Each finite group is a symmetry group of some map (an “Atom”-bifurcation)

Maps are studied, i.e., cell decompositions of closed two-dimensional surfaces, or two-dimensional atoms which encode bifurcations of Liouville fibrations of non-degenerate integrable Hamiltonian systems. Any finite group G is proved to be a symmetry group of an orientable map (of an atom). Moreover, one such map X(G) is constructed algorithmically. Upper bounds are obtained for the minimal genus Mg(G) of an orientable map with the given symmetry group G and for the minimal number of vertices, edges, and sides of such maps.     

On natural isomorphisms of cycle permutation graphs

Two problems are approached in this paper. Given a permutation onn elements, which permutations onn elements yield cycle permutation graphs isomorphic to the cycle permutation graph yielded by the given permutation? And, given two cycle permutation graphs, are they isomorphic? Here the author deals only with natural isomorphisms, those isomorphisms which map the outer and inner cycles of one cycle permutation graph to the outer and inner cycles of another cycle permutation graph. A theorem is stated which then allows the construction of the set of permutations which yield cycle permutation graphs isomorphic to a given cycle permutation graph by a natural isomorphism. Another theorem is presented which finds the number of such permutations through the use of groups of symmetry of certain drawings of cycles in the plane. These drawings are also used to determine whether two given cycle permutation graphs are isomorphic by a natural isomorphism. These two methods are then illustrated by using them to solve the first problem, restricted to natural isomorphism, for a certain class of cycle permutation graphs.     

The reversibility and the center problem

In this work we study the narrow relation between reversibility and the center problem and also between reversibility and the integrability problem. It is well known that an analytic system having either a non-degenerate or nilpotent center at the origin is analytically reversible or orbitally analytically reversible, respectively. In this paper we prove the existence of a smooth map that transforms an analytic system having a degenerate center at the origin with either an analytic first integral or a C^∞ inverse integrating factor into a reversible linear system (after rescaling the time). Moreover, if the degenerate center has an analytic or a C^∞ reversing symmetry, then the transformed system by the map also has a reversing symmetry. From the knowledge of a first integral near the center we give a procedure to detect reversing symmetries.     

Cones with a mapping cone symmetry in the finite-dimensional case

In the finite-dimensional case, we present a new approach to the theory of cones with a mapping cone symmetry, first introduced by Størmer. Our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the Jamio?kowski-Choi isomorphism is an isometry. We consider a slightly modified class of cones, although not substantially different from the original mapping cones by Størmer. Using the new approach, several known results are proved faster and often in more generality than before. For example, the dual of a mapping cone turns out to be a mapping cone as well, without any additional assumptions. The main result of the paper is a characterization of cones with a mapping cone symmetry, saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive. A similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone. Our result is proved without the additional assumptions of symmetry and equality between the domain and the target space. We show how it gives a number of older results as a corollary, including an exemplary application.     

Reversible maps and their symmetry lines

In this paper, we present some methods to determine whether a planar map is reversible. Using these methods, we show that four automorphisms are reversible including Cremona map, cubic Hénon map, Knuth map and McMillan map. Some of them are not polynomial automorphism. We give the recurrence formulas of their symmetry lines, draw their phase portraits and symmetry lines with MATLAB software. Some special properties of their symmetry lines are explained and their beauties are also visually displayed.     

Degenerate relative equilibria, curvature of the momentum map, and homoclinic bifurcation

A fundamental class of solutions of symmetric Hamiltonian systems is relative equilibria. In this paper the nonlinear problem near a degenerate relative equilibrium is considered. The degeneracy creates a saddle-center and attendant homoclinic bifurcation in the reduced system transverse to the group orbit. The surprising result is that the curvature of the pullback of the momentum map to the Lie algebra determines the normal form for the homoclinic bifurcation. There is also an induced directional geometric phase in the homoclinic bifurcation. The backbone of the analysis is the use of singularity theory for smooth mappings between manifolds applied to the pullback of the momentum map. The theory is constructive and generalities are given for symmetric Hamiltonian systems on a vector space of dimension (2n+2) with an n-dimensional abelian symmetry group. Examples for n=1,2,3 are presented to illustrate application of the theory.     

