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The exponential and Cayley maps on SE(3) are the prevailing coordinate maps used in Lie group integration schemes for rigid body and flexible body systems. Such geometric integrators are the Munthe–Kaas and generalized- α schemes, which involve the differential and its directional derivative of the respective coordinate map. Relevant closed form expressions, which were reported over the last two decades, are scattered in the literature, and some are reported without proof. This paper provides a reference summarizing all relevant closed-form relations along with the relevant proofs, including the right-trivialized differential of the exponential and Cayley map and their directional derivatives (resembling the Hessian). The latter gives rise to an implicit generalized- α scheme for rigid/flexible multibody systems in terms of the Cayley map with improved computational efficiency.

Given a finite group Γ , a regular t -balanced Cayley map ( RBCM t for short) is a regular Cayley map CM ( G , Ω , ρ ) such that ρ ( ω ) − 1 = ρ t ( ω ) for all ω ∈ Ω . In this paper, we clarify a connection between quotients of polynomial rings and RBCM t ’s on abelian groups, so as to propose a new approach for classifying RBCM t ’s. We obtain many new results, in particular, a complete classification for RBCM t ’s on abelian 2-groups.

The Cayley Isomorphism property for combinatorial objects was introduced by L. Babai in 1977. Since then it has been intensively studied for binary relational structures: graphs, digraphs, colored graphs etc. In this paper we study this property for oriented Cayley maps. A Cayley map is a Cayley graph provided by a cyclic rotation of its connection set. If the underlying graph is connected, then the map is an embedding of a Cayley graph into an oriented surface with the same cyclic rotation around every vertex.Two Cayley maps are called Cayley isomorphic if there exists a map isomorphism between them which is a group isomorphism too. We say that a finite group $H$ is a CIM-group if any two Cayley maps over $H$ are isomorphic if and only if they are Cayley isomorphic.The paper contains two main results regarding CIM-groups. The first one provides necessary conditons for being a CIM-group. It shows that a CIM-group should be one of the following$$\mathbb{Z}_m\times\mathbb{Z}_2^r, \\mathbb{Z}_m\times\mathbb{Z}_{4},\\mathbb{Z}_m\times\mathbb{Z}_{8}, \ \mathbb{Z}_m\times Q_8, \\mathbb{Z}_m\rtimes\mathbb{Z}_{2^e}, e=1,2,3,$$ where $m$ is an odd square-free number and $r$ a non-negative integer. Our second main result shows that the groups $\mathbb{Z}_m\times\mathbb{Z}_2^r$, $\mathbb{Z}_m\times\mathbb{Z}_{4}$, $\mathbb{Z}_m\times Q_8$ contained in the above list are indeed CIM-groups.

We show that a map defined by Pfurner, Schrocker and Husty, mapping points in 7-dimensional projective space to the Study quadric, is equivalent to the composition of an extended inverse Cayley map with the direct Cayley map, where the Cayley map in question is associated to the adjoint representation of the group SE(3). We also verify that subgroups and symmetric subspaces of SE(3) lie on linear spaces in dual quaternion representation of the group. These two ideas are combined with the observation that the Pfurner-Schrocker-Husty map preserves these linear subspaces. This means that the interpolation method proposed by Pfurner et al. can be restricted to subgroups and symmetric subspaces of SE(3).

The embeddings of the $\frak{so}(3)$ Lie algebra and the Lie group ${\rm SO}(3)$ in higher dimensions is an important construction from both mathematical and physical viewpoint. Here we present results based on a program package for building the generating matrices of the irreducible embeddings of the $\frak{so}(3)$ Lie algebra within $\frak{so}(n)$ in arbitrary dimension $n \geq 3, n \neq 4k+2, k \in \mathbb{N}$ relying on the algorithm developed recently by Campoamor-Strursberg [3]. For the remaining cases $n = 4k+2$ embeddings of $\frak{so}(3)$ into $\frak{so}(n)$ are also constructed. Besides, we investigate the characteristic polynomials of these $\frak{so}(n)$ elements. We show that the Cayley map applied to $\cal{C} \in \frak{so}(n)$ is well defined and generates a subset of ${\rm SO}(n)$. Furthermore, we obtain explicit formulas for the images of the Cayley map. The so obtained ${\rm SO}(n)$ matrices are expressed as polynomials of ${\cal C}$ whose coefficients are rational functions of the norm of the vector-parameter ${\bf c}$. The composition laws are extracted for the cases $n = 4, 6$ and for the first case it is shown that via the Cayley map the isomorphism ${\rm SU}(2) \cong {\rm im}{{\rm Cay}_{{\rm im}{j_4}}}\cup\{-\mathcal{I}_4\}$ holds. Also, for $n = 4$ explicit formulas for the the angular velocity matrices are derived. Comparisons between the results obtained via the exponential map and the Cayley map are made as well. In contrast to the case of the Cayley map, the results for the exponential map include either irrational or transcendental functions of the module of the vector-parameter.

A Cayley map M = C M ( G , X , p ) is t -balanced if p ( x ) − 1 = p t ( x − 1 ) for all x ∈ X . Recently, Conder et al. classified the regular anti-balanced Cayley maps for abelian groups and Kwak et al. classified the regular t -balanced Cayley maps for dihedral groups [8] and dicyclic groups [9] . Oh [11] classified the regular t -balanced Cayley maps for semi-dihedral groups. In this paper, we classify the regular t -balanced Cayley maps for cyclic groups for any t .

In this paper, we introduce a Cayley hypergraph which is a generalization of a Cayley graph, and a Cayley hypermap which is a generalization of a Cayley map. We investigate some properties of Cayley hypergraphs and Cayley hypermaps, describe the automorphism group of a Cayley hypermap and determine when a Cayley hypermap is regular.

