Flattening Functions on Flowers

Let $T$ be an orientation-preserving Lipschitz expanding map of the circle \T. A pre-image selector is a map \tau:\T\to\T with finitely many discontinuities, each of which is a jump discontinuity, and such that $\tau(x)\in T^{-1}(x)$ for all x\in\T. The closure of the image of a pre-image selector is called a flower, and a flower with $p$ connected components is called a $p$-flower. We say that a real-valued Lipschitz function can be Lipschitz flattened on a flower whenever it is Lipschitz cohomologous to a constant on that flower. The space of Lipschitz functions which can be flattened on a given $p$-flower is shown to be of codimension $p$ in the space of all Lipschitz functions, and the linear constraints determining this subspace are derived explicitly. If a Lipschitz function $f$ has a maximizing measure $S$ which is Sturmian (i.e. is carried by a 1-flower), it is shown that $f$ can be Lipschitz flattened on some 1-flower carrying $S$.Comment: Accepted for publication and confirmed for december 200

The Analyticity of a Generalized Ruelle's Operator

In this work we propose a generalization of the concept of Ruelle operator for one dimensional lattices used in thermodynamic formalism and ergodic optimization, which we call generalized Ruelle operator, that generalizes both the Ruelle operator proposed in [BCLMS] and the Perron Frobenius operator defined in [Bowen]. We suppose the alphabet is given by a compact metric space, and consider a general a-priori measure to define the operator. We also consider the case where the set of symbols that can follow a given symbol of the alphabet depends on such symbol, which is an extension of the original concept of transition matrices from the theory of subshifts of finite type. We prove the analyticity of the Ruelle operator and present some examples

Osteomimicry of mammary adenocarcinoma cells in vitro; increased expression of bone matrix proteins and proliferation within a 3D collagen environment.

Bone is the most common site of metastasis for breast cancer, however the reasons for this remain unclear. We hypothesise that under certain conditions mammary cells possess osteomimetic capabilities that may allow them to adapt to, and flourish within, the bone microenvironment. Mammary cells are known to calcify within breast tissue and we have recently reported a novel in vitro model of mammary mineralization using murine mammary adenocarcinoma 4T1 cells. In this study, the osteomimetic properties of the mammary adenocarcinoma cell line and the conditions required to induce mineralization were characterized extensively. It was found that exogenous organic phosphate and inorganic phosphate induce mineralization in a dose dependent manner in 4T1 cells. Ascorbic acid and dexamethasone alone have no effect. 4T1 cells also show enhanced mineralization in response to bone morphogenetic protein 2 in the presence of phosphate supplemented media. The expression of several bone matrix proteins were monitored throughout the process of mineralization and increased expression of collagen type 1 and bone sialoprotein were detected, as determined by real-time RT-PCR. In addition, we have shown for the first time that 3D collagen glycosaminoglycan scaffolds, bioengineered to represent the bone microenvironment, are capable of supporting the growth and mineralization of 4T1 adenocarcinoma cells. These 3D scaffolds represent a novel model system for the study of mammary mineralization and bone metastasis. This work demonstrates that mammary cells are capable of osteomimicry, which may ultimately contribute to their ability to preferentially metastasize to, survive within and colonize the bone microenvironment

On the zero-temperature limit of Gibbs states

We exhibit Lipschitz (and hence H\"older) potentials on the full shift $\{0,1\}^{\mathbb{N}}$ such that the associated Gibbs measures fail to converge as the temperature goes to zero. Thus there are "exponentially decaying" interactions on the configuration space $\{0,1\}^{\mathbb Z}$ for which the zero-temperature limit of the associated Gibbs measures does not exist. In higher dimension, namely on the configuration space $\{0,1\}^{\mathbb{Z}^{d}}$, $d\geq3$, we show that this non-convergence behavior can occur for finite-range interactions, that is, for locally constant potentials.Comment: The statement of Theorem 1.2 is more accurate and some new comment follow i

Natural equilibrium states for multimodal maps

This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, Collet-Eckmann. For a map in this class, we prove existence and uniqueness of equilibrium states for the geometric potentials $-t \log|Df|$, for the largest possible interval of parameters $t$. We also study the regularity and convexity properties of the pressure function, completely characterising the first order phase transitions. Results concerning the existence of absolutely continuous invariant measures with respect to the Lebesgue measure are also obtained

The Lagrange and Markov spectra from the dynamical point of view

This text grew out of my lecture notes for a 4-hours minicourse delivered on October 17 \& 19, 2016 during the research school "Applications of Ergodic Theory in Number Theory" -- an activity related to the Jean-Molet Chair project of Mariusz Lema\'nczyk and S\'ebastien Ferenczi -- realized at CIRM, Marseille, France. The subject of this text is the same of my minicourse, namely, the structure of the so-called Lagrange and Markov spectra (with an special emphasis on a recent theorem of C. G. Moreira).Comment: 27 pages, 6 figures. Survey articl

