Let F be a finitely generated group. Denote by an(T) (resp.<rn(r)) the number of subgroups of F of index n (resp. of index at most n). This paper deals with the connection between the algebraic structure of the group F and the arithmetic properties of the sequence an(F), n — 1,2,3,..., e.g., the growth of the sequence an(T) ("the subgroup growth") or the properties of the function Cr(s) = X2£Li...

Let G = (V, E) be a connected simple graph. A labeling f : V → Z2 induces an edge labeling f∗ : E → Z2 defined by f∗(xy) = f(x) + f(y) for each xy ∈ E. For i ∈ Z2, let vf (i) = |f−1(i)| and ef (i) = |f∗−1(i)|. A labeling f is called friendly if |vf (1)− vf (0)| ≤ 1. The full friendly index set of G consists all possible differences between the number of edges labeled by 1 and the number of edge...

The object of this note is to give a very short proof of the following theorem of Ivanov and Schupp. Let H be a finitely generated subgroup of a free group F and the index [F : H ] infinite. Then there exists a nontrivial normal subgroup N of F such that N ∩ H = {1}. As noted in the abstract above the object of this note is to give a new proof of the following theorem of S.V.Ivanov and P.E.Schu...

BACKGROUND Non-invasive imaging biomarkers of cellular proliferation hold great promise for quantifying response to personalized medicine in oncology. An emerging approach to assess tumor proliferation utilizes the positron emission tomography (PET) tracer 3'-deoxy-3'[(18)F]-fluorothymidine, [(18)F]-FLT. Though several studies have associated serial changes in [(18)F]-FLT-PET with elements of t...

This paper considers Q-Fano 3-folds X with ρ = 1. The aim is to determine the maximal Fano index f of X. We prove that f ≤ 19, and that in case of equality, the Hilbert series of X equals that of weighted projective space P(3, 4, 5, 7). We also consider all possibility of X for f ≥ 9. 0. Introduction We say that X is a Q-Fano variety if it has only terminal singularities, the anticanonical Weil...

Let F = (Kn,P) be a circulant homogeneous factorisation of index k, that means P is a partition of the arc set of the complete digraph Kn into k circulant factor digraphs such that there exists σ ∈ Sn permuting the factor circulants transitively amongst themselves. Suppose further such an element σ normalises the cyclic regular automorphism group of these circulant factor digraphs, we say F is ...

The Padmakar-Ivan (PI) index is a first-generation topological index (TI) based on sums over all edges between numbers of edges closer to one endpoint and numbers of edges closer to the other endpoint. Edges at equal distances from the two endpoints are ignored. An analogous definition is valid for the Wiener index W, with the difference that sums are replaced by products. A few other TIs are d...

We consider Lévy flights characterized by the step index f in a quenched random force field. By means of a dynamic renormalization group analysis we find that the dynamic exponent z for f < 2 locks onto f , independent of dimension and independent of the presence of weak quenched disorder. The critical dimension, however, depends on the step index f for f < 2 and is given by dc = 2f − 2. For d ...

By considering a least squares approximation of a given square integrable function f : [0, 1] → IR by a shifted L-statistic function (a shifted linear combination of order statistics), we define an index which measures the global influence of the kth largest variable on f . We show that this influence index has appealing properties and we interpret it as an average value of the difference quoti...

A self-map f on the CW complex Z is a periodic homotopy idempotent if for some r 0 and p > 0 the iterates fr and fr+p are homotopic. Geoghegan and Nicas defined the rotation index RI(f) of such a map. They proved that for r = p = 1, the homotopy idempotent f splits if and only if RI(f) = 1, while for r = 0, the index RI(f) divides p2. We extend this to arbitrary p and r, and generalize various ...