Euclidean Space Research Articles

TSS-ConvNet for electrical impedance tomography image reconstruction

Objective. The objective of this study was to propose a novel data-driven method for solving ill-posed inverse problems, particularly in certain conditions such as time-difference electrical impedance tomography for detecting the location and size of bubbles inside a pipe. Approach. We introduced a new layer architecture composed of three paths: spatial, spectral, and truncated spectral paths. The spatial path processes information locally, whereas the spectral and truncated spectral paths provide the network with a global receptive field. This unique architecture helps eliminate the ill-posedness and nonlinearity inherent in the inverse problem. The three paths were designed to be interconnected, allowing for an exchange of information on different receptive fields with varied learning abilities. Our network has a bottleneck architecture that enables it to recover signal information from noisy redundant measurements. We named our proposed model truncated spatial-spectral convolutional neural network (TSS-ConvNet). Main results. Our model demonstrated superior accuracy with relatively high resolution on both simulation and experimental data. This indicates that our approach offers significant potential for addressing ill-posed inverse problems in complex conditions effectively and accurately. Significance. The TSS-ConvNet overcomes the receptive field limitation found in most existing models that only utilize local information in Euclidean space. We trained the network on a large dataset covering various configurations with random parameters to ensure generalization over the training samples.

Locating-dominating number of certain infinite families of convex polytopes with applications

A convex hull of finitely many points in the Euclidean space Rd is known as a convex polytope. Graphically, they are planar graphs i.e. embeddable on R2. Minimum dominating sets possess diverse applications in computer science and engineering. Locating-dominating sets are a natural extension of dominating sets. Studying minimizing locating-dominating sets of convex polytopes reveal interesting distance-dominating related topological properties of these geometrical planar graphs. In this paper, exact value of the locating-dominating number is shown for one infinite family of convex polytopes. Moreover, tight upper bounds on γl−d are shown for two more infinite families. Tightness in the upper bounds is shown by employing an updated integer linear programming (ILP) model for the locating-dominating number γl−d of a fixed graph. Results are explained with help of some examples. The second part of the paper solves an open problem in Khan (2023) [28] which asks to find a domination-related parameter which delivers a correlation coefficient of ρ>0.9967 with the total π-electronic energy of lower benzenoid hydrocarbons. We show that the locating-dominating number γl−d delivers such a strong prediction potential. The paper is concluded with putting forward some open problems in this area.

Global Newlander–Nirenberg Theorem for Domains with C2 Boundary

The Newlander–Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the standard complex structure in the complex Euclidean space. In this paper, we consider two natural generalizations of the Newlander–Nirenberg theorem under the presence of a C2 strictly pseudoconvex boundary. When a given formally integrable complex structure X is defined on the closure of a bounded strictly pseudoconvex domain with C2 boundary D⊂Cn, we show the existence of global holomorphic coordinate systems defined on D‾ that transform X into the standard complex structure provided that X is sufficiently close to the standard complex structure. Moreover, we show that such closeness is stable under a small C2 perturbation of ∂D. As a consequence, when a given formally integrable complex structure is defined on a one-sided neighborhood of some point in a C2 real hypersurface M⊂Cn, we prove the existence of local one-sided holomorphic coordinate systems provided that M is strictly pseudoconvex with respect to the given complex structure. We also obtain results when the structures are finite smooth.

ASSOCIATED CURVES ACCORDING TO BISHOP FRAME IN 4-DIMENSIONAL EUCLIDEAN SPACE

Many studies have been doneaccording to different frames of the theory of curves in Euclidean Space. Many scientists have studied frames such as the Frenet frame, Bishop frame, and Adapted frame in this theory. These frames help us in the characterization of curves.In this study, associated curves with the Frenet curve according to the Bishop frame in 4-dimensional Euclidean space are investigated. Direction and rectifying curves of the Frenet curve according to this frame are given.

CONHyperKGE: Using Contrastive Learning in Hyperbolic Space for Knowledge Graph Embedding

The embedding of Knowledge Graphs (KGs) in hyperbolic space has recently received great attention in the field of deep learning because it can provide more accurate and concise representations of hierarchical structures compared to Euclidean spaces and complex spaces. Although hyperbolic space embeddings have shown significant improvements over Euclidean spaces and complex space embeddings in handling the task of KG embedding, they still face challenges related to the uneven distribution and insufficient alignment of high-dimensional sparse data. To address this issue, we propose the CONHyperKGE model, which leverages contrastive learning to optimize the embedding distribution in hyperbolic space. This approach enables better capture of hierarchical structures, improved handling of symmetry, and enhanced treatment of sparse matrices. Our proposed method is evaluated on four standard KG Embedding (KGE) datasets: WN18RR, FB15k-237, Kinship, and UMLS. After extensive experimental verification, our method has improved its performance on all four datasets. Notably, on the low-dimensional Kinship dataset, our method achieves an average Mean Reciprocal Rank (MRR) improvement of 2% over the original method, while on the high-dimensional WN18RR dataset, an average MRR improvement of 1% is observed compared to the original method.

