Euclidean Space Research Articles

On the existence of geodesics connecting Lagrangian graphs in Euclidean space

On the existence of geodesics connecting Lagrangian graphs in Euclidean space

Graph Kernel Neural Networks.

The convolution operator at the core of many modern neural architectures can effectively be seen as performing a dot product between an input matrix and a filter. While this is readily applicable to data such as images, which can be represented as regular grids in the Euclidean space, extending the convolution operator to work on graphs proves more challenging, due to their irregular structure. In this article, we propose to use graph kernels, i.e., kernel functions that compute an inner product on graphs, to extend the standard convolution operator to the graph domain. This allows us to define an entirely structural model that does not require computing the embedding of the input graph. Our architecture allows to plug-in any type of graph kernels and has the added benefit of providing some interpretability in terms of the structural masks that are learned during the training process, similar to what happens for convolutional masks in traditional convolutional neural networks (CNNs). We perform an extensive ablation study to investigate the model hyperparameters' impact and show that our model achieves competitive performance on standard graph classification and regression datasets.

Towards Metric DBSCAN: Exact, Approximate, and Streaming Algorithms

DBSCAN is a popular density-based clustering algorithm that has many different applications in practice. However, the running time of DBSCAN in high-dimensional space or general metric space (\em e.g., clustering a set of texts by using edit distance) can be as large as quadratic in the input size. Moreover, most of existing accelerating techniques for DBSCAN are only available for low-dimensional Euclidean space. In this paper, we study the DBSCAN problem under the assumption that the inliers (the core points and border points) have a low intrinsic dimension (which is a realistic assumption for many high-dimensional applications), where the outliers can locate anywhere in the space without any assumption. First, we propose a k-center clustering based algorithm that can reduce the time-consuming labeling and merging tasks of DBSCAN to be linear. Further, we propose a linear time approximate DBSCAN algorithm, where the key idea is building a novel small-size summary for the core points. Also, our algorithm can be efficiently implemented for streaming data and the required memory is independent of the input size. Finally, we conduct our experiments and compare our algorithms with several popular DBSCAN algorithms. The experimental results suggest that our proposed approach can significantly reduce the computational complexity in practice.

Schatten properties of Calderón–Zygmund singular integral commutator on stratified Lie groups

We provide full characterization of the Schatten properties of [Mb,T], the commutator of Calderón–Zygmund singular integral T with symbol b(Mbf(x):=b(x)f(x)) on stratified Lie groups G. We show that, when p is larger than the homogeneous dimension Q of G, the Schatten Lp norm of the commutator is equivalent to the Besov semi-norm BpQ/p of the function b; but when p≤Q, the commutator belongs to Lp if and only if b is a constant. For the endpoint case at the critical index p=Q, we further show that the Schatten LQ,∞ norm of the commutator is equivalent to the Sobolev norm W˙1,Q of b. Our method at the endpoint case differs from existing methods of Fourier transforms or trace formula for Euclidean spaces or Heisenberg groups, respectively, and hence can be applied to various settings beyond.

Graph convolutional network for lithological classification and mapping using stream sediment geochemical data and geophysical data

Lithological mapping plays a vital role in acquiring a comprehensive understanding of subsurface rocks and geological structures. Traditional lithological mapping relies heavily on manual field surveys, with the accuracy of the mapping greatly constrained by the expertise of mappers, and thus this process is commonly slow and costly. In the present study, the graph convolutional network (GCN) method was proposed for efficient and automatic lithological mapping, using various geological survey data (e.g. geochemical and geophysical data). The GCN is not limited by the spatial distribution of sampling data, and hence it can be directly applied to the integrated resampling data that do not adhere to the regular spatial distribution patterns of Euclidean space. To evaluate the potential application of the GCN method in lithological classification and mapping, a case study was conducted in the Maqin region (China). Geochemical and geophysical data were utilized as the basis for lithological classification, as they can provide valuable information on geochemical compositions and physical properties of geological bodies, respectively. The study process was primarily divided into three parts. First, basic pre-processing was conducted on the geochemical data and the geophysical data, including centred log-ratio transformation and resampling. Subsequently, the resampled points were divided into training and testing sets in the proportion of 5.5:1. In the training set, 20% of the points are randomly selected as the validation set. Ultimately, using each resampled point as a node and the integrated geochemical and geophysical data as features, a GCN model and an undirected graph were constructed based on an adjacency distance of 1 km. The graph was employed by the constructed GCN model for lithological classification. The results reveal that the majority of lithological units can be accurately classified with an overall accuracy of 86%, indicating that the GCN model exhibits good applicability in lithological classification.

