scholarly journals Pure rolling motion of hyperquadrics in pseudo-Euclidean spaces

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
André Marques ◽  
Fátima Silva Leite
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Affine Space ◽  
General Framework ◽  
Nonholonomic Constraints ◽  
Rolling Motion ◽  
Explicit Solutions ◽  
Euclidean Spaces ◽  
Pure Rolling ◽  
Definition Of

<p style='text-indent:20px;'>This paper is devoted to rolling motions of one manifold over another of equal dimension, subject to the nonholonomic constraints of no-slip and no-twist, assuming that these motions occur inside a pseudo-Euclidean space. We first introduce a definition of rolling map adjusted to this situation, which generalizes the classical definition of Sharpe [<xref ref-type="bibr" rid="b26">26</xref>] for submanifolds of an Euclidean space. We also prove some important properties of these rolling maps. After presenting the general framework, we analyse the particular rolling of hyperquadrics embedded in pseudo-Euclidean spaces. The central topic is the rolling of a pseudo-hyperbolic space over the affine space associated with its tangent space at a point. We derive the kinematic equations, as well as the corresponding explicit solutions for two specific cases, and prove the existence of a rolling map along any curve in that rolling space. Rolling of a pseudo-hyperbolic space on another and rolling of pseudo-spheres are equally treated. Finally, for the central theme, we write the kinematic equations as a control system evolving on a certain Lie group and prove its controllability. The choice of the controls corresponds to the choice of a rolling curve.</p>

Corresponding Polyhedra in the Three Spaces of Constant Curvature

1953 ◽  
Vol 5 ◽  
pp. 40-45
Author(s):  
Ernst Roeser
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Constant Curvature ◽  
Dihedral Angles ◽  
Euclidean Spaces ◽  
Platonic Solids ◽  
Spaces Of Constant Curvature ◽  
Elliptic Space ◽  
The Face

The five Platonic solids can be drawn in elliptic or hyperbolic space just as well as in Euclidean space. Their numerical properties are, of course, the same in all three. So are the various angles subtended at the centre. But the face-angles and dihedral angles are greater in elliptic space, smaller in hyperbolic. It is a special feature of the non-Euclidean spaces that we cannot change the size of a solid without changing its shape.

Control of Sliding and Rolling at Natural Joints

1984 ◽  
Vol 106 (4) ◽  
pp. 368-375 ◽  
Author(s):  
C. Wongchaisuwat ◽  
H. Hemami ◽  
H. J. Buchner
Keyword(s):  
Knee Joint ◽  
Relative Motion ◽  
Control Strategy ◽  
Control Structure ◽  
Nonholonomic Constraints ◽  
Rolling Motion ◽  
Human Knee ◽  
Human Knee Joint ◽  
Pure Rolling ◽  
Holonomic And Nonholonomic Constraints

The planar motion of the human knee joint is modeled, involving the relative motion of the geometry of the contacting surface between the tibia and the femur. The pure gliding motion and the pure rolling motion are formulated including the holonomic and nonholonomic constraints that must be satisfied. A control strategy with two classes of inputs: muscle forces that stabilize and bring about the motion and the ligament forces that maintain the constraints is presented. Finally, the effectiveness of this control structure is demonstrated via digital computer simulations in the pure gliding motion and the pure rolling motion of the knee.

scholarly journals Horosphere slab separation theorems in manifolds without conjugate points

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Sameh Shenawy
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Existence And Uniqueness ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Conjugate Points ◽  
Separation Theorems ◽  
Farthest Points ◽  
Simply Connected ◽  
Convex Subsets

Abstract Let $\mathcal {W}^{n}$ W n be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let $W\in \mathcal {W}^{n}$ W ∈ W n and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in $W\in \mathcal {W}$ W ∈ W are considered.

scholarly journals A note on surfaces with prescribed oriented Euclidean Gauss map

2005 ◽  
Vol 2005 (4) ◽  
pp. 537-543
Author(s):  
Ricardo Sa Earp ◽  
Eric Toubiana
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Gauss Map ◽  
Compatibility Relation ◽  
Hyperbolic Gauss Map ◽  
Conformal Immersion

We present another proof of a theorem due to Hoffman and Osserman in Euclidean space concerning the determination of a conformal immersion by its Gauss map. Our approach depends on geometric quantities, that is, the hyperbolic Gauss mapGand formulae obtained in hyperbolic space. We use the idea that the Euclidean Gauss map and the hyperbolic Gauss map with some compatibility relation determine a conformal immersion, proved in a previous paper.

Diagnosis Rule Discovery Based on Causality in the Context of Fault Diagnosis in Rotating Machinery

2014 ◽  
Vol 532 ◽  
pp. 113-117
Author(s):  
Zhou Jin ◽  
Ru Jing Wang ◽  
Jie Zhang
Keyword(s):  
Fault Diagnosis ◽  
General Framework ◽  
Complex Structure ◽  
Rotating Machinery ◽  
Massive Data ◽  
Formal Definition ◽  
Rule Discovery ◽  
Rule Based ◽  
Simple Implementation ◽  
Definition Of

The rotating machineries in a factory usually have the characteristics of complex structure and highly automated logic, which generated a large amounts of monitoring data. It is an infeasible task for uses to deal with the massive data and locate fault timely. In this paper, we explore the causality between symptom and fault in the context of fault diagnosis in rotating machinery. We introduce data mining into fault diagnosis and provide a formal definition of causal diagnosis rule based on statistic test. A general framework for diagnosis rule discovery based on causality is provided and a simple implementation is explored with the purpose of providing some enlightenment to the application of causality discovery in fault diagnosis of rotating machinery.

