Asymptotic integration of nonlinear systems of differential equations whose phase portrait is foliated on invariant tori

We consider the class of autonomous systems $\dot x=f(x)$, where $x \in {\bf R}^{2n}$, $f \in C^1({\bf R}^{2n})$ whose phase portrait is a Cartesian product of $n$ two-dimensional {\em centres}. We also consider perturbations of this system, namely $\dot x=f(x)+g(t,x)$, where $g \in C^1({\bf R}\times{\bf R}^{2n})$ and $g$ is asymptotically small, that is $g\Rightarrow 0$ as $t\to +\infty$ uniformly with respect to $x$. The rate of decrease of $g$ is assumed to be $t^{-p}$ where $p>1$. We prove under this conditions the existence of bounded solutions of the perturbed system and discuss their convergence to solutions of the unperturbed system. This convergence depends on $p$. Moreover, we show that the original unperturbed system may be reduced to the form $\dot r=0$, $\dot\theta=A(r)$, and taking $r\in {\bf R}^m_{+}$, $\theta\in {\bf T}^n$, where ${\bf T}^n$ denotes the $n$-dimensional torus, we investigate the more general case of systems whose phase portrait is foliated on invariant tori. We notice that integrable Hamiltonian systems are of the same nature. We give also several examples, showing that the conditions of our theorems cannot be improved

Lieb-Thirring inequalities on some manifolds

We prove Lieb-Thirring inequalities with improved constants on the two-dimensional sphere and the two-dimensional torus. In the one-dimensional periodic case we obtain a simultaneous bound for the negative trace and the number of negative eigenvalues

Case studies in reconstruction efficiency of current distribution in CICC's by self field measurements

The measurements of the self magnetic field by means of Hall sensors (HS) in the vicinity of a superconducting cable-in-conduit conductor (CICC) is often used to study current distribution effects. It is possible that current imbalance may affect the performance of CICC's and therefore knowledge of the current distribution is needed. Recently a model was presented to approximate the current distribution inside a conductor. Basically, the inverse problem must be solved in which the input data usually are the experimentally measured values of the local magnetic field, the location and orientation of the HS's and the geometry of the line or segment currents. All these, together with the adopted algorithm, determine the accuracy of the reconstruction procedure. In the present study the impact of two basic orientations of the HS: polar-symmetric and plane-parallel on the current reconstruction efficiency is performed for the analytical model developed in Twente. For the case study, a 36 strands CICC and a mock-up conductor are considered. The influence of the experimental errors and geometrical errors on the model output is also investigated

Stoponium Search at Photon Linear Collider

In some supersymmetric extensions of the Standard Model fairly light superpartner of t-quark is predicted, which may form bound states ({\it stoponiums}) under certain conditions. We study prospects of search for stoponium at the future Photon Linear Collider. It is found that this machine could be the best machine for discovery and study of these resonances at some scenarios of supersymmetric extension of the Standard Model. In particular, if the $hh$ decay channel is dominant stoponium could be observed at the beginning of PLC run with collision energy tuned at the stoponium mass. If this channel is kinematically closed stoponium could be discovered in $gg$, $\gamma\gamma$ and $ZZ$ decay channels but higher statistics are needed. Effects of the stoponium-Higgs mixing and degeneracy are briefly discussed.Comment: 11 pages, 2 figures added, corrections taken into account result in increasing of signal significanc

Sharp estimates for the number of degrees of freedom for the damped-driven 2D Navier--Stokes equations

We derive upper bounds for the number of asymptotic degrees (determining modes and nodes) of freedom for the two-dimensional Navier--Stokes system and Navier-Stokes system with damping. In the first case we obtain the previously known estimates in an explicit form, which are larger than the fractal dimension of the global attractor. However, for the Navier--Stokes system with damping our estimates for the number of the determining modes and nodes are comparable to the sharp estimates for the fractal dimension of the global attractor. Our investigation of the damped-driven 2D Navier--Stokes system is inspired by the Stommel--Charney barotropic model of ocean circulation where the damping represents the Rayleigh friction. We remark that our results equally apply to the Stommel--Charney model

On the Domain of Analyticity and Small Scales for the Solutions of the Damped-driven 2D Navier-Stokes Equations

We obtain a logarithmically sharp estimate for the space-analyticity radius of the solutions of the damped-driven 2D Navier-Stokes equations with periodic boundary conditions and relate this to the small scales in this system. This system is inspired by the Stommel--Charney barotropic ocean circulation model

A Quality and Cost Approach for Comparison of Small-World Networks

We propose an approach based on analysis of cost-quality tradeoffs for comparison of efficiency of various algorithms for small-world network construction. A number of both known in the literature and original algorithms for complex small-world networks construction are shortly reviewed and compared. The networks constructed on the basis of these algorithms have basic structure of 1D regular lattice with additional shortcuts providing the small-world properties. It is shown that networks proposed in this work have the best cost-quality ratio in the considered class.Comment: 27 pages, 16 figures, 1 tabl