Liaison Linkages

The complete classification of hexapods - also known as Stewart Gough platforms - of mobility one is still open. To tackle this problem, we can associate to each hexapod of mobility one an algebraic curve, called the configuration curve. In this paper we establish an upper bound for the degree of this curve, assuming the hexapod is general enough. Moreover, we provide a construction of hexapods with curves of maximal degree, which is based on liaison, a technique used in the theory of algebraic curves.Comment: 40 pages, 6 figure

High-field/high-frequency EPR spectroscopy in protein research: principles and examples

During the last decades, the combined efforts of biologists, chemists, and physicists in developing high-field/high-frequency EPR techniques and applying them to functional proteins have demonstrated that this type of magnetic resonance spectroscopy is particularly powerful for characterizing the structure and dynamics of stable and transient states of proteins in action on biologically relevant time scales ranging from nanoseconds to hours. The review article describes how high-field EPR methodology, in conjunction with site-specific isotope and spin-labeling strategies, is capable of providing new insights into fundamental biological processes. Specifically, we discuss the theoretical and instrumental background of continuous-wave and pulse high-field EPR and the multiple-resonance extensions EDNMR, ENDOR, TRIPLE, ESEEM, PELDOR, and RIDME. Some emphasis is placed on a balanced description of both the historical spadework and the achieved performance of advanced EPR at 95 GHz and 360 GHz. This culminates in a coherent treatment of state-of-the-art research of high-field EPR in terms of both instrumentation development and application to representative protein complexes such as cofactor binding sites in photosynthesis

High-field/High-frequency EPR Spectroscopy in Protein Research: Principles and Examples

During the last decades, the combined efforts of biologists, chemists, and physicists in developing high-field/high-frequency EPR techniques and applying them to functional proteins have demonstrated that this type of magnetic resonance spectroscopy is particularly powerful for characterizing the structure and dynamics of stable and transient states of proteins in action on biologically relevant time scales ranging from nanoseconds to hours. The review article describes how high-field EPR methodology, in conjunction with site-specific isotope and spin-labeling strategies, is capable of providing new insights into fundamental biological processes. Specifically, we discuss the theoretical and instrumental background of continuous-wave and pulse high-field EPR and the multiple-resonance extensions EDNMR, ENDOR, TRIPLE, ESEEM, PELDOR, and RIDME. Some emphasis is placed on a balanced description of both the historical spadework and the achieved performance of advanced EPR at 95 GHz and 360 GHz. This culminates in a coherent treatment of state-of-the-art research of high-field EPR in terms of both instrumentation development and application to representative protein complexes such as cofactor binding sites in photosynthesis

Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds

Motivated by the vast success of deep convolutional networks, there is a great interest in generalizing convolutions to non-Euclidean manifolds. A major complication in comparison to flat spaces is that it is unclear in which alignment a convolution kernel should be applied on a manifold. The underlying reason for this ambiguity is that general manifolds do not come with a canonical choice of reference frames (gauge). Kernels and features therefore have to be expressed relative to arbitrary coordinates. We argue that the particular choice of coordinatization should not affect a network's inference -- it should be coordinate independent. A simultaneous demand for coordinate independence and weight sharing is shown to result in a requirement on the network to be equivariant under local gauge transformations (changes of local reference frames). The ambiguity of reference frames depends thereby on the G-structure of the manifold, such that the necessary level of gauge equivariance is prescribed by the corresponding structure group G. Coordinate independent convolutions are proven to be equivariant w.r.t. those isometries that are symmetries of the G-structure. The resulting theory is formulated in a coordinate free fashion in terms of fiber bundles. To exemplify the design of coordinate independent convolutions, we implement a convolutional network on the M\"obius strip. The generality of our differential geometric formulation of convolutional networks is demonstrated by an extensive literature review which explains a large number of Euclidean CNNs, spherical CNNs and CNNs on general surfaces as specific instances of coordinate independent convolutions.Comment: The implementation of orientation independent M\"obius convolutions is publicly available at https://github.com/mauriceweiler/MobiusCNN

Knowledge Graph Embedding: An Overview

Many mathematical models have been leveraged to design embeddings for representing Knowledge Graph (KG) entities and relations for link prediction and many downstream tasks. These mathematically-inspired models are not only highly scalable for inference in large KGs, but also have many explainable advantages in modeling different relation patterns that can be validated through both formal proofs and empirical results. In this paper, we make a comprehensive overview of the current state of research in KG completion. In particular, we focus on two main branches of KG embedding (KGE) design: 1) distance-based methods and 2) semantic matching-based methods. We discover the connections between recently proposed models and present an underlying trend that might help researchers invent novel and more effective models. Next, we delve into CompoundE and CompoundE3D, which draw inspiration from 2D and 3D affine operations, respectively. They encompass a broad spectrum of techniques including distance-based and semantic-based methods. We will also discuss an emerging approach for KG completion which leverages pre-trained language models (PLMs) and textual descriptions of entities and relations and offer insights into the integration of KGE embedding methods with PLMs for KG completion

Hyperbolic Geometry in Computer Vision: A Novel Framework for Convolutional Neural Networks

Real-world visual data exhibit intrinsic hierarchical structures that can be represented effectively in hyperbolic spaces. Hyperbolic neural networks (HNNs) are a promising approach for learning feature representations in such spaces. However, current methods in computer vision rely on Euclidean backbones and only project features to the hyperbolic space in the task heads, limiting their ability to fully leverage the benefits of hyperbolic geometry. To address this, we present HCNN, the first fully hyperbolic convolutional neural network (CNN) designed for computer vision tasks. Based on the Lorentz model, we generalize fundamental components of CNNs and propose novel formulations of the convolutional layer, batch normalization, and multinomial logistic regression (MLR). Experimentation on standard vision tasks demonstrates the effectiveness of our HCNN framework and the Lorentz model in both hybrid and fully hyperbolic settings. Overall, we aim to pave the way for future research in hyperbolic computer vision by offering a new paradigm for interpreting and analyzing visual data. Our code is publicly available at https://github.com/kschwethelm/HyperbolicCV