Comparison of averages of flows and maps

It is shown that in transient chaos there is no direct relation between averages in a continuos time dynamical system (flow) and averages using the analogous discrete system defined by the corresponding Poincare map. In contrast to permanent chaos, results obtained from the Poincare map can even be qualitatively incorrect. The reason is that the return time between intersections on the Poincare surface becomes relevant. However, after introducing a true-time Poincare map, quantities known from the usual Poincare map, such as conditionally invariant measure and natural measure, can be generalized to this case. Escape rates and averages, e.g. Liapunov exponents and drifts can be determined correctly using these novel measures. Significant differences become evident when we compare with results obtained from the usual Poincare map.Comment: 4 pages in Revtex with 2 included postscript figures, submitted to Phys. Rev.

Behavior of the Escape Rate Function in Hyperbolic Dynamical Systems

For a fixed initial reference measure, we study the dependence of the escape rate on the hole for a smooth or piecewise smooth hyperbolic map. First, we prove the existence and Holder continuity of the escape rate for systems with small holes admitting Young towers. Then we consider general holes for Anosov diffeomorphisms, without size or Markovian restrictions. We prove bounds on the upper and lower escape rates using the notion of pressure on the survivor set and show that a variational principle holds under generic conditions. However, we also show that the escape rate function forms a devil's staircase with jumps along sequences of regular holes and present examples to elucidate some of the difficulties involved in formulating a general theory.Comment: 21 pages. v2 differs from v1 only by additions to the acknowledgment

Diffusion in normal and critical transient chaos

In this paper we investigate deterministic diffusion in systems which are spatially extended in certain directions but are restricted in size and open in other directions, consequently particles can escape. We introduce besides the diffusion coefficient D on the chaotic repeller a coefficient ${\hat D}$ which measures the broadening of the distribution of trajectories during the transient chaotic motion. Both coefficients are explicitly computed for one-dimensional models, and they are found to be different in most cases. We show furthermore that a jump develops in both of the coefficients for most of the initial distributions when we approach the critical borderline where the escape rate equals the Liapunov exponent of a periodic orbit.Comment: 4 pages Revtex file in twocolumn format with 2 included postscript figure

Spectral degeneracy and escape dynamics for intermittent maps with a hole

We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron-Frobenius (transfer) operators when a small perturbation is applied near the intermittent fixed point. In particular, we describe precisely the scaling of the second eigenvalue with the perturbation size, provide upper and lower bounds, and demonstrate $L^1$ convergence of the positive part of the second eigenfunction to the ACIM as the perturbation goes to zero. This perturbation and associated eigenvalue scalings and convergence results are all compatible with Ulam's method and provide a formal explanation for the numerical behaviour of Ulam's method in this nonuniformly hyperbolic setting. The main results of the paper are illustrated with numerical computations.Comment: 34 page

Open Mushrooms: Stickiness revisited

We investigate mushroom billiards, a class of dynamical systems with sharply divided phase space. For typical values of the control parameter of the system $\rho$, an infinite number of marginally unstable periodic orbits (MUPOs) exist making the system sticky in the sense that unstable orbits approach regular regions in phase space and thus exhibit regular behaviour for long periods of time. The problem of finding these MUPOs is expressed as the well known problem of finding optimal rational approximations of a real number, subject to some system-specific constraints. By introducing a generalized mushroom and using properties of continued fractions, we describe a zero measure set of control parameter values $\rho\in(0,1)$ for which all MUPOs are destroyed and therefore the system is less sticky. The open mushroom (billiard with a hole) is then considered in order to quantify the stickiness exhibited and exact leading order expressions for the algebraic decay of the survival probability function $P(t)$ are calculated for mushrooms with triangular and rectangular stems.Comment: 21 pages, 11 figures. Includes discussion of a three-dimensional mushroo

Providing high-quality care remotely to patients with rare bone diseases during COVID-19 pandemic

During the COVID-19 outbreak, the European Reference Network on Rare Bone Diseases (ERN BOND) coordination team and Italian rare bone diseases healthcare professionals created the "COVID-19 Helpline for Rare Bone Diseases" in an attempt to provide high-quality information and expertise on rare bone diseases remotely to patients and healthcare professionals. The present position statement describes the key characteristics of the Helpline initiative, along with the main aspects and topics that recurrently emerged as central for rare bone diseases patients and professionals. The main topics highlighted are general recommendations, pulmonary complications, drug treatment, trauma, pregnancy, children and elderly people, and patient associations role. The successful experience of the "COVID-19 Helpline for Rare Bone Diseases" launched in Italy could serve as a primer of gold-standard remote care for rare bone diseases for the other European countries and globally. Furthermore, similar COVID-19 helplines could be considered and applied for other rare diseases in order to implement remote patients' care

Escape Rates and Physically Relevant Measures for Billiards with Small Holes

We study the billiard map corresponding to a periodic Lorentz gas in 2-dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, we establish the existence of a common escape rate and normalized limiting distribution. This limiting distribution is conditionally invariant and is the natural analogue of the SRB measure of a closed system. Finally, we prove that as the size of the hole tends to zero, the limiting distribution converges to the smooth invariant measure of the billiard map.Comment: 39 pages, 4 figure

Glycerol treatment as recovery procedure for cryopreserved human skin allografts positive for bacteria and fungi

Human donor skin allografts are suitable and much used temporary biological (burn) wound dressings. They prepare the excised wound bed for final autografting and form an excellent substrate for revascularisation and for the formation of granulation tissue. Two preservation methods, glycerol preservation and cryopreservation, are commonly used by tissue banks for the long-term storage of skin grafts. The burn surgeons of the Queen Astrid Military Hospital preferentially use partly viable cryopreserved skin allografts. After mandatory 14-day bacterial and mycological culture, however, approximately 15% of the cryopreserved skin allografts cannot be released from quarantine because of positive culture. To maximize the use of our scarce and precious donor skin, we developed a glycerolisation-based recovery method for these culture positive cryopreserved allografts. The inactivation and preservation method, described in this paper, allowed for an efficient inactivation of the colonising bacteria and fungi, with the exception of spore-formers, and did not influence the structural and functional aspects of the skin allografts