A new method for lattice reduction using directional and hyperplanar shearing

A geometric method of lattice reduction based on cycles of directional and hyperplanar shears is presented. The deviation from cubicity at each step of the reduction is evaluated by a parameter called `basis rhombicity' which is the sum of the absolute values of the elements of the metric tensor associated with the basis. The levels of reduction are quite similar to those obtained with the Lenstra–Lenstra–Lovász (LLL) algorithm, at least up to the moderate dimensions that have been tested (lower than 20). The method can be used to reduce unit cells attached to given hyperplanes.

Cauchy and Goursat problems for the generalized spin zero rest-mass fields on Minkowski spacetime

In this paper, we study the Cauchy and Goursat problems of the spin-$n/2$ zero rest-mass equations on Minkowski spacetime by using the conformal geometric method. In our strategy, we prove the wellposedness of the Cauchy problem in Einstein's cylinder. Then we establish pointwise decays of the fields and prove the energy equalities of the conformal fields between the null conformal boundaries $\scri^\pm$ and the hypersurface $\Sigma_0=\left\{ t=0 \right\}$. Finally, we prove the wellposedness of the Goursat problem in the partial conformal compactification by using the energy equalities and the generalisation of H\"ormander's result.

The Lagrange–Charpit Theory of the Hamilton–Jacobi Problem

The Lagrange–Charpit theory is a geometric method of determining a complete integral by means of a constant of the motion of a vector field defined on a phase space associated to a nonlinear PDE of first order. In this article, we establish this theory on the symplectic structure of the cotangent bundle $$T^{*}Q$$ T ∗ Q of the configuration manifold Q. In particular, we use it to calculate explicitly isotropic submanifolds associated with a Hamilton–Jacobi equation.

Generation of filament winding patterns for elbows with various cross-sections

As a connecting component of tubes, the elbow is indispensable to pipe-fitting in composite products. Previous studies have addressed methods for generating winding paths based on parametric equations on the elbow. However, these methods are unsuitable for elbows whose surfaces are difficult to describe using mathematical expressions. In this study, a geometric method was proposed for generating winding patterns for various elbow types. With this method, the mandrel surface is first converted into uniform and high-quality quadrilateral elements; an algorithm is then provided for calculating the minimum winding angle for bridging-free. Next, an angle for non-bridging was defined as the design-winding angle to generate the uniform and slippage-free basic winding paths on the quadrilateral elements in non-geodesic directions. Finally, after a series of uniform points were calculated on the selected vertical edge according to the elbow type, the pattern paths were generated with the uniform points and basic paths. The proposed method is advantageously not limited to the elbow’s shape.

Validation of an Echidna Forelimb Musculoskeletal Model Using XROMM and diceCT

In evolutionary biomechanics, musculoskeletal computer models of extant and extinct taxa are often used to estimate joint range of motion (ROM) and muscle moment arms (MMAs), two parameters which form the basis of functional inferences. However, relatively few experimental studies have been performed to validate model outputs. Previously, we built a model of the short-beaked echidna (Tachyglossus aculeatus) forelimb using a traditional modelling workflow, and in this study we evaluate its behaviour and outputs using experimental data. The echidna is an unusual animal representing an edge-case for model validation: it uses a unique form of sprawling locomotion, and possesses a suite of derived anatomical features, in addition to other features reminiscent of extinct early relatives of mammals. Here we use diffusible iodine-based contrast-enhanced computed tomography (diceCT) alongside digital and traditional dissection to evaluate muscle attachments, modelled muscle paths, and the effects of model alterations on the MMA outputs. We use X-ray Reconstruction of Moving Morphology (XROMM) to compare ex vivo joint ROM to model estimates based on osteological limits predicted via single-axis rotation, and to calculate experimental MMAs from implanted muscles using a novel geometric method. We also add additional levels of model detail, in the form of muscle architecture, to evaluate how muscle torque might alter the inferences made from MMAs alone, as is typical in evolutionary studies. Our study identifies several key findings that can be applied to future models. 1) A light-touch approach to model building can generate reasonably accurate muscle paths, and small alterations in attachment site seem to have minimal effects on model output. 2) Simultaneous movement through multiple degrees of freedom, including rotations and translation at joints, are necessary to ensure full joint ROM is captured; however, single-axis ROM can provide a reasonable approximation of mobility depending on the modelling objectives. 3) Our geometric method of calculating MMAs is consistent with model-predicted MMAs calculated via partial velocity, and is a potentially useful tool for others to create and validate musculoskeletal models. 4) Inclusion of muscle architecture data can change some functional inferences, but in many cases reinforced conclusions based on MMA alone.

Geometric and topological approaches to shape variation in Ginkgo leaves

Leaf shape is a key plant trait that varies enormously. The range of applications for data on this trait requires frequent methodological development so that researchers have an up-to-date toolkit with which to quantify leaf shape. We generated a dataset of 468 leaves produced by Ginkgo biloba , and 24 fossil leaves produced by evolutionary relatives of extant Ginkgo . We quantified the shape of each leaf by developing a geometric method based on elastic curves and a topological method based on persistent homology. Our geometric method indicates that shape variation in modern leaves is dominated by leaf size, furrow depth and the angle of the two lobes at the leaf base that is also related to leaf width. Our topological method indicates that shape variation in modern leaves is dominated by leaf size and furrow depth. We have applied both methods to modern and fossil material: the methods are complementary, identifying similar primary patterns of variation, but also revealing different aspects of morphological variation. Our topological approach distinguishes long-shoot leaves from short-shoot leaves, both methods indicate that leaf shape influences or is at least related to leaf area, and both could be applied in palaeoclimatic and evolutionary studies of leaf shape.

The reduced Dijkgraaf–Witten invariant of double twist knots in the Bloch group of 𝔽p

In 2004, Neumann showed that the complex hyperbolic volume of a hyperbolic 3-manifold [Formula: see text] can be obtained as the image of the Dijkgraaf–Witten invariant of [Formula: see text] by a certain 3-cocycle. After that, Zickert gave an analogue of Neumann’s work for free fields containing finite fields. The author formulated a geometric method to calculate a weaker version of Zickert’s analogue, called the reduced Dijkgraaf–Witten invariant, for finite fields and gave a formula for twist knot complements and [Formula: see text] in his previous work. In this paper, we show concretely how to calculate the reduced Dijkgraaf–Witten invariants of double twist knot complements and [Formula: see text], and give a formula of them for [Formula: see text].

A Geometric Method for Accelerated Sphere Tracing of Implicit Surfaces

Sphere tracing is a common raytracing technique used for rendering implicit surfaces defined by a signed distance function (SDF). However, these distance functions are often expensive to compute, prohibiting several real-time applications despite recent efforts to accelerate it. This paper presents a method to precompute a slightly augmented distance field that hugely accelerates rendering. This novel method called quadric tracing supports two configurations: (i) accelerating raytracing without losing precision, so the original SDF is still needed; (ii) entirely replacing the SDF and tracing an interpolated surface. Quadric tracing can offer 20% to 100% speedup in rendering static scenes and thereby amortizing the slowdown caused by the complexity of the geometry.