H-convex standard fundamental domain of a subgroup of a modular group

The two major ways of obtaining fundamental domains for discrete subgroups of SL(2,?) are the Dirichlet Polygon construction (see Lehner in Discontinuous Groups and Automorphic Functions, American Mathematical Society, Providence, 1964) and Ford’s construction (see Ford in Automorphic Functions, McGraw–Hill, New York, 1929). Each of these two methods yield a hyperbolically convex fundamental domain for any discrete subgroup of SL(2,?).However, the Dirichlet polygon construction and Ford’s construction are not well adapted for the actual construction of a hyperbolically convex fundamental domain due to their nature of construction and their reliance on knowing almost all elements of the group under discussion.

A third-and most important and practical-method of obtaining a fundamental domain is through the use of a right coset decomposition as described below. Let Γ₂ be a subgroup of Γ₁ and

$\Gamma_{1}=\Gamma_{2}\cdot \{L_{1},L_{2},\ldots,L_{m}\}.$

If \(\mathbb{F}\) is a fundamental domain of the bigger group Γ₁, then the set

$\mathcal{R}_{\Gamma}=\Biggl(\overline{\bigcup_{k=1}^{m}L_{k}(\mathbb{F})}\,\Biggr)^{o}$

(1)

is a fundamental domain of Γ₂. One can ask at this juncture, is it possible to choose the right cosets suitably so that the set ?_Γ is hyperbolically convex? We will answer this question affirmatively for

$\Gamma_{1}=\Gamma(1)\quad \mbox{and}\quad \mathbb{F}=\biggl\{\tau \in \mathbb{H}:|\tau|>1\ \&;\ |\mathrm{Re}(\tau)|<\frac{1}{2}\biggr\}.$

     

Renal replacement lipomatosis: MR findings in one case

Replacement lipomatosis of the kidney is the result of severe atrophy or destruction of the renal parenchyma, often caused by calculous disease with secondary marked proliferation of renal sinus, renal hilus, and perirenal fatty tissue. Different diagnostic tools have been used to define this entity, with ultrasonography (US), i.v. pyelography (IVP) and computed tomography (CT) used most commonly. We report MR urography and MR appearance of replacement lipomatosis in a 38-year-old man. We think that it combines the advantages of IVP, US and CT.     

(E)-2,6,10-Trimethyldodec-8-en-2-ol: An Undescribed Sesquiterpenoid from Copaiba Oil

The use of copaiba oil has been reported since the 16th century in Amazon traditional medicine, especially as an anti-inflammatory ingredient and for wound healing. The use of copaiba oil continues today, and it is sold in various parts of the world, including the United States. Copaiba oil contains mainly sesquiterpenes, bioactive compounds that are popular for their positive effect on human health. As part of our ongoing research endeavors to identify the chemical constituents of broadly consumed herbal supplements or their adulterants, copaiba oil was investigated. In this regard, copaiba oil was subjected to repeated silica gel column chromatography to purify the compounds. As a result, one new and seven known sesquiterpenes/sesquiterpenoids were isolated and identified from the copaiba oil. The new compound was elucidated as (E)-2,6,10-trimethyldodec-8-en-2-ol. Structure elucidation was achieved by 1D- and 2D NMR and GC/Q-ToF mass spectral data analyses. The isolated chemical constituents in this study could be used as chemical markers to evaluate the safety or quality of copaiba oil.     

Effect of shear strain on the α-ε phase transition of iron: a new approach in the rotational diamond anvil cell

The effect of shear strain on the iron α-ε phase transformation has been studied using a rotational diamond anvil cell (RDAC). The initial transition is observed to take place at the reduced pressure of 10.8?GPa under pressure and shear operation. Complete phase transformation was observed at 15.4?GPa. The rotation of an anvil causes limited pressure elevation and makes the pressure distribution symmetric in the sample chamber before the phase transition. However, it causes a significant pressure increase at the centre of the sample and brings about a large pressure gradient during the phase transformation. The resistance to the phase interface motion is enhanced due to strain hardening during the pressure and shear operations on iron and this further increases the transition pressure. The work of macroscopic shear stress and the work of the pressure and shear stress at the defect tips account for the pressure reduction of the iron phase transition.     

A remark on the fundamental lemma of Jacquet

We conjecture a generalization of the fundamental lemma of Jacquet in the context of ${\mathit{GL}}_n$ over a quadratic extension. We provide a heuristic argument for our expectation and prove our conjecture for ${\mathit{GL}}_2$. To cite this article: O. Offen, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Flavonoid glycosides and methylinositol from Ebenus haussknechtii

Two flavonoid glycosides (compounds 1 and 3) of which one is reported for the first time and a methylinositol (compound 2) were isolated from the aerial parts of Ebenus haussknechtii (Leguminosae). The structures were established as quercetin-7-O-[alpha-L-rhamnopyranosyl(1 --> 6)-beta-D-galactopyranoside] (1), morin-3-O-[4-[5-(4-hydroxyphenyl)pentanoyl]-alpha-L-rhamnopyranosyl(1 --> 6)-beta-D-galactopyranosyl]-7-4'-di-O-methyleter (3), and methylinositol (2) on the basis of chemical and spectroscopic means. The antimicrobial activities of the extracts have also been examined.     

A facile method of producing femtoliter metal cups by pulsed laser ablation

Cuplike structures of Au, Ag, Cu, Zn, Nb, Cd, Al, In, and Sn in the size range of 300 nm to a few micrometers with an internal volume of a few femtoliters have been produced by the laser ablation of metal targets in a vacuum, by optimizing, in each case, the laser fluence and the substrate temperature. The metal droplets impinging on the substrate seem to undergo a hydraulic jump driven by the surface tension forces before solidifying into cups. The cups are robust and can be functionalized with biomarkers, filled with nanoparticle sols, oxidized to crucibles, or detached from the substrate without causing any deformation. We envisage their potential applications as femtoliter metal containers.     

Effect of elevated concentrations of strontium and iron on the structural and dielectric characteristics of Ba(1−x−y)Sr(x)Ti Fe(y)O3 prepared through sol–gel technique

Nano-composite Ba_1−xSr_(x)TiO₃ (BST), where x=0.01–0.50 and doped with different concentrations of iron Ba_(1−x−y)Sr_(x)TiFe _(y)O₃ (BSTF), where x=0.01 and y=0.01–0.05 powders were prepared by sol–gel method.     

Synthesis of arylated xanthones by site-selective Suzuki–Miyaura reactions of the bis(triflate) of 1,3-dihydroxyxanthone

Arylated xanthones were prepared by site-selective Suzuki–Miyaura reactions of the bis(triflate) of 1,3-dihydroxyxanthone. The first attack occurs at the sterically less encumbered position 3. The site-selectivity is in contrast to the bis(triflate) of 1,3-dihydroxyanthraquinone.