Emperor Julian's three-book treatise Contra Galilaeos survives solely in those Christian sources that quoted it in order to respond to its forceful attack on Christianity. The bulk of these survivals comes from Cyril of Alexandria's twenty-book Contra Iulianum. The recent publication of the first modern critical edition of Cyril's work creates the occasion for a fresh study of the remnants of Julian's text that can be recovered from it. This is especially true for Books 11–20 of Cyril's treatise that are themselves lost and survive only in quotations in later Greek and Syriac sources. The present article undertakes a reassessment of the Julianic material preserved via the Syriac transmission of Contra Iulianum, including several passages hitherto unknown or ignored in earlier studies of Julian's treatise. It provides the Syriac text and English translation of eight passages and contextualizes them in the wider argumentative aim of Contra Galilaeos.

We consider finite simple graphs. Given a graph H and a positive integer $n,$ the Turán number of H for the order $n,$ denoted $\mathrm {ex}(n,H),$ is the maximum size of a graph of order n not containing H as a subgraph. Erdős asked: ‘For which graphs H is it true that every graph on n vertices and $\mathrm {ex}(n,H)+1$ edges contains at least two H’s? Perhaps this is always true.’ We solve this problem in the negative by proving that for every integer $k\ge 4$ there exists a graph H of order k and at least two orders n such that there exists a graph of order n and size $\mathrm {ex}(n,H)+1$ which contains exactly one copy of $H.$ Denote by $C_4$ the $4$ -cycle. We also prove that for every integer n with $6\le n\le 11$ there exists a graph of order n and size $\mathrm {ex}(n,C_4)+1$ which contains exactly one copy of $C_4,$ but, for $n=12$ or $n=13,$ the minimum number of copies of $C_4$ in a graph of order n and size $\mathrm {ex}(n,C_4)+1$ is two.

The exact crossing number is only known for a small number of families of graphs. Many of the families for which crossing numbers have been determined correspond to cartesian products of two graphs. Here, the cartesian product of the sunlet graph, denoted ${\mathcal{S}}_{n}$, and the star graph, denoted $K_{1,m}$, is considered for the first time. It is proved that the crossing number of ${\mathcal{S}}_{n}\Box K_{1,2}$ is $n$, and the crossing number of ${\mathcal{S}}_{n}\Box K_{1,3}$ is $3n$. An upper bound for the crossing number of ${\mathcal{S}}_{n}\Box K_{1,m}$ is also given.

During follicular development, granulosa cells undergo functional and structural changes affecting their steroidogenic activity. Oestrogen synthesis mainly occurs in the endoplasmic reticulum and relies on aromatase activity to convert androgens that arise from theca cells. In the present study, indicators of mitochondria-related steroidogenic capacity, as steroidogenic acute regulatory (StAR) protein expression and mitochondrial membrane potential (MMP), have been evaluated in bovine granulosa cells (GCs) and related to follicle growth and atresia. Atresia was estimated by morphological examination of follicle walls and cumulus–oocyte complexes (COC) and assessed by terminal deoxynucleotidyl transferase (TdT)-mediated dUTP nick end labeling (TUNEL) assay for apoptosis detection. Bovine ovarian follicles were macroscopically classified according to their atresia grade and grouped into small, medium or large follicles. After follicle opening, the COCs were morphologically classified for follicle atresia and the GCs were collected. Granulosa cells were fixed for immunofluorescence (IF) and TUNEL assay, frozen for western blotting (WB) or freshly maintained for MMP analyses. StAR protein expression was assessed using both IF and WB analyses. The follicle atresia grade could be efficiently discriminated based on either follicle wall or COC morphological evaluations. Granulosa cells collected from small non-atretic follicles showed a higher (P <0.01) MMP and WB-based StAR protein expression than small atretic follicles. For IF analysis, StAR protein expression in large atretic follicles was higher (P <0.05) than that in large non-atretic follicles. These results suggest a role played by mitochondria in GC steroidogenic activity, which declines in healthy follicles along with their growth. In large follicles, steroidogenic activity increases with atresia and is possibly associated with progesterone production.

The formation of EHB stars is linked to the lives of AGB stars by indications that such EHB/sdB stars might form in globular clusters with multiple populations linked to AGB evolution. Observations of massive globular clusters, such as ω-Centauri (Bedin et al.2004, Piotto et al.2005) suggest that single EHB stars might form from He-enhanced progenitors (D’Antona et al.2005, D’Antona & Caloi 2008, Lee et al.2005) in environments enriched by AGB ejecta. The studies conducted by Han et al. (2002), Han et al. (2003), and Han et al. (2007) have been able to provide a strong case for the binary formation of EHB/sdB stars in the Galactic field, though binary formation channels in globular clusters is uncertain. Simulations presented here are an extension of the simulations of Han et al. (2002) and Han et al. (2003), for low metallicities to examine the binary EHB population in globular clusters.

A limited list of new results of the searches for the new magnetic stars among late B and early A stars is in this work. Continual observations with spectroscopic devices of the 6m Russian telescope BTA led to successful detection of about 10 new magnetic stars that occupy different parts of evolutional tracks for the stars of 2–3 solar masses. Measurements of the longitudinal magnetic field show weak and medium strength magnetic field in all program stars.

Let $F$ be a non-separable $\text{LF}$-space homeomorphic to the direct sum ${{\sum }_{n\in \text{N}}}\,{{\ell }_{2}}\left( {{\tau }_{n}} \right)$, where ${{\aleph }_{0}}<{{\tau }_{1}}<{{\tau }_{2}}<\cdot \cdot \cdot $. It is proved that every open subset $U$ of $F$ is homeomorphic to the product $\left| K \right|\,\times \,F$ for some locally finite-dimensional simplicial complex $K$ such that every vertex $v\,\in \,{{K}^{\left( 0 \right)}}$ has the star $\text{St}\left( v,\,K \right)$ with card $\text{St}{{\left( v,K \right)}^{\left( 0 \right)}}<\tau =\sup {{\tau }_{n}}$ (and card ${{K}^{\left( 0 \right)}}\le \tau $), and, conversely, if $K$ is such a simplicial complex, then the product $\left| K \right|\,\times \,F$ can be embedded in $F$ as an open set, where $\left| K \right|$ is the polyhedron of $K$ with the metric topology.

We design a O(n3) polynomial time algorithm for finding a (k-1)- regular subgraph in a k-regular graph without any induced star K 1,3(claw-free). A polynomial time algorithm for finding a cubic subgraph in a 4-regular locally connected graph is also given. A family of k-regular graphs with an induced star K 1,3 (k even, k ≥ 6), not containing any (k-1)-regular subgraph is also constructed.

A star is a connected graph in which every node but possibly one has valency 1. Let G be a graph and C a spanning subgraph of G in which every component is a star. With each component α of C let us associate a weight wα. Let Пα wα be the weight associated with the entire subgraph G the star polynomial of G is ΣПα wα where the summation is taken over all spanning subgraphs of G consisting of stars. In this paper an algorithm for finding star polynomials of graphs is given. The star polynomials of various classes of graphs are then found, and some results about node-disjoint decomposition of complete graphs and complete bipartite graphs are deduced.