Rigidity theorems for asymptotically Euclidean Q-singular spaces

The abundant discordance between evolutionary relationships across the genome has rekindled interest in methods for comparing and averaging trees on a shared leaf set. However, compared to tree topology, where much progress has been made, handling branch lengths has been more challenging. Species tree branch lengths can be measured in various units, often different from gene trees. Moreover, rates of evolution change across the genome, the species tree, and specific branches of gene trees. These factors compound the stochasticity of coalescence times and estimation noise, making branch lengths highly heterogeneous across the genome. For many downstream applications in phylogenomic analyses, branch lengths are as important as the topology, and yet, existing tools to compare and combine weighted trees are limited. In this paper, we address the question of matching one tree to another, accounting for their branch lengths. We define a series of computational problems called Topology-Constrained Metric Matching (TCMM) that seek to transform the branch lengths of a query tree based on a reference tree. We show that TCMM problems can be solved efficiently using a linear algebraic formulation coupled with dynamic programming preprocessing. While many applications can be imagined for this framework, we explore two applications in this paper: embedding leaves of gene trees in Euclidean space to find outliers potentially indicative of estimation errors, and summarizing gene tree branch lengths onto the species tree. In these applications, our method, when paired with existing methods, increases their accuracy at limited computational expense.

Abstract The objective of the addendum was to propose a hypothetical charge for the particles in four-dimensional Euclidean space. A thorough examination of the four possible charges is conducted, with each charge being deliberated within its respective three-dimensional subspace. The number of baryonic projections of a purported 4D object is six. It has been established that there are two groups of imaginary fermionic particles. These particles have been identified as four light and four fermions. The charges and masses of the composite objects are derived from four three-dimensional subspaces.

Accurately predicting drug-target interactions (DTIs) is crucial for drug discovery, repositioning. However, most deep learning-based DTI models are designed in Euclidean space, making it difficult to effectively represent the hierarchical and scale-free characteristics of biological data. Due to its unique negatively curved geometric properties, hyperbolic space can more effectively represent hierarchical relationships within data. Therefore, we propose a multimanifold learning framework that integrates multimodal features in hyperbolic and Euclidean spaces for drug-target interaction prediction. Specifically, we employ a Hyperbolic Graph Neural Network (HGNN) to extract features from molecular graphs of small-molecular drugs, thereby effectively capturing the hierarchical structural information within these graphs. To integrate heterogeneous information, a Multi-Manifold Feature Fusion Module combines structural features from the HGNN, chemical fingerprints, and semantic embeddings derived from pretrained language models. Extensive experiments on benchmark data sets demonstrate that our framework achieves superior performance compared with state-of-the-art Euclidean-based methods. The experimental results demonstrate that hyperbolic geometry offers significant advantages in extracting hierarchical features from non-Euclidean data and also highlight the promising potential of multimanifold feature fusion in the field of drug-target interaction prediction.

Genetic sequences play a central role in biological and medical research, and mathematics provides powerful means for their representation and analysis. Conventional approaches, such as the fuzzy polynucleotide space [0, 1]12, model codons as 12-dimensional vectors, but this comes at the cost of high dimensionality. In this study, we introduce two new models, Vector-Fuzzy-I and Vector-Fuzzy-II, that map codons and genetic sequences into the 4-dimensional Euclidean space ℝ4 using vector algebra and fuzzy set theory. In the first model, sequence structure is represented by successive vector addition, while in the second, it is represented by positional frequencies normalized by nucleotide locations. These low-dimensional representations are unique, preserve sequence order, and allow effective measurement of similarity and difference via Euclidean metrics. Compared with the fuzzy polynucleotide space, the proposed models achieve dimensionality reduction while enhancing the resolution of sequence differentiation. Our approach offers new mathematical perspectives for sequence analysis in theoretical biology.

Abstract Consider a point process in Euclidean space obtained by perturbing the integer lattice with independent and identically distributed random vectors. Under mild assumptions on the law of the perturbations, we construct a translation-invariant perfect matching between this point process and the lattice, such that the matching distance has the same tail behavior as the hole probability of the point process, which is a natural lower bound.

