The Lie-Poisson structure of the reduced n-body problem

The classical n-body problem in d-dimensional space is invariant under the Galilean symmetry group. We reduce by this symmetry group using the method of polynomial invariants. As a result we obtain a reduced system with a Lie-Poisson structure which is isomorphic to sp(2n-2), independently of d. The reduction preserves the natural form of the Hamiltonian as a sum of kinetic energy that depends on velocities only and a potential that depends on positions only. Hence we proceed to construct a Poisson integrator for the reduced n-body problem using a splitting method.Comment: 26 pages, 2 figure

Projective dynamics and first integrals

We present the theory of tensors with Young tableau symmetry as an efficient computational tool in dealing with the polynomial first integrals of a natural system in classical mechanics. We relate a special kind of such first integrals, already studied by Lundmark, to Beltrami's theorem about projectively flat Riemannian manifolds. We set the ground for a new and simple theory of the integrable systems having only quadratic first integrals. This theory begins with two centered quadrics related by central projection, each quadric being a model of a space of constant curvature. Finally, we present an extension of these models to the case of degenerate quadratic forms.Comment: 39 pages, 2 figure

Pauli graphs, Riemann hypothesis, Goldbach pairs

Let consider the Pauli group $\mathcal{P}_q=$ with unitary quantum generators $X$ (shift) and $Z$ (clock) acting on the vectors of the $q$-dimensional Hilbert space via $X|s> =|s+1>$ and $Z|s> =\omega^s |s>$, with $\omega=\exp(2i\pi/q)$. It has been found that the number of maximal mutually commuting sets within $\mathcal{P}_q$ is controlled by the Dedekind psi function $\psi(q)=q \prod_{p|q}(1+\frac{1}{p})$ (with $p$ a prime) \cite{Planat2011} and that there exists a specific inequality $\frac{\psi (q)}{q}>e^{\gamma}\log \log q$, involving the Euler constant $\gamma \sim 0.577$, that is only satisfied at specific low dimensions $q \in \mathcal {A}=\{2,3,4,5,6,8,10,12,18,30\}$. The set $\mathcal{A}$ is closely related to the set $\mathcal{A} \cup \{1,24\}$ of integers that are totally Goldbach, i.e. that consist of all primes $p2$) is equivalent to Riemann hypothesis. Introducing the Hardy-Littlewood function $R(q)=2 C_2 \prod_{p|n}\frac{p-1}{p-2}$ (with $C_2 \sim 0.660$ the twin prime constant), that is used for estimating the number $g(q) \sim R(q) \frac{q}{\ln^2 q}$ of Goldbach pairs, one shows that the new inequality $\frac{R(N_r)}{\log \log N_r} \gtrapprox e^{\gamma}$ is also equivalent to Riemann hypothesis. In this paper, these number theoretical properties are discusssed in the context of the qudit commutation structure.Comment: 11 page

On the convex central configurations of the symmetric (ℓ + 2)-body problem

For the 4-body problem there is the following conjecture: Given arbitrary positive masses, the planar 4-body problem has a unique convex central configuration for each ordering of the masses on its convex hull. Until now this conjecture has remained open. Our aim is to prove that this conjecture cannot be extended to the (ℓ + 2)-body problem with ℓ ⩾ 3. In particular, we prove that the symmetric (2n + 1)-body problem with masses m1 = … = m2n−1 = 1 and m2n = m2n+1 = m sufficiently small has at least two classes of convex central configuration when n = 2, five when n = 3, and four when n = 4. We conjecture that the (2n + 1)-body problem has at least n classes of convex central configurations for n > 4 and we give some numerical evidence that the conjecture can be true. We also prove that the symmetric (2n + 2)-body problem with masses m1 = … = m2n = 1 and m2n+1 = m2n+2 = m sufficiently small has at least three classes of convex central configuration when n = 3, two when n = 4, and three when n = 5. We also conjecture that the (2n + 2)-body problem has at least [(n +1)/2] classes of convex central configurations for n > 5 and we give some numerical evidences that the conjecture can be true

Statistical Mechanics of Vacancy and Interstitial Strings in Hexagonal Columnar Crystals

Columnar crystals contain defects in the form of vacancy/interstitial loops or strings of vacancies and interstitials bounded by column ``heads'' and ``tails''. These defect strings are oriented by the columnar lattice and can change size and shape by movement of the ends and forming kinks along the length. Hence an analysis in terms of directed living polymers is appropriate to study their size and shape distribution, volume fraction, etc. If the entropy of transverse fluctuations overcomes the string line tension in the crystalline phase, a string proliferation transition occurs, leading to a supersolid phase. We estimate the wandering entropy and examine the behaviour in the transition regime. We also calculate numerically the line tension of various species of vacancies and interstitials in a triangular lattice for power-law potentials as well as for a modified Bessel function interaction between columns as occurs in the case of flux lines in type-II superconductors or long polyelectrolytes in an ionic solution. We find that the centered interstitial is the lowest energy defect for a very wide range of interactions; the symmetric vacancy is preferred only for extremely short interaction ranges.Comment: 22 pages (revtex), 15 figures (encapsulated postscript

