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

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)

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

Q: Is ketamine effective and safe for treatment-resistant depression?

Evidence-based answer: Maybe, but it’s too soon to tell. There is limited evidence that ketamine by itself is effective in the very short term. Single-dose intravenous (IV) ketamine is more likely than placebo (odds ratio = 11-13) to produce improvement (> 50%) in standardized depression scores in 1 to 3 days, lasting up to a week. Twice- or thriceweekly IV ketamine improves symptom scores by 20%-25% over 2 weeks (strength of recommendation [SOR]: B, meta-analysis of small, low-quality, randomized controlled trials [RCTs] and a single small RCT). Augmentation of sertraline with daily oral ketamine moderately improves symptom scores for 6 weeks in patients with moderate depression (SOR: B, small, low-quality RCTs). Augmentation of oral antidepressants (duloxetine, escitalopram, sertraline, venlafaxine) with intranasal esketamine spray improves response and remission rates at 4 weeks (16% for both outcomes) in patients with predominantly treatment-resistant major depression (SOR: A, meta-analysis of RCTs). Ketamine therapy is associated with confusion, emotional blunting, headache, dizziness, and blurred vision (SOR: A, meta-analyses). Nasal esketamine spray produces the adverse effects of dizziness, vertigo, and blurred vision severe enough to cause discontinuation in 4% of patients; it also can produce transient elevation of blood pressure (SOR: A, meta-analyses).Amanda Zorn, MD; Sean Linn, PharmD; Mat Jenkinson, PharmD; Jon O. Neher, MD (Valley Family Medicine, Residency, University of Washington at Valley, Renton), Sarah Safranek, MLIS (Health Sciences Librarian Emeritus, University of Washington Medical School, Seattle)Includes bibliographical reference

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

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

How many inflections are there in the Lyapunov spectrum?

Iommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection

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