Euclidean Space Research Articles

Extragradient method with feasible inexact projection to variational inequality problem

The variational inequality problem in finite-dimensional Euclidean space is addressed in this paper, and two inexact variants of the extragradient method are proposed to solve it. Instead of computing exact projections on the constraint set, as in previous versions extragradient method, the proposed methods compute feasible inexact projections on the constraint set using a relative error criterion. The first version of the proposed method provided is a counterpart to the classic form of the extragradient method with constant steps. In order to establish its convergence we need to assume that the operator is pseudo-monotone and Lipschitz continuous, as in the standard approach. For the second version, instead of a fixed step size, the method presented finds a suitable step size in each iteration by performing a line search. Like the classical extragradient method, the proposed method does just two projections into the feasible set in each iteration. A full convergence analysis is provided, with no Lipschitz continuity assumption of the operator defining the variational inequality problem.

Dynamic Euclidean bottleneck matching

A fundamental question in computational geometry is for a set of input points in the Euclidean space, that is subject to discrete changes (insertion/deletion of points at each time step), whether it is possible to maintain an exact/approximate minimum weight perfect matching and/or bottleneck matching (a perfect matching that minimizes the length of the longest matched edge), in sublinear update time. In this work, we answer this question in the affirmative for points on a real line and for points in the plane with a bounded geometric spread.For a set P of n points on a line, we show that there exists a dynamic algorithm that maintains an exact bottleneck matching of P and supports insertion and deletion in O(log⁡n) time. Moreover, we show that a modified version of this algorithm maintains an exact minimum-weight perfect matching with O(log⁡n) update (insertion and deletion) time. Next, for a set P of n points in the plane, we show that a (62)-factor approximate bottleneck matching of Pk, at each time step k, can be maintained in O(log⁡Δ) amortized time per insertion and O(log⁡Δ+|Pk|) amortized time per deletion, where Δ is the geometric spread of P (the ratio between the diameter of P and the distance between the closest pair of points in P).

On decay properties for solutions of the Zakharov–Kuznetsov equation

This work mainly focuses on spatial decay properties of solutions to the Zakharov–Kuznetsov equation. For the two- and three-dimensional cases, it was established that if the initial condition u0 verifies 〈σ⋅x〉ru0∈L2(σ⋅x≥κ), for some r∈N, κ∈R, being σ be a suitable non-null vector in the Euclidean space, then the corresponding solution u(t) generated from this initial condition verifies 〈σ⋅x〉ru(t)∈L2σ⋅x>κ−νt, for any ν>0. Additionally, depending on the magnitude of the weight r, it was also deduced some localized gain of regularity. In this regard, we first extend such results to arbitrary dimensions, decay power r>0 not necessarily an integer, and we give a detailed description of the gain of regularity propagated by solutions. The deduction of our results depends on a new class of pseudo-differential operators, which is useful for quantifying decay and smoothness properties on a fractional scale. Secondly, we show that if the initial data u0 has a decay of exponential type on a particular half space, that is, ebσ⋅xu0∈L2(σ⋅x≥κ), then the corresponding solution satisfies ebσ⋅xu(t)∈Hpσ⋅x>κ−t, for all p∈N, and time t≥δ, where δ>0. To our knowledge, this is the first study of such property. As a further consequence, we also obtain well-posedness results in anisotropic weighted Sobolev spaces in arbitrary dimensions.Finally, as a by-product of the techniques considered here, we show that our results are also valid for solutions of the Korteweg–de Vries equation.

Regularity of almost-surely injective projections in Euclidean spaces

Regularity of almost-surely injective projections in Euclidean spaces

Uniquely Realisable Graphs in Analytic Normed Planes

Abstract A framework $(G,p)$ in Euclidean space $\mathbb{E}^{d}$ is globally rigid if it is the unique realisation, up to rigid congruences, of $G$ with the edge lengths of $(G,p)$. Building on key results of Hendrickson [28] and Connelly [14], Jackson and Jordán [29] gave a complete combinatorial characterisation of when a generic framework is global rigidity in $\mathbb{E}^{2}$. We prove an analogous result when the Euclidean norm is replaced by any norm that is analytic on $\mathbb{R}^{2} \setminus \{0\}$. Specifically, we show that a graph $G=(V,E)$ has an open set of globally rigid realisations in a non-Euclidean analytic normed plane if and only if $G$ is 2-connected and $G-e$ contains 2 edge-disjoint spanning trees for all $e\in E$. We also prove that the analogous necessary conditions hold in $d$-dimensional normed spaces.

