Euclidean Space Research Articles

The support of mixed area measures involving a new class of convex bodies

Mixed volumes in n-dimensional Euclidean space are functionals of n-tuples of convex bodies K,L,C1,…,Cn−2. The Alexandrov–Fenchel inequalities are fundamental inequalities between mixed volumes of convex bodies. As very special cases they cover or imply many important inequalities between basic geometric functionals. A complete characterization of the equality cases in the Alexandrov–Fenchel inequality remains a challenging open problem. Major recent progress was made by Yair Shenfeld and Ramon van Handel [13,14], in particular they resolved the problem in the cases where C1,…,Cn−2 are polytopes, zonoids or smooth bodies (under some dimensional restriction). In [6] we introduced the class of polyoids, which are defined as limits of finite Minkowski sums of polytopes having a bounded number vertices. Polyoids encompass polytopes, zonoids and triangle bodies, and they can be characterized by means of generating measures. Based on this characterization and Shenfeld and van Handel's contribution (and under a dimensional restriction), we extended their result to polyoids (or smooth bodies). Our previous result was stated in terms of the support of the mixed area measure associated with the unit ball Bn and C1,…,Cn−2. This characterization result is completed in the present work which more generally provides a geometric description of the support of the mixed area measure of an arbitrary (n−1)-tuple of polyoids (or smooth bodies). The result thus (partially) confirms a long-standing conjecture by Rolf Schneider in the case of polyoids, and hence in particular it covers the case of zonoids and triangle bodies.

Some results on the strict Fréchet differentiability of the metric projection operator in Hilbert spaces

In this paper, we first present a simpler proof of a result on the strict Fréchet differentiability of the metric projection operator onto closed balls centred at the origin in Hilbert spaces, which are given by Li in [Strict Fréchet differentiability of the metric projection operator in Hilbert spaces; 2024. Available from: https://arxiv.org/abs/2312.14362]. Then, based on this result, we prove the strict Fréchet differentiability of the metric projection operator onto closed balls with centre at arbitrarily given point in Hilbert spaces. Finally, we study the strict Fréchet differentiability of the metric projection operator onto a second-order cone in Euclidean spaces.

Riemannian geometry for efficient analysis of protein dynamics data

An increasingly common viewpoint is that protein dynamics datasets reside in a nonlinear subspace of low conformational energy. Ideal data analysis tools should therefore account for such nonlinear geometry. The Riemannian geometry setting can be suitable for a variety of reasons. First, it comes with a rich mathematical structure to account for a wide range of geometries that can be modeled after an energy landscape. Second, many standard data analysis tools developed for data in Euclidean space can be generalized to Riemannian manifolds. In the context of protein dynamics, a conceptual challenge comes from the lack of guidelines for constructing a smooth Riemannian structure based on an energy landscape. In addition, computational feasibility in computing geodesics and related mappings poses a major challenge. This work considers these challenges. The first part of the paper develops a local approximation technique for computing geodesics and related mappings on Riemannian manifolds in a computationally feasible manner. The second part constructs a smooth manifold and a Riemannian structure that is based on an energy landscape for protein conformations. The resulting Riemannian geometry is tested on several data analysis tasks relevant for protein dynamics data. In particular, the geodesics with given start- and end-points approximately recover corresponding molecular dynamics trajectories for proteins that undergo relatively ordered transitions with medium-sized deformations. The Riemannian protein geometry also gives physically realistic summary statistics and retrieves the underlying dimension even for large-sized deformations within seconds on a laptop.

Geodesic metrics for RBF approximation of some physical quantities measured on sphere

The radial basis function (RBF) approximation is a rapidly developing field of mathematics. In the paper, we are concerned with the measurement of scalar physical quantities at nodes on sphere in the 3D Euclidean space and the spherical RBF interpolation of the data acquired. We employ a multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 considered in Cartesian coordinates. Attention is paid to geodesic metrics that define the distance of two points on a sphere. The choice of a particular geodesic metric function is an important part of the construction of interpolation formula.We show the existence of an interpolation formula of the type considered. The approximation formulas of this type can be useful in the interpretation of measurements of various physical quantities. We present an example concerned with the sampling of anisotropy of magnetic susceptibility having extensive applications in geosciences and demonstrate the advantages and drawbacks of the formulas chosen, in particular the strong dependence of interpolation results on condition number of the matrix of the system considered and on round-off errors in general.