The Separability Problem in Molecular Quantum Systems: Information-Theoretic Framework for Atoms in Molecules.

Even though molecules are fundamentally quantum entities, the concept of a molecule retains certain classical attributes concerning its constituents. This includes the empirical separability of a molecule into its three-dimensional, rigid structure in Euclidean space, a framework often obtained through experimental methods like X-Ray crystallography. In this work, we delve into the mathematical implications of partitioning a molecule into its constituent parts using the widely recognized Atoms-In-Molecules (AIM) schemes, aiming to establish their validity within the framework of Information Theory concepts. We have uncovered information-theoretical justifications for employing some of the most prevalent AIM schemes in the field of Chemistry, including Hirshfeld (stockholder partitioning), Bader's (topological dissection), and the quantum approach (Hilbert's space definition). In the first approach we have applied the generalized principle of minimum relative entropy derived from the Sharma-Mittal two-parameter functional, avoiding the need for an arbitrary selection of reference promolecular atoms. Within the ambit of topological-information partitioning, we have demonstrated that the Fisher information of Bader's atoms conform to a comprehensive theory based on the Principle of Extreme Physical Information avoiding the need of employing the Schwinger's principle, which has been proven to be problematic. For the quantum approach we have presented information-theoretic justifications for conducting Löwdin symmetric transformations on the density matrix to form atomic Hilbert spaces generating orthonormal atomic orbitals with maximum occupancy for a given wavefunction.

Weighted Alexandrov–Fenchel type inequalities for hypersurfaces in Rn$\mathbb {R}^n$

AbstractIn this paper, we prove the following geometric inequalities in the Euclidean space , which are weighted Alexandrov–Fenchel type inequalities, provided that is a star‐shaped and ‐convex hypersurface. Equality holds if and only if is a coordinate sphere in . As an application, by letting in the above inequality, we obtain a lower bound for the outer radius in terms of the curvature integrals for star‐shaped and ‐convex hypersurfaces.

Some Aspects of Maxwell’s Equations, Klein-Gordon Equations, and Heat and Mass Transfer Equations in an n-Dimensional Maximally Symmetric Space-Time from the Classical and Quantum Mechanical Standpoints

This manuscript examines Maxwell’s equations, Klein-Gordon equations, and heat and mass transfer equations in n-dimensional maximally symmetric space-time. It investigates these equations in spherical and hyperbolic spaces embedded in higher-dimensional Euclidean and Minkowski spaces. The study focuses on the implications of these geometries and symmetries on the behaviour of the equations, highlighting how specific transformations and parametrizations impact their solutions. The findings reveal the underlying connections between geometric symmetries and physical laws, providing insights into their possible applications in theoretical physics. We touch upon both classical and quantum mechanical aspects of density and velocity evolutions with time in the universe. Quantum mechanical aspects of single and two-particle state evolution and statistical moments of the matter four-current are derived from the quantum Boltzmann equation and Feynman’s path integral method for fields applied to gravity interacting with electrons and positrons.

Outer billiards in the spaces of oriented geodesics of the three‐dimensional space forms

AbstractLet be the three‐dimensional space form of constant curvature , that is, Euclidean space , the sphere , or hyperbolic space . Let be a smooth, closed, strictly convex surface in . We define an outer billiard map on the four‐dimensional space of oriented complete geodesics of , for which the billiard table is the subset of consisting of all oriented geodesics not intersecting . We show that is a diffeomorphism when is quadratically convex. For , has a Kähler structure associated with the Killing form of . We prove that is a symplectomorphism with respect to its fundamental form and that can be obtained as an analogue to the construction of Tabachnikov of the outer billiard in defined in terms of the standard symplectic structure. We show that does not preserve the fundamental symplectic form on associated with the cross product on , for . We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points.

Metrics for Domain Shift Characterization: Comparisons and New Directions

Domain adaptation is an important area of research, as it aims to remedy the effects of the domain shift due to differences in the distribution between the source domain used for training and the target domain where prediction takes place. However, methods for characterizing the domain shift across datasets are lacking. In this work, we propose a domain shift metric called SpOT, which stands for spherical optimal transport, by operating on the spherical manifold. We realize our approach with a spherical network, used to obtain features, and an orthogonal projection loss, used to impose orthogonality in the feature space. The resulting spherical features have better inter-class separation and lower intra-class variation compared to features in Euclidean space. This type of feature clustering makes each domain representation more compact and more suitable for further analysis. The domain shift between the datasets is calculated using the optimal transport on the spherical features, which has a sound theoretical basis. Our results are further supported by experiments that show the correlation of SpOT with a new gain of transfer measure across domain adaptation datasets.