Phase operators, temporally stable phase states, mutually unbiased bases and exactly solvable quantum systems

We introduce a one-parameter generalized oscillator algebra A(k) (that covers the case of the harmonic oscillator algebra) and discuss its finite- and infinite-dimensional representations according to the sign of the parameter k. We define an (Hamiltonian) operator associated with A(k) and examine the degeneracies of its spectrum. For the finite (when k < 0) and the infinite (when k > 0 or = 0) representations of A(k), we construct the associated phase operators and build temporally stable phase states as eigenstates of the phase operators. To overcome the difficulties related to the phase operator in the infinite-dimensional case and to avoid the degeneracy problem for the finite-dimensional case, we introduce a truncation procedure which generalizes the one used by Pegg and Barnett for the harmonic oscillator. This yields a truncated generalized oscillator algebra A(k,s), where s denotes the truncation order. We construct two types of temporally stable states for A(k,s) (as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of A(k,s)). Two applications are considered in this article. The first concerns physical realizations of A(k) and A(k,s) in the context of one-dimensional quantum systems with finite (Morse system) or infinite (Poeschl-Teller system) discrete spectra. The second deals with mutually unbiased bases used in quantum information.Comment: Accepted for publication in Journal of Physics A: Mathematical and Theoretical as a pape

Predicting the Impact of Climate Change on Threatened Species in UK Waters

Global climate change is affecting the distribution of marine species and is thought to represent a threat to biodiversity. Previous studies project expansion of species range for some species and local extinction elsewhere under climate change. Such range shifts raise concern for species whose long-term persistence is already threatened by other human disturbances such as fishing. However, few studies have attempted to assess the effects of future climate change on threatened vertebrate marine species using a multi-model approach. There has also been a recent surge of interest in climate change impacts on protected areas. This study applies three species distribution models and two sets of climate model projections to explore the potential impacts of climate change on marine species by 2050. A set of species in the North Sea, including seven threatened and ten major commercial species were used as a case study. Changes in habitat suitability in selected candidate protected areas around the UK under future climatic scenarios were assessed for these species. Moreover, change in the degree of overlap between commercial and threatened species ranges was calculated as a proxy of the potential threat posed by overfishing through bycatch. The ensemble projections suggest northward shifts in species at an average rate of 27 km per decade, resulting in small average changes in range overlap between threatened and commercially exploited species. Furthermore, the adverse consequences of climate change on the habitat suitability of protected areas were projected to be small. Although the models show large variation in the predicted consequences of climate change, the multi-model approach helps identify the potential risk of increased exposure to human stressors of critically endangered species such as common skate (Dipturus batis) and angelshark (Squatina squatina)

Sigma frequency dependent motor learning in Williams syndrome

Abstract There are two basic stages of fine motor learning: performance gain might occur during practice (online learning), and improvement might take place without any further practice (offline learning). Offline learning, also called consolidation, has a sleep-dependent stage in terms of both speed and accuracy of the learned movement. Sleep spindle or sigma band characteristics affect motor learning in typically developing individuals. Here we ask whether the earlier found, altered sigma activity in a neurodevelopmental disorder (Williams syndrome, WS) predicts motor learning. TD and WS participants practiced in a sequential finger tapping (FT) task for two days. Although WS participants started out at a lower performance level, TD and WS participants had a comparable amount of online and offline learning in terms of the accuracy of movement. Spectral analysis of WS sleep EEG recordings revealed that motor accuracy improvement is intricately related to WS-specific NREM sleep EEG features in the 8–16 Hz range profiles: higher 11–13.5 Hz z-transformed power is associated with higher offline FT accuracy improvement; and higher oscillatory peak frequencies are associated with lower offline accuracy improvements. These findings indicate a fundamental relationship between sleep spindle (or sigma band) activity and motor learning in WS

Motor Skills Enhance Procedural Memory Formation and Protect against Age-Related Decline

The ability to consolidate procedural memories declines with increasing age. Prior knowledge enhances learning and memory consolidation of novel but related information in various domains. Here, we present evidence that prior motor experience-in our case piano skills-increases procedural learning and has a protective effect against age-related decline for the consolidation of novel but related manual movements. In our main experiment, we tested 128 participants with a sequential finger-tapping motor task during two sessions 24 hours apart. We observed enhanced online learning speed and offline memory consolidation for piano players. Enhanced memory consolidation was driven by a strong effect in older participants, whereas younger participants did not benefit significantly from prior piano experience. In a follow up independent control experiment, this compensatory effect of piano experience was not visible after a brief offline period of 30 minutes, hence requiring an extended consolidation window potentially involving sleep. Through a further control experiment, we rejected the possibility that the decreased effect in younger participants was caused by training saturation. We discuss our results in the context of the neurobiological schema approach and suggest that prior experience has the potential to rescue memory consolidation from age-related cognitive decline