Asymptotic expansions about infinity for solutions of nonlinear differential equations with coherently decaying forcing functions

This paper studies, in fine details, the long-time asymptotic behavior of decaying solutions of a general class of dissipative systems of nonlinear differential equations in complex Euclidean spaces. The forcing functions decay, as time tends to infinity, in a coherent way expressed by combinations of the exponential, power, logarithmic and iterated logarithmic functions. The decay may contain sinusoidal oscillations not only in time but also in the logarithm and iterated logarithm of time. It is proved that the decaying solutions admit corresponding asymptotic expansions, which can be constructed concretely. In the case of the real Euclidean spaces, the real-valued decaying solutions are proved to admit real-valued asymptotic expansions. Our results unite and extend the theory investigated in many previous works.

HydraProt: A New Deep Learning Tool for Fast and Accurate Prediction of Water Molecule Positions for Protein Structures.

Water molecules are integral to the structural stability of proteins and vital for facilitating molecular interactions. However, accurately predicting their precise position around protein structures remains a significant challenge, making it a vibrant research area. In this paper, we introduce HydraProt (deep Hydration of Proteins), a novel methodology for predicting precise positions of water molecule oxygen atoms around protein structures, leveraging two interconnected deep learning architectures: a 3D U-net and a Multi-Layer Perceptron (MLP). Our approach starts by introducing a coarse voxel-based representation of the protein, which allows for rapid sampling of candidate water positions via the 3D U-net. These water positions are then assessed by embedding the water-protein relationship in the Euclidean space by means of an MLP. Finally, a postprocessing step is applied to further refine the MLP predictions. HydraProt surpasses existing state-of-the-art approaches in terms of precision and recall and has been validated on large data sets of protein structures. Notably, our method offers rapid inference runtime and should constitute the method of choice for protein structure studies and drug discovery applications. Our pretrained models, data, and the source code required to reproduce these results are accessible at https://github.com/azamanos/HydraProt.

High accuracy quasi-interpolation using a new class of generalized multiquadrics

A new generalization of multiquadric functions φ(x)=c2d+||x||2d, where x∈Rn, c∈R, d∈N, is presented to increase the accuracy of quasi-interpolation further. With the restriction to Euclidean spaces of odd dimensionality, the generalization can be used to generate a quasi-Lagrange operator that reproduces all polynomials of degree 2d−1. In contrast to the classical multiquadric, the convergence rate of the quasi-interpolation operator can be significantly improved by a factor h2d−n−1, where h>0 represents the grid spacing. Among other things, we compute the generalized Fourier transform of this new multiquadric function. Finally, an infinite regular grid is employed to analyse the properties of the aforementioned generalization in detail. We also present numerical results to demonstrate the advantages of our new multiquadric functions.

Embeddings of Graphs: Tessellate and Decussate Structures

We address the problem of finding a unique graph embedding that best describes a graph’s “topology” i.e., a canonical embedding (spatial graph). This question is of particular interest in the chemistry of materials. Graphs that admit a tiling in 3-dimensional Euclidean space are termed tessellate, those that do not decussate. We give examples of decussate and tessellate graphs that are finite and 3-periodic. We conjecture that a graph has at most one tessellate embedding. We give reasons for considering this the default “topology” of periodic graphs.

Diffusive stability and self-similar decay for the harmonic map heat flow

In this paper we study the harmonic map heat flow on the euclidean space Rd and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space B˙p,∞dp(Rd) where d<p<∞. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form ‖∇u(t)‖L∞(Rd)≤Ct−12.Additionally, under the assumption of a stronger spatial localization of the initial conditions, we show that the temporal decay happens in a self-similar way. We also explain that similar results hold for the biharmonic map heat flow and the semilinear heat equation with a power-type nonlinearity.

Cofinite integral braces and flat manifolds

Braces are ring-like structures equivalent to groups G with a G-module structure such that the identity map is a 1-cocycle. By comparison, crystallographic groups give rise to 1-cocycles into the Euclidean space on which they act. Associated with every torsion-free crystallographic group is a closed connected flat Riemannian manifold. The Calabi construction for these manifolds is reinterpreted and extended to cofinite integral braces, which can be conceived as analogues and refinements of crystallographic groups. It is shown that all three-dimensional Bieberbach groups and most of the two-dimensional crystallographic groups are adjoint groups of cofinite integral braces. For arbitrary cofinite (e.g., finite) braces, the transfer map from the adjoint group into the socle is shown to be a brace morphism, which leads to surprising connections between group-theoretic and brace-theoretic invariants.

