Classical Dynamical Systems from q-algebras:"cluster" variables and explicit solutions

A general procedure to get the explicit solution of the equations of motion for N-body classical Hamiltonian systems equipped with coalgebra symmetry is introduced by defining a set of appropriate collective variables which are based on the iterations of the coproduct map on the generators of the algebra. In this way several examples of N-body dynamical systems obtained from q-Poisson algebras are explicitly solved: the q-deformed version of the sl(2) Calogero-Gaudin system (q-CG), a q-Poincare' Gaudin system and a system of Ruijsenaars type arising from the same (non co-boundary) q-deformation of the (1+1) Poincare' algebra. Also, a unified interpretation of all these systems as different Poisson-Lie dynamics on the same three dimensional solvable Lie group is given.Comment: 19 Latex pages, No figure

The Ljapunov-Schmidt reduction for some critical problems

This is a survey about the application of the Ljapunov-Schmidt reduction for some critical problems

Bubble concentration on spheres for supercritical elliptic problems

We consider the supercritical Lane-Emden problem (P_\eps)\qquad -\Delta v= |v|^{p_\eps-1} v \ \hbox{in}\ \mathcal{A} ,\quad u=0\ \hbox{on}\ \partial\mathcal{A} where $\mathcal A$ is an annulus in \rr^{2m}, $m\ge2$ and p_\eps={(m+1)+2\over(m+1)-2}-\eps, \eps>0. We prove the existence of positive and sign changing solutions of (P_\eps) concentrating and blowing-up, as \eps\to0, on $(m-1)-$dimensional spheres. Using a reduction method (see Ruf-Srikanth (2010) J. Eur. Math. Soc. and Pacella-Srikanth (2012) arXiv:1210.0782)we transform problem (P_\eps) into a nonhomogeneous problem in an annulus \mathcal D\subset \rr^{m+1} which can be solved by a Ljapunov-Schmidt finite dimensional reduction

Gaudin Models and Bending Flows: a Geometrical Point of View

In this paper we discuss the bihamiltonian formulation of the (rational XXX) Gaudin models of spin-spin interaction, generalized to the case of sl(r)-valued spins. In particular, we focus on the homogeneous models. We find a pencil of Poisson brackets that recursively define a complete set of integrals of the motion, alternative to the set of integrals associated with the 'standard' Lax representation of the Gaudin model. These integrals, in the case of su(2), coincide wih the Hamiltonians of the 'bending flows' in the moduli space of polygons in Euclidean space introduced by Kapovich and Millson. We finally address the problem of separability of these flows and explicitly find separation coordinates and separation relations for the r=2 case.Comment: 27 pages, LaTeX with amsmath and amssym

New Formula for the Eigenvectors of the Gaudin Model in the sl(3) Case

We propose new formulas for eigenvectors of the Gaudin model in the \sl(3) case. The central point of the construction is the explicit form of some operator P, which is used for derivation of eigenvalues given by the formula $| w_1, w_2) = \sum_{n=0}^\infty P^n/n! | w_1, w_2,0>$, where $w_1$, $w_2$ fulfil the standard well-know Bethe Ansatz equations

Blow-up solutions for linear perturbations of the Yamabe equation

For a smooth, compact Riemannian manifold (M,g) of dimension N \geg 3, we are interested in the critical equation $$\Delta_g u+(N-2/4(N-1) S_g+\epsilon h)u=u^{N+2/N-2} in M, u>0 in M,$$ where \Delta_g is the Laplace--Beltrami operator, S_g is the Scalar curvature of (M,g), $h\in C^{0,\alpha}(M)$, and $\epsilon$ is a small parameter