The Generic Failure of Lower-Semicontinuity for the Linear Distortion Functional

AbstractWe consider the convexity properties of distortion functionals, particularly the linear distortion, defined for homeomorphisms of domains in Euclidean n-spaces, $$n\ge 3$$ n ≥ 3 . The inner and outer distortion functionals are lower semi-continuous in all dimensions and so for the curve modulus or analytic definitions of quasiconformality it ifollows that if $$ \{ f_{n} \}_{n=1}^{\infty } $$ { f n } n = 1 ∞ is a sequence of K-quasiconformal mappings (here K depends on the particular distortion functional but is the same for every element of the sequence) which converges locally uniformly to a mapping f, then this limit function is also K-quasiconformal. Despite a widespread belief that this was also true for the geometric definition of quasiconformality (defined through the linear distortion $$H({f_{n}})$$ H ( f n ) ), T. Iwaniec gave a specific and surprising example to show that the linear distortion functional is not always lower-semicontinuous on uniformly converging sequences of quasiconformal mappings. Here we show that this failure of lower-semicontinuity is common, perhaps generic in the sense that under mild restrictions on a quasiconformal f, there is a sequence $$ \{f_{n} \}_{n=1}^{\infty } $$ { f n } n = 1 ∞ with $$ {f_{n}}\rightarrow {f}$$ f n → f locally uniformly and with $$\limsup _{n\rightarrow \infty } H( {f_{n}})<H( {f})$$ lim sup n → ∞ H ( f n ) < H ( f ) . Our main result shows this is true for affine mappings. Addressing conjectures of Gehring and Iwaniec we show the jump up in the limit can be arbitrarily large and give conjecturally sharp bounds: for each $$\alpha <\sqrt{2}$$ α < 2 there is $${f_{n}}\rightarrow {f}$$ f n → f locally uniformly with f affine and $$\begin{aligned} \alpha \; \limsup _{n\rightarrow \infty } H( {f_{n}}) < H( {f}) \end{aligned}$$ α lim sup n → ∞ H ( f n ) < H ( f ) We conjecture $$\sqrt{2}$$ 2 to be best possible.

SGG-DGCN: Wind Turbine Anomaly Identification by Using Deep Graph Convolutional Networks with Similarity Graph Generation Strategy

In order to minimize wind turbine failures, fault diagnosis of wind turbines is becoming increasingly important, deep learning methods excel at multivariate monitoring and data modeling, but they are often limited to Euclidean space and struggle to capture the complex coupling between wind turbine sensors. To address this problem, we convert SCADA data into graph data, where sensors act as nodes and their topological connections act as edges, to represent these complex relationships more efficiently. Specifically, a wind turbine anomaly identification method based on deep graph convolutional neural network using similarity graph generation strategy (SGG-DGCN) is proposed. Firstly, a plurality of similarity graphs containing similarity information between nodes are generated by different distance metrics. Then, the generated similarity graphs are fused using the proposed similarity graph generation strategy. Finally, the fused similarity graphs are fed into the DGCN model for anomaly identification. To verify the effectiveness of the proposed SGG-DGCN model, we conducted a large number of experiments. The experimental results show that the proposed SGG-DGCN model has the highest accuracy compared with other models. In addition, the results of ablation experiment also demonstrate that the proposed SGG strategy can effectively improve the accuracy of WT anomaly identification. Conflict of Interest Statement The authors declare no conflicts of interest.

Explicit Parameterizations of Ortho-Symplectic Matrices in R4

Starting from the very first principles we derive explicit parameterizations of the ortho-symplectic matrices in the real four-dimensional Euclidean space. These matrices depend on a set of four real parameters which splits naturally as a union of the real line and the three-dimensional space. It turns out that each of these sets is associated with a separate Lie algebra which after exponentiations generates Lie groups that commute between themselves. Besides, by making use of the Cayley and Fedorov maps, we have arrived at alternative realizations of the ortho-symplectic matrices in four dimensions. Finally, relying on the fundamental structure results in Lie group theory we have derived one more explicit parameterization of these matrices which suggests that the obtained earlier results can be viewed as a universal method for building the representations of the unitary groups in arbitrary dimension.