Analytic Modelling of 2-D Transient Heat Conduction with Heat Source Under Mixed Boundary Constraints by Symplectic Superposition

Abstract Many studies have been conducted on 2-D transient heat conduction, but analytic modelling is still uncommon for the cases with complex boundary constraints due to the mathematical challenge. With an unusual symplectic superposition method, this paper reports new analytic solutions to 2-D isotropic transient heat conduction problems with heat source over a rectangular region under mixed boundary constraints at an edge. With the Laplace transform, the Hamiltonian governing equation is derived. The applicable mathematical treatments, e.g., the variable separation and the symplectic eigenvector expansion in the symplectic space, are implemented for the fundamental solutions whose superposition yields the ultimate solutions. Benchmark results obtained by the present method are tabulated, with verification by the finite element solutions. Instead of the conventional Euclidean space, the present symplectic-space solution framework has the superiority on rigorous derivations without pre-determining solution forms, which may be extended to more issues with the complexity caused by mixed boundary constraints.

Semilinear elliptic equations on manifolds with nonnegative Ricci curvature

In this paper, we prove classification results for solutions to subcritical and critical semilinear elliptic equations with a nonnegative potential on noncompact manifolds with nonnegative Ricci curvature. We show in the subcritical case that all nonnegative solutions vanish identically. Moreover, under some natural assumptions, in the critical case we prove a strong rigidity result, namely we classify all nontrivial solutions showing that they exist only if the potential is constant and the manifold is isometric to the Euclidean space.

New asymptotic stability theory for real order systems and applications

Metzler asymptotic stability of real order systems gives hidden paths of nontrivial solutions to geometric constant solutions as time tends to ∞ in positive and negative orthants of Euclidean space when associated with random initial-time. Suppose that there is a governing law that describes the flow of geometric variables described by a real-order system in Euclidean space and a geometric solution zero vector that satisfies such a system. We introduce the concepts of orthant Metzler asymptotic stability that give rise to the notion of so-called new L-asymptotic stability. We prove that there exist order-dependent conditions that give a new L-asymptotic stability. The main results of this paper are comparison theorems that give L-asymptotic stability, which is essential in finding convergence of non-negative and non-positive solutions to geometric constant solutions. We establish upper and lower theoretical α-order Mittag-Leffler bounds of the Euclidean norm measure of non-trivial solutions to such systems, which characterize the Mittag-Leffler rate of decay. As applications of some theoretical results, we provide three examples that include a feedback method to control cancer in a cancer chemotherapy model to illustrate the effectiveness.

Learning Fixed Points of Recurrent Neural Networks by Reparameterizing the Network Model.

In computational neuroscience, recurrent neural networks are widely used to model neural activity and learning. In many studies, fixed points of recurrent neural networks are used to model neural responses to static or slowly changing stimuli, such as visual cortical responses to static visual stimuli. These applications raise the question of how to train the weights in a recurrent neural network to minimize a loss function evaluated on fixed points. In parallel, training fixed points is a central topic in the study of deep equilibrium models in machine learning. A natural approach is to use gradient descent on the Euclidean space of weights. We show that this approach can lead to poor learning performance due in part to singularities that arise in the loss surface. We use a reparameterization of the recurrent network model to derive two alternative learning rules that produce more robust learning dynamics. We demonstrate that these learning rules avoid singularities and learn more effectively than standard gradient descent. The new learning rules can be interpreted as steepest descent and gradient descent, respectively, under a non-Euclidean metric on the space of recurrent weights. Our results question the common, implicit assumption that learning in the brain should be expected to follow the negative Euclidean gradient of synaptic weights.

Joint Probability Densities on Riemannian Manifolds are Symmetric Tensor Densities

This paper presents the tensor properties of joint probability densities on a Riemannian manifold. Initially, we develop a binary data matrix to record the values of a large number of particles confining in a closed system at a certain time in order to retrieve the joint probability densities of related variables. By introducing the particle-oriented coordinate and the generalized inner product as a multi-linear operation on the basis of this coordinate, we extract the set of joint probabilities and prove them to meet covariant tensor properties on a general Riemannian space of variables. Based on the Taylor expansion of scalar fields in Riemannian manifolds, it has been shown that the symmetrized iterative covariant derivatives of the cumulative probability function defined on Riemannian manifolds also give the set of related joint probability densities equivalent to the aforementioned multi-linear method. We show these covariant tensors reduce to classical ordinary partial derivatives in ordinary Euclidean space with Cartesian coordinates and give the formal definition of joint probabilities by partial derivatives of the cumulative distribution function. The equivalence between the symmetrized covariant derivative and the generalized inner product has been concluded. Some examples of well-known physical tensors clarify that many deterministic physical variables are presented as tensor densities with an interpretation similar to probability densities.

