Book embeddings of Reeb graphs

Let $X$ be a simplicial complex with a piecewise linear function $f:X\to\mathbb{R}$. The Reeb graph $Reeb(f,X)$ is the quotient of $X$, where we collapse each connected component of $f^{-1}(t)$ to a single point. Let the nodes of $Reeb(f,X)$ be all homologically critical points where any homology of the corresponding component of the level set $f^{-1}(t)$ changes. Then we can label every arc of $Reeb(f,X)$ with the Betti numbers $(\beta_1,\beta_2,\dots,\beta_d)$ of the corresponding $d$-dimensional component of a level set. The homology labels give more information about the original complex $X$ than the classical Reeb graph. We describe a canonical embedding of a Reeb graph into a multi-page book (a star cross a line) and give a unique linear code of this book embedding.Comment: 12 pages, 5 figures, more examples will be at http://kurlin.or

The Efficiency of Glucocorticoid Therapy in Secondary-progressive Course of Multiple Sclerosis

We have investigated the efficacy of pulse-therapy with glucocorticoids (GC) at different time stages (in debuts, at the recurrent stage and at the stage of progression) of secondary progressive course (SPC) of multiple sclerosis (MS) in 70 patients (57 women and 13 men) at the ages from 28 to 67 years (mean age 45±2.5 years). The duration of the disease accounted for 7 up to 34 years (average duration was 19.8±2.3 years). We have conducted 438 courses of GC therapy: at the onsets – 11, at the recurrent stage (RS) – 178 and at the stage of secondary progression-249.The efficacy of hormonal therapy was assessed taking into account the following criteria: the dynamics of regression of neurological symptoms under the influence of the first course of GC therapy at the stage of onsets; a comparative evaluation of remission\u27s duration after admission and without taking GC at the onsets; duration of RS depending on the duration of remissions after the first course of GC therapy; a comparative evaluation of remissions\u27 duration after the 1st (at the stage of onset and/or on the RS), and the period of stabilization on the SPS before the last courses of GC; the variants of secondary progression under the influence of GC courses; scores according the EDSS disability scale after the 1st and before the last course of GC therapy; the rate of progression under the influence of the repeated courses of GC therapy.We have defined the three categories of efficacy at the repeated courses of pulse therapy with GC: the moderate efficacy, the low efficacy, the lack of efficacy. We have not observed the high efficacy in patients with SPC.The patients with MFR <1.0, among which the pulse therapy with GC promoted the prolongation of RS, relieved the severe (less often) and moderate (more often) relapses, the outcome of which was accompanied by a moderate and stable neurologic deficit, were subsumed under the subgroup with moderate efficacy (21 individuals). The most favorable progressive variant of progression prevailed in these patients after transformation of RS into SPS.The patients with different rate of MFR (9 – with MFR <1.0 and 12 – with MFR>1.0), with short (more often) and moderate (less often) RS, during which the accumulation of neurological deficit due to the frequent and heavy relapses had occurred, were subsumed under the subgroup with low efficiency (21 individuals). After transformation into SPS, the recurrent variant of progression prevailed in these patients.The patients who were characterized by short RS, by predominance of severe and prolonged relapses, the MFR value greater than 1.0, the steady accumulation of a pronounced and persistent neurologic deficit, a high rate of progression and high scores on the EDSS disability scale more than 6.5 points) were subsumed under the subgroup with the lack of efficacy (28 individuals). After transformation in the SPC, the most unfavorable variant of progression prevailed (21 patients); significantly less frequent were the recurrent (5 patients) and a combination of a steady and recurrent (2 patients) progression. The persistent lack of efficacy of the hormonal therapy in this subgroup of patients was most likely associated with the genetically determined low individual sensitivity to GC.Therefore, the results of our study showed that the efficacy of GC therapy in SPC of MS is determined by the complex interaction of clinical indicators having the prognostic value, as well as by the number of the genetic factors, which require their further observation

Dark Matter in the Standard Model Extension with Singlet Quark

We analyze the possibility of hadron Dark Matter carriers consisting of singlet quark and the light standard one. It is shown that stable singlet quarks generate effects of new physics which do not contradict to restrictions from precision electroweak data. The neutral and charged pseudoscalar low-lying states are interpreted as the Dark Matter particle and its mass-degenerated partner. We evaluated their masses and lifetime of the charged component, and describe asymptotics of the potential of low-energy interactions of these particles with nucleons and with each other. Some peculiarities of Sommerfeld enhancement effect in the annihilation process are also discussed.Comment: 13 pages, 2 tabl