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Depth completion aims to reconstruct dense depth maps from sparse LiDAR measurements guided by RGB images. Although BPNet enhanced depth structure perception through a bilateral propagation module and achieved state-of-the-art performance at the time, there is still room for improvement in leveraging 3D geometric priors and adaptively fusing heterogeneous modalities. To this end, we proposed GAC-Net, a Geometric–Attention Fusion Network that enhances geometric representation and cross-modal fusion. Specifically, we designed a dual-branch PointNet++-S encoder, where two PointNet++ modules with different receptive fields are applied to extract scale-aware geometric features from the back-projected sparse point cloud. These features are then fused using a channel attention mechanism to form a robust global 3D representation. A Channel Attention-Based Feature Fusion Module (CAFFM) was further introduced to adaptively integrate this geometric prior with RGB and depth features. Experiments on the KITTI depth completion benchmark demonstrated the effectiveness of GAC-Net, achieving an RMSE of 680.82 mm, ranking first among all peer-reviewed methods at the time of submission.

To address the challenges of accurately segmenting irregular building boundaries in complex urban environments faced by existing remote sensing change detection methods, this paper proposes a building change detection network based on multilevel geometric representation optimization using frame fields called BuildingCDNet. The proposed method employs a multi-scale feature aggregation encoder–decoder architecture, leveraging contextual information to capture the characteristics of buildings of varying sizes in the imagery. Cross-attention mechanisms are incorporated to enhance the feature correlations between the change pairs. Additionally, the frame field is introduced into the network to model the complex geometric structure of the building target. By learning the local orientation information of the building structure, the frame field can effectively capture the geometric features of complex building features. During the training process, a multi-task learning strategy is used to align the predicted frame field with the real building outline, while learning the overall segmentation, edge outline, and corner point features of the building. This improves the accuracy of the building polygon representation. Furthermore, a discriminative loss function is constructed through multi-task learning to optimize the polygonal structured information of the building targets. The proposed method achieves state-of-the-art results on two commonly used datasets.

Let$p$and$\\ell$be distinct primes, and let$\\overline{\\unicode[STIX]{x1D70C}}$be an orthogonal or symplectic representation of the absolute Galois group of an$\\ell$-adic field over a finite field of characteristic$p$. We define and study a liftable deformation condition of lifts of$\\overline{\\unicode[STIX]{x1D70C}}$‘ramified no worse than$\\overline{\\unicode[STIX]{x1D70C}}$’, generalizing the minimally ramified deformation condition for$\\operatorname{GL}_{n}$studied in Clozel  et al. [ Automorphy for some$l$- adic lifts of automorphic mod$l$Galois representations , Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181; MR 2470687 (2010j:11082)]. The key insight is to restrict to deformations where an associated unipotent element does not change type when deforming. This requires an understanding of nilpotent orbits and centralizers of nilpotent elements in the relative situation, not just over fields.

A new geometric representation of qubit and qutrit states based on probability simplexes is used to describe the separability and entanglement properties of density matrices of two qubits. The Peres–Horodecki positive partial transpose (ppt) -criterion and the concurrence inequalities are formulated as the conditions that the introduced probability distributions must satisfy to present entanglement. A four-level system, where one or two states are inaccessible, is considered as an example of applying the elaborated probability approach in an explicit form. The areas of three Triadas of Malevich’s squares for entangled states of two qubits are defined through the qutrit state, and the critical values of the sum of their areas are calculated. We always find an interval for the sum of the square areas, which provides the possibility for an experimental checkup of the entanglement of the system in terms of the probabilities.

