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

Generalized binomial distribution in photon statistics

The photon-number distribution between two parts of a given volume is found for an arbitrary photon statistics. This problem is related to the interaction of a light beam with a macroscopic device, for example a diaphragm, that separates the photon flux into two parts with known probabilities. To solve this problem, a Generalized Binomial Distribution (GBD) is derived that is applicable to an arbitrary photon statistics satisfying probability convolution equations. It is shown that if photons obey Poisson statistics then the GBD is reduced to the ordinary binomial distribution, whereas in the case of Bose-Einstein statistics the GBD is reduced to the Polya distribution. In this case, the photon spatial distribution depends on the phase-space volume occupied by the photons. This result involves a photon bunching effect, or collective behavior of photons that sharply differs from the behavior of classical particles. It is shown that the photon bunching effect looks similar to the quantum interference effect.Comment: 8 pages, 4 figure

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