Chaos and dynamical trends in barred galaxies: bridging the gap between N-body simulations and time-dependent analytical models

Self-consistent N-body simulations are efficient tools to study galactic dynamics. However, using them to study individual trajectories (or ensembles) in detail can be challenging. Such orbital studies are important to shed light on global phase space properties, which are the underlying cause of observed structures. The potentials needed to describe self-consistent models are time-dependent. Here, we aim to investigate dynamical properties (regular/chaotic motion) of a non-autonomous galactic system, whose time-dependent potential adequately mimics certain realistic trends arising from N-body barred galaxy simulations. We construct a fully time-dependent analytical potential, modeling the gravitational potentials of disc, bar and dark matter halo, whose time-dependent parameters are derived from a simulation. We study the dynamical stability of its reduced time-independent 2-degrees of freedom model, charting the different islands of stability associated with certain orbital morphologies and detecting the chaotic and regular regions. In the full 3-degrees of freedom time-dependent case, we show representative trajectories experiencing typical dynamical behaviours, i.e., interplay between regular and chaotic motion for different epochs. Finally, we study its underlying global dynamical transitions, estimating fractions of (un)stable motion of an ensemble of initial conditions taken from the simulation. For such an ensemble, the fraction of regular motion increases with time.Comment: 17 pages, 11 figures (revised version, accepted for publication in Mon. Not. R. Astron. Soc.

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XVI. On some relations between the optical and the electrical qualities of metals

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The clinical course of bone metastases from breast cancer.

All patients with carcinoma of the breast seen in this Unit since 1970 were reviewed to study the incidence, prognosis, morbidity and response to treatment of bone metastases. The biological characteristics of the primary tumour were compared in patients relapsing first in bone or liver. Sixty-nine percent of patients dying with breast cancer had bone metastases and bone was the commonest site of first distant relapse. Bone relapse was more common in receptor positive or well differentiated (grade 1) tumours. The median survival was 24 months in those with disease apparently confined to the skeleton compared with 3 months after first relapse in liver. Ten percent of patients with breast cancer developed hypercalcaemia. All had metastatic disease and 85% had widespread skeletal involvement. Fifteen percent of patients with disease confined to the skeleton developed hypercalcaemia. The response in bone to primary endocrine therapy, and chemotherapy, was apparently less than the overall response achieved. A large proportion had apparently static disease reflecting the insensitivity of the UICC assessment criteria. The duration of survival in these patients was similar to responding patients, suggesting a tumour response may occur in the absence of discernable radiological evidence of healing

3(Amino-1,1-hydroxypropylidene) bisphosphonate (APD) for hypercalcaemia of breast cancer.

The effect of a single dose of APD on hypercalcaemia has been studied in advanced breast cancer. Twenty-five patients were rehydrated intravenously for 48 h. Twenty-three remained hypercalcaemic and received 5-15 mg APD as a 2 h infusion. Eighteen patients achieved normocalcaemia, 15 after a dose of less than or equal to 15 mg. One patient died within 24 h from rapidly advancing disease and 4 remained hypercalcaemic. Urinary calcium excretion increased during rehydration as glomerular function improved and tubular reabsorption of calcium fell. After APD, calcium excretion fell to normal in 22/24 patients reflecting inhibition of bone resorption. Hydroxyproline excretion remained high. The effect of a single dose of APD on hypercalcaemia lasted a median of 11 days (range 7-17). Transient fever occurred in 2 patients, but there were no other side effects. The possibility of long-term control of osteolysis using a 2 weekly schedule of APD administration is now being studied