Effect of Using the Same vs Different Order for Second Readings of Screening Mammograms on Rates of Breast Cancer Detection A Randomized Clinical Trial

Importance Interpreting screening mammograms is a difficult repetitive task that can result in missed cancers and false-positive recalls. In the United Kingdom, 2 film readers independently evaluate each mammogram to search for signs of cancer and examine digital mammograms in batches. However, a vigilance decrement (reduced detection rate with time on task) has been observed in similar settings. Objective To determine the effect of changing the order for the second film reader of batches of screening mammograms on rates of breast cancer detection. Design, Setting, and Participants A multicenter, double-blind, cluster randomized clinical trial conducted at 46 specialized breast screening centers from the National Health Service Breast Screening Program in England for 1 year (all between December 20, 2012, and November 3, 2014). Three hundred sixty readers participated (mean, 7.8 readers per center)—186 radiologists, 143 radiography advanced practitioners, and 31 breast clinicians, all fully qualified to report mammograms in the NHS breast screening program. Interventions The 2 readers examined each batch of digital mammograms in the same order in the control group and in the opposite order to one another in the intervention group. Main Outcomes and Measures The primary outcome was cancer detection rate; secondary outcomes were rates of recall and disagreements between readers. Results Among 1 194 147 women (mean age, 59.3; SD, 7.49) who had screening mammograms (596 642 in the intervention group; 597 505 in the control group), the images were interpreted in 37 688 batches (median batch size, 35; interquartile range [IQR]; 16-46), with each reader interpreting a median of 176 batches (IQR, 96-278). After completion of all subsequent diagnostic tests, a total of 10 484 cases (0.88%) of breast cancer were detected. There was no significant difference in cancer detection rate with 5272 cancers (0.88%) detected in the intervention group vs 5212 cancers (0.87%) detected in the control group (difference, 0.01% points; 95% CI, −0.02% to 0.04% points; recall rate, 24 681 [4.14%] vs 24 894 [4.17%]; difference, −0.03% points; 95% CI, −0.10% to 0.04% points; or rate of reader disagreements, 20 471 [3.43%] vs 20 793 [3.48%]; difference, −0.05% points; 95% CI, −0.11% to 0.02% points). Conclusions and Relevance Interpretation of batches of mammograms by qualified screening mammography readers using a different order vs the same order for the second reading resulted in no significant difference in rates of detection of breast cancer. Trial Registration isrctn.org Identifier: ISRCTN4660337

Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: A hundred decimal digits for the dimension of &ITE&IT2

We prove that the algorithm of [13] for approximating the Hausdorff dimension of dynamically defined Cantor sets, using periodic points of the underlying dynamical system, can be used to establish completely rigorous high accuracy bounds on the dimension. The effectiveness of these rigorous estimates is illustrated for Cantor sets consisting of continued fraction expansions with restricted digits. For example the Hausdorff dimension of the set $E_2$ (of those reals whose continued fraction expansion only contains digits 1 and 2) can be rigorously approximated, with an accuracy of over 100 decimal places, using points of period up to 25. The method for establishing rigorous dimension bounds involves the holomorphic extension of mappings associated to the allowed continued fraction digits, an appropriate disc which is contracted by these mappings, and an associated transfer operator acting on the Hilbert Hardy space of analytic functions on this disc. We introduce methods for rigorously bounding the approximation numbers for the transfer operators, showing that this leads to effective estimates on the Taylor coefficients of the associated determinant, and hence to explicit bounds on the Hausdorff dimension.Comment: Advances in Mathematics, to appea

Rigorous Computation of Diffusion Coefficients for Expanding Maps

For real analytic expanding interval maps, a novel method is given for rigorously approximating the diffusion coefficient of real analytic observables. As a theoretical algorithm, our approximation scheme is shown to give quadratic exponential convergence to the diffusion coefficient. The method for converting this rapid convergence into explicit high precision rigorous bounds is illustrated in the setting of Lanford’s map x↦2x+12x(1−x)(mod1)

Access and utilisation of primary health care services comparing urban and rural areas of Riyadh Providence, Kingdom of Saudi Arabia

The Kingdom of Saudi Arabia (KSA) has seen an increase in chronic diseases. International evidence suggests that early intervention is the best approach to reduce the burden of chronic disease. However, the limited research available suggests that health care access remains unequal, with rural populations having the poorest access to and utilisation of primary health care centres and, consequently, the poorest health outcomes. This study aimed to examine the factors influencing the access to and utilisation of primary health care centres in urban and rural areas of Riyadh province of the KSA