This paper investigates the iterative process of constructing Cevian triangles in a finite Euclidean plane. First, we prove that starting from an initial triangle, the process of iteratively taking Cevian triangles converges to a unique fixed point. Second, we show this convergence process is surjective onto the interior of the triangle; that is, for any target point in the interior, one can find an initial point whose sequence of iterated Cevian triangles converges to that target. Finally, we examine the limiting configuration of an infinite iteration and characterize a novel property intrinsic to the discrete nature of the finite geometric space, setting it apart from the classical real Euclidean case.

Abstract The discrete data encoded in the power moments of a positive measure, fast decaying at infinity on Euclidean space, are incomplete for recovery, leading to the concept of moment indeterminateness. On the other hand, classical integral transforms (Fourier‐Laplace, Fantappiè, Poisson) of such measures are complete, often invertible via an effective inverse operation. The gap between the two non‐uniqueness/uniqueness phenomena is manifest in the dual picture, when trying to extend the measure, regarded as a positive linear functional, from the polynomial algebra to the full space of continuous functions. This point of view was advocated by Marcel Riesz a century ago, in the single real variable setting. Notable advances in functional analysis have their root in Riesz's celebrated four notes devoted to the moment problem. A key technical ingredient being there the monotone approximation by polynomials of kernels of integral transforms. With inherent new obstacles, we reappraise in the context of several real variables M. Riesz's variational principle. The result is an array of necessary and sufficient moment indeterminateness criteria, some raising real algebra questions, as well as others involving intriguing analytic problems, all gravitating around the concept of moment separating function.

Embedding graphs in continuous spaces is a key factor for automatic information extraction in diverse tasks (e.g., learning, inferring, predicting). The reliability of graph embeddings directly depends on how much the geometry of the manifold in continuous space matches the graph structure. State-of-the-art of manifold-based graph embedding algorithms assume that the projection on a tangential space of each point in the manifold (corresponding to a node in the graph) would locally resemble a Euclidean space. Although this condition helps in achieving efficient analytical solutions to the embedding problem, it is not an adequate set-up to work with modern real life graphs, that are characterized by weighted connections across nodes often computed over sparse datasets with missing records. In this work, we introduce a new class of manifold, named soft manifold, that can solve this situation. Soft manifolds are mathematical structures with spherical symmetry where the tangent spaces to each point are hypocycloids whose shape is defined according to the velocity of information propagation across the data points. Experimental results on reconstruction tasks on synthetic and real datasets show how the proposed approach enable more accurate and reliable characterization of graphs in continuous spaces with respect to the state-of-the-art.

Existing methods to summarize posterior inference for mixture models focus on identifying a point estimate of the implied random partition for clustering and validating it using clustering-based loss functions (Wade and Ghahramani, 2018; Dahl et al., 2022). We propose a novel approach for summarizing posterior inference in nonparametric Bayesian mixture models, prioritizing estimation of the mixing measure as an inference target. Moreover, we propose to validate the estimation of the partition via a new perspective which is based on the implied density estimate. One of the key features is the model-agnostic nature of the approach, which remains valid under arbitrarily complex dependence structures in the underlying sampling model. Using a decision-theoretic framework, the proposed methods identify a point estimate using loss functions defined as discrepancies between mixing measures. Estimating the mixing measure then implies inference on the mixture density and the random partition. To define a discrepancy between mixing measures we exploit the discrete nature of the mixing measure and use a version of sliced Wasserstein distance. We introduce two variants for Gaussian mixtures. The first, mixed sliced Wasserstein, applies generalized geodesic projections on a product of Euclidean space and the manifold of symmetric positive definite matrices. The second, sliced mixture Wasserstein, leverages the linearity of Gaussian mixture measures to define a projection.

Let [Formula: see text] be a [Formula: see text]-smooth, [Formula: see text]-dimensional Riemannian manifold that is diffeomorphic to [Formula: see text] and admits a properly discontinuous, cocompact, isometric, fixed-point-free action by a group. This work proves the existence of a [Formula: see text] equivariant isometric embedding of [Formula: see text] into some Euclidean space [Formula: see text] with [Formula: see text] which is the same as the optimal dimension bound in Matthias Günther’s results.

