An Exactly Conservative Integrator for the n-Body Problem

The two-dimensional n-body problem of classical mechanics is a non-integrable Hamiltonian system for n > 2. Traditional numerical integration algorithms, which are polynomials in the time step, typically lead to systematic drifts in the computed value of the total energy and angular momentum. Even symplectic integration schemes exactly conserve only an approximate Hamiltonian. We present an algorithm that conserves the true Hamiltonian and the total angular momentum to machine precision. It is derived by applying conventional discretizations in a new space obtained by transformation of the dependent variables. We develop the method first for the restricted circular three-body problem, then for the general two-dimensional three-body problem, and finally for the planar n-body problem. Jacobi coordinates are used to reduce the two-dimensional n-body problem to an (n-1)-body problem that incorporates the constant linear momentum and center of mass constraints. For a four-body choreography, we find that a larger time step can be used with our conservative algorithm than with symplectic and conventional integrators.Comment: 17 pages, 3 figures; to appear in J. Phys. A.: Math. Ge

Palaeoecological aspects of an Ukrainian Upper Holocene chernozem

A physicochemical, palynological and chronological analysis of a soil profile corresponding to a Haplic Chernozem (FAO, 2015) developed in the Ukrainian steppe allow the interpretation of recent environmental changes that have conditioned its formation. Uniformed under the blackening process, its surface horizon dark color contrasts with yellowish color loessic parental material; it is decarbonated and the organic carbon content on the surface is 2.24%.The texture is silty in surface but sandy in lower horizon denoting a clear wind selection and an energy change in aeolian sedimentation processes. The clays present similar values in the three analyzed horizons. The presence of two discontinuities in the profile has made it possible to distinguish a very sandy blackened horizon of 2500 +/- 25cal BP chronologies from another silty surface and black chromas of age 1336–1256 cal BP. A total of fifteen pollen types have been identified; superficial horizon (A11) has a high presence of pollen of the Amaranthaceae type, the Poaceae are the most abundant and Quercus gender is identified. The sandy horizon (2A/B) shows Poaceas, Pinus, and Oleaceaspresence together pollen of Rosaceas type (14%). Pollen data reveal vegetative changes in the three horizons with the presence of even non-existent species today, linked both to recent anthropic-climatic and holocene-type changes on a millennial scale since the last glacial period

Fictional Characters, Transparency, and Experiential Sharing

How can providing less textual information about a fictional character make his or her mind more transparent and accessible to the reader? This is the question that emerges from an empirical study of reader response conducted by Kotovych et al. Taking my cue from this study, I discuss the role of implied information in readers’ interactions with characters in prose fiction. This is the textual strategy I call ‘character-centered implicature.’ I argue that the inferential work cued by implicature creates an intersubjective dynamic analogous to what philosophers Zahavi and Rochat discuss under the heading of ‘experiential sharing.’ Effectively, readers complement the textual evocation of mind by drawing on their own past experiences, which leads to a distinctive first-person plural (‘we’) perspective—a sharing of cognitive resources that is responsible for the perceived transparency of the character’s mind. While this experiential sharing may result in empathetic perspective-taking, not all instances of empathy for fictional characters involve sharing