A note on holonomy of gerbes over orbifolds

In this paper, we generalize the construction of the inverse transgression map done by Adem, A., Ruan, Y. and Zhang, B. in [A stringy product on twisted orbifold K-theory. Morfismos, 11, 33 64 （2007）] and give a different proof to the statement that the image of the inverse transgression map for a gerbe with connection over an orbifold is an inner local system on its inertia orbifold.     

Index four orientable embeddings and case zero of the Heawood conjecture

By imposing a special symmetry, we are able to construct index four triangular embeddings of graphs in compact orientable 2-manifolds. Because of the complexity of the current graphs required, such embeddings have heretofore been unattainable, but the imposed symmetry reduces the problem to constructing a special kind of index two current graph. We illustrate the method with a solution for case zero of the Heawood conjecture, using an abelian group, thus completing a constructive proof of the Heawood map color theorem, and eliminating the need for Galois field theory and nonabelian groups in its solution. The method has also been used in the determination of the genus of K_n,n,n,n.     

Efficient three-level scheme for parabolic equations in cylindrical coordinates in a region with a small hole

An efficient three-level scheme for parabolic equations in cylindrical coordinates is constructed in a region with a small hole. No axial symmetry is assumed. The convergence rate of the scheme is estimated under minimum requirements on the initial data. The estimates are uniform with respect to a small parameter—the inner diameter of the region. The order of convergence is τ + h ², τ^1/2 + h, τ + h, depending on the smoothness of the data.     

Derivations on group algebras with coding theory applications

This paper classifies the derivations of group algebras in terms of the generators and defining relations of the group. If RG is a group ring, where R is commutative and S is a set of generators of G then necessary and sufficient conditions on a map from S to RG are established, such that the map can be extended to an R-derivation of RG. Derivations are shown to be trivial for semisimple group algebras of abelian groups. The derivations of finite group algebras are constructed and listed in the commutative case and in the case of dihedral groups. In the dihedral case, the inner derivations are also classified. Lastly, these results are applied to construct well known binary codes as images of derivations of group algebras.     

Reduction of Courant algebroids and generalized complex structures

We present a theory of reduction for Courant algebroids as well as Dirac structures, generalized complex, and generalized Kähler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction. The enhanced symmetry group of a Courant algebroid leads us to define extended actions and a generalized notion of moment map. Key examples of generalized Kähler reduced spaces include new explicit bi-Hermitian metrics on CP².     

Effect of wall alignment in a very short rotating annulus

This paper reports numerical results of the study of effects of cylinders wall alignment in a small aspect ratio Taylor–Couette system. The investigation concerns bifurcations of steady vortical structures when the cylindrical walls defining the gap are not perfectly parallel. The imperfection is introduced by opening the outer fixed cylinder with a certain angle with regard to the vertical to form a tapered very short liquid column and keeping the inner rotating cylinder wall vertical. The numerical results obtained for the velocity components have revealed that bifurcation from a particular mode to another one occurs at a range of specific values of the inclination angle of the outer cylinder. The band width of the angle at which bifurcation occurred depended on the Reynolds number Re and was found to become narrower as Re increased. It is shown that geometrically broken symmetry can yield flow symmetry for specific combinations of geometrical and dynamical parameters.     

On p-harmonic maps and convex functions

We prove that, in general, given a p-harmonic map F : M → N and a convex function ${H : N \rightarrow \mathbb{R}}$ , the composition ${H\circ F}$ is not p-subharmonic, if p ≠ 2. This answers in the negative an open question arisen from a paper by Lin and Wei. By assuming some rotational symmetry on manifolds and functions, we reduce the problem to an ordinary differential inequality. The key of the proof is an asymptotic estimate for the p-harmonic map under suitable assumptions on the manifolds.     