Entanglement and classical nonseparability convertible from orthogonal polarizations

The nonclassicality of a macroscopic single-mode optical superposition state is potentially convertible into entanglement, when the state is mixed with the vacuum on a beam splitter. Considering light beams with polarization degree of freedom in Euclidean space as coherent product states in a bipartite Hilbert space, we propose a method to convert the two orthogonal polarizations into simultaneous entanglement and classical nonseparability through nonclassicality in the superpositions of coherent and displaced Fock states. Equivalent Bell state emerges from the resulted superpositions and the proportion of mixed entanglement and nonseparablity is determined by the displacement amplitudes along the polarization directions. We characterize the state nonclassicality via features in Wigner distributions and propose an experimental method for generating these states and measuring them via homodyne tomography.

Symplectic analytical solutions for free vibration of elastically line-hinged orthotropic rectangular plates with rotationally restrained edges

The elastically line-hinged orthotropic rectangular plates with rotationally restrained edges are commonly used in engineering applications such as deployable structures. However, it is intractable to obtain the analytical solutions for the free vibration problems of such structures owing to the challenges in processing the high-order partial differential equations. Here, we make a first attempt to deal with such issues within the Hamilton system-based symplectic space. The problem of a plate is transformed into the symplectic space from the Euclidean space, and the subplates that may be analyzed analytically by the symplectic superposition method are then obtained with the division of the entire plate. The complex boundary and connection forms are achieved by enforcing mechanical quantities with undetermined expansion coefficients on the edges of the subplates. By integrating the solutions of the subplates, the final solution of an elastically line-hinged orthotropic rectangular plate with rotationally restrained edges is accessible. The proposed solution scheme is performed with rational derivations, with no requirement for pre-defined solution forms. Comprehensive results with validations under various boundary and hinge connection cases are presented. Moreover, detailed parametric investigations are conducted, which could facilitate the engineering design of deployable structures.

A New Approach for Hardy Spaces on Euclidean Space

A New Approach for Hardy Spaces on Euclidean Space

Unscented Schmidt-Kalman filter on Lie groups with application to spacecraft attitude estimation

This paper addresses the estimation problem of nonlinear systems evolving on Lie groups with unknown parameters. More precisely, some parameters in the equations of motion or sensor measurements are unknown, such as gravitational anomalies and measurement biases, and are infeasible to estimate with available observations. The unscented Schmidt-Kalman filter (USKF) approach in Euclidean space is incorporated with exponential maps from Lie algebra to Lie groups, to develop USKF algorithms on Lie groups. Two types of USKFs are derived, respectively, from left-invariant and right-invariant state estimation errors. The two USKFs, not only account for the effect of unknown parameters but also provide estimates preserving the geometry of state manifold. They are advantageous over the extended Schmidt-Kalman filter for nonlinear systems in the sense of avoiding the computation of Jacobian and achieving higher or comparable estimation accuracy depending on the magnitude of parameters uncertainties. The proposed method is then applied to a spacecraft attitude estimation problem based on quaternion representation, where the magnitude of the gyroscope bias noise is unknown. Simulations are conducted to illustrate the effectiveness of the proposed algorithms in comparison with other methods.

Heat Kernel Asymptotics for Scaling Limits of Isoradial Graphs

AbstractWe consider the asymptotics of the discrete heat kernel on isoradial graphs for the case where the time and the edge lengths tend to zero simultaneously. Depending on the asymptotic ratio between time and edge lengths, we show that two different regimes arise: (i) a Gaussian regime and (ii) a Poissonian regime, which resemble the short-time asymptotics of the heat kernel on (i) Euclidean spaces and (ii) graphs, respectively.