Reconstruction of a Ruled Surface in 3-dimensional Euclidean Space

In this paper, we first provide a summary of certain results related to the differential geometry of ruled surfaces. Next, we introduce the signature curve for ruled surfaces in Euclidean three-space. Additionally, we present a straightforward algorithm for the reconstruction of a ruled surface, which is both efficient and entirely local, requiring only the initial motion direction and starting point. Finally, we illustrate the efficiency and accuracy of the algorithm through several examples.&#x0D; Bu makalede, öncelikle regle yüzeylerin diferesiyel geometrisi ile ilgili bazı sonuçların bir özetini sunuyoruz. Daha sonra, Öklid üç uzayında regle yüzeyler için işaret eğrisini tanıtıyoruz. Ek olarak, bir regle yüzeyin yeniden yapılandırılması için hem verimli hem de tamamen yerel olan, yalnızca ilk hareket yönü ve başlangıç noktası gerektiren basit bir algoritma sunuyoruz. Son olarak, algoritmanın verimliliğini ve doğruluğunu birkaç örnekle gösteriyoruz.

Entanglement entropy in scalar quantum electrodynamics

We find the entanglement entropy of a subregion of the vacuum state in scalar quantum electrodynamics, working perturbatively to the two-loop level. Doing so leads us to derive the Maxwell-Proca propagator in conical Euclidean space. The area law of entanglement entropy is recovered in both the massive and massless limits of the theory, as is expected. These results yield the renormalization group flow of entanglement entropy, and we find that loop contributions suppress entanglement entropy. We highlight these results in the light of the renormalization group flow of couplings and correlators, which are increased in scalar quantum electrodynamics, so that the potential tension between the increase in correlations between two points of spacetime and the decrease in entanglement entropy between two regions of spacetime with energy is discussed. We indeed show that the vacuum of a subregion of spacetime purifies with energy in scalar quantum electrodynamics, which is related to the concept of screening. Published by the American Physical Society 2024

Global attention based GNN with Bayesian collaborative learning for glomerular lesion recognition

Background:Glomerular lesions reflect the onset and progression of renal disease. Pathological diagnoses are widely regarded as the definitive method for recognizing these lesions, as the deviations in histopathological structures closely correlate with impairments in renal function. Methods:Deep learning plays a crucial role in streamlining the laborious, challenging, and subjective task of recognizing glomerular lesions by pathologists. However, the current methods treat pathology images as data in regular Euclidean space, limiting their ability to efficiently represent the complex local features and global connections. In response to this challenge, this paper proposes a graph neural network (GNN) that utilizes global attention pooling (GAP) to more effectively extract high-level semantic features from glomerular images. The model incorporates Bayesian collaborative learning (BCL), enhancing node feature fine-tuning and fusion during training. In addition, this paper adds a soft classification head to mitigate the semantic ambiguity associated with a purely hard classification. Results:This paper conducted extensive experiments on four glomerular datasets, comprising a total of 491 whole slide images (WSIs) and 9030 images. The results demonstrate that the proposed model achieves impressive F1 scores of 81.37%, 90.12%, 87.72%, and 98.68% on four private datasets for glomerular lesion recognition. These scores surpass the performance of the other models used for comparison. Furthermore, this paper employed a publicly available BReAst Carcinoma Subtyping (BRACS) dataset with an 85.61% F1 score to further prove the superiority of the proposed model. Conclusion:The proposed model not only facilitates precise recognition of glomerular lesions but also serves as a potent tool for diagnosing kidney diseases effectively. Furthermore, the framework and training methodology of the GNN can be adeptly applied to address various pathology image classification challenges.

Hyperbolic Deep Learning in Computer Vision: A Survey

Deep representation learning is a ubiquitous part of modern computer vision. While Euclidean space has been the de facto standard manifold for learning visual representations, hyperbolic space has recently gained rapid traction for learning in computer vision. Specifically, hyperbolic learning has shown a strong potential to embed hierarchical structures, learn from limited samples, quantify uncertainty, add robustness, limit error severity, and more. In this paper, we provide a categorization and in-depth overview of current literature on hyperbolic learning for computer vision. We research both supervised and unsupervised literature and identify three main research themes in each direction. We outline how hyperbolic learning is performed in all themes and discuss the main research problems that benefit from current advances in hyperbolic learning for computer vision. Moreover, we provide a high-level intuition behind hyperbolic geometry and outline open research questions to further advance research in this direction.

ClockstaRX: Testing Molecular Clock Hypotheses With Genomic Data.

