectral gap of the internal conformal field theory in the same manner as in decompactification or emergent string limits, regardless of supersymmetry or whether the internal sector is geometric. As a byproduct, we elucidate the role of the species scale in (de)compactifications and show compatibility between effective field theory and worldsheet approaches in geometric settings with curvature.

Bifurcation for a sharp interface generation problem

As opposed to the widely studied bifurcation phenomena for maps or PDE problems, we are concerned with bifurcation for stationary points of a nonlocal variational functional defined not on functions but on sets of finite perimeter, and involving a nonlocal term. This sharp interface model (1.2), arised as the \Gamma -limit of the FitzHugh–Nagumo energy functional in a (flat) square torus in {\mathbb{R}}^{2} of size T , possesses lamellar stationary points of various widths with well-understood stability ranges and exhibits many interesting phenomena of pattern formation as well as wave propagation. We prove that when the lamella loses its stability, bifurcation occurs, leading to a two-dimensional branch of nonplanar stationary points. Thinner nonplanar structures, achieved through a smaller T , or multiple layered lamellae in the same-sized torus, are more stable. To the best of our knowledge, bifurcation for nonlocal problems in a geometric measure theoretic setting is an entirely new result.

Lars Brink and SIC-POVMs

The notion of SIC-POVMs comes from quantum theory, and they were not on the horizon when I was Lars Brink’s student in the early 1980s. In the summer of 2022, I told Lars that I know how to use number theoretical insights to construct SIC-POVMs in any Hilbert space of dimension [Formula: see text], and that the construction provides a geometric setting for some deep number theoretical conjectures. I will give a sketch of this development, of what it was like to be Lars’ student, and of what his reaction to our construction was.

Triharmonic Legendre curves on Lorentzian Heisenberg Space

This paper investigates triharmonic Legendre curves in the context of Lorentzian Heisenberg space (H3,g), a framework that merges concepts from differential geometry and theoretical physics. Triharmonic curves, defined by their stationarity with respect to the first variation of the energy functional, offer rich geometric properties. We derive the fundamental equations governing these curves and explore their behavior under the unique curvature of Lorentzian Heisenberg space (H3,g). The study reveals how the interplay between the curve’s triharmonic nature and the space’s Lorentzian structure leads to distinctive geometric features, including relativity and geometric modeling. The findings enhance the understanding of how triharmonic curves behave in complex geometrical settings, contributing to the broader discourse on the significance of curvature in higher-dimensional spaces. This research opens avenues for further exploration of triharmonic properties in other geometric framework.

Data-driven physics-based modeling of pedestrian dynamics.

Pedestrian crowds encompass a complex interplay of intentional movements aimed at reaching specific destinations, fluctuations due to personal and interpersonal variability, and interactions with each other and the environment. Previous work demonstrated the effectiveness of Langevin-like equationsin capturing the statistical properties of pedestrian dynamics in simple settings, such as almost straight trajectories. However, modeling more complex dynamics, such as when multiple routes and origin destinations are involved, remains a significant challenge. In this work, we introduce a novel and generic framework to describe the dynamics of pedestrians in any geometric setting, significantly extending previous works. Our model is based on Langevin dynamics with two timescales. The fast timescale corresponds to the stochastic fluctuations present when a pedestrian is walking. The slow timescale is associated with the dynamics that a pedestrian plans to follow, thus a smoother path without stochastic fluctuations. Employing a data-driven approach inspired by statistical field theories, we learn the complex potentials directly from the data, namely a high-statistics database of real-life pedestrian trajectories. This approach makes the model generic as the potentials can be read from any trajectory data set and the underlying Langevin structure enables physics-based insights. We validate our model through a comprehensive statistical analysis, comparing simulated trajectories with actual pedestrian measurements across five complementary settings of increasing complexity, including a real-life train platform scenario, underscoring its practical societal relevance. We show that our model, by learning the effective potential, captures fluctuation statistics in the dynamics of individual pedestrians, both in dilute (no interaction with other pedestrians) as well as in dense crowds conditions (in presence of interactions). Our results can be reproduced with our generic open-source Python implementation [Pouw et al. (2024) [Software] doi:10.5281/zenodo.13362271] and validated with the supplemented data set [Pouw et al. (2024) [Dataset] doi:10.5281/zenodo.13784588]. Beyond providing fundamental insights and predictive capabilities in pedestrian dynamics, our model could be used to investigate generic active dynamics such as vehicular traffic and collective animal behavior.