A cellular embedding of a connected graph G is said to be Hamiltonian if every face of the embedding is bordered by a Hamiltonian cycle (a cycle containing all the vertices of G) and it is an m-gonal embedding if every face of the embedding has the same length m. In this paper, we establish a theory of generalized Cayley maps, including a new extension of voltage graph techniques, to show that for each even n there exists a Hamiltonian embedding of Kn such that the embedding is a Cayley map and that there is no n-gonal Cayley map of Kn if n≥5 is a prime. In addition, we show that there is no Hamiltonian Cayley map of Kn if n=pe, p an odd prime and e>1.

The inherent high symmetry of Cayley maps makes them an excellent source of orientably regular maps, and the regularity of a Cayley map has been shown to be equivalent to the existence of a skew-morphism of its underlying group that has a generating orbit closed under inverses. We set to investigate the properties of the so-called t -balanced skew-morphisms of abelian groups with the aim of providing the basis for a complete classification of t -balanced regular Cayley maps of abelian groups. In the case of cyclic groups, we show that the only t -balanced regular Cayley maps for the groups Z 2 r , Z 2 p r and Z 4 p r , p an odd prime, r ? 1 , are the well understood balanced and antibalanced Cayley maps.

In this paper, properties of reflexible Cayley maps for abelian groups are investigated, and as a result, it is shown that a regular Cayley map of valency greater than 2 for a cyclic group is reflexible if and only if it is anti-balanced.

Let $$\mathcal {M}={{\mathrm{CM}}}(D_n,X,p)$$M=CM(Dn,X,p) be a regular Cayley map on the dihedral group $$D_n$$Dn of order $$2n, n \ge 2,$$2n,nź2, and let $$\psi $$ź be the skew-morphism associated with $$\mathcal {M}$$M. In this paper it is shown that the kernel $${{\mathrm{Ker}}}\psi $$Kerź of the skew-morphism $$\psi $$ź is a dihedral subgroup of $$D_n$$Dn and if $$n \ne 3,$$nź3, then the kernel $${{\mathrm{Ker}}}\psi $$Kerź is of order at least 4. Moreover, all $$\mathcal {M}$$M are classified for which $${{\mathrm{Ker}}}\psi $$Kerź is of order 4. In particular, besides four sporadic maps on 4, 4, 8 and 12 vertices, respectively, two infinite families of non-t-balanced Cayley maps on $$D_n$$Dn are obtained.

A graph is called Frobenius if it is a connected orbital regular graph of a Frobenius group. A Frobenius map is a regular Cayley map whose underlying graph is Frobenius. In this paper, we show that almost all low-rank Frobenius graphs admit regular embeddings and enumerate non-isomorphic Frobenius maps for a given Frobenius graph. For some Frobenius groups, we classify all Frobenius maps derived from these groups. As a result, we construct some Frobenius maps with trivial exponent groups as a partial answer of a question raised by Nedela and Škoviera (Exponents of orientable maps, Proc. London Math. Soc. 75(3) (1997) 1–31).

The automorphism groups i>Aut(i>C(i>G, i>X)) and i>Aut(i>CM(i>G, i>X, i>p)) of a Cayley graph i>C(i>G, i>X) and a Cayley map i>CM(i>G, i>X, i>p) both contain an isomorphic copy of the underlying group i>G acting via left translations. In our paper, we show that both automorphism groups are rotary extensions of the group i>G by the stabilizer subgroup of the vertex 1i>G. We use this description to derive necessary and sufficient conditions to be satisfied by a finite group in order to be the (full) automorphism group of a Cayley graph or map and classify all the finite groups that can be represented as the (full) automorphism group of some Cayley graph or map.

In this paper we deal with high oscillatory systems and numerical methods for the approximation of their solutions. Some classical schemes developed in the literature are recalled and a recent approach based on the expression of the oscillatory solution by means of the exponential map is considered. Moreover we introduce a new method based on the Cayley map and provide some numerical tests in order to compare the different approaches.

A comparison is made, between three methods for visualizing the Minkowski space, in view of their application to relativistic kinematics.The velocity manifold method, in which a vector is represented by a point on the upper sheet of the unit-radius hyperboloid, presents two advantages: there is no distortion of angles, and the method is independent of the choice of any particular frame or observer. It is, however, limited to the representation of positive timelike vectors.The Cayley map method allows the representation of spacelike as well as timelike vectors, but it has two drawbacks: it distorts the angles, and it depends on the choice of a reference system.In the third method, the use of a unified space–time formalism permits one to visualize any kind of vector. This method generalizes the velocity manifold method, and presents the same two advantages: it provides us with a conformal representation, which is independent of any particular observer.We also present a rather concise comparison between two covariant methods of calculation, considered from the point of view of relativistic kinematics. One method is the familiar tensor calculus; the other is the spinor calculus, based on the representation of four-vectors by matrices.

A Cayley map M=CM(G,S,p) is t-balanced if p(x)^-^1=p^t(x^-^1) for every [email protected]?S. In this paper, we classify the regular t-balanced Cayley maps on semi-dihedral groups for any t. As a result, for every positive integer m, 4m-valent regular (2m+1)-balanced Cayley maps on semi-dihedral groups are constructed.

The purpose of this paper is to study self-dual embeddings of balanced Cayley maps. Given a Cayley map, necessary and sufficient conditions are given in terms of its underlying group for the map to be isomorphic to its dual embedding. Applications include self-dual embeddings of 2 n -dimensional cubes.