Trajectory optimization of unmanned aerial vehicles in the electromagnetic environment

AbstractWe consider a type of routing problems common in defence and security, in which we control a fleet of unmanned aerial vehicles (UAVs) that have to reach one or more target locations without being detected by an adversary. Detection can be carried out by a variety of sensors (radio receivers, cameras, personnel, etc) placed by the adversary around the target sites. We model the act of detecting a UAV from first principles by noting that sensors work by monitoring frequencies in the electromagnetic spectrum for signals or noise emitted. By this, we are able to provide a flexible and versatile nonlinear optimisation framework in which the problem is modeled as a novel trajectory optimisation problem with paths of the UAVs as continuous arcs in an Euclidean space. The flexibility of our approach is exhibited by the fact that we can easily consider various relevant objectives, among them minimising the overall probability of detection and maximising the location error that the adversary experiences when trying to locate our UAVs. Our model is also versatile enough to consider the act of jamming, in which one or more of our UAVs intentionally send out signals to interfere with the operations of the adversary’s sensors. Numerical results show the flexibility of our framework, and that we can solve realistic instances of this problem type.

The Brownian transport map

Contraction properties of transport maps between probability measures play an important role in the theory of functional inequalities. The actual construction of such maps, however, is a non-trivial task and, so far, relies mostly on the theory of optimal transport. In this work, we take advantage of the infinite-dimensional nature of the Gaussian measure and construct a new transport map, based on the Föllmer process, which pushes forward the Wiener measure onto probability measures on Euclidean spaces. Utilizing the tools of the Malliavin and stochastic calculus in Wiener space, we show that this Brownian transport map is a contraction in various settings where the analogous questions for optimal transport maps are open. The contraction properties of the Brownian transport map enable us to prove functional inequalities in Euclidean spaces, which are either completely new or improve on current results. Further and related applications of our contraction results are the existence of Stein kernels with desirable properties (which lead to new central limit theorems), as well as new insights into the Kannan–Lovász–Simonovits conjecture. We go beyond the Euclidean setting and address the problem of contractions on the Wiener space itself. We show that optimal transport maps and causal optimal transport maps (which are related to Brownian transport maps) between the Wiener measure and other target measures on Wiener space exhibit very different behaviors.

An optimal control problem with state constraints in a spatio-temporal economic growth model on networks

We introduce a spatial economic growth model where space is described as a network of interconnected geographic locations and we study a corresponding finite-dimensional optimal control problem on a graph with state constraints. Economic growth models on networks are motivated by the nature of spatial economic data, which naturally possess a graph-like structure: this fact makes these models well-suited for numerical implementation and calibration. The network setting is different from the one adopted in the related literature, where space is modeled as a subset of a Euclidean space, which gives rise to infinite dimensional optimal control problems. After introducing the model and the related control problem, we prove existence and uniqueness of an optimal control and a regularity result for the value function, which sets up the basis for a deeper study of the optimal strategies. Then, we focus on specific cases where it is possible to find, under suitable assumptions, an explicit solution of the control problem. Finally, we discuss the cases of networks of two and three geographic locations.

Using Large Language Models to Understand Suicidality in a Social Media-Based Taxonomy of Mental Health Disorders: Linguistic Analysis of Reddit Posts.

Rates of suicide have increased by over 35% since 1999. Despite concerted efforts, our ability to predict, explain, or treat suicide risk has not significantly improved over the past 50 years. The aim of this study was to use large language models to understand natural language use during public web-based discussions (on Reddit) around topics related to suicidality. We used large language model-based sentence embedding to extract the latent linguistic dimensions of user postings derived from several mental health-related subreddits, with a focus on suicidality. We then applied dimensionality reduction to these sentence embeddings, allowing them to be summarized and visualized in a lower-dimensional Euclidean space for further downstream analyses. We analyzed 2.9 million posts extracted from 30 subreddits, including r/SuicideWatch, between October 1 and December 31, 2022, and the same period in 2010. Our results showed that, in line with existing theories of suicide, posters in the suicidality community (r/SuicideWatch) predominantly wrote about feelings of disconnection, burdensomeness, hopeless, desperation, resignation, and trauma. Further, we identified distinct latent linguistic dimensions (well-being, seeking support, and severity of distress) among all mental health subreddits, and many of the resulting subreddit clusters were in line with a statistically driven diagnostic classification system-namely, the Hierarchical Taxonomy of Psychopathology (HiTOP)-by mapping onto the proposed superspectra. Overall, our findings provide data-driven support for several language-based theories of suicide, as well as dimensional classification systems for mental health disorders. Ultimately, this novel combination of natural language processing techniques can assist researchers in gaining deeper insights about emotions and experiences shared on the web and may aid in the validation and refutation of different mental health theories.