Phylogenomic data provide valuable opportunities for studying evolutionary rates and timescales. These analyses require theoretical and statistical tools based on molecular clocks. We present ClockstaRX, a flexible platform for exploring and testing evolutionary rate signals in phylogenomic data. Here, information about evolutionary rates in branches across gene trees is placed in Euclidean space, allowing data transformation, visualization, and hypothesis testing. ClockstaRX implements formal tests for identifying groups of loci and branches that make a large contribution to patterns of rate variation. This information can then be used to test for drivers of genomic evolutionary rates or to inform models for molecular dating. Drawing on the results of a simulation study, we recommend forms of data exploration and filtering that might be useful prior to molecular-clock analyses.

Hyperbolic Graph Diffusion Model

Diffusion generative models (DMs) have achieved promising results in image and graph generation. However, real-world graphs, such as social networks, molecular graphs, and traffic graphs, generally share non-Euclidean topologies and hidden hierarchies. For example, the degree distributions of graphs are mostly power-law distributions. The current latent diffusion model embeds the hierarchical data in a Euclidean space, which leads to distortions and interferes with modeling the distribution. Instead, hyperbolic space has been found to be more suitable for capturing complex hierarchical structures due to its exponential growth property. In order to simultaneously utilize the data generation capabilities of diffusion models and the ability of hyperbolic embeddings to extract latent hierarchical distributions, we propose a novel graph generation method called, Hyperbolic Graph Diffusion Model (HGDM), which consists of an auto-encoder to encode nodes into successive hyperbolic embeddings, and a DM that operates in the hyperbolic latent space. HGDM captures the crucial graph structure distributions by constructing a hyperbolic potential node space that incorporates edge information. Extensive experiments show that HGDM achieves better performance in generic graph and molecule generation benchmarks, with a 48% improvement in the quality of graph generation with highly hierarchical structures.

Approximate Distance Oracle for Fault-Tolerant Geometric Spanners

In this paper, we present approximate distance and shortest-path oracles for fault-tolerant Euclidean spanners motivated by the routing problem in real-world road networks. A fault-tolerant Euclidean spanner for a set of points in Euclidean space is a graph in which, despite the deletion of small number of any points, the distance between any two points in the damaged graph is an approximation of their Euclidean distance. Given a fault-tolerant Euclidean spanner and a small approximation factor, our data structure allows us to compute an approximate distance between two points in the damaged spanner in constant time when a query involves any two points and a small set of failed points. Additionally, by incorporating additional data structures, we can return a path itself in time almost linear in the length of the returned path. Both data structures require near-linear space.

DiSCO: Diffusion Schrödinger Bridge for Molecular Conformer Optimization

The generation of energetically optimal 3D molecular conformers is crucial in cheminformatics and drug discovery. While deep generative models have been utilized for direct generation in Euclidean space, this approach encounters challenges, including the complexity of navigating a vast search space. Recent generative models that implement simplifications to circumvent these challenges have achieved state-of-the-art results, but this simplified approach unavoidably creates a gap between the generated conformers and the ground-truth conformational landscape. To bridge this gap, we introduce DiSCO: Diffusion Schrödinger Bridge for Molecular Conformer Optimization, a novel diffusion framework that enables direct learning of nonlinear diffusion processes in prior-constrained Euclidean space for the optimization of 3D molecular conformers. Through the incorporation of an SE(3)-equivariant Schrödinger bridge, we establish the roto-translational equivariance of the generated conformers. Our framework is model-agnostic and offers an easily implementable solution for the post hoc optimization of conformers produced by any generation method. Through comprehensive evaluations and analyses, we establish the strengths of our framework, substantiating the application of the Schrödinger bridge for molecular conformer optimization. First, our approach consistently outperforms four baseline approaches, producing conformers with higher diversity and improved quality. Then, we show that the intermediate conformers generated during our diffusion process exhibit valid and chemically meaningful characteristics. We also demonstrate the robustness of our method when starting from conformers of diverse quality, including those unseen during training. Lastly, we show that the precise generation of low-energy conformers via our framework helps in enhancing the downstream prediction of molecular properties. The code is available at https://github.com/Danyeong-Lee/DiSCO.

A special fuzzy star-shaped number space with the sendograph metric

As a natural generalization of fuzzy numbers, fuzzy star-shaped numbers play a crucial role in fuzzy mathematics. Let Y be a non-degenerate compact convex subset of the n-dimensional Euclidean space, and let FSz0(Y) be the family consisting of all fuzzy star-shaped numbers with respect to z0 whose supports contained in Y. Using methods from infinite-dimensional topology, we mainly show that the space FSz0(Y) with the topology induced by the sendograph metric is homeomorphic to the separable Hilbert space ℓ2.