Abstract Airfoil shape optimization is crucial for improving aerodynamic performance in advanced aircraft designs. Given the extensive functional evaluations required for optimization, surrogate modeling is widely used to alleviate computational burden. However, greater flexibility in airfoil parameterization often requires a larger number of design variables, leading to the challenge known as the curse of dimensionality in surrogate modeling. In recent years, generative models such as generative adversarial networks and variational autoencoders have shown potential to represent large design spaces with compact design variables. However, these models still exhibit limited feasibility and intuitiveness due to their high model capacity, which in turn degrades the efficiency of design optimization. To address this issue, we have developed a novel airfoil parameterization method using a variational autoencoder. The proposed method improves feasibility by using architecture modeling to separate the generation of thickness and camber distributions, resulting in smooth and nonintersecting airfoils. It also improves intuitiveness by using a physics loss function that aligns latent dimensions with geometric features of the airfoils. Notably, extensive comparative analyses validate the effectiveness of our method in terms of flexibility, parsimony, feasibility, and intuitiveness, leading to increased efficiency in aerodynamic design optimization. Graphical Abstract Graphical Abstract

We study and classify a class of representations (called generalized geometric representations) of a Coxeter group of finite rank. These representations can be viewed as a natural generalization of the geometric representation. The classification is achieved by using characters of the integral homology group of certain graphs closely related to the Coxeter graph. On this basis, we also provide an explicit description of those representations on which the defining generators of the Coxeter group act by reflections.

The complex three-dimensional (3D) geometry of the native tricuspid aortic valve (AV) is represented by select parametric curves allowing for a general construction and representation of the 3D-AV structure including the cusps, commissures and sinuses. The proposed general mathematical description is performed by using three independent parametric curves, two for the cusp and one for the sinuses. These curves are used to generate different surfaces that form the structure of the AV. Additional dependent curves are also generated and utilized in this process, such as the joint curve between the cusps and the sinuses. The model's feasibility to generate patient-specific parametric geometry is examined against 3D-transesophageal echocardiogram (3D-TEE) measurements from a non-pathological AV. Computational finite-element (FE) mesh can then be easily constructed from these surfaces. Examples are given for constructing several 3D-AV geometries by estimating the needed parameters from echocardiographic measurements. The average distance (error) between the calculated geometry and the 3D-TEE measurements was only 0.78±0.63mm. The proposed general 3D parametric method is very effective in quantitatively representing a wide range of native AV structures, with and without pathology. It can also facilitate a methodical quantitative investigation over the effect of pathology and mechanical loading on these major AV parameters.

Abstract Albert Einstein's special theory of relativity is encompassed under his general theory of relativity in everything but length contraction and time dilation. In the special theory of relativity, these phenomena are based on relative motion. The general theory of relativity is based on spacetime. This paper corrects that disconnect and shows that spacetime can explain length contraction and time dilation. In this paper, I mathematically step the spacetime interval to the Lorentz transformation. I do this by introducing the spacetime speed triangle which is a geometric representation of spacetime created from the spacetime interval. Not only does the spacetime speed triangle clean up the relationship between the special and general theories but it also brings a connection between spacetime and quantum mechanics. This connection is the spacetime speed triangle is a similar triangle to the energy-momentum relation triangle. This similarity with the matter waves equations brings in other quantum mechanics variables to spacetime.

In this paper, we study asymptotic properties of nonlinear support vector machines (SVM) in high-dimension, low-sample-size settings. We propose a bias-corrected SVM (BC-SVM) which is robust against imbalanced data in a general framework. In particular, we investigate asymptotic properties of the BC-SVM having the Gaussian kernel and compare them with the ones having the linear kernel. We show that the performance of the BC-SVM is influenced by the scale parameter involved in the Gaussian kernel. We discuss a choice of the scale parameter yielding a high performance and examine the validity of the choice by numerical simulations and actual data analyses.

This study proposes a new method to generate a three-dimensional (3D) geometric representation of an indoor environment by refining and processing an indoor point cloud data (PCD) captured through backpack laser scanners. The proposed algorithm comprises two parts to generate the 3D geometric representation: data refinement and data processing. In the refinement section, the inputted indoor PCD are roughly segmented by applying random sample consensus (RANSAC) to raw data based on an estimated normal vector. Next, the 3D geometric representation is generated by calculating and separating tangent points on segmented PCD. This study proposes a robust algorithm that utilizes the topological feature of the indoor PCD created by a hierarchical data process. The algorithm minimizes the size and the uncertainty of raw PCD caused by the absence of a global navigation satellite system and equipment errors. The result of this study shows that the indoor environment can be converted into 3D geometric representation by applying the proposed algorithm to the indoor PCD.