scholarly journals The kinematic image of RR, PR, and RP dyads

Robotica ◽  
2018 ◽  
Vol 36 (10) ◽  
pp. 1477-1492 ◽  
Author(s):  
Tudor-Dan Rad ◽  
Daniel F. Scharler ◽  
Hans-Peter Schröcker
Keyword(s):  
Three Dimensional ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Dual Quaternion ◽  
Projective Transformations ◽  
Revolute Joints ◽  
Prismatic Joints ◽  
Coordinate Changes ◽  
Necessary And Sufficient ◽  
Three Space

SUMMARYWe provide necessary and sufficient conditions for all projective transformations of the projectivized dual quaternion model of rigid body displacements that are induced by coordinate changes in moving and/or fixed frame. These transformations fix the quadrics of a pencil and preserve the two families of rulings of an exceptional three-dimensional quadric. Moreover, we fully characterize the constraint varieties of dyads with revolute and prismatic joints in the dual quaternion model. The constraint variety of a dyad with two revolute joints is a regular ruled quadric in a three-space that contains a “null quadrilateral.” If a revolute joint is replaced by a prismatic joint, this quadrilateral collapses into a pair of conjugate complex null lines and a real line but these properties are not sufficient to characterize such dyads. We provide a complete characterization by introducing a new invariant, the “Study fibre projectivity,” and we present examples that demonstrate its potential to explain hitherto not sufficiently well-understood phenomena.

scholarly journals Multigenerator Gabor Frames on Local Fields

2018 ◽  
Vol 33 (2) ◽  
pp. 307
Author(s):  
Owais Ahmad ◽  
Neyaz Ahmad Sheikh
Keyword(s):  
Sufficient Conditions ◽  
Complete Characterization ◽  
Necessary And Sufficient Conditions ◽  
Local Fields ◽  
Gabor Frames ◽  
Gabor System ◽  
Necessary And Sufficient

The main objective of this paper is to provide complete characterization of multigenerator Gabor frames on a periodic set $\Omega$ in $K$. In particular, we provide some necessary and sufficient conditions for the multigenerator Gabor system to be a frame for $L^2(\Omega)$. Furthermore, we establish the complete characterizations of multigenerator Parseval Gabor frames.

On *-clean group rings

2014 ◽  
Vol 14 (01) ◽  
pp. 1550004 ◽  
Author(s):  
Yuanlin Li ◽  
M. M. Parmenter ◽  
Pingzhi Yuan
Keyword(s):  
Local Ring ◽  
Prime Order ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Group Rings ◽  
Clean Ring ◽  
Commutative Local Ring ◽  
Irreducible Factorization ◽  
Necessary And Sufficient

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vaš asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C3, C4, S3 and Q8. As a consequence, the authors provided many examples of group rings which are clean, but not *-clean. In this paper, we continue this investigation and we give a complete characterization of when the group algebra 𝔽Cp is *-clean, where 𝔽 is a field and Cp is the cyclic group of prime order p. Our main result is related closely to the irreducible factorization of a pth cyclotomic polynomial over the field 𝔽. Among other results we also obtain a complete characterization of when RCn (3 ≤ n ≤ 6) is *-clean where R is a commutative local ring.

scholarly journals On Embedding of the Morse-Smale Diffeomorphisms in a Topological Flow

2020 ◽  
Vol 66 (2) ◽  
pp. 160-181
Author(s):  
V. Z. Grines ◽  
E. Ya. Gurevich ◽  
O. V. Pochinka
Keyword(s):  
Invariant Manifolds ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Topological Classification ◽  
Higher Dimension ◽  
Topological Information ◽  
Topology Of Manifolds ◽  
Necessary And Sufficient

This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse-Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse-Smale diffeomorphisms to embed in topological flows. The progress achieved in solving of Paliss problem in dimension three is associated with the recently obtained complete topological classification of Morse-Smale diffeomorphisms on three-dimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

