Euclidean Space Research Articles

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

AbstractWe 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.

Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3

Abstract An osculating curve is a type of curve in space that holds significance in the study of differential geometry. In this article, we investigate certain geometric invariants of osculating curves on smooth and regularly immersed surfaces under conformal transformations in Euclidean space R 3 {{\mathbb{R}}}^{3} . The primary objective of this article is to explore conditions sufficient for the conformal invariance of the osculating curve under both conformal transformations and isometries. We also compute the tangential and normal components of the osculating curves, demonstrating that they remain invariant under the isometry of the surfaces in R 3 {{\mathbb{R}}}^{3} .

Gauge-Optimal Approximate Learning for Small Data Classification.

Small data learning problems are characterized by a significant discrepancy between the limited number of response variable observations and the large feature space dimension. In this setting, the common learning tools struggle to identify the features important for the classification task from those that bear no relevant information and cannot derive an appropriate learning rule that allows discriminating among different classes. As a potential solution to this problem, here we exploit the idea of reducing and rotating the feature space in a lower-dimensional gauge and propose the gauge-optimal approximate learning (GOAL) algorithm, which provides an analytically tractable joint solution to the dimension reduction, feature segmentation, and classification problems for small data learning problems. We prove that the optimal solution of the GOAL algorithm consists in piecewise-linear functions in the Euclidean space and that it can be approximated through a monotonically convergent algorithm that presents-under the assumption of a discrete segmentation of the feature space-a closed-form solution for each optimization substep and an overall linear iteration cost scaling. The GOAL algorithm has been compared to other state-of-the-art machine learning tools on both synthetic data and challenging real-world applications from climate science and bioinformatics (i.e., prediction of the El Niño Southern Oscillation and inference of epigenetically induced gene-activity networks from limited experimental data). The experimental results show that the proposed algorithm outperforms the reported best competitors for these problems in both learning performance and computational cost.

Multi-Input data ASsembly for joint Analysis (MIASA): A framework for the joint analysis of disjoint sets of variables.

The joint analysis of two datasets [Formula: see text] and [Formula: see text] that describe the same phenomena (e.g. the cellular state), but measure disjoint sets of variables (e.g. mRNA vs. protein levels) is currently challenging. Traditional methods typically analyze single interaction patterns such as variance or covariance. However, problem-tailored external knowledge may contain multiple different information about the interaction between the measured variables. We introduce MIASA, a holistic framework for the joint analysis of multiple different variables. It consists of assembling multiple different information such as similarity vs. association, expressed in terms of interaction-scores or distances, for subsequent clustering/classification. In addition, our framework includes a novel qualitative Euclidean embedding method (qEE-Transition) which enables using Euclidean-distance/vector-based clustering/classification methods on datasets that have a non-Euclidean-based interaction structure. As an alternative to conventional optimization-based multidimensional scaling methods which are prone to uncertainties, our qEE-Transition generates a new vector representation for each element of the dataset union [Formula: see text] in a common Euclidean space while strictly preserving the original ordering of the assembled interaction-distances. To demonstrate our work, we applied the framework to three types of simulated datasets: samples from families of distributions, samples from correlated random variables, and time-courses of statistical moments for three different types of stochastic two-gene interaction models. We then compared different clustering methods with vs. without the qEE-Transition. For all examples, we found that the qEE-Transition followed by Ward clustering had superior performance compared to non-agglomerative clustering methods but had a varied performance against ultrametric-based agglomerative methods. We also tested the qEE-Transition followed by supervised and unsupervised machine learning methods and found promising results, however, more work is needed for optimal parametrization of these methods. As a future perspective, our framework points to the importance of more developments and validation of distance-distribution models aiming to capture multiple-complex interactions between different variables.

Three Cube Packing for All Dimensions

Let Vn(d) denote the least number, such that every collection of n d-cubes with total volume 1 in d-dimensional (Euclidean) space can be packed parallelly into some d-box of volume Vn(d). We show that V3(d)=r1−dd if d≥11 and V3(d)=1r+1rd+1r−rd+1 if 2≤d≤10, where r is the only solution of the equation 2(d−1)kd+dkd−1=1 on 22,1 and (k+1)d(1−k)d−1dk2+d+k−1=kddkd+1+dkd+kd+1 on 22,1, respectively. The maximum volume is achieved by hypercubes with edges x, y, z, such that x=2rd+1−1/d, y=z=rx if d≥11, and x=rd+(1r−r)d+1−1/d, y=rx, z=(1r−r)x if 2≤d≤10. We also proved that only for dimensions less than 11 are there two different maximum packings, and for all dimensions greater than 10, the maximum packing has the same two smallest cubes.