Convergence of vector bundles with metrics of Sasaki-type

If a sequence of Riemannian manifolds, $X_i$, converges in the pointed Gromov-Hausdorff sense to a limit space, $X_\infty$, and if $E_i$ are vector bundles over $X_i$ endowed with metrics of Sasaki-type with a uniform upper bound on rank, then a subsequence of the $E_i$ converges in the pointed Gromov-Hausdorff sense to a metric space, $E_\infty$. The projection maps $\pi_i$ converge to a limit submetry $\pi_\infty$ and the fibers converge to its fibers; the latter may no longer be vector spaces but are homeomorphic to $\R^k/G$, where $G$ is a closed subgroup of $O(k)$ ---called the {\em wane group}--- that depends on the basepoint and that is defined using the holonomy groups on the vector bundles. The norms $\mu_i=\|\cdot\|_i$ converges to a map $\mu_{\infty}$ compatible with the re-scaling in $\R^k/G$ and the $\R$-action on $E_i$ converges to an $\R-$action on $E_{\infty}$ compatible with the limiting norm. In the special case when the sequence of vector bundles has a uniform lower bound on holonomy radius (as in a sequence of collapsing flat tori to a circle), the limit fibers are vector spaces. Under the opposite extreme, e.g. when a single compact $n$-dimensional manifold is re-scaled to a point, the limit fiber is $\R^n/H$ where $H$ is the closure of the holonomy group of the compact manifold considered. An appropriate notion of parallelism is given to the limiting spaces by considering curves whose length is unchanged under the projection. The class of such curves is invariant under the $\R$-action and each such curve preserves norms. The existence of parallel translation along rectifiable curves with arbitrary initial conditions is also exhibited. Uniqueness is not true in general, but a necessary condition is given in terms of the aforementioned wane groups $G$.Comment: 44 pages, 1 figure, in V.2 added Theorem E and Section 4 on parallelism in the limit space

Progress on DC-DC Converters for a Silicon Tracker for the sLHC Upgrade

There is a need for DC-DC converters which can operate in the extremely harsh environment of the sLHC Si Tracker. The environment requires radiation qualification to a total ionizing radiation dose of 50 Mrad and a displacement damage fluence of 5 x 1014 /cm2 of 1 MeV equivalent neutrons. In addition a static magnetic field of 2 Tesla or greater prevents the use of any magnetic components or materials. In February 2007 an Enpirion EN5360 was qualified for the sLHC radiation dosage but the converter has an input voltage limited to a maximum of 5.5V. From a systems point of view this input voltage was not sufficient for the application. Commercial LDMOS FETs have developed using a 0.25 μm process which provided a 12 volt input and were still radiation hard. These results are reported here and in previous papers. Plug in power cards with ×10 voltage ratio are being developed for testing the hybrids with ABCN chips. These plug-in cards have air coils but use commercial chips that are not designed to be radiation hard. This development helps in evaluating system noise and performance. GaN FETs are tested for radiation hardness to ionizing radiation and displacement damage and preliminary results are given

Classical approximation to quantum cosmological correlations

We investigate up to which order quantum effects can be neglected in calculating cosmological correlation functions after horizon exit. As a toy model, we study $\phi^3$ theory on a de Sitter background for a massless minimally coupled scalar field $\phi$. We find that for tree level and one loop contributions in the quantum theory, a good classical approximation can be constructed, but for higher loop corrections this is in general not expected to be possible. The reason is that loop corrections get non-negligible contributions from loop momenta with magnitude up to the Hubble scale H, at which scale classical physics is not expected to be a good approximation to the quantum theory. An explicit calculation of the one loop correction to the two point function, supports the argument that contributions from loop momenta of scale $H$ are not negligible. Generalization of the arguments for the toy model to derivative interactions and the curvature perturbation leads to the conclusion that the leading orders of non-Gaussian effects generated after horizon exit, can be approximated quite well by classical methods. Furthermore we compare with a theorem by Weinberg. We find that growing loop corrections after horizon exit are not excluded, even in single field inflation.Comment: 44 pages, 1 figure; v2: corrected errors, added references, conclusions unchanged; v3: added section in which we compare with stochastic approach; this version matches published versio