An inequality for the convolutions on unimodular locally compact groups and the optimal constant of Young's inequality

Let $\mu$ be the Haar measure of a unimodular locally compact group $G$ and $m (G)$ as the infimum of the volumes of all open subgroups of $G$. The main result of this paper is that \begin{align*} \int_{G}^{} f \circ \left( \phi_1 * \phi_2 \right) \left( g \right) dg \leq \int_{\mathbb{R}}^{} f \circ \left( \phi_1^* * \phi_2^* \right) \left( x \right) dx \end{align*} holds for any measurable functions $\phi_1, \phi_2 \colon G \to \mathbb{R}_{\geq 0}$ with $\mu ( \mathrm{supp} \; \phi_1 ) + \mu ( \mathrm{supp} \; \phi_2 ) \leq m(G)$ and any convex function $f \colon \mathbb{R}_{\geq 0} \to \mathbb{R}$ with $f(0) = 0$. Here $\phi^*$ is the rearrangement of $\phi$. Let $Y_O(P,G)$ and $Y_R(P,G)$ denote the optimal constants of Young's and the reverse Young's inequality, respectively, under the assumption of $\mu ( \mathrm{supp} \; \phi_1 ) + \mu ( \mathrm{supp} \; \phi_2 ) \leq m(G)$. Then we have $Y_O(P,G) \leq Y_O(P,\mathbb{R})$ and $Y_R(P,G) \geq Y_R(P,\mathbb{R})$ as a corollary. Thus, we obtain that $m (G) = \infty$ if and only if $H (p,G) \leq H (p, \mathbb{R})$ in the case of $p' := p/(p-1) \in 2 \mathbb{Z}$, where $H (p,G)$ is the optimal constant of the Hausdorff--Young inequality.Comment: 22 pages, 5 figure

ANTIBACTERIAL ACTIVITY OF BIFIDOBACTERIA ISOLATED FROM INFANT FAECES

Antibacterial activity of bifidobacteria isolated from Mongolian infant faeces was elucidated on pathogenic intestinal bacteria for the development of a new antibacterial bifidobacteria, the permission for which was granted by the Mongolian Medical Ethics Committee Approval (MMECA). A total of forty-nine single colonies were obtained from 3 samples by using a BL medium enrichment. Among them, 29 isolates had Gram−positive, catalase−negative properties, and maul−like or Y−shaped morphology, and then, 20 Bifidobacterium breve and 9 Bifidobacterium longum strains were detected by the B. breve and B. longum specific primers. Organic acids produced by the isolated bifidobacteria in their cell-free supernatants were quantitatively analyzed by a spectrophotometric absorbance at 340 nm, suggesting that D−lactic, L−lactic, and acetic acids were produced, and the pH of the supernatants was at 3.86−4.55. The isolated bifidobacteria showed antibacterial activity toward Escherichia coli and Salmonella typhimurium as high as that of a standard bifidobacteria, however, lower activity against Staphylococcus aureus. The antibacterial activity was probably due to the production of organic acids

Sensing of Metal Ions by Hybrid Systems of a Chiral Schiff Base Zn(II) Complex and Spiropyran

We have constructed three (hybrid) systems for quantitative fluorescence sensing of metal ions by using chiral Schiff base Zn(II) complexes. System 1 is a hybrid system composed of a trans-type chiral Schiff base Zn(II) complex and merocyanine (Mc), which is open-form of spiropyran (Sp) after photoisomerization. Depending on intermolecular interactions and quenching, increase (Zn2+) or decrease (Cu2+ and Gd3+) of fluorescence intensity of Zn(II) complex could be observed as functions of concentration of metal ions. System 2 is a sole component of a salen-type chiral Schiff base Zn(II) complex which can coordinate metal ions. After coordination of Zn2+, Cu2+, and Gd3+ ions, decreasing of fluorescence intensity could be found for all the cases for system 2 under the same condition to system 1. System 3 is a hybrid system being consisted of a salen-type chiral Schiff base Zn(II) complex and Mc. Decrease or increase of fluorescence intensity is in agreement with intermolecular interactions, namely affinity for the Zn(II) complex is Zn2+ > Cu2+ > Gd3+, whereas it is reverse order for Mc. The quantitative spectroscopic nature (detected as absorption, CD, and fluorescence) of these systems is predominantly attributed to both organic ligands and Zn(II) center in a probe comple

Genetic alterations in gliosarcoma and giant cell glioblastoma

The majority of glioblastomas develop rapidly with a short clinical history (primary glioblastoma IDH wild-type), whereas secondary glioblastomas progress from diffuse astrocytoma or anaplastic astrocytoma. IDH mutations are the genetic hallmark of secondary glioblastomas. Gliosarcomas and giant cell glioblastomas are rare histological glioblastoma variants, which usually develop rapidly. We determined the genetic patterns of 36 gliosarcomas and 19 giant cell glioblastomas. IDH1 and IDH2 mutations were absent in all 36 gliosarcomas and in 18 of 19 giant cell glioblastomas analyzed, indicating that they are histological variants of primary glioblastoma. Furthermore, LOH 10q (88%) and TERT promoter mutations (83%) were frequent in gliosarcomas. Copy number profiling using the 450k methylome array in 5 gliosarcomas revealed CDKN2A homozygous deletion (3 cases), trisomy chromosome 7 (2 cases), and monosomy chromosome 10 (2 cases). Giant cell glioblastomas had LOH 10q in 50% and LOH 19q in 42% of cases. ATRX loss was detected immunohistochemically in 19% of giant cell glioblastomas, but absent in 17 gliosarcomas. These and previous results suggest that gliosarcomas are a variant of, and genetically similar to, primary glioblastomas, except for a lack of EGFR amplification, while giant cell glioblastoma occupies a hybrid position between primary and secondary glioblastomas. This article is protected by copyright. All rights reserved

Malignant Pigmented Villonodular Synovitis in the Knee - Report of a Case with Rapid Clinical Progression

Malignant pigmented villonodular synovitis (PVNS) (or malignant giant cell tumor of tendon sheath (GCTTS) is an extremely rare condition defined as a malignant lesion occurring with concomitant or previously documented PVNS at the same site. To date, only less than 20 cases have been reported in English literatures. We report a case of malignant PVNS in the knee in a 56-year-old woman with unpredictable rapid progression. This case raised a caution that when atypical components in specimens of recurrent benign PVNS are detected, even if low-grade or tiny, both pathologists and surgeons should consider the risk of malignant PVNS, which could display aggressive clinical progression

Spontaneous complete necrosis of hepatocellular carcinoma caused by feeding vessel occlusion outside the tumour capsule

A 64-year-old man began treatment for chronic hepatitis C with peg-interferon and ribavirin. His hepatitis C virus ribonucleic acid (HCV-RNA) results turned negative. Just after the treatment, a computed tomography (CT) scan revealed a hypovascular mass in the segment 8. We performed a right hemihepatectomy as HCC. Upon macroscopic examination, the tumour was yellow and firm with a fibrous capsule. A wedge-shaped necrotic area was located at the top of an artery and a portal vein of segment 8. Necrosis was observed not only in the tumour but in the adjacent parenchyma. A histopathological examination showed that the tumour had been completely replaced by necrosis. This tumour was surrounded by capillary vessels and fed by several thick arteries, but organized thrombi were not detected. No viable cells were found. The histopathological diagnosis was a spontaneous complete necrosis of HCC caused by the occlusion of feeding vessels outside the capsule