scholarly journals Characterisation of upper gradients on the weighted Euclidean space and applications

2021 ◽  
Author(s):  
Danka Lučić ◽  
Enrico Pasqualetto ◽  
Tapio Rajala
Keyword(s):  
Euclidean Space ◽  
Sobolev Space ◽  
Radon Measure ◽  
Lipschitz Functions ◽  
Euclidean Spaces ◽  
Upper Gradient

AbstractIn the context of Euclidean spaces equipped with an arbitrary Radon measure, we prove the equivalence among several different notions of Sobolev space present in the literature and we characterise the minimal weak upper gradient of all Lipschitz functions.

scholarly journals Parallel Robot Translational Performance Evaluation through Direction-Selective Index (DSI)

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
G. Boschetti ◽  
R. Rosa ◽  
A. Trevisani
Keyword(s):  
Performance Evaluation ◽  
Performance Index ◽  
General Framework ◽  
Dynamic Performance ◽  
Parallel Manipulators ◽  
Parallel Robot ◽  
Performance Indexes ◽  
Definition Of ◽  
Performance Variations

Performance indexes usually provide global evaluations of robot performances mixing their translational and/or rotational capabilities. This paper proposes a definition of performance index, called direction-selective index (DSI), which has been specifically developed for parallel manipulators and can provide uncoupled evaluations of robot translational capabilities along relevant directions. The DSI formulation is first presented within a general framework, highlighting its relationship with traditional manipulability definitions, and then applied to a family of parallel manipulators (4-RUU) of industrial interest. The investigation is both numerical and experimental and allows highlighting the two chief advantages of the proposed DSIs over more conventional manipulability indexes: not only are DSIs more accurate in predicting the workspace regions where manipulators can best perform translational movements along specific directions, but also they allow foreseeing satisfactorily the dynamic performance variations within the workspace, though being purely kinematic indexes. The experiments have been carried out on an instrumented 4-RUU commercial robot.

scholarly journals From Euclidean Spaces to Metric Spaces

2019 ◽  
Vol 8 (1) ◽  
Author(s):  
Ryan Joseph Rogers ◽  
Ning Zhong
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Euclidean Spaces ◽  
Definition Of

In this note, we provide the definition of a metric space and establish that, while all Euclidean spaces are metric spaces, not all metric spaces are Euclidean spaces. It is then natural and interesting to ask which theorems that hold in Euclidean spaces can be extended to general metric spaces and which ones cannot be extended. We survey this topic by considering six well-known theorems which hold in Euclidean spaces and rigorously exploring their validities in general metric spaces.

scholarly journals A General Framework for Implicit and Explicit Debiasing of Distributional Word Vector Spaces

2020 ◽  
Vol 34 (05) ◽  
pp. 8131-8138
Author(s):  
Anne Lauscher ◽  
Goran Glavaš ◽  
Simone Paolo Ponzetto ◽  
Ivan Vulić
Keyword(s):  
General Framework ◽  
Semantic Information ◽  
Evaluation Framework ◽  
Vector Spaces ◽  
Racial Biases ◽  
Definition Of ◽  
Cross Lingual ◽  
Experimental Findings ◽  
Implicit And Explicit ◽  
Embedding Methods

Distributional word vectors have recently been shown to encode many of the human biases, most notably gender and racial biases, and models for attenuating such biases have consequently been proposed. However, existing models and studies (1) operate on under-specified and mutually differing bias definitions, (2) are tailored for a particular bias (e.g., gender bias) and (3) have been evaluated inconsistently and non-rigorously. In this work, we introduce a general framework for debiasing word embeddings. We operationalize the definition of a bias by discerning two types of bias specification: explicit and implicit. We then propose three debiasing models that operate on explicit or implicit bias specifications and that can be composed towards more robust debiasing. Finally, we devise a full-fledged evaluation framework in which we couple existing bias metrics with newly proposed ones. Experimental findings across three embedding methods suggest that the proposed debiasing models are robust and widely applicable: they often completely remove the bias both implicitly and explicitly without degradation of semantic information encoded in any of the input distributional spaces. Moreover, we successfully transfer debiasing models, by means of cross-lingual embedding spaces, and remove or attenuate biases in distributional word vector spaces of languages that lack readily available bias specifications.

scholarly journals About Chirality in Minkowski Spacetime

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1320 ◽  
Author(s):  
Michel Petitjean
Keyword(s):  
Metric Space ◽  
Time Reversal ◽  
Minkowski Spacetime ◽  
Euclidean Spaces ◽  
Mathematical Definition ◽  
Definition Of ◽  
Mathematical Definitions ◽  
Lorentz Boosts

In this paper, we show that Lorentz boosts are direct isometries according to the recent mathematical definitions of direct and indirect isometries and of chirality, working for any metric space. Here, these definitions are extended to the Minkowski spacetime. We also show that the composition of parity inversion and time reversal is an indirect isometry, which is the opposite of what could be expected in Euclidean spaces. It is expected that the extended mathematical definition of chirality presented here can contribute to the unification of several definitions of chirality in space and in spacetime, and that it helps clarify the ubiquitous concept of chirality.

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