Piecewise Linear Models for the Quasiperiodic Transition to Chaos

We formulate and study analytically and computationally two families of piecewise linear degree one circle maps. These families offer the rare advantage of being non-trivial but essentially solvable models for the phenomenon of mode-locking and the quasi-periodic transition to chaos. For instance, for these families, we obtain complete solutions to several questions still largely unanswered for families of smooth circle maps. Our main results describe (1) the sets of maps in these families having some prescribed rotation interval; (2) the boundaries between zero and positive topological entropy and between zero length and non-zero length rotation interval; and (3) the structure and bifurcations of the attractors in one of these families. We discuss the interpretation of these maps as low-order spline approximations to the classic ``sine-circle'' map and examine more generally the implications of our results for the case of smooth circle maps. We also mention a possible connection to recent experiments on models of a driven Josephson junction.Comment: 75 pages, plain TeX, 47 figures (available on request

In vitro drug sensitivity of normal peripheral blood lymphocytes and childhood leukaemic cells from bone marrow and peripheral blood.

In vitro drug sensitivity of leukaemic cells might be influenced by the contamination of such a sample with non-malignant cells and the sample source. To study this, sensitivity of normal peripheral blood (PB) lymphocytes to a number of cytostatic drugs was assessed with the MTT assay. We compared this sensitivity with the drug sensitivity of leukaemic cells of 38 children with acute lymphoblastic leukaemia. We also studied a possible differential sensitivity of leukaemic cells from bone marrow (BM) and PB. The following drugs were used: Prednisolone, dexamethasone, 6-mercaptopurine, 6-thioguanine, cytosine arabinoside, vincristine, vindesine, daunorubicin, doxorubicin, mafosfamide (Maf), 4-hydroperoxy-ifosfamide, teniposide, mitoxantrone, L-asparaginase, methotrexate and mustine. Normal PB lymphocytes were significantly more resistant to all drugs tested, except to Maf. Leukaemic BM and PB cells from 38 patients (unpaired samples) showed no significant differences in sensitivity to any of the drugs. Moreover, in 11 of 12 children with acute leukaemia of whom we investigated simultaneously obtained BM and PB (paired samples), their leukaemic BM and PB cells showed comparable drug sensitivity profiles. In one patient the BM cells were more sensitive to most drugs than those from the PB, but the actual differences in sensitivity were small. We conclude that the contamination of a leukaemic sample with normal PB lymphocytes will influence the results of the MTT assay. The source of the leukaemic sample, BM or PB, does not significantly influence the assay results

Polydispersity and ordered phases in solutions of rodlike macromolecules

We apply density functional theory to study the influence of polydispersity on the stability of columnar, smectic and solid ordering in the solutions of rodlike macromolecules. For sufficiently large length polydispersity (standard deviation $\sigma>0.25$) a direct first-order nematic-columnar transition is found, while for smaller $\sigma$ there is a continuous nematic-smectic and first-order smectic-columnar transition. For increasing polydispersity the columnar structure is stabilized with respect to solid perturbations. The length distribution of macromolecules changes neither at the nematic-smectic nor at the nematic-columnar transition, but it does change at the smectic-columnar phase transition. We also study the phase behaviour of binary mixtures, in which the nematic-smectic transition is again found to be continuous. Demixing according to rod length in the smectic phase is always preempted by transitions to solid or columnar ordering.Comment: 13 pages (TeX), 2 Postscript figures uuencode

The Exact Ground State of the Frenkel-Kontorova Model with Repeated Parabolic Potential: I. Basic Results

The problem of finding the exact energies and configurations for the Frenkel-Kontorova model consisting of particles in one dimension connected to their nearest-neighbors by springs and placed in a periodic potential consisting of segments from parabolas of identical (positive) curvature but arbitrary height and spacing, is reduced to that of minimizing a certain convex function defined on a finite simplex.Comment: 12 RevTeX pages, using AMS-Fonts (amssym.tex,amssym.def), 6 Postscript figures, accepted by Phys. Rev.

Mononuclear cells contaminating acute lymphoblastic leukaemic samples tested for cellular drug resistance using the methyl-thiazol-tetrazolium assay.

