Complexity and integrability in 4D bi-rational maps with two invariants

In this letter we give fourth-order autonomous recurrence relations with two invariants, whose degree growth is cubic or exponential. These examples contradict the common belief that maps with sufficiently many invariants can have at most quadratic growth. Cubic growth may reflect the existence of non-elliptic fibrations of invariants, whereas we conjecture that the exponentially growing cases lack the necessary conditions for the applicability of the discrete Liouville theorem.Comment: 16 pages, 2 figure

Formulation and performance of variational integrators for rotating bodies

Variational integrators are obtained for two mechanical systems whose configuration spaces are, respectively, the rotation group and the unit sphere. In the first case, an integration algorithm is presented for Euler’s equations of the free rigid body, following the ideas of Marsden et al. (Nonlinearity 12:1647–1662, 1999). In the second example, a variational time integrator is formulated for the rigid dumbbell. Both methods are formulated directly on their nonlinear configuration spaces, without using Lagrange multipliers. They are one-step, second order methods which show exact conservation of a discrete angular momentum which is identified in each case. Numerical examples illustrate their properties and compare them with existing integrators of the literature

Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a discretization technique of Veselov. The general idea for these variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators

Nonlinear Supersymmetry as a Hidden Symmetry

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Dimer Models and Integrable Systems

We explore various aspects of the correspondence between dimer models and integrable systems recently introduced by Goncharov and Kenyon. Dimer models give rise to relativistic integrable systems that match those arising from 5d N=1 gauge theories studied by Nekrasov. We apply the correspondence to dimer models associated to the Y^{p,0} geometries, showing that they give rise to the relativistic generalization of the periodic Toda chain originally studied by Ruijsenaars. The correspondence reduces the calculation of all conserved charges to a straightforward combinatorial problem of enumerating non-intersecting paths in the dimer model. We show how the usual periodic Toda chain emerges in the non-relativistic limit and how the Lax operator corresponds to the Kasteleyn matrix of the dimer model. We discuss how the dimer models for general Y^{p,q} manifolds give rise to other relativistic integrable systems, generalizing the periodic Toda chain and construct the integrable systems for general Y^{p,p} explicitly. The impurities introduced in the construction of Y^{p,q} quivers are identified with impurities in twisted sl(2) XXZ spin chains. Finally we discuss how the physical concept of higgsing a dimer model provides an efficient method for producing new integrable systems starting from known ones. We illustrate this idea by constructing the integrable systems for higgsings of Y^{4,0}.Comment: 29 pages, 16 figures. v2: typos fixe

Suicide among persons with childhood leukaemia in Slovenia

Pri osebah, ki so v otroštvu zbolele za rakom, so pogosto prisotne telesne in psihosocialne posledice bolezni ter njenega zdravljenja. Mnoge raziskave so pokazale, da je pri osebah z izkušnjo raka v otroštvu depresivnost in samomorilno vedenje močneje izraženo. V naši raziskavi smo proučili pojavljanje samomorov pri osebah, ki so v otroštvu zbolele za levkemijo, v primerjavi s splošno populacijo v Sloveniji, v obdobju 1978–2010. Pričakovano število samomorov smo izračunali na osnovi kontrolne skupine posameznikov iz splošne populacije, ki je bila s skupino preiskovancev, tj. oseb, ki so v otroštvu zbolele za levkemijo, izenačena po spolu, starosti ob začetku opazovanja, letu začetka opazovanja in dolžini opazovanja. Raziskava je pokazala, da med tistimi, ki so v otroštvu zboleli za levkemijo, v letih 1978–2010 nobena oseba ni storila samomora, kar se statistično značilno ne razlikuje od pričakovanega števila samomorov (0,448) v primerljivi splošni populaciji v Sloveniji. Ugotovitve raziskave nakazujejo, da kljub znano bolj izraženem samomorilnem vedenju med preživelimi raka v otroštvu v Sloveniji v primerjavi s splošno populacijo pojavljanje samomorov pri osebah, zbolelih za levkemijo v otroštvu, ni pogostejše kot v splošni populaciji.Persons with childhood leukaemia often suffer from physical and psychosocial consequences of the disease and its treatment. Several studies have shown that depression and suicidal behaviour are expressed strongly in persons with a childhood cancer experience. In our study, we researched the occurrence of suicides among persons with childhood leukaemia compared to the general population in Slovenia in the period 1978–2010. The expected number of suicides was calculated based on the control group of individuals from the general population with the same gender, age at the beginning of observation, starting year and duration of observation as the research group, thus group of persons with childhood cancer. The study showed that none of the persons with childhood cancer committed suicide in the period 1978-2010, which is not statistically different from the expected number of suicides (0.448) in comparison with the general population in Slovenia. The findings of this study indicate that, despite the significantly increased expression of suicidal behaviour among survivors of childhood leukaemia in Slovenia compared to the general population, suicides do not occur more often among people with childhood leukaemia than among the general population

Markov numbers, Mather’s beta function and stable norm

Fock (1997 (arXiv:dg-ga/9702018v3); Fock et al 2007 Handbook of Teichmüller Theory (Zürich: European Mathematical Society)) introduced an interesting function related to Markov numbers. We explain its relation to Federer–Gromov's stable norm and Mather's -function, and use this to study its properties. We prove that and its natural generalisations are differentiable at every irrational x and non-differentiable otherwise, by exploiting the relation with length of simple closed geodesics on the punctured or one-holed tori with the hyperbolic metric and the results by Bangert (1994 Calculus Variations Partial Differ. Equ. 2 49–63) and McShane–Rivin (1995 C. R. Acad. Sci. Paris I 320)