Some geometric applications of the beta distribution

Let be the angle between a line and a random k-space in Euclidean n-space R ⁿ. Then the random variable cos² has the beta distribution. This result is applied to show (1) in R ⁿthere are exponentially many (in n) lines going through the origin so that any two of them are nearly perpendicular, (2) any N-point set of diameter d in R ⁿlies between two parallel hyperplanes distance 2d{(log N)/(n-1)}^1/2 apart and (3) an improved version of a lemma of Johnson and Lindenstrauss (1984, Contemp. Math., 26, 189–206). A simple estimate of the area of a spherical cap, and an area-formula for a neighborhood of a great circle on a sphere are also given.     

Wallace’s theorem and Miquel’s theorem in higher dimensions

We present a simple proof for the generalizations of a result due to Wallace and a result due to Miquel to higher dimensions.     

Palladium-catalyzed direct oxidative vinylation of thiophenes and furans under weakly basic conditions

The palladium-catalyzed oxidative coupling of thiophenes and furans with alkenes proceeds in the presence of copper and lithium salts as oxidant and additive, respectively, under weakly basic or almost neutral conditions to afford the corresponding vinylated heteroarenes. Under such conditions, diphenyl(hydroxy)methyl and acetal functions on the heteroarene substrates are tolerable. The former function on a thiophene ring can be substituted by palladium-catalyzed arylation via C-C bond cleavage after the vinylation to produce 2-aryl-5-vinylthiophenes.     

Surface structures of K on Mo(1 1 0) surface investigated by RHEED

Phase transitions of K on Mo(1 1 0) have been studied by RHEED technique. As Ba and Cs structures on the bcc(1 1 0) surface, surface structures of K were hexagonal from RT to 250 °C for θ > 0.9 ML. The hexagonal structure successively expanded from α to γ structure with Nishiyama-Wassermamm (N-W) orientation relationship. The nearest neighbor spacing in the α structure at RT was 4.50 Å, which is very closed to the atomic distance of K in metal, and stretched to 5.14 Å in the γ structure at T = 200 °C. At temperatures greater than T = 250 °C, the γ structure oriented in N-W and Kurdjumov-Sachs (K-S) relationships at the same time and stayed up to the temperature of 450 °C. These two orientations of γ structure also appeared in all temperature range for 0.4 < θ < 0.9 ML.     

When does the union of random spherical caps become connected?

DropN random caps all of the same angular radius on a unit sphere. LetU denote the part of the surface covered by these caps. We prove that if , then the probability thatU is connected tends to 1 asN→∞, while ifc<1, then the probability thatU is connected tends to 0 asN→∞.     

Embedding a polytope in a lattice

We present a special similarity ofR ⁴ⁿ which maps lattice points into lattice points. Applying this similarity, we prove that if a (4n−1)-polytope is similar to a lattice polytope (a polytope whose vertices are all lattice points) inR ⁴ⁿ , then it is similar to a lattice polytope inR ⁴ⁿ⁻¹, generalizing a result of Schoenberg [4]. We also prove that ann-polytope is similar to a lattice polytope in someR ^N if and only if it is similar to a lattice polytope inR ²ⁿ⁺¹, and if and only if sin²(<ABC) is rational for any three verticesA, B, C of the polytope.     

Accumulation of photofrin in lesions of airway stenosis rabbit models

Airway stenosis in childhood is resistant to conventional treatments. Endoscope-assisted photodynamic therapy (PDT) is a potent candidate for the therapeutic modality owing to the easy approach to the tracheal lesion and low degree of invasiveness. The aim of the present study was to examine whether a photosensitizer preferentially accumulates in the lesion of airway stenosis in order to explore the possible applicability of PDT. The tracheal mucosa of rabbits was scraped off, and the rabbits were intravenously administered with Photofrin. The tissue concentration of Photofrin was quantitatively measured by fluorometric analysis. Granulation formation was seen in the mucosa-deprived lesion, causing airway stenosis. Photofrin concentration in the granulation tissue was four-fold higher than that in the intact trachea and 10-fold higher than that in the liver, spleen, skin and muscle. Photofrin preferentially accumulated in the lesion of airway stenosis. A preliminary experiment on PDT using transtracheal illumination showed an amelioration of airway stenosis, resulting in reduction in respiratory stridor.     

Diffusion-weighted echo-planar imaging of the head and neck using 3-T MRI: Investigation into the usefulness of liquid perfluorocarbon pads and choice of optimal fat suppression method

Purpose

To investigate whether image quality can be improved using liquid perfluorocarbon pads (Sat Pad) and clarify the optimal fat-suppression method among chemical shift selective (CHESS), water excitation (WEX), and short TI inversion recovery (STIR) methods in diffusion-weighted imaging (DWI) of the head and neck using 3-T magnetic resonance imaging. Correlations between results of visual inspection and quantitative analysis were also examined.

Material and Methods

This study was approved by our Institutional Review Board and informed consent was waived. DWI was performed on 25 subjects with/without Sat Pad and using three fat-suppression methods (6 patterns). Image quality was evaluated visually (4-point scales and lesion-depiction capability) and by quantitative analysis (signal-to-noise ratio (SNR) and contrast-to-noise ratio (CNR)). Two-way repeated-measures analysis of variance (ANOVA) was used to detect significant differences in scores of visual evaluation, SNR, and CNR.

Results

Mean visual evaluation scores were significantly higher with Sat Pad using STIR than without Sat Pad for all fat-suppression methods (P < 0.05). DWI with Sat Pad using STIR tended to be useful for depicting lesions. DWI using STIR showed reduced W-SNR (W: whole area of depicted structure) and CNR (between semispinalis capitis muscle and subcutaneous fat) due to fewer artifacts and uniform fat suppression.

Conclusion

Combining Sat Pad with STIR provides good image quality for visual inspections. When numerous artifacts are present and fat suppression is insufficient, higher SNR and CNR do not always provide good diagnostic image quality.