Stabilization of a weak viscoelastic wave equation in Riemannian geometric setting with an interior delay under nonlinear boundary dissipation

Stabilization of a weak viscoelastic wave equation in Riemannian geometric setting with an interior delay under nonlinear boundary dissipation

Affine manifolds: The differential geometry of the multi-dimensionally consistent TED equation

Affine manifolds: The differential geometry of the multi-dimensionally consistent TED equation

The geometric exceptional set in Manin’s conjecture for Batyrev and Tschinkel’s example

The geometric exceptional set in Manin’s conjecture for Batyrev and Tschinkel’s example

Competitive Data-Structure Dynamization

Data-structure dynamization is a general approach for making static data structures dynamic. It is used extensively in geometric settings and in the guise of so-called merge (or compaction) policies in big-data databases such as LevelDB and Google Bigtable. Previous theoretical work is based on worst-case analyses for uniform inputs—insertions of one item at a time and non-varying read rate. In practice, merge policies must not only handle batch insertions and varying read/write ratios, they can take advantage of such non-uniformity to reduce cost on a per-input basis. To model this, we initiate the study of data-structure dynamization through the lens of competitive analysis via two new online set-cover problems. For each, the input is a sequence of disjoint sets of weighted items. The sets are revealed one at a time. The algorithm must respond to each with a set cover that covers all items revealed so far. It obtains the cover incrementally from the previous cover by adding one or more sets and optionally removing existing sets. For each new set the algorithm incurs build cost equal to the weight of the items in the set. In the first problem the objective is to minimize total build cost plus total query cost , where the algorithm incurs a query cost at each time \(t\) equal to the current cover size. In the second problem, the objective is to minimize the build cost while keeping the query cost from exceeding \(k\) (a given parameter) at any time. We give deterministic online algorithms for both variants, with competitive ratios of \(\Theta(\log^{*}n)\) and \(k\) , respectively. The latter ratio is optimal for the second variant.

Combining UAV Multi-Source Remote Sensing Data with CPO-SVR to Estimate Seedling Emergence in Breeding Sunflowers

In order to accurately obtain the seedling emergence rate of breeding sunflower and to assess the quality of sowing as well as the merit of sunflower varieties, a method of extracting the sunflower seedling emergence rate using multi-source remote sensing information from unmanned aerial vehicles is proposed. Visible and multispectral images of sunflower seedlings were acquired using a UAV. The thresholding method was used to segment the excess green image of the visible image into vegetation and non-vegetation, to obtain the center point of the vegetation to generate a buffer, and to mask the visible image to achieve weed removal. The components of color models such as the hue–saturation value (HSV), green-relative color space (YCbCr), cyan-magenta-yellow-black (CMYK), and CIELAB color space (L*A*B) models were compared and analyzed. The A component of the L*A*B model was preferred for the optimization of K-means clustering to segment sunflower seedlings and mulch using the genetic algorithm, and the segmentation accuracy was improved by 4.6% compared with the K-means clustering algorithm. All told, 10 geometric features of sunflower seedlings were extracted using segmented images, and 10 vegetation indices and 48 texture features of sunflower seedlings were calculated based on multispectral images. The Pearson’s correlation coefficient method was used to filter the three types of features, and the geometric feature set, the vegetation index set, the texture feature set, and the preferred feature set were constructed. The construction of a sunflower plant number estimation model using the crested porcupine optimizer–support vector machine is proposed and compared with the sunflower plant number estimation models constructed based on decision tree regression, BP neural network, and support vector machine regression. The results show that the accuracy of the model based on the preferred feature set is higher than that of the other three feature sets, indicating that feature screening can improve the accuracy and stability of models; assessed using the CPO-SVR model, the accuracy of the preferred feature set was the highest, with an R² of 0.94, an RMSE of 5.16, and an MAE of 3.03. Compared to the SVR model, the value of the R2 is improved by 3.3%, the RMSE decreased by 18.3%, and the MAE decreased by 18.1%. The results of the study can be cost-effective, accurate, and reliable in terms of obtaining the seedling emergence rate of sunflower field breeding.

Geometric target margin strategy of proton craniospinal irradiation for pediatric medulloblastoma.