Weighted topological pressure revisited

Abstract Feng and Huang [Variational principle for weighted topological pressure. J. Math. Pures Appl. (9)106 (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems and established its variational principle. Tsukamoto [New approach to weighted topological entropy and pressure. Ergod. Th. & Dynam. Sys.43 (2023), 1004–1034] redefined those invariants quite differently for the simplest case and showed via the variational principle that the two definitions coincide. We generalize Tsukamoto’s approach, redefine the weighted topological entropy and pressure for higher dimensions, and prove the variational principle. Our result allows for an elementary calculation of the Hausdorff dimension of affine-invariant sets such as self-affine sponges and certain sofic sets that reside in Euclidean space of arbitrary dimension.

Representing Infinite Periodic Hyperbolic Delaunay Triangulations Using Finitely Many Dirichlet Domains

AbstractThe Delaunay triangulation of a set of points P on a hyperbolic surface is the projection of the Delaunay triangulation of the set $$\widetilde{P}$$ P ~ of lifted points in the hyperbolic plane. Since $$\widetilde{P}$$ P ~ is infinite, the algorithms to compute Delaunay triangulations in the plane do not generalize naturally. Using a Dirichlet domain, we exhibit a finite set of points that captures the full triangulation. We prove that an edge of a Delaunay triangulation has a combinatorial length (a notion we define in the paper) smaller than $$12g-6$$ 12 g - 6 with respect to a Dirichlet domain. To achieve this, we introduce new tools, of intrinsic interest, that capture the properties of length-minimizing curves in the context of closed curves. We then use these to derive structural results on Delaunay triangulations and exhibit certain distance minimizing properties of both the edges of a Delaunay triangulation and of a Dirichlet domain. The bounds produced in this paper depend only on the topology of the surface. They provide mathematical foundations for hyperbolic analogs of the algorithms to compute periodic Delaunay triangulations in Euclidean space.

Correlation functions of scalar field theories from homotopy algebras

We present expressions for correlation functions of scalar field theories in perturbation theory using quantum A∞ algebras. Our expressions are highly explicit and can be used for theories both in Euclidean space and in Minkowski space including quantum mechanics. Correlation functions at a given order of perturbation theory can be calculated algebraically without using canonical quantization or the path integral, and we demonstrate it explicitly for φ3 theory. We show that the Schwinger-Dyson equations are satisfied as an immediate consequence of the form of the expressions based on quantum A∞ algebras.

On Liouville’s Theorem for Conformal Maps

A theorem of Liouville asserts that the simplest angle-preserving transformations on Euclidean space—translations, dilations, reflections, and inversions—generate all angle-preserving transformations when the dimension is at least 3. This note gives a proof which uses only elementary multivariable calculus and simplifies a differential-geometric argument of Flanders.

Bent-half space model problem for Lame equation with surface tension

The study of fluid flow is a very fascinating area of fluid dynamics. Fluid motion has received more and more attention in recent years and numerous researchers have looked into this topic. However, they rarely used a mathematical analysis approach to analyse fluid motion; instead, they used numerical analysis. This serves as a significant justification for the researcher's decision to study fluid flow from the perspective of mathematical analysis. In this paper, we consider the ${\mathcal R}$-boundedness of the solution operator families of the Lam\'e equation with surface tension in bent half-space model problem by taking into account the surface tension in a bounded domain of {\it N}-dimensional Euclidean space ($N \geq 2$). The motion of the model problem can be described by linearizing an equation system of a model problem. This research is a continuation of [13]. They investigated the ${\mathcal R}$-boundedness of the solution operator families in the half-space case for the model problem of the Lam\'e equation with surface tension. First of all, by using Laplace transformation we consider the resolvent of the model problem, then treat the problem in bent half-space case. By using Weis's operator-valued Fourier multiplier theorem, we know that ${\mathcal R}$-boundedness implies the maximal $L_p$-$L_q$ regularity for the initial boundary value. This regularity is an essential tool for the partial differential equation problem.

Prediction of dynamical systems from time-delayed measurements with self-intersections

In the context of predicting the behaviour of chaotic systems, Schroer, Sauer, Ott and Yorke conjectured in 1998 that if a dynamical system defined by a smooth diffeomorphism T of a Riemannian manifold X admits an attractor with a natural measure μ of information dimension smaller than k, then k time-delayed measurements of a one-dimensional observable h are generically sufficient for μ-almost sure prediction of future measurements of h. In a previous paper we established this conjecture in the setup of injective Lipschitz transformations T of a compact set X in Euclidean space with an ergodic T-invariant Borel probability measure μ. In this paper we prove the conjecture for all (also non-invertible) Lipschitz systems on compact sets with an arbitrary Borel probability measure, and establish an upper bound for the decay rate of the measure of the set of points where the prediction is subpar. This partially confirms a second conjecture by Schroer, Sauer, Ott and Yorke related to empirical prediction algorithms as well as algorithms estimating the dimension and number of required delayed measurements (the so-called embedding dimension) of an observed system. We also prove general time-delay prediction theorems for locally Lipschitz or Hölder systems on Borel sets in Euclidean space.

