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It is found explicitly 5 Liouville integrals in addition to total angular momentum which Poisson commute with Hamiltonian of 3-body Newtonian Gravity in ${ℝ}^3$ along the remarkable figure-8-shape trajectory discovered by Moore-Chenciner-Montgomery. It is verified that they become constants of motion along this trajectory. Hence, 3-body choreographic motion on figure-8-shape trajectory in ${ℝ}^3$ Newtonian gravity (Moore, 1993), as well as in ${ℝ}^2$ modified Newtonian gravity by Fujiwara et al., is maximally superintegrable. It is conjectured that any 3-body potential theory that admits Figure-8-shape choreographic motion is superintegrable along the trajectory.

For one 3-body and two 5-body planar choreographies on the same algebraic lemniscate by Bernoulli we found explicitly a maximal possible set of (particular) Liouville integrals, 7 and 15, respectively, (including the total angular momentum), which Poisson commute with the corresponding Hamiltonian along the trajectory. Thus, these choreographies are particularly maximally superintegrable. It is conjectured that the total number of (particular) Liouville integrals is maximal possible for any odd number of bodies $(2n+1)$ moving choreographically (without collisions) along given algebraic lemniscate, thus, the corresponding trajectory is particularly, maximally superintegrable. Some of these Liouville integrals are presented explicitly. The limit $n\to \infty$ is studied: it is predicted that one-dimensional liquid with nearest-neighbor interactions occurs, it moves along algebraic lemniscate and it is characterized by infinitely many constants of motion.

Business contracts play a central role in governing commercial interactions between organizations. It is increasingly recognized that business contract conditions need to be closely linked to internal and external business processes, both to reduce the risk of contract violations and to ensure compliance with legislative regimes. Recent research has proposed contract languages allowing the specification of obligations, permissions and prohibitions in business contracts. Business processes that cross-organizational boundaries can be specified in choreography and coordination languages but these do not provide appropriate abstractions for contract constraints. In this paper, we examine the transformation of contract constraints in a business contract language into expressions in a choreography language. An example cross-organizational process is presented, along with a specification of the process in a choreography language and a specification of a set of contract conditions for the process in a business contract language. The contract terms are then translated into choreography expressions that govern the process to ensure compliance. Subsequent discussion explores a number of business and technology issues related to the results. We conclude that cross-organizational business processes can be monitored and enforced according to business contract specifications through the transformation of a contract definition to constraints on process behavior.

As the adoption of the Service Oriented Architecture paradigm has dramatically increased over the past few years, proper coordination of loosely coupled services becomes an important issue when building state-of-the-art applications. This coordination is typically organized through orchestration (requiring a central coordinating entity) or through choreographies. While the latter approach allows for a fully distributed coordination, the need also arises for a distributed conformance check, ensuring that each participant of the choreography behaves according to the general choreography.

In this paper, a formalism is presented to ensure this conformance at design time, with possible extensions to deploy time and to runtime conformance checking. This formalism is referred to as the piX-model and it will be shown that the approach taken is inherently less complex, both in time and space, than the conventional π-calculus-based approach, whilst offering the same conformance guarantees. This gain in performance allows for a small design turnaround time, and also opens the avenue to runtime conformance checking by resource constrained devices.

Robot dance is an important topic in robotics. Conventional robot dance systems mainly rely on beats or rhythms of music; however, these conventional systems suffer from limited dance styles and less action novelty. In this paper, we instead develop a humanoid robot dance system driven by musical structures and emotions. In the proposed system, a musical phrase and a dance phrase are considered as the basic structural units of music and dance, respectively. A musical phrasing algorithm based on music theories is created to divide a piece of music into a sequence of phrases. When the emotion of each phrase has been recognized, an emotion sequence can be established. Meanwhile, a hidden Markov model (HMM) matches a dance phrase sequence to the emotion sequence. In particular, several concepts of the “chance method” created by choreographer Merce Cunningham are adopted to guide our robot dance system; thus, a dance phrase is choreographed by randomly selecting and combining a number of actions from a predesigned action library. Based on the approach, one music can generate diverse robotic dance motions, showing the novelty and diversity of robot dance. The experiments on our humanoid robot “Alpha1 Pro” show that our robot can do a good job dancing to music according to musical structures and emotions and can be well accepted by various people.

Using the variational method Chenciner and Montgomery proved the existence of a new periodic solution of figure-eight shape to the planar three-body problem. Since then, number of periodic solutions have been found as minimizers of variational formulation of the n-body problem in various different settings.

In this paper we survey the theory and present some numerical algorithm for finding periodic solutions by based on the theory.