Convex Dynamics and Applications

This paper proves a theorem about bounding orbits of a time dependent dynamical system. The maps that are involved are examples in convex dynamics, by which we mean the dynamics of piecewise isometries where the pieces are convex. The theorem came to the attention of the authors in connection with the problem of digital halftoning. \textit{Digital halftoning} is a family of printing technologies for getting full color images from only a few different colors deposited at dots all of the same size. The simplest version consist in obtaining grey scale images from only black and white dots. A corollary of the theorem is that for \textit{error diffusion}, one of the methods of digital halftoning, averages of colors of the printed dots converge to averages of the colors taken from the same dots of the actual images. Digital printing is a special case of a much wider class of scheduling problems to which the theorem applies. Convex dynamics has roots in classical areas of mathematics such as symbolic dynamics, Diophantine approximation, and the theory of uniform distributions.Comment: LaTex with 9 PostScript figure

Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving

We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree (in any number of variables) but are nevertheless completely general. The underlying ideas, which we take the time to describe in an elementary way, come from tropical geometry. We thus reduce a hard algebraic problem to high-precision linear optimization, proving new upper and lower complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding

Anatomy of a wrong diagnosis: false Sinus Venosus Atrial Septal Defect

In contrast with transthoracic echocardiography, transesophageal echocardiography provides a sure way to make the diagnosis of sinus venosus atrial septal defect; on the other hand this abnormality is more complex than that seen with the secundum atrial septal defect, and inexperienced operators may fail to recognize properly the defect. In front of a high reported sensitivity using transesophageal echocardiography, specificity is difficult to assess, due to possible underreporting of diagnostic errors. We describe a false positive diagnosis of sinus venosus atrial septal defect, in the setting of enlarged right chambers of the heart because of pressure overload. Modified anatomy of the heart, together with the presence of a prominent linear structure(probably Eustachian Valve) and an incomplete examination in this case made image interpretation very prone to misinterpretation. In this anatomical setting transesophageal longitudinal "bicaval" view may be sub-optimal for examining the atrial septum, potentially showing false images that need to be known for correct image interpretation. Nonetheless, a scan plane taken more accurately at the superior level would have demonstrated/excluded the pathognomonic feature of sinus venosus atrial septal defect in the high atrial septum, between the fatty limbus and the inferior aspect of the right pulmonary artery; moreover TEE allows morphological information about the posterior structures of the heart that need to be investigated in detail for a complete diagnosis

Stochastic stability at the boundary of expanding maps

We consider endomorphisms of a compact manifold which are expanding except for a finite number of points and prove the existence and uniqueness of a physical measure and its stochastical stability. We also characterize the zero-noise limit measures for a model of the intermittent map and obtain stochastic stability for some values of the parameter. The physical measures are obtained as zero-noise limits which are shown to satisfy Pesin?s Entropy Formula

Infinitely Many Stochastically Stable Attractors

Let f be a diffeomorphism of a compact finite dimensional boundaryless manifold M exhibiting infinitely many coexisting attractors. Assume that each attractor supports a stochastically stable probability measure and that the union of the basins of attraction of each attractor covers Lebesgue almost all points of M. We prove that the time averages of almost all orbits under random perturbations are given by a finite number of probability measures. Moreover these probability measures are close to the probability measures supported by the attractors when the perturbations are close to the original map f.Comment: 14 pages, 2 figure

Fast-slow partially hyperbolic systems versus Freidlin-Wentzell random systems

We consider a simple class of fast-slow partially hyperbolic dynamical systems and show that the (properly rescaled) behaviour of the slow variable is very close to a Friedlin--Wentzell type random system for times that are rather long, but much shorter than the metastability scale. Also, we show the possibility of a "sink" with all the Lyapunov exponents positive, a phenomenon that turns out to be related to the lack of absolutely continuity of the central foliation.Comment: To appear in Journal of Statistical Physic

Simultaneous Continuation of Infinitely Many Sinks Near a Quadratic Homoclinic Tangency

We prove that the $C^3$ diffeomorphisms on surfaces, exhibiting infinitely many sinksnear the generic unfolding of a quadratic homoclinic tangency of a dissipative saddle, can be perturbed along an infinite dimensional manifold of $C^3$ diffeomorphisms such that infinitely many sinks persist simultaneously. On the other hand, if they are perturbed along one-parameter families that unfold generically the quadratic tangencies, then at most a finite number of those sinks have continuation