Implicit and efficient schemes for a parabolic equation in a spherical layer

An implicit and an efficient three-level scheme for a parabolic equation in spherical coordinates is constructed in a spherical layer. No axial symmetry is assumed. The convergence rates of the schemes are estimated under minimum requirements on the initial data. The estimates are uniform with respect to the inner diameter of the domain. The order of convergence is τ^α/2 + h ^α, α = 1, 2, depending on the smoothness of the data. The results remain valid for a domain without a hole.     

A study of type I intermittency of a circular differential equation under a discontinuous right-hand side

In this paper we study a circular differential equation under a discontinuous periodic input, developing a quadratic differential equations system on S¹ and a linear differential equations system in the Minkowski space M³. The symmetry groups of these two systems are, respectively, PSOo(2,1) and SOo(2,1). The Poincaré circle map is constructed exactly, and a critical value αc of the parameter is identified. Depending on α of the input amplitude the equation may exhibit periodic, subharmonic or quasiperiodic motions. When α varies from α>αc to α<αc, there undergoes an inverse tangent bifurcation; consequently, the resultant Poincaré circle map offers one route to the quasiperiodicity via a type I intermittency. In the parameter range of α<αc the orbit generated by the Poincaré circle map is either m-periodic or quasiperiodic when n/m is a rational or an irrational number.     

The J-invariant and Tits algebras for groups of inner type E 6

A connection between the indices of the Tits algebras of a split linear algebraic group G and the degree one parameters of its motivic J-invariant was introduced by Quéguiner-Mathieu, Semenov and Zainoulline through use of the second Chern class map in the Riemann-Roch theorem without denominators. In this paper we extend their result to higher Chern class maps and provide applications to groups of inner type E ₆.     

Concentrated boundary data and axially symmetric harmonic maps

We examine the existence problem for harmonic maps between the three-dimensional ball and the two-sphere. We utilize results on the classification of harmonic maps into hemispheres and a result on the regularity of the weak limit of energy minimizers over the class of axially symmetric maps to establish the existence of asmooth harmonic extension for boundary data suitably “concentrated” away from the axis of symmetry. In addition, we establish convergence results for the harmonic map heat flow problem for suitably “concentrated” axially symmetric initial and boundary data.     

The symplectic reduced spaces of a Poisson actionLes espaces réduits symplectiques d'une action de Poisson

During the last thirty years, symplectic or Marsden–Weinstein reduction has been a major tool in the construction of new symplectic manifolds and in the study of mechanical systems with symmetry. This procedure has been traditionally associated to the canonical action of a Lie group on a symplectic manifold, in the presence of a momentum map. In this Note we show that the symplectic reduction phenomenon has much deeper roots. More specifically, we will find symplectically reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. In other words, the right category to obtain symplectically reduced spaces is that of Poisson manifolds acted upon canonically by a Lie group. To cite this article: J.-P. Ortega, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 999–1004.     

Sinusoidal and Gaussian kicking of a dissipative oscillator map

We propose a two-dimensional map that coincides with the Poincaré map of a kicked oscillator in the zero-dissipation limit. We investigate the main properties of the map under the influence of general odd perturbations. We study the map numerically under variation of the control parameters for two different models of the perturbation, sinusoidal and Gaussian, which give the system complementary properties: the forcing symmetry dominates in the first system, and the resonance symmetry dominates in the second system.     

New constructions of twistor lifts for harmonic maps

We show that given a harmonic map φ from a Riemann surface to a classical compact simply connected inner symmetric space, there is a J ₂-holomorphic twistor lift of φ (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae in terms of holomorphic data which describes their extended solutions. In particular, this gives explicit formulae for the twistor lifts of all harmonic maps of finite uniton number from a surface to the above symmetric spaces.