Two-dimensional Hilbert-Huang transform-based thermographic data processing for non-destructive material defect detection

ABSTRACT Previous research has established the Hilbert-Huang transform (HHT) as a powerful tool for processing and analysing one-dimensional time series signals. Its extension into multi-dimensional Euclidean spaces further demonstrates its versatility. In this study, we leverage a two-dimensional (2D) HHT to enhance non-destructive testing, utilising active infrared thermographic data to improve the detection of material defects. The proposed methodology incorporates a 2D HHT, which synergises multidimensional ensemble empirical mode decomposition with the Riesz transform (RT). This combination is adept at uncovering both global (i.e. intrinsic mode functions) and local (i.e. phase angle and spatial frequency) information. In particular, RT is integrated with a Gaussian filter, which allows for a more thorough analysis of the inherent characteristics of defects through the monogenic signal produced by RT. The experimental application of this method on a mosaic sample, intentionally embedded with defects, has yielded promising results. The approach effectively distinguishes critical defect signals from the surroundings, leading to more accurate and clearer results.

Conformal trajectories in three-dimensional space forms

We introduce the notion of conformal trajectories in three-dimensional Riemannian manifolds [Formula: see text]. Given a conformal vector field [Formula: see text], a conformal trajectory of [Formula: see text] is a regular curve [Formula: see text] in [Formula: see text] satisfying [Formula: see text], for some fixed nonzero constant [Formula: see text]. In this paper, we study the conformal trajectories in the space forms [Formula: see text], [Formula: see text] and [Formula: see text]. For (non-Killing) conformal vector fields in [Formula: see text] (respectively, in [Formula: see text]), we prove that conformal trajectories have constant curvature and its torsion is a linear combination of trigonometric (respectively, hyperbolic) functions on the arc-length parameter. In the case of Euclidean space [Formula: see text], we obtain the same result for the radial vector field and characterizing all conformal trajectories.

Pure point diffraction and entropy beyond the Euclidean space

For Euclidean pure point diffractive Delone sets of finite local complexity and with uniform patch frequencies it is well known that the patch counting entropy computed along the closed centred balls is zero. We consider such sets in the setting of σ-compact locally compact Abelian groups and show that the topological entropy of the associated Delone dynamical system is zero. For this we provide a suitable version of the variational principle. We furthermore construct counterexamples, which show that the patch counting entropy of such sets can be non-zero in this context. Other counterexamples will show that the patch counting entropy of such a set cannot be computed along a limit and even be infinite in this setting.

A Property that Characterizes the Enneper Surface and Helix Surfaces

The main goal of this paper is to show that helix surfaces and the Enneper surface are the only surfaces in the 3-dimensional Euclidean space R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^{3}$$\\end{document} whose isogonal lines are generalized helices and pseudo-geodesic lines.

Deep network embedding with dimension selection

Network embedding is a general-purpose machine learning technique that converts network data from non-Euclidean space to Euclidean space, facilitating downstream analyses for the networks. However, existing embedding methods are often optimization-based, with the embedding dimension determined in a heuristic or ad hoc way, which can cause potential bias in downstream statistical inference. Additionally, existing deep embedding methods can suffer from a nonidentifiability issue due to the universal approximation power of deep neural networks. We address these issues within a rigorous statistical framework. We treat the embedding vectors as missing data, reconstruct the network features using a sparse decoder, and simultaneously impute the embedding vectors and train the sparse decoder using an adaptive stochastic gradient Markov chain Monte Carlo (MCMC) algorithm. Under mild conditions, we show that the sparse decoder provides a parsimonious mapping from the embedding space to network features, enabling effective selection of the embedding dimension and overcoming the nonidentifiability issue encountered by existing deep embedding methods. Furthermore, we show that the embedding vectors converge weakly to a desired posterior distribution in the 2-Wasserstein distance, addressing the potential bias issue experienced by existing embedding methods. This work lays down the first theoretical foundation for network embedding within the framework of missing data imputation.

Coupling Hyperbolic GCN with Graph Generation for Spatial Community Detection and Dynamic Evolution Analysis

Spatial community detection is a method that divides geographic spaces into several sub-regions based on spatial interactions, reflecting the regional spatial structure against the background of human mobility. In recent years, spatial community detection has attracted extensive research in the field of geographic information science. However, mining the community structures and their evolutionary patterns from spatial interaction data remains challenging. Most existing methods for spatial community detection rely on representing spatial interaction networks in Euclidean space, which results in significant distortion when modeling spatial interaction networks; since spatial community detection has no ground truth, this results in the detection and evaluation of communities being difficult. Furthermore, most methods usually ignore the dynamics of these spatial interaction networks, resulting in the dynamic evolution of spatial communities not being discussed in depth. Therefore, this study proposes a framework for community detection and evolutionary analysis for spatial interaction networks. Specifically, we construct a spatial interaction network based on network science theory, where geographic units serve as nodes and interaction relationships serve as edges. In order to fully learn the structural features of the spatial interaction network, we introduce a hyperbolic graph convolution module in the community detection phase to learn the spatial and non-spatial attributes of the spatial interaction network, obtain vector representations of the nodes, and optimize them based on a graph generation model to achieve the final community detection results. Considering the dynamics of spatial interactions, we analyze the evolution of the spatial community over time. Finally, using taxi trajectory data as an example, we conduct relevant experiments within the fifth ring road of Beijing. The empirical results validate the community detection capabilities of the proposed method, which can effectively describe the dynamic spatial structure of cities based on human mobility and provide an effective analytical method for urban spatial planning.