Some Results on Calibrated Submanifolds in Euclidean Space of Cohomogeneity One and Two

In curve theory, Mannheim curves are well-established objects that have been extensively analyzed in both Euclidean and Minkowski 3-space. A curve $\varphi $ is classified as a Mannheim curve if there exists a corresponding relationship with another space curve $\varphi ^{\star }$ such that, at corresponding points on each curve, the principal normal lines of $\varphi $ align with the binormal lines of $\varphi ^{\star }$. In this study, we focus on examining the differential geometric properties of spacelike Mannheim curves within Minkowski 3-space, utilizing a novel approach. Through this method, we derive the conditions under which a spacelike curve qualifies as a Mannheim curve and introduce new examples of Mannheim curves that are not covered in the classical framework.

Curvature plays a central role in the proper function of many biological processes. With active matter being a standard framework for understanding many aspects of the physics of life, it is natural to ask what effect curvature has on the collective behavior of active matter. In this paper, we use the classical theory of surfaces to explore the active motion of self-propelled agents confined to move on a smooth curved two-dimensional surface embedded in Euclidean space. Even without interactions and alignment, the motion is nontrivially affected by the presence of curvature, leading to effects akin, e.g., to gravitational lensing and tidal forces. Such effects can lead to intermittent trapping of particles and profoundly affect their flocking behavior. We show that these effects are governed by a geometric torque that, in the absence of noise and interactions, compels particles to move along geodesics.

The paper is devoted to providing Michael–Simon-type L^{p} -logarithmic-Sobolev inequalities on complete, not necessarily compact n -dimensional submanifolds \Sigma of the Euclidean space \mathbb{R}^{n+m} . Our first result, stated for p=2 , is sharp, it is valid on general submanifolds, and it involves the mean curvature of \Sigma . It implies in particular the main result of S. Brendle [Comm. Pure Appl. Math. 75 (2022), 449–454]. In addition, it turns out that equality can occur if and only if \Sigma is isometric to the Euclidean space \mathbb{R}^{n} and the extremizer is a Gaussian. Our second result is a general L^{p} -logarithmic-Sobolev inequality for p\geq 2 on Euclidean submanifolds with constants that are codimension-free in the case of minimal submanifolds. In order to prove the above results – especially, to deal with the equality cases – we elaborate the theory of optimal mass transport on submanifolds between measures that are not necessarily compactly supported. Two applications are provided to sharp hypercontractivity estimates of Hopf–Lax semigroups on submanifolds. The first hypercontractivity estimate is for general submanifolds with bounded mean curvature vector, while the second is for self-similar shrinkers endowed with the natural Gaussian measure. The equality cases are characterized here as well.

Let H \mathbb {H} denote the three-dimensional Heisenberg group. In this paper, we study vertical curves in H \mathbb {H} and fibers of maps H → R 2 \mathbb {H}\to \mathbb {R}^2 from a metric perspective. We say that a set in H \mathbb {H} is a vertical curve if it satisfies a cone condition with respect to a homogeneous cone with axis ⟨ Z ⟩ \langle Z \rangle , the center of H \mathbb {H} . This is analogous to the cone condition used to define intrinsic Lipschitz graphs. In the first part of the paper, we prove that connected vertical curves are locally bi-Hölder equivalent to intervals. We also show that the class of vertical curves coincides with the class of intersections of intrinsic Lipschitz graphs satisfying a transversality condition. Unlike intrinsic Lipschitz graphs, the Hausdorff dimension of a vertical curve can vary; we construct vertical curves with Hausdorff dimension either strictly larger or strictly smaller than 2. Consequently, there are intersections of intrinsic Lipschitz graphs with Hausdorff dimension either strictly larger or strictly smaller than 2. In the second part of the paper, we consider smooth functions β \beta from the unit ball B B in H \mathbb {H} to R 2 \mathbb {R}^2 . We show that, in contrast to the situation in Euclidean space, there are maps such that β \beta is arbitrarily close to the projection π \pi from H \mathbb {H} to the horizontal plane, but the average H 2 \mathcal {H}^2 measure of a fiber of β \beta in B B is arbitrarily small.

