In the papers [B1] and [B2] we have established some conditions for the finitude of the Hausdorff h-measure of some set. Now, we shall determine a better minorant for this measure. 1 Definitions We denote by R the euclidean n dimensional space and by d(E) the diameter of a set E ⊂ R. Definition 1 If r0 > 0 is a fixed number, a continuous function h(r), defined on [0, r0) , nondecreasing and suc...

Before we state our main theorems, we begin with some notation: given a finite subset A of some commutative ring, we let A + A denote the set of sums a + b, where a, b ∈ A; and, we let A.A denote the set of products ab, a, b ∈ A. When three or more sums or products are used, we let kA denote the k-fold sumset A+A+ · · ·+A, and let A denote the k-fold product set A.A...A. Lastly, by d ∗ A we mea...

Let S ∑ be a measurable space and let h : S S R be a real symmetric Borel/∑ ∑measurable function on S S. Let B be a non-empty measurable subset in S and let μ be a probability measure supported on the restriction of the measurable space S ∑ to B. Let B have finite h-diameter h ess sup h u v : u v B ∞ Let U U1 U2 .... be a sequence of independent random points taking values in B according to μ a...

Erdős and Hajnal conjectured that for every graph H there is a constant ε = ε(H) > 0 such that every graph G that does not have H as an induced subgraph contains a clique or a stable set of order Ω(|V (G)|ε). The conjecture would be false if we set ε = 1; however, in an asymptotic setting, we obtain this strengthened form of Erdős and Hajnal’s conjecture for almost every graph H, and in particu...

In this study, aerial parts of Lippia citriodora Kunth were harvested at three phenological stages (i.e. vegetative, full flowering and fruit set). The essential oil isolated by hydro distillation for 3 h using a Clevenger-type apparatus and the oil was analyzed by GC and GC-MS. The yields of oils (w/w%) in different stages was in the order of: full flowering (0.9%), vegetative (0.48%) and frui...

The genus Hypericum is one of the most important medicinal plants that contain 17 species in Iran, three of them are endemics. This paper reports the essential oil composition of eight Hypericum species from Iran. The essential oil analysis of a number of the studied plants has already been reported but their report from Iran may be valuable for scientists.  Samples collected from different pla...

 The category of the title is called $mathcal{W}$. This has all free objects $F(I)$ ($I$ a set). For an object class $mathcal{A}$, $Hmathcal{A}$ consists of all homomorphic images of $mathcal{A}$-objects. This note continues the study of the $H$-closed monoreflections $(mathcal{R}, r)$ (meaning $Hmathcal{R} = mathcal{R}$), about which we show ({em inter alia}): $A in mathcal{A}$ if and  only if...

α-acyloxycarboxamides are synthesized from three component Passerini reaction between indane-1,2,3-trione, isocyanides, and thiophenecarboxylic acids in quantitative yields. The structures of the final products were confirmed by IR, 1H and 13C NMR spectroscopy, mass spectrometry, and elemental analysis. The B3LYP/HF calculations for computation of 1H an...

Let g be a fixed element of a finite group G. We introduce the g-noncommuting graph of G whose vertex set is whole elements of the group G and two vertices x,y are adjacent whenever [x,y] g  and  [y,x] g. We denote this graph by . In this paper, we present some graph theoretical properties of g-noncommuting graph. Specially, we investigate about its planarity and regularity, its clique number a...