A distributed geometric rewiring model

We propose a distributed rewiring model which starts with a planar graph embedded into the Euclidean space and then behaves as a distributed system, where each node is provided with a set of dynamic links. The proposed rewiring evolves through cycles, where nodes explore the network to identify possible shortcuts and rewire their dynamic links. The rewiring decisions are subject to Euclidean and geodesic distance constrains. The emerging networks were assessed through topological and robustness analyses. We found that the networks display a variety of characteristics observed in complex networks encompassing phenomena such as preferential attachment, the distinctive traits of small-world networks, the presence of community structures, and robustness against degradation process. We consider that our proposal can be applied in the design of those self-managed systems in which there is a limitation on communication resources that can be represented by the Euclidean distance and, however, the components themselves can deploy strategies to optimize the transport of information and develop tolerance before contingencies.

Inhomogeneous long-range percolation in the weak decay regime

We study a general class of percolation models in Euclidean space including long-range percolation, scale-free percolation, the weight-dependent random connection model and several other previously investigated models. Our focus is on the weak decay regime, in which inter-cluster long-range connection probabilities fall off polynomially with small exponent, and for which we establish several structural properties. Chief among them are the continuity of the bond percolation function and the transience of infinite clusters.

Maximal directional derivatives in Laakso space

We investigate the connection between maximal directional derivatives and differentiability for Lipschitz functions defined on Laakso space. We show that maximality of a directional derivative for a Lipschitz function implies differentiability only for a [Formula: see text]-porous set of points. On the other hand, the distance to a fixed point is differentiable everywhere except for a [Formula: see text]-porous set of points. This behavior is completely different to the previously studied settings of Euclidean spaces, Carnot groups and Banach spaces. Hence, the techniques used in these spaces do not generalize to metric measure spaces.

Homology and K-theory of dynamical systems IV. Further structural results on groupoid homology

Abstract We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk [A homology theory for étale groupoids. J. Reine Angew. Math.521 (2000), 25–46], with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-duality type result for principal groupoids whose orbits are copies of an Euclidean space. We conclude with a few example computations for systems associated to nilpotent groups such as self-similar actions, and we generalize previous homological calculations by Burke and Putnam for systems which are analogues of solenoids arising from algebraic numbers. For the latter systems, we prove the HK conjecture, even when the resulting groupoid is not ample.

On the Properties of Quasi-Banach Function Spaces

In this paper we explore some basic properties of quasi-Banach function spaces which are important in applications. Namely, we show that they possess a generalised version of Riesz–Fischer property, that embeddings between them are always continuous, and that the dilation operator is bounded on them. We also provide a characterisation of separability for quasi-Banach function spaces over the Euclidean space. Furthermore, we extend the classical Riesz–Fischer theorem to the context of quasinormed spaces and, as a consequence, obtain an alternative proof of completeness of quasi-Banach function spaces.

Constructible Sandwich Cut

In mathematical measure theory, the “Ham-Sandwich” theorem states that any n objects in an n-dimensional Euclidean space can be simultaneously divided in half with a single cut by an (n-1)-dimensional hyperplane. While it guarantees its existence, the theorem does not provide a way of finding this halving hyperplane, as it is only an existence result. In this paper, we look at the problem in dimension 2, more in the style of Euclid and the antique Greeks, that is from a constructible point of view, with straight edge and compass. For two arbitrary regions in the plane, there is certainly no hope for constructing the halving line by straight edge and compass. At the opposite end, for two circles the problem is very easy, as the halving line is simply the line which passes through the centers of both circles. It turns out that for the case of two equilateral triangles the problem is already interesting and challenging from a constructible point of view, and this is the main goal of our paper. We will investigate the “deltoid” of a triangle, that is, three hyperbolas which form an “envelope” of all halving lines of a given triangle, and how to construct halving lines with certain given properties.