Orbital approach to microstate free entropy

Motivated by Voiculescu's liberation theory, we introduce the orbital free entropy $\chi_orb$ for non-commutative self-adjoint random variables (also for "hyperfinite random multi-variables"). Besides its basic properties the relation of $\chi_orb$ with the usual free entropy $\chi$ is shown. Moreover, the dimension counterpart $\delta_{0,orb}$ of $\chi_orb$ is discussed, and we obtain the relation of $\delta_{0,orb}$ with the original free entropy dimension $\delta_0$ with applications to $\delta_0$ itself.Comment: 38 pages; Section 5 was largely improved and Section 6 was adde

Strong time operators associated with generalized Hamiltonians

Let the pair of operators, $(H, T)$, satisfy the weak Weyl relation: $Te^{-itH} = e^{-itH}(T + t)$, where $H$ is self-adjoint and $T$ is closed symmetric. Suppose that g is a realvalued Lebesgue measurable function on \RR such that $g \in C^2(R K)$ for some closed subset K \subset \RR with Lebesgue measure zero. Then we can construct a closed symmetric operator $D$ such that $(g(H), D)$ also obeys the weak Weyl relation.Comment: 10 page

Formation of racemic compound crystals by mixing of two enantiomeric crystals in the solid state. Liquid transport of molecules from crystal to crystal

Mixing of powdered (-)- and (+)-enantiomer crystals in the solid state gives crystals of the racemic compound. This racemic crystal formation was followed by IR spectral measurement of a 1 :1 mixture of (-)- and (+)-enantiomer crystals as a Nujol mull. As the formation of racemic crystals proceeds, the OH absorptions of the enantiomer disappear gradually and new OH absorptions due to the racemic compound appear. The formation of racemic crystals from enantiomer crystals has been studied for various kinds of chiral compounds: 2,29-dihydroxy-1,19-binaphthyl (1) and its derivatives, 10,109-dihydroxy-9,99-biphenanthryl (4), 2,29-dihydroxy-4,49,6,69-tetramethylbiphenyl (5) and its derivatives, 4,49-dihydroxy-2,29,3,39,6,69- hexamethylbiphenyl (8), 1,6-di(o-chlorophenyl)-1,6-diphenylhexa-2,4-diyne-1,6-diol (11) and its derivatives, trans-4,5-bis[hydroxy(diphenyl)methyl]-2,2-dimethyl-1,3-dioxacyclopentane (17) and itsderivatives, tartaric acid (20) dimethyl tartrate (21), malic acid (22), mandelic acid (23), and norephedrine (24). These molecular movements and blending occur rapidly in the presence of liquids such as liquid paraffin (Nujol), seed oils such as olive, coconut, rapeseed and soybean oil, artificial oil such as silicone oil and water, although the same movement also occurs in the absence of the liquid. For example, keeping a mixture of powdered (-)-1 (1a) and (+)-1 (1b) at room temperature for 48 h gives racemic crystals (1c). However, molecular aggregation sometimes occurs in solution but not in the solid state. Forexample, recrystallization of (-)-16 (16a) and (+)-16 (16b) from solvent gives racemic crystals of 16c, although mixing of these two components as powders in the presence of liquid does not give 16c. In order to determine the mechanism of the molecular movement in the solid state, X-ray crystal structures of optically active and racemic compounds and also the molecular movements from optically active crystal to racemic crystal have been studied

The Selective Arterial Calcium Injection Test is a Valid Diagnostic Method for Invisible Gastrinoma with Duodenal Ulcer Stenosis : A Case Report

The localization and diagnosis of microgastrinomas in a patient with multiple endocrine neoplasia type 1 is difficult preoperatively. The selective arterial calcium injection (SACI) test is a valid diagnostic method for the preoperative diagnosis of these invisible microgastrinomas. We report a rare case of multiple invisible duodenal microgastrinomas with severe duodenal stenosis diagnosed preoperatively by using the SACI test. A 50-year-old man was admitted to our hospital with recurrent duodenal ulcers. His serum gastrin level was elevated to 730 pg/ml. It was impossible for gastrointestinal endoscopy to pass through to visualize the inferior part of the duodenum, because recurrent duodenal ulcers had resulted in severe duodenal stenosis. The duodenal stenosis also prevented additional endoscopic examinations such as endoscopic ultrasonography. Computed tomography did not show any tumors in the duodenum and pancreas. The SACI test provided the evidence for a gastrinoma in the vascular territory of the inferior pancreatic-duodenal artery. We diagnosed a gastrinoma in the peri- ampullary lesion, so we performed Subtotal Stomach-Preserving Pancreatico- duodenectomy with regional lymphadenectomy. Histopathological findings showed multiple duodenal gastrinomas with lymph node metastasis and nonfunctioning pancreatic neuroendocrine tumors. Twenty months after surgery, the patient is alive with no evidence of recurrence and a normal gastrin level. In conclusion, the SACI test can enhance the accuracy of preoperative localization and diagnosis of invisible microgastrinomas, especially in the setting of severe duodenal stenosis