The methyl-thiazol-tetrazolium (MTT) assay is a drug resistance assay which cannot discriminate between malignant and non-malignant cells. We previously reported that samples with > or = 80% leukaemic cells at the start of culture give similar results in the MTT assay and the differential staining cytotoxicity assay, in which a discrimination between malignant and non-malignant cells can be made. However, the percentage of leukaemic cells may change during culture, which might affect the results of the MTT assay. We studied 106 untreated childhood acute lymphoblastic leukemia (ALL) samples with > or = 80% leukaemic cells at the start of culture. This percentage decreased below 80% in 28%, and below 70% in 13%, of the samples after 4 days of culture. A decrease below 70% occurred more often in case of 80-89% leukaemic cells (9/29) than in case of > or = 90% leukaemic cells at the start of culture (5/77, P = 0.0009). Samples with < 70% leukaemic cells after culture were significantly more resistant to 6 out of 13 drugs, and showed a trend towards being more resistant to two more drugs, than samples with > or = 80% leukaemic cells. No such differences were seen between samples with 70-79% and samples with > or = 80% leukaemic cells after culture. We next studied in another 30 ALL samples whether contaminating mononuclear cells could be removed by using immunoamagnetic beads. Using a beads to target cell ratio of 10:1, the percentage of leukaemic cells increased from mean 72% (s.d. 9.3%) to mean 87% (s.d. 6.7%), with an absolute increase of 2-35%. The recovery of leukaemic cells was mean 82.1% (range 56-100%, s.d. 14.0%). The procedure itself did not influence the results of the MTT assay in three samples containing only leukaemic cells. We conclude that it is important to determine the percentage of leukaemic cells at the start and at the end of the MTT assay and similar drug resistance assays. Contaminating mononuclear cells can be successfully removed from ALL samples using immunomagnetic beads. This approach may increase the number of leukaemic samples which can be evaluated for cellular drug resistance with the MTT assay or a similar cell culture drug resistance assay

Boundaries of Disk-like Self-affine Tiles

Let $T:= T(A, {\mathcal D})$ be a disk-like self-affine tile generated by an integral expanding matrix $A$ and a consecutive collinear digit set ${\mathcal D}$, and let $f(x)=x^{2}+px+q$ be the characteristic polynomial of $A$. In the paper, we identify the boundary $\partial T$ with a sofic system by constructing a neighbor graph and derive equivalent conditions for the pair $(A,{\mathcal D})$ to be a number system. Moreover, by using the graph-directed construction and a device of pseudo-norm $\omega$, we find the generalized Hausdorff dimension $\dim_H^{\omega} (\partial T)=2\log \rho(M)/\log |q|$ where $\rho(M)$ is the spectral radius of certain contact matrix $M$. Especially, when $A$ is a similarity, we obtain the standard Hausdorff dimension $\dim_H (\partial T)=2\log \rho/\log |q|$ where $\rho$ is the largest positive zero of the cubic polynomial $x^{3}-(|p|-1)x^{2}-(|q|-|p|)x-|q|$, which is simpler than the known result.Comment: 26 pages, 11 figure

The global burden attributable to low bone mineral density

Introduction: The Global Burden of Disease Study 2010 estimated the worldwide health burden of 291 diseases and injuries and 67 risk factors by calculating disability-adjusted life years (DALYs). Osteoporosis was not considered as a disease, and bone mineral density (BMD) was analysed as a risk factor for fractures, which formed part of the health burden due to falls. Objectives: To calculate (1) the global distribution of BMD, (2) its population attributable fraction (PAF) for fractures and subsequently for falls, and (3) the number of DALYs due to BMD. Methods: A systematic review was performed seeking population-based studies in which BMD was measured by dual-energy X-ray absorptiometry at the femoral neck in people aged 50 years and over. Age- and sex-specific mean ± SD BMD values (g/cm2) were extracted from eligible studies. Comparative risk assessment methodology was used to calculate PAFs of BMD for fractures. The theoretical minimum risk exposure distribution was estimated as the age- and sex-specific 90th centile from the Third National Health and Nutrition Examination Survey (NHANES III). Relative risks of fractures were obtained from a previous meta-analysis. Hospital data were used to calculate the fraction of the health burden of falls that was due to fractures. Results: Global deaths and DALYs attributable to low BMD increased from 103 000 and 3 125 000 in 1990 to 188 000 and 5 216 000 in 2010, respectively. The percentage of low BMD in the total global burden almost doubled from 1990 (0.12%) to 2010 (0.21%). Around one-third of falls-related deaths were attributable to low BMD. Conclusions: Low BMD is responsible for a growing global health burden, only partially representative of the real burden of osteoporosis