Tur\'an Graphs, Stability Number, and Fibonacci Index

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Tur\'an graphs frequently appear in extremal graph theory. We show that Tur\'an graphs and a connected variant of them are also extremal for these particular problems.Comment: 11 pages, 3 figure

Complements and signed digit representations: Analysis of a multi-exponentiation-algorithm of Wu, Lou, Lai and Chang

Wu, Lou, Lai and Chang proposed a multi-exponentiation algorithm using binary complements and the non-adjacent form. The purpose of this paper is to show that neither the analysis of the algorithm given by its original proposers nor that by other authors are correct. In fact it turns out that the complement operation does not have significant influence on the performance of the algorithm and can therefore be omitted

Trees with Given Stability Number and Minimum Number of Stable Sets

We study the structure of trees minimizing their number of stable sets for given order $n$ and stability number $\alpha$. Our main result is that the edges of a non-trivial extremal tree can be partitioned into $n-\alpha$ stars, each of size $\lceil \frac{n-1}{n-\alpha} \rceil$ or $\lfloor \frac{n-1}{n-\alpha}\rfloor$, so that every vertex is included in at most two distinct stars, and the centers of these stars form a stable set of the tree.Comment: v2: Referees' comments incorporate

Topological self-similarity on the random binary-tree model

Asymptotic analysis on some statistical properties of the random binary-tree model is developed. We quantify a hierarchical structure of branching patterns based on the Horton-Strahler analysis. We introduce a transformation of a binary tree, and derive a recursive equation about branch orders. As an application of the analysis, topological self-similarity and its generalization is proved in an asymptotic sense. Also, some important examples are presented

Towards the system-wide implementation of the International Classification of Functioning, Disability, and Health in routine clinical practice: Empirical findings of a pilot study from Mainland China

Objective: The aims of this study were to evaluate the feasibility of using the International Classification of Functioning, Disability and Health (ICF) Generic Set in routine clinical practice, and of creating a functioning score based on it, and, subsequently, to examine its sensitivity to change.Methods: In this prospective cohort study, data from 761 adult inpatients from 21 Chinese hospitals were analysed. Each patient was assessed at admission and discharge. Feasibility was evaluated by analysing mean assessment time. The Rasch model was used to create a metric of functioning. Sensitivity to change was analysed with mixed-effects regression and by calculating standardized effect size based on Cohen's f2.Results: Mean duration of assessment was 5.3 min, with a significant decrease between admission and discharge. After removal of the item remunerative employment, the remaining ICF Generic Set categories fitted the Rasch model well. With a mean improvement in functioning of 12.1(95% confidence interval (95% CI):11.5-12.6), this metric proved sensitive to change, both in terms of statistical significance (p < 0.001) and standardized effect size (Cohen's f2=2.35).Discussion: The ICF Generic Set is feasible for use in routine clinical practice and is promising to serve as the basis for the development of a functioning score that is sensitive to change.Chinese Mainland Affairs Offic

Super congruences and Euler numbers

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$ $$\sum_{k=0}^{(p-1)/2}\binom{2k}{k}^2/16^k=(-1)^{(p-1)/2}+p^2E_{p-3} (mod p^3)$$, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for $\pi^2$, $\pi^{-2}$ and the constant $K:=\sum_{k>0}(k/3)/k^2$ (with (-) the Jacobi symbol), two of which are $$\sum_{k=1}^\infty(10k-3)8^k/(k^3\binom{2k}{k}^2\binom{3k}{k})=\pi^2/2$$ and $$\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$

Differentiation of mouse bone marrow derived stem cells toward microglia-like cells

<p>Abstract</p> <p>Background</p> <p>Microglia, the macrophages of the brain, have been implicated in the causes of neurodegenerative diseases and display a loss of function during aging. Throughout life, microglia are replenished by limited proliferation of resident microglial cells. Replenishment by bone marrow-derived progenitor cells is still under debate. In this context, we investigated the differentiation of mouse microglia from bone marrow (BM) stem cells. Furthermore, we looked at the effects of FMS-like tyrosine kinase 3 ligand (Flt3L), astrocyte-conditioned medium (ACM) and GM-CSF on the differentiation to microglia-like cells.</p> <p>Methods</p> <p>We assessed <it>in vitro-</it>derived microglia differentiation by marker expression (CD11b/CD45, F4/80), but also for the first time for functional performance (phagocytosis, oxidative burst) and <it>in situ </it>migration into living brain tissue. Integration, survival and migration were assessed in organotypic brain slices.</p> <p>Results</p> <p>The cells differentiated from mouse BM show function, markers and morphology of primary microglia and migrate into living brain tissue. Flt3L displays a negative effect on differentiation while GM-CSF enhances differentiation.</p> <p>Conclusion</p> <p>We conclude that <it>in vitro-</it>derived microglia are the phenotypic and functional equivalents to primary microglia and could be used in cell therapy.</p

Mycobacterium tuberculosis complex genetic diversity: mining the fourth international spoligotyping database (SpolDB4) for classification, population genetics and epidemiology

BACKGROUND: The Direct Repeat locus of the Mycobacterium tuberculosis complex (MTC) is a member of the CRISPR (Clustered regularly interspaced short palindromic repeats) sequences family. Spoligotyping is the widely used PCR-based reverse-hybridization blotting technique that assays the genetic diversity of this locus and is useful both for clinical laboratory, molecular epidemiology, evolutionary and population genetics. It is easy, robust, cheap, and produces highly diverse portable numerical results, as the result of the combination of (1) Unique Events Polymorphism (UEP) (2) Insertion-Sequence-mediated genetic recombination. Genetic convergence, although rare, was also previously demonstrated. Three previous international spoligotype databases had partly revealed the global and local geographical structures of MTC bacilli populations, however, there was a need for the release of a new, more representative and extended, international spoligotyping database. RESULTS: The fourth international spoligotyping database, SpolDB4, describes 1939 shared-types (STs) representative of a total of 39,295 strains from 122 countries, which are tentatively classified into 62 clades/lineages using a mixed expert-based and bioinformatical approach. The SpolDB4 update adds 26 new potentially phylogeographically-specific MTC genotype families. It provides a clearer picture of the current MTC genomes diversity as well as on the relationships between the genetic attributes investigated (spoligotypes) and the infra-species classification and evolutionary history of the species. Indeed, an independent Naïve-Bayes mixture-model analysis has validated main of the previous supervised SpolDB3 classification results, confirming the usefulness of both supervised and unsupervised models as an approach to understand MTC population structure. Updated results on the epidemiological status of spoligotypes, as well as genetic prevalence maps on six main lineages are also shown. Our results suggests the existence of fine geographical genetic clines within MTC populations, that could mirror the passed and present Homo sapiens sapiens demographical and mycobacterial co-evolutionary history whose structure could be further reconstructed and modelled, thereby providing a large-scale conceptual framework of the global TB Epidemiologic Network. CONCLUSION: Our results broaden the knowledge of the global phylogeography of the MTC complex. SpolDB4 should be a very useful tool to better define the identity of a given MTC clinical isolate, and to better analyze the links between its current spreading and previous evolutionary history. The building and mining of extended MTC polymorphic genetic databases is in progress