150 research outputs found
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
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