Existence of periodic orbits near heteroclinic connections

We consider a potential $W:R^m\rightarrow R$ with two different global minima $a_-, a_+$ and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian system \begin{equation} \ddot{u}=W_u(u), \hskip 2cm (1) \end{equation} has a family of $T$-periodic solutions $u^T$ which, along a sequence $T_j\rightarrow+\infty$, converges locally to a heteroclinic solution that connects $a_-$ to $a_+$. We then focus on the elliptic system \begin{equation} \Delta u=W_u(u),\;\; u:R^2\rightarrow R^m, \hskip 2cm (2) \end{equation} that we interpret as an infinite dimensional analogous of (1), where $x$ plays the role of time and $W$ is replaced by the action functional $$J_R(u)=\int_R\Bigl(\frac{1}{2}\vert u_y\vert^2+W(u)\Bigr)dy.$$ We assume that $J_R$ has two different global minimizers $\bar{u}_-, \bar{u}_+:R\rightarrow R^m$ in the set of maps that connect $a_-$ to $a_+$. We work in a symmetric context and prove, via a minimization procedure, that (2) has a family of solutions $u^L:R^2\rightarrow R^m$, which is $L$-periodic in $x$, converges to $a_\pm$ as $y\rightarrow\pm\infty$ and, along a sequence $L_j\rightarrow+\infty$, converges locally to a heteroclinic solution that connects $\bar{u}_-$ to $\bar{u}_+$.Comment: 36 pages, 4 figure

On the existence of connecting orbits for critical values of the energy

We consider an open connected set Ω and a smooth potential U which is positive in Ω and vanishes on â\u88\u82Ω. We study the existence of orbits of the mechanical system u¨=Ux(u), that connect different components of â\u88\u82Ω and lie on the zero level of the energy. We allow that â\u88\u82Ω contains a finite number of critical points of U. The case of symmetric potential is also considered

Generalization of a method by Mossotti for initial orbit determination

Here we revisit an initial orbit determination method introduced by O. F. Mossotti employing four geocentric sky-plane observations and a linear equation to compute the angular momentum of the observed body. We then extend the method to topocentric observations, yielding a quadratic equation for the angular momentum. The performance of the two versions are compared through numerical tests with synthetic asteroid data using different time intervals between consecutive observations and different astrometric errors. We also show a comparison test with Gauss's method using simulated observations with the expected cadence of the VRO-LSST telescope.Comment: 22 pages, 9 figure

The sacral chordoma margin

[Objective]: Aim of the manuscript is to discuss how to improve margins in sacral chordoma. [Background]: Chordoma is a rare neoplasm, arising in half cases from the sacrum, with reported local failure in >50% after surgery. [Methods]: A multidisciplinary meeting of the “Chordoma Global Consensus Group” was held in Milan in 2017, focusing on challenges in defining and achieving optimal margins in chordoma with respect to surgery, definitive particle radiation therapy (RT) and medical therapies. This review aims to report on the outcome of the consensus meeting and to provide a summary of the most recent evidence in this field. Possible new ways forward, including on-going international clinical studies, are discussed. [Results]: En-bloc tumor-sacrum resection is the cornerstone of treatment of primary sacral chordoma, aiming to achieve negative microscopic margins. Radical definitive particle therapy seems to offer a similar outcome compared to surgery, although confirmation in comparative trials is lacking; besides there is still a certain degree of technical variability across institutions, corresponding to different fields of treatment and different tumor coverage. To address some of these questions, a prospective, randomized international study comparing surgery versus definitive high-dose RT is ongoing. Available data do not support the routine use of any medical therapy as (neo)adjuvant/cytoreductive treatment. [Conclusion]: Given the significant influence of margins status on local control in patients with primary localized sacral chordoma, the clear definition of adequate margins and a standard local approach across institutions for both surgery and particle RT is vital for improving the management of these patients

Bone sarcomas: ESMO–EURACAN–GENTURIS–ERN PaedCan Clinical Practice Guideline for diagnosis, treatment and follow-up

Production costs have been covered by ESMO from central funds

Classical Methods of Orbit Determination

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