Combinatorial classes T that are recursively defined using combinations of the standard multiset, sequence, directed cycle and cycle constructions, and their restrictions, have generating series T(z) with a positive radius of convergence; for most of these a simple test can be used to quickly show that the form of the asymptotics is the same as that for the class of rooted trees: C · ρ−n · n−3/...

The Jacobian conjecture [Keller, Monatsh. Math. Phys., 1939] gives rise to a problem in combinatorial linear algebra: Is the vector space generated by rooted trees spanned by forest shuffle vectors? In order to make headway on this problem we must study the algebraic and combinatorial properties of rooted trees. We prove three theorems about the vector space generated by binary rooted trees: Sh...

In order to obtain the full asymptotic expansion for Pólya trees, i.e. rooted unlabelled and non-plane trees, Flajolet and Sedgewick observed that their specification could be seen as a slight disturbance of the functional equation satisfied by the Cayley tree function. Such an approach highlights the complicated formal expressions with some combinatorial explanation. They initiated this proces...

Identifying and accessing all pulp canals is particularly challenging in the endodontic treatment of teeth with atypical canal configurations. This report describes the nonsurgical endodontic treatment of a maxillary first premolar with three root canals. The prevalence of this unusual morphology is reviewed and suggestions are offered to help dentists identify and treat such teeth. This case r...

Morphological characteristics of rooted cuttings and seedlings of four open-pollinated loblolly pine families cultured at the G.H.W. Weyerhaeuser Nursery near Washington, North Carolina were assessed after lifting. Cuttings were visually assigned to nine grading classes and six putative cull classes. The nine grading classes represented a factorial of three root collar diameter classes (small [...

The maximum-likelihood (ML) solution to a simple phylogenetic estimation problem is obtained analytically The problem is estimation of the rooted tree for three species using binary characters with a symmetrical rate of substitution under the molecular clock. ML estimates of branch lengths and log-likelihood scores are obtained analytically for each of the three rooted binary trees. Estimation ...

Given a set B of finite rooted graphs and a radius r as an input, we prove that it is undecidable to determine whether there exists a sequence (Gi) of finite bounded degree graphs such that the rooted r-radius neighbourhood of a random node of Gi is isomorphic to a rooted graph in B with probability tending to 1. Our proof implies a similar result for the case where the sequence (Gi) is replace...

The simultaneous occurrence of various multiple dental anomalies has been reported previously, particularly in cases of chromosomal abnormalities that often manifest with multisystem involvement.' 2 Additionally, multiple dental anomalies have been reported in individuals and within families, without evidence of other systemic manifestations. Ekman-Westborg and Julin reported a case of macrodon...

BACKGROUND Mandibular first molars typically have two roots but sometimes a supernumerary root presents distolingually called as radix entomolaris (RE). AIM The present study evaluated the prevalence of permanent mandibular first molars featuring a distolingual root in Haryana (North India). MATERIALS AND METHODS Five hundred patients possessing bilateral mandibular first molars were select...

We introduce and study a modal logic wK4F which is closely related to the logic S4F that is important in the context of epistemic logics for representing and reasoning about an agent’s knowledge. It is obtained by adding the axiom F to the modal logic wK4, or dropping from S4F the T axiom. We show that wK4F is sound and complete with respect to the class of all minimal topological spaces i.e. t...