Equivalent conditions for the validity of the Helmholtz decomposition of Muckenhoupt ApA_{p}-weighted LpL^{p}-spaces

summary:We discuss the validity of the Helmholtz decomposition of the Muckenhoupt $A_{p}$-weighted $L^{p}$-space $(L^{p}_{w}(\Omega ))^{n}$ for any domain $\Omega$ in $\mathbb {R}^{n}$, $n \in \mathbb {Z}$, $n\geq 2$, $1<p<\infty$ and Muckenhoupt $A_{p}$-weight $w \in A_{p}$. Set $p':={p}/{(p-1)}$ and $w':=w^{-{1}/{(p-1)}}$. Then the Helmholtz decomposition of $(L^{p}_{w}(\Omega ))^{n}$ and $(L^{p'}_{w'}(\Omega ))^{n}$ and the variational estimate of $L^{p}_{w,\pi }(\Omega )$ and $L^{p'}_{w',\pi }(\Omega )$ are equivalent. Furthermore, we can replace $L^{p}_{w,\pi }(\Omega )$ and $L^{p'}_{w',\pi }(\Omega )$ by $L^{p}_{w,\sigma }(\Omega )$ and $L^{p'}_{w',\sigma }(\Omega )$, respectively. The proof is based on the reflexivity and orthogonality of $L^{p}_{w,\pi }(\Omega )$ and $L^{p}_{w,\sigma }(\Omega )$ and the Hahn-Banach theorem. As a corollary of our main result, we obtain the extrapolation theorem with the aid of the Helmholtz projection of $(L^{p}_{w}(\Omega ))^{n}$

Synthesis and crystal structures of triruthenium clusters containing phosphinidene ligands

The manipulation of trinuclear ruthenium clusters [Ru3(CO)9(μ-H)2(μ3-PMes)] (1) with PH2Ph, in refluxing hexane, produces [Ru3(CO)8(PH2Ph)(μ-H)2(μ3-PMes)] (2) where PH2Ph coordinates as a terminal ligand. Cluster 2 gets converted to [Ru3 (CO)9(μ3-PMes)(μ3-PPh)] (3) when heated in a toluene solution in the presence of CO. This study provides the solutions of the X-ray structures for clusters 2 and 3

Monte Carlo study on low-temperature phase diagrams of the J1J_1-J2J_2 classical XYXY kagome antiferromagnet

Frustrated magnets with degenerate ground states exhibit exotic ground states and rich phase structures when perturbations and/or thermal fluctuations lift the degeneracy. In two-dimensional models with short-range interactions, continuous symmetries cannot spontaneously break at finite temperatures, leading to the suppression of conventional magnetic long-range ordering (LRO). In this paper, we numerically study the classical $J_1$-$J_2$ $XY$ antiferromagnet on the kagome lattice as a prototype model of such frustrated magnets, where $J_2$ denotes the next-nearest-neighbor exchange interaction. We map out the $J_2$-$T$ phase diagram of this model employing extensive classical Monte Carlo (MC) simulations. The obtained phase diagram features Berezinskii-Kosterlitz-Thouless (BKT) transitions of $q=0$, $\sqrt{3}\times\sqrt{3}$ magnetic orders, and octupole orders, in addition to finite-temperature phase transitions of both ferrochiral and antiferrochiral long-range orders. Additionally, we find a non-trivial first-order transition for antiferromagnetic $J_2/J_1 < 0$. The origin of this transition is discussed in the context of non-local loop structures present in local $120^\circ$ spin structures

Development Process of Watershed Partnership: a Case Study of Limboto Watershed - Gorontalo, Indonesia

Establishment of watershed partnership is required to achieve integrated watershed management in Indonesia. Taking “partnership of Limboto watershed” as a case study, this study aimed to explore the development process of watershed partnership and its state and role in a watershed. Data and information were collected through interviews, group discussions and questionnaire survey. The results showed that in the early stages, through a good preparation process, the partnership gained recognition and support from stakeholders. Informal, open, and voluntary processes were adopted by the coordinator of the partnership for the establishment and initial operation phases which gave the partnership the experience of being inclusive and an active period which was considered as a good success. However, in the next phase, the partnership was unable to cope with the formalization of the watershed partnership as demanded by the decentralized institution by having a new structure without a definite and responsible institution, secretariat and budget source due to lack of interest of the expected local government to take on the responsibility. Formalization resulted in the disfunction of the partnership. Though, many participants considered the partnership had a positive role, especially concerning exchange of information and improvement of concern about the community. This partnership did not yet function for integrated watershed management, but more to educate and inspire people on watershed conservation through project-based activities

VEGF is indirectly associated with NO production

Background? Increased levels of vascular endothelial growth factor (VEGF) have been observed in patients with metabolic syndrome (MetS). Nitric oxide (NO) formation is reduced in MetS, but its relationship to VEGF production remains poorly defined. We evaluated the association between VEGF/NO synthesis and insulin sensitivity in obese subjects and investigated the secretory response of VEGF to an acute elevation of glucose.Materials and methods? Seven healthy normal-weight subjects, seven obese subjects without MetS and seven obese subjects with MetS were recruited. Anthropometry, body composition and cardiometabolic functions (blood pressure, glucose, insulin, triglycerides, total cholesterol, HDL-C and VEGF) were measured, and a novel stable isotope method was used to assess in vivo rates of NO production. A frequent sampling intravenous glucose tolerance test was performed to study the dynamics of VEGF release.Results? Fasting VEGF levels were significantly higher in the two obese groups compared to the control group (P for trend = 0·02), but the difference was not significant after adjustment for age. Vascular endothelial growth factor levels were associated with systolic blood pressure (? = 0·54; P = 0·01) and NO production (? = ?0·44; P = 0·04). Vascular endothelial growth factor levels increased in response to acute hyperglycaemia in normal-weight and obese subjects (P &lt; 0·001).Conclusions? Vascular endothelial growth factor levels rapidly increase during hyperglycaemia and are inversely related to NO production at steady state. The potential link between the acute secretion of VEGF and atherosclerotic risk in subjects with poorly controlled glycaemia as well as the potential of lowering elevated VEGF levels by increasing NO production and/or availability warrants further investigation

Ruptured anterior paraclinoid aneurysms

The purpose of this study was to evaluate cases of subarachnoid hemorrhage (SAH) from ruptured anterior (dorsal) paraclinoid aneurysms. Anterior paraclinoid aneurysms are defined as aneurysms arising from the anterolateral wall of the proximal internal carotid artery without any relationship to an arterial branch. Between 1991 and 2008, a total of 159 patients with 169 paraclinoid aneurysms were treated at the Shinshu University Hospital and its affiliated hospitals. A retrospective analysis was carried out using charts, operation records, operation videos, and neuroimagings. Twenty six patients had anterior paraclinoid aneurysm. Six patients presented with SAH. Three aneurysms were saccular and the others were blister-like aneurysms based on operative findings. Neck laceration or premature rupture frequently happened during the clipping surgery even though the aneurysm was saccular type. The treatment of a ruptured anterior paraclinoid aneurysm is quite difficult. Trapping and bypass would be recommended for such fragile aneurysms.ArticleNEUROSURGICAL REVIEW. 34(1):49-54 (2011)journal articl