Structure of Primitive Pythagorean Triples in Generating Trees

A Pythagorean triple is a triple of positive integers $(a,b,c)$ such that $a^2+b^2=c^2$. If $a,b$ are coprime, then it is called a primitive Pythagorean triple. It is known that every primitive Pythagorean triple can be generated from the triple $(3,4,5)$ using multiplication by unique number and order of three specific $3\times3$ matrices, which yields a ternary tree of triplets. Two such trees were described by Berggren and Price, respectively. A different approach is to view the primitive Pythagorean triples as points in the three-dimensional Euclidean space. In this paper, we prove that the triple of descendants of any primitive Pythagorean triple in Berggren's or Price's tree forms a triangle (and therefore defines a plane), and we present our results related to these triangles (and these planes).

Distance‐Based Smoothing of Curves on Surface Meshes

AbstractThe smoothing of surface curves is an essential tool in mesh processing, important to applications that require segmenting and cutting surfaces such as surgical planning. Surface curves are typically designed by professionals to match certain surface features. For this reason, the smoothed curves should be close to the original and easily adjustable by the user in interactive tools. Previous methods achieve this desired behavior, e.g., by utilizing energy‐minimizing splines or generalizations of Bézier splines, which require a significant number of control points and may not provide interactive frame rates or numerical stability. This paper presents a new algorithm for robust smoothing of discrete surface curves on triangular surface meshes. By using a scalar penalty potential as the fourth coordinate, the given surface mesh is embedded into the 4D Euclidean space. Our method is based on finding geodesics in this lifted surface, which are then projected back onto the original 3D surface. The benefits of this approach include guaranteed convergence and good approximation of the initial curve. We propose a family of penalty potentials with one single parameter for adjusting the trade‐off between smoothness and similarity. The implementation of our method is straightforward as we rely on existing methods for computing geodesics and penalty fields. We evaluate our implementation and confirm its robustness and efficiency.

On the asymptotic behaviour of the fractional Sobolev seminorms: A geometric approach

We study the well-known asymptotic formulas for fractional Sobolev functions à la Bourgain–Brezis–Mironescu and Maz'ya–Shaposhnikova, in a geometric approach. We show that the key to these asymptotic formulas are Rademacher's theorem and volume growth at infinity respectively. Examples fitting our framework includes Euclidean spaces, Riemannian manifolds, Alexandrov spaces, finite dimensional Banach spaces, and some ideal sub-Riemannian manifolds.

Nonlinear dynamic analysis of shear- and torsion-free rods using isogeometric discretization and outlier removal

AbstractIn this paper, we present a discrete formulation of nonlinear shear- and torsion-free rods introduced by Gebhardt and Romero (Acta Mechanica 232(10):3825–3847, 2021) that uses isogeometric discretization and robust time integration. Omitting the director as an independent variable field, we reduce the number of degrees of freedom and obtain discrete solutions in multiple copies of the Euclidean space $$\left( \mathbb {R}^3\right) $$ R 3 , which is larger than the corresponding multiple copies of the manifold $$\left( \mathbb {R}^3 \varvec{\times } S^2\right) $$ R 3 × S 2 obtained with standard Hermite finite elements. For implicit time integration, we choose the same integration scheme as Gebhardt and Romero in (2021) that is a hybrid form of the midpoint and the trapezoidal rules. In addition, we apply a recently introduced approach for outlier removal by Hiemstra et al. (Comput Methods Appl Mech Eng 387:114115, 2021) that reduces high-frequency content in the response without affecting the accuracy, ensuring robustness of our nonlinear discrete formulation. We illustrate the efficiency of our nonlinear discrete formulation for static and transient rods under different loading conditions, demonstrating good accuracy in space, time and the frequency domain. Our numerical example coincides with a relevant application case, the simulation of mooring lines.

The Schwarzian Derivative and the Degree of a Classical Minimal Surface

Using the Schwarzian derivative we construct a sequence ( P l ) l ⩾ 2 of meromorphic differentials on every non-flat oriented minimal surface in Euclidean 3-space. The differentials ( P l ) l ⩾ 2 are invariant under all deformations of the surface arising via the Weierstrass representation and depend on the induced metric and its derivatives only. A minimal surface is said to have degree n if its n-th differential is a polynomial expression in the differentials of lower degree. We observe that several well-known minimal surfaces have small degree, including Enneper’s surface, the helicoid/catenoid and the Scherk—as well as the Schwarz family. Furthermore, it is shown that locally and away from umbilic points every minimal surface can be approximated by a sequence of minimal surfaces of increasing degree.

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.