Qualitative and asymptotic analysis of repulsion systems with diverse applications

Repulsion systems under the discrete dynamical systems perspective are considered. The goal is to study the nonlinear dynamics of an arrangement of individual or cells under various types of repulsion laws. Repulsion is a natural mechanism, so we consider the following laws: global, open, cyclic and bounded repulsion. A first scenario under consideration is when particles are freely located on an Euclidean space and the second scenario where they are restricted to remain in a bounded set. We consider those two scenarios under the previous types of repulsion. The analysis culminates in a unification of two mechanisms: attraction and repulsion via a specific social behaviour problem.

Temporal entanglement entropy as a probe of renormalization group flow

The recently introduced concept of timelike entanglement entropy has sparked a lot of interest. Unlike the traditional spacelike entanglement entropy, timelike entanglement entropy involves tracing over a timelike subsystem. In this work, we propose an extension of timelike entanglement entropy to Euclidean space (“temporal entanglement entropy”), and relate it to the renormalization group (RG) flow. Specifically, we show that tracing over a period of Euclidean time corresponds to coarse-graining the system and can be connected to momentum space entanglement. We employ Holography, a framework naturally embedding RG flow, to illustrate our proposal. Within cutoff holography, we establish a direct link between the UV cutoff and the smallest resolvable time interval within the effective theory through the irrelevant \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T\\overline{T }$$\\end{document} deformation. Increasing the UV cutoff results in an enhanced capability to resolve finer time intervals, while reducing it has the opposite effect. Moreover, we show that tracing over a larger Euclidean time interval is formally equivalent to integrating out more UV degrees of freedom (or lowering the temperature). As an application, we point out that the temporal entanglement entropy can detect the critical Lifshitz exponent z in non-relativistic theories which is not accessible from spatial entanglement at zero temperature and density.

The theory of screws derived from a module over the dual numbers

AbstractThe theory of screws clarifies many analogies between apparently unrelated notions in mechanics, including the duality between forces and angular velocities. It is known that the real 6-dimensional space of screws can be endowed with an operator $$\mathcal {E}$$ E , $$\mathcal {E}^2=0$$ E 2 = 0 , that converts it into a rank 3 free module over the dual numbers. In this paper we prove the converse, namely, given a rank 3 free module over the dual numbers, endowed with orientation and a suitable scalar product ($$\mathbb {D}$$ D -module geometry), we show that it is possible to define, in a canonical way, a Euclidean space so that each element of the module is represented by a screw vector field over it. The new approach has the effectiveness of motor calculus while being independent of any reduction point. It gives insights into the transference principle by showing that affine space geometry is basically vector space geometry over the dual numbers. The main results of screw theory are then recovered by using this point of view.

EEG functional connectivity as a Riemannian mediator: An application to malnutrition and cognition.

Mediation analysis assesses whether an exposure directly produces changes in cognitive behavior or is influenced by intermediate "mediators". Electroencephalographic (EEG) spectral measurements have been previously used as effective mediators representing diverse aspects of brain function. However, it has been necessary to collapse EEG measures onto a single scalar using standard mediation methods. In this article, we overcome this limitation and examine EEG frequency-resolved functional connectivity measures as a mediator using the full EEG cross-spectral tensor (CST). Since CST samples do not exist in Euclidean space but in the Riemannian manifold of positive-definite tensors, we transform the problem, allowing for the use of classic multivariate statistics. Toward this end, we map the data from the original manifold space to the Euclidean tangent space, eliminating redundant information to conform to a "compressed CST." The resulting object is a matrix with rows corresponding to frequencies and columns to cross spectra between channels. We have developed a novel matrix mediation approach that leverages a nuclear norm regularization to determine the matrix-valued regression parameters. Furthermore, we introduced a global test for the overall CST mediation and a test to determine specific channels and frequencies driving the mediation. We validated the method through simulations and applied it to our well-studied 50+-year Barbados Nutrition Study dataset by comparing EEGs collected in school-age children (5-11 years) who were malnourished in the first year of life with those of healthy classmate controls. We hypothesized that the CST mediates the effect of malnutrition on cognitive performance. We can now explicitly pinpoint the frequencies (delta, theta, alpha, and beta bands) and regions (frontal, central, and occipital) in which functional connectivity was altered in previously malnourished children, an improvement to prior studies. Understanding the specific networks impacted by a history of postnatal malnutrition could pave the way for developing more targeted and personalized therapeutic interventions. Our methods offer a versatile framework applicable to mediation studies encompassing matrix and Hermitian 3D tensor mediators alongside scalar exposures and outcomes, facilitating comprehensive analyses across diverse research domains.