Vectors

1996 ◽  
Author(s):  
Gerhard Oertel
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Unit Vector ◽  
Cross Product ◽  
Nonzero Vector ◽  
Algebraic Representation ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Three Dimensional Space

The reader, even if familiar with vectors, will find it useful to work through this chapter because it introduces notation that will be used throughout this book. We will take vectors to be entities that possess magnitude, orientation, and sense in three-dimensional space. Graphically, we will represent them as arrows with the sense from tail to head, magnitude proportional to the length, and orientation indicated by the angles they form with a given set of reference directions. Two different kinds of symbol will be used to designate vectors algebraically, boldface letters (and the boldface number zero for a vector of zero magnitude), and subscripted letters to be introduced later. The first problems deal with simple vector geometry and its algebraic representation. Multiplying a vector by a scalar affects only its magnitude (length) without changing its direction. Problem 1. State the necessary and sufficient conditions for the three vectors A, B, and C to form a triangle. (Problems 1–9, 12–14, 19–23, and 25 from Sokolnikoff & Redheffer, 1958.) Problem 2. Given the sum S = A + B and the difference D = A – B, find A and B in terms of S and D (a) graphically and (b) algebraically. Problem 3. (a) State the unit vector a with the same direction as a nonzero vector A. (b) Let two nonzero vectors A and B issue from the same point, forming an angle between them; using the result of (a), find a vector that bisects this angle. Problem 4. Using vector methods, show that a line from one of the vertices of a parallelogram to the midpoint of one of the nonadjacent sides trisects one of the diagonals. Two vectors are said to form with each other two distinct products: a scalar, the dot product, and a vector, the cross product.

scholarly journals Post-Lie algebra structures for nilpotent Lie algebras

2018 ◽  
Vol 28 (05) ◽  
pp. 915-933
Author(s):  
Dietrich Burde ◽  
Christof Ender ◽  
Wolfgang Alexander Moens
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Nilpotent Lie Algebras ◽  
Dimension Formula ◽  
Necessary And Sufficient

We study post-Lie algebra structures on [Formula: see text] for nilpotent Lie algebras. First, we show that if [Formula: see text] is nilpotent such that [Formula: see text], then also [Formula: see text] must be nilpotent, of bounded class. For post-Lie algebra structures [Formula: see text] on pairs of [Formula: see text]-step nilpotent Lie algebras [Formula: see text] we give necessary and sufficient conditions such that [Formula: see text] defines a CPA-structure on [Formula: see text], or on [Formula: see text]. As a corollary, we obtain that every LR-structure on a Heisenberg Lie algebra of dimension [Formula: see text] is complete. Finally, we classify all post-Lie algebra structures on [Formula: see text] for [Formula: see text], where [Formula: see text] is the three-dimensional Heisenberg Lie algebra.

A (2+1)-dimensional shallow water equation and its explicit lump solutions

2019 ◽  
Vol 33 (07) ◽  
pp. 1950038 ◽  
Author(s):  
Solomon Manukure ◽  
Yuan Zhou
Keyword(s):  
Shallow Water Equation ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Contour Plots ◽  
Dimensional Equation ◽  
New Equation ◽  
Necessary And Sufficient ◽  
Hirota Bilinear

We introduce a new (2+1)-dimensional equation by modifying the potential form of the Calogero–Bogoyavlenskii–Schiff (CBS) equation. By applying the Hirota bilinear method, we construct explicit lump solutions to this new equation and establish necessary and sufficient conditions that guarantee that the solutions are analytic and rationally localized in all directions in space. We also depict the evolution of the profiles of some selected lump solutions with three-dimensional and contour plots. It is immediately observed that the lump solutions generated are solitary wave type solutions as is the case with the KP equation.