In proton craniospinal irradiation (CSI) for skeletally immature pediatric patients, a treatment plan should be developed to ensure that the dose is uniformly delivered to all vertebrae, considering the effects on bone growth balance. The technical (t) clinical target volume (CTV) is conventionally set by manually expanding the CTV from the entire intracranial space and thecal sac, based on the physician's experience. However, there are differences in contouring methods among physicians. Therefore, we aimed to propose a new geometric target margin strategy. Nine pediatric patients with medulloblastoma who underwent proton CSI were enrolled. We measured the following water equivalent lengths for each vertebra in each patient: body surface to the dorsal spinal canal, vertebral limbus, ventral spinal canal and spinous processes. A simulated tCTV (stCTV) was created by assigning geometric margins to the spinal canal using the measurement results such that the vertebral limb and dose distribution coincided with a margin assigned to account for the uncertainty of the proton beam range. The stCTV with a growth factor (correlation between body surface area and age) and tCTV were compared and evaluated. The median values of each index for cervical, thoracic and lumber spine were: the Hausdorff distance, 9.14, 9.84 and 9.77mm; mean distance-to-agreement, 3.26, 2.65 and 2.64mm; Dice coefficient, 0.84, 0.81 and 0.82 and Jaccard coefficient, 0.50, 0.60 and 0.62, respectively. The geometric target margin setting method used in this study was useful for creating an stCTV to ensure consistent and uniform planning.

Spherical objects in dimensions two and three

This paper classifies spherical objects in various geometric settings in dimensions two and three, including both minimal and partial crepant resolutions of Kleinian singularities, as well as arbitrary flopping 3 -fold contractions with only Gorenstein terminal singularities. The main result is much more general: in each such setting, we prove that all objects in the associated null category \mathcal{C} with no negative Ext groups are the image, under the action of an appropriate braid or pure braid group, of some object in the heart of a bounded t-structure. The corollary is that all objects x which admit no negative Exts, and for which \mathrm{Hom}_{\mathcal{C}}(x,x)=\mathbb{C} , are the images of the simples. A variation on this argument goes further and classifies all bounded t-structures on \mathcal{C} . There are multiple geometric, topological and algebraic consequences, primarily to autoequivalences and stability conditions. Our main new technique also extends into representation theory, and we establish that in the derived category of a finite-dimensional algebra which is silting discrete, every object with no negative Ext groups lies in the heart of a bounded t-structure. As a consequence, every semibrick complex can be completed to a simple-minded collection.

Numerical Study on the Effect of Geometrical Changes on the Deformation Behaviour of Recycled Aluminium Alloy AA6061 undergoing High Velocity Impact using Hyperworks-Radioss

Numerical analysis is important to predict material behaviour without require an experimental implementation. This manuscript focuses on a numerical prediction establishment of a direct recycled aluminium alloys AA6061 undergoing Taylor Cylinder Impact test using Johnson-Cook model in HyperWorks Radioss. The numerical setup was first validated against the experimental data at the velocity range of 179 to 212 m/s. Good agreement between simulation and experimental data was obtained within the range that exhibits a mushrooming shape fracture mode. A parametric study was then conducted to study the deformation behaviour of the selected recycled aluminum alloys within the validated range at various geometrical settings. The analysis was made by focusing on the post-impact configuration of the projectile at different impact velocities in terms of residual length, deformed diameter, and the final length-to-diameter ratio. It was found that a broader projectile experienced a less significant reduction in its final length (Lf/Lo goes from 0.87 to 0.9 for projectile diameter 9mm to 34mm) and a smaller increase in the deformed diameter compared to a thinner projectile (Df/Do goes from 1.18 to 1.12 for projectile diameter 9mm to 34mm). It was found that a thinner projectile experienced more diameter expansion than length reduction post impact. In addition, a longer projectile experienced more residual length reduction (Lf/Lo goes from 0.92 to 0.87 for projectile length 2mm to 14mm) and more radial deformation compared to the one with a smaller initial length (Df/Do goes from 1.06 to 1.18 for projectile length 2mm to 14mm). All projectiles showed more significant changes on the deformed diameter compared to the changes in residual length post-impact. The results helped the understanding of a critical aspect of the deformation behaviour of recycled aluminum alloy AA6061 more effectively compared to experimental work implementation.