Gestational weight gain information: seeking and sources among pregnant women

BACKGROUND: Promoting healthy gestational weight gain (GWG) is important for preventing obstetric and perinatal morbidity, along with obesity in both mother and child. Provision of GWG guidelines by health professionals predicts women meeting GWG guidelines. Research concerning women\u27s GWG information sources is limited. This study assessed pregnant women\u27s sources of GWG information and how, where and which women seek GWG information. METHODS: Consecutive women (n = 1032) received a mailed questionnaire after their first antenatal visit to a public maternity hospital in Melbourne, Australia. Recalled provision of GWG guidelines by doctors and midwives, recalled provided GWG goals, and the obtaining of GWG information and information sources were assessed. RESULTS: Participants (n = 368; 35.7 % response) averaged 32.5 years of age and 20.8 weeks gestation, with 33.7 % speaking a language other than English. One in ten women recalled receiving GWG guidelines from doctors or midwives, of which half were consistent with Institute of Medicine guidelines. More than half the women (55.4 %) had actively sought GWG information. Nulliparous (OR 7.07, 95 % CI = 3.91-12.81) and obese (OR 1.96, 95 % CI = 1.05-3.65) women were more likely to seek information. Underweight (OR 0.29, 95 % CI = 0.09-0.97) women and those working part time (OR 0.52, 95 % CI = 0.28-0.97) were less likely to seek information. Most frequently reported GWG sources included the internet (82.7 %), books (55.4 %) and friends (51.5 %). The single most important sources were identified as the internet (32.8 %), general practitioners (16.9 %) and books (14.9 %). CONCLUSION: More than half of women were seeking GWG guidance and were more likely to consult non-clinician sources. The small numbers given GWG targets, and the dominance of non-clinical information sources, reinforces that an important opportunity to provide evidence based advice and guidance in the antenatal care setting is currently being missed

Postpartum circulating cell-free insulin DNA levels are higher in women with previous gestational diabetes mellitus who develop type 2 diabetes in later life

Background. Women with previous gestational diabetes mellitus (GDM) have evidence of postpartum β-cell dysfunction, which increases their risk of developing type 2 diabetes (T2DM) later in life. Elevated levels of circulating cell-free preproinsulin (INS) DNA correlate with dying β-cells in both mice and humans. The aim of this study was to determine if cell-free circulating INS DNA levels are higher in women with previous GDM who develop T2DM. Methods. We used droplet digital (dd) PCR to measure the levels of cell-free circulating methylated and unmethylated INS DNA in plasma from 97 women with normal glucose tolerance (NGT), 12 weeks following an index GDM pregnancy. Women were assessed for up to 10 years for the development of T2DM. Results. In the follow-up period, 22% of women developed T2DM. Compared with NGT women, total cell-free INS DNA levels were significantly higher in women who developed T2DM (P=0.02). There was no difference in cell-free circulating unmethylated and methylated INS DNA levels between NGT women and women who developed T2DM (P=0.09 and P=0.07, respectively). Conclusions. In women with a previous index GDM pregnancy, postpartum levels of cell-free circulating INS DNA are significantly higher in those women who later developed T2DM

Postpartum circulating cell-free insulin DNA levels are higher in women with previous gestational diabetes mellitus who develop type 2 diabetes in later life

Background. Women with previous gestational diabetes mellitus (GDM) have evidence of postpartum β-cell dysfunction, which increases their risk of developing type 2 diabetes (T2DM) later in life. Elevated levels of circulating cell-free preproinsulin (INS) DNA correlate with dying β-cells in both mice and humans. The aim of this study was to determine if cell-free circulating INS DNA levels are higher in women with previous GDM who develop T2DM. Methods. We used droplet digital (dd) PCR to measure the levels of cell-free circulating methylated and unmethylated INS DNA in plasma from 97 women with normal glucose tolerance (NGT), 12 weeks following an index GDM pregnancy. Women were assessed for up to 10 years for the development of T2DM. Results. In the follow-up period, 22% of women developed T2DM. Compared with NGT women, total cell-free INS DNA levels were significantly higher in women who developed T2DM (P=0.02). There was no difference in cell-free circulating unmethylated and methylated INS DNA levels between NGT women and women who developed T2DM (P=0.09 and P=0.07, respectively). Conclusions. In women with a previous index GDM pregnancy, postpartum levels of cell-free circulating INS DNA are significantly higher in those women who later developed T2DM