A bstract We construct generalized sets of asymptotic conditions for both three-dimensional Maxwell Chern-Simons gravity and a novel extension that incorporates torsion through a deformation of the Maxwell algebra. These boundary conditions include the most general temporal components of the gauge fields that consistently preserve the corresponding asymptotic Maxwell algebras with identical classical central charges, while allowing for the inclusion of chemical potentials conjugate to the conserved charges. We show that both sets of asymptotic configurations admit nontrivial solutions carrying not only mass and angular momentum but also an additional global spin-2 charge. In the torsionless case, the theory admits locally flat cosmological spacetimes, whereas in the presence of torsion, it generalizes to BTZ-like black hole geometries. For each case, the thermodynamic properties are consistently derived in terms of the gauge fields and the topology of the Euclidean manifold, shown to correspond to a solid torus. Furthermore, we obtain a general expression for the entropy, depending on both the horizon area and its spin-2 analogues, which can be written as a reparametrization-invariant integral of the induced spin-2 fields on the spacelike section of the horizon.

To address data scarcity and distribution shifts in motor imagery electroencephalogram (MI-EEG) based brain computer interface, we propose a 1-dimensional convolution-based deep transfer learning model with embedded Feature Alignment block (1DC-DTL-FA) in this article. It integrates multi-stage feature extraction, classification, and FA block. Unlike complex models, it utilizes Neural Architecture Search (NAS) to automatically locate the optimal FA position in Euclidean space Evaluated on BCI 2000 and BCI IV2a datasets, 1DC-DTL-FA achieved superior accuracies of 89.80% and 82.96%. The results demonstrate that this simple architecture effectively handles complex feature extraction and online alignment, outperforming state-of-the-art models in MI-EEG decoding.

We show that surfaces with assigned director field immersed in the three-dimensional Euclidean space can be defined intrinsically by four tensor fields, of which two are of order two, one of order one, and one of order zero. We then show how a surface and its assigned director field can be reconstructed from these four tensors and prove that the reconstruction operator is continuous between ad hoc functional spaces with as little regularity as possible. These results have applications in the Cosserat theory of nonlinearly elastic shells, the strain energy of which are defined precisely in terms of the four tensor fields defined in this paper.

In this paper we compare the rotation of a rigid body in the real three-dimensional Euclidean space E3 and its representation in the complex plane (Klein space), on one hand, with the transformation of polarization states of light (SOPs) by the phase-shifters figured in the complex plane and on the Poincaré sphere, on the other hand. Both the Klein space, in classical mechanics, and the Poincaré sphere, in polarization theory, are abstract spaces, whose construction is based on the classical stereographical projection between Riemann sphere and the simple complex plane C1. They are classical abstract spaces, even if they have been largely used for representing quantum spinorial physical realities too. At the interface of classical/quantum physics persist some misaperceptions about what is intrinsically of quantum origin and nature, and what is imported from the classical domain. In this context we examine some misunderstandings that take place in the field of these spaces. I shall focus on the double angle relationship between the rotation of representative points of the SOPs on the Poincaré sphere with respect to the corresponding rotations of the azimuthal and ellipticity angles of the “form of the SOPs”, at a transformation of state given by a phase shifter. This is a classical result, that is transferred on the sphere from the complex plane, on the basis of the stereographic bijective connection between the points on the sphere and those in the complex plane. In any textbook of quantum mechanics “the double angle/half angle problem” is presented as a pure quantum spinorial one, avoiding its classical face and origin. A quantum spinorial approach, obviously, recovers the classical results, together with the specific spinorial ones, but with regards to the double angle/half angle issue it is superfluous. We shall also briefly examine the classical and quantum spinorial content of what we know today under the global name of Pauli spin matrices. Often in papers or textbooks of physics the results are presented in a mélange in which it is difficult to establish from which point on one needs to appeal to spinorial or quantum aspects.