scholarly journals Second order infinitesimal bending of curves

Filomat ◽  
2017 ◽  
Vol 31 (13) ◽  
pp. 4127-4137 ◽  
Author(s):  
Marija Najdanovic ◽  
Ljubica Velimirovic
Keyword(s):  
Vector Fields ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Second Order ◽  
Dimensional Euclidean Space ◽  
Second Variation ◽  
Explicit Formulas ◽  
Infinitesimal Bending ◽  
Necessary And Sufficient ◽  
The Second Variation

We investigate a second order infinitesimal bending of curves in a three-dimensional Euclidean space in this paper. We give the necessary and sufficient conditions for the vector fields to be infinitesimal bending fields of the corresponding order, as well as explicit formulas which determine these fields. We examine the first and the second variation of some geometric magnitudes which describe a curve, specially a change of the curvature. Two illustrative examples (a circle and a helix) are studied not only analytically but also by drawing curves using computer program Mathematica.

scholarly journals Dynamics of constrained many body problems in constant curvature two-dimensional manifolds

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170425 ◽  
Author(s):  
Andrzej J. Maciejewski ◽  
Maria Przybylska
Keyword(s):  
Constant Curvature ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Rigid Bodies ◽  
The Body ◽  
Two Dimensional ◽  
Many Body ◽  
Finite Dimensional ◽  
Necessary And Sufficient

In this paper, we investigate systems of several point masses moving in constant curvature two-dimensional manifolds and subjected to certain holonomic constraints. We show that in certain cases these systems can be considered as rigid bodies in Euclidean and pseudo-Euclidean three-dimensional spaces with points which can move along a curve fixed in the body. We derive the equations of motion which are Hamiltonian with respect to a certain degenerated Poisson bracket. Moreover, we have found several integrable cases of described models. For one of them, we give the necessary and sufficient conditions for the integrability. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

Stability of a thermoelastic mixture with second sound

2018 ◽  
Vol 24 (6) ◽  
pp. 1692-1706 ◽  
Author(s):  
Margareth S. Alves ◽  
Marcio V. Ferreira ◽  
Jaime E. Muñoz Rivera ◽  
O. Vera Villagrán
Keyword(s):  
Initial Data ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Second Sound ◽  
Dimensional Model ◽  
One Dimensional ◽  
The One ◽  
Necessary And Sufficient

We consider the one-dimensional model of a thermoelastic mixture with second sound. We give a complete characterization of the asymptotic properties of the model in terms of the coefficients of the model. We establish the necessary and sufficient conditions for the model to be exponential or polynomial stable and also the conditions for which there exist initial data for where the energy is conserved.

scholarly journals Infinitesimal Transformations of Locally Conformal Kähler Manifolds

2019 ◽  
Author(s):  
Yevhen Cherevko ◽  
Volodymyr Berezovski ◽  
Irena Hinterleitner ◽  
Dana Smetanová
Keyword(s):  
Partial Differential Equations ◽  
Sufficient Conditions ◽  
Lie Derivative ◽  
Conformal Transformations ◽  
Projective Transformations ◽  
Integrability Conditions ◽  
Necessary And Sufficient ◽  
Locally Conformal ◽  
Lee Form ◽  
Infinitesimal Transformations

The article is devoted to infinitesimal transformations. We have obtained that LCK-manifolds do not admit nontrivial infinitesimal projective transformations. Then we study infinitesimal conformal transformations of LCK-manifolds. We have found the expression for the Lie derivative of a Lee form. Also we have obtained the system of partial differential equations for the transformations, and explored its integrability conditions. Hence we have got the necessary and sufficient conditions in order that the an LCK-manifold admits a group of conformal motions. Also we have calculated the number of parameters which the group depends on. We have proved that a group of conformal motions admitted by an LCK-manifold is isomorphic to a homothetic group admitted by corresponding K\"{a}hlerian metric. We also established that an isometric group of an LCK-manifold is isomorphic to a some subgroup of homothetic group of the coresponding local K\"{a}hlerian metric.

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