A hybrid Euclidean–Lorentzian universe

The limited velocity in a geometry of Lorentzian signature seem to prevent the universe to reach thermodynamic equilibrium as suggested by the cosmic microwave background. Thus it was suggested that the universe which was initially minuscule has reached a more considerable radius in a short duration by a process known as cosmic inflation. However, to drive such a process have led to the suggestion of an ad-hoc scalar field the inflaton, which has no purpose in nature other than driving the cosmic inflation field and then stopping it once the universe reached the right size. In a recent paper it was shown that rapid expansion can occur without postulating an inflation by following Hawking’s suggestion and assuming that primordially the metric of the universe had an Euclidean signature, in which case velocity is not limited and thermalization and rapid expansion are derived without the need to assume an ad-hoc field. However, while in the previous work emphasis was put on the dynamics and physical statistics of the particles in a Euclidean space versus Lorentzian space in which both spaces were given. No mathematical model was given regarding the development of current Lorentzian space-time from the early Euclidean space-time, and how fundamental problems such as the space-time singularity and the homogeneity of the CMB can be solved in the hybrid Euclidean–Lorentzian picture. This lacuna is to be rectified in this paper.

Routing on heavy path WSPD spanners

In this article, we present a construction of a spanner on a set of n points in Rd that we call a heavy path WSPD spanner. The construction is parameterized by a constant s>2 called the separation ratio. The size of the graph is O(sdn) and the spanning ratio is at most 1+2/s+2/(s−1). We also show that this graph has a hop spanning ratio of at most 2lg⁡n+1.We present a memoryless local routing algorithm for heavy path WSPD spanners. The routing algorithm requires a vertex v of the graph to store O(deg⁡(v)log⁡n) bits of information, where deg⁡(v) is the degree of v. The routing ratio is at most 1+4/s+1/(s−1) and at least 1+4/s in the worst case. The number of edges on the routing path is bounded by 2lg⁡n+1.We then show that the heavy path WSPD spanner can be constructed in metric spaces of bounded doubling dimension. These metric spaces have been studied in computational geometry as a generalization of Euclidean space. We show that, in a metric space with doubling dimension λ, the heavy path WSPD spanner has size O(sλn) where s is the separation ratio. The spanning ratio and hop spanning ratio are the same as in the Euclidean case.Finally, we show that the local routing algorithm works in the bounded doubling dimension case. The vertices require the same amount of storage, but the routing ratio becomes at most 1+(2+ττ−1)/s+1/(s−1) in the worst case, where τ≥11 is a constant related to the doubling dimension.

HyperCARS: Using Hyperbolic Embeddings for Generating Hierarchical Contextual Situations in Context-Aware Recommender Systems

Contextual situations, such as having dinner at a restaurant on Friday with the spouse, became a useful mechanism to represent context in context-aware recommender systems (CARS). Prior research has shown important advantages of using latent embedding representation approaches to model contextual information in the Euclidean space leading to better recommendations. However, these traditional approaches have major challenges with the construction of proper embeddings of hierarchical structures of contextual information, as well as with interpretations of the obtained representations. To address these problems, we propose the HyperCARS method that models hierarchical contextual situations in the latent hyperbolic space. HyperCARS combines hyperbolic embeddings with hierarchical clustering to construct contextual situations, which allows loose coupling of the contextual modeling component with recommendation algorithms and, therefore, provides flexibility to use a broad range of previously developed recommendation algorithms. We demonstrate empirically that HyperCARS better captures and interprets hierarchical contextual representations, leading to better context-aware recommendations. Because hyperbolic embeddings can also be used in many other applications besides CARS, we also propose the latent embeddings representation framework that systematically classifies prior work on embeddings and identifies novel research streams for hyperbolic embeddings across information systems applications.

Multiplicative rectifying submanifolds of multiplicative Euclidean space

In this paper, we initiate the study of the multiplicative Euclidean submanifolds, exhibiting the multiplicative counterparts of the Gauss–Weingarten formulas. Some particular types such as multiplicative conic, spherical, and totally geodesic submanifolds are analyzed. In relation to such multiplicative submanifolds, we introduce multiplicative rectifying submanifolds and characterize by means of multiplicative concurrent vector fields. The multiplicative rectifying hypersurfaces are also obtained.