Reproducing Kernels for Iterated Parabolic Operators on the Upper Half Space with Application to Polyharmonic Bergman Spaces

We consider parabolic operators of fractional order and their iterates on the upper half space of the euclidean space. We deal with Hilbert spaces of solutions of those parabolic equations. We shall show, in this note, the existence of reproducing kernels and give a formula by using their fundamental solutions. As an application, we also discuss the polyharmonic Bergman spaces and give their reproducing kernels by using the Poisson kernel on the upper half space.     

Aerobic oxidation of alcohols using bismuth bromide as a catalyst

We developed an environmentally friendly method for aerobic oxidation of alcohols using a commercially available, relatively benign bismuth salt as a catalyst. We found that the catalytic combination of BiBr₃ with nitric acid is key for enhancing the reactivity. The reaction proceeds well under air, making the use of pure oxygen unnecessary. Each of the primary or secondary alcohols tested was oxidized to the corresponding aldehydes or ketones using this protocol.     

Silica gel-mediated hydrohalogenation of unactivated alkenes using hydrohalogenic acids under organic solvent-free conditions

Silica gel-mediated hydrochlorination of unactivated alkenes using 35% hydrochloric acid under organic solvent-free conditions proceeded to give the corresponding chlorides in good yields. Hydrobromination or hydriodination using 47% hydrobromic acid or 55% hydriodic acid afforded the corresponding halides, respectively. Silica gel could be recycled five times without any significant loss of activities.     

Internal radiocesium contamination of adults and children in Fukushima 7 to 20 months after the Fukushima NPP accident as measured by extensive whole-body-counter surveys

The Fukushima Dai-ichi NPP accident contaminated the soil of densely-populated regions in Fukushima Prefecture with radioactive cesium, which poses significant risks of internal and external exposure to the residents. If we apply the knowledge of post-Chernobyl accident studies, internal exposures in excess of a few mSv/y would be expected to be frequent in Fukushima.Extensive whole-body-counter surveys (n = 32,811) carried out at the Hirata Central Hospital between October, 2011 and November, 2012, however show that the internal exposure levels of residents are much lower than estimated. In particular, the first sampling-bias-free assessment of the internal exposure of children in the town of Miharu, Fukushima, shows that the ¹³⁷Cs body burdens of all children (n = 1,383, ages 6–15, covering 95% of children enrolled in town-operated schools) were below the detection limit of 300 Bq/body in the fall of 2012. These results are not conclusive for the prefecture as a whole, but are consistent with results obtained from other municipalities in the prefecture, and with prefectural data.     

Noncolliding Squared Bessel Processes

We consider a particle system of the squared Bessel processes with index ν>−1 conditioned never to collide with each other, in which if −1<ν<0 the origin is assumed to be reflecting. When the number of particles is finite, we prove for any fixed initial configuration that this noncolliding diffusion process is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel called the correlation kernel. When the number of particles is infinite, we give sufficient conditions for initial configurations so that the system is well defined. There the process with an infinite number of particles is determinantal and the correlation kernel is expressed using an entire function represented by the Weierstrass canonical product, whose zeros on the positive part of the real axis are given by the particle-positions in the initial configuration. From the class of infinite-particle initial configurations satisfying our conditions, we report one example in detail, which is a fixed configuration such that every point of the square of positive zero of the Bessel function J _ν is occupied by one particle. The process starting from this initial configuration shows a relaxation phenomenon converging to the stationary process, which is determinantal with the extended Bessel kernel, in the long-term limit.     

Novel enantioselective direct aldol-type reaction promoted by a chiral phosphine oxide as an organocatalyst

Chiral phosphine oxide successfully catalyzed the direct aldol-type reactions of cyclohexanone derivatives and benzaldehyde derivatives in high stereoselectivities. The reaction mechanism involves the in situ formation of trichlorosilyl enol ethers. The present reaction could be extended to the cross-aldol reactions between two aldehydes.     

Synthesis of 7‐acetyloxy‐3,7‐dimethy1‐7,8‐dihydro‐6H‐isochromene‐6,8‐dione and its analogues

7‐Alkanoyloxy‐3,7‐dimefhyl‐7,8‐dihydro‐6H‐isochromene‐6,8‐diones 12‐15 were synthesized in 69‐16% yields from the reaction of 2,4‐dihydroxy‐3‐methyl‐6‐(2‐oxopropyl)benzaldehyde 11 with p‐toluenesulfonic acid in various carboxylic acids such as acetic acid, propionic acid, butyric acid and heptanoic acid followed by oxidation with lead tetraacetate. On the other hand, (±)‐daldinin A 5 (oleate) was not obtained using oleic acid as a medium. In the cases of heptanoic acid and oleic acid, esters 16 and 17 were produced in 23 and 9% yields, respectively. 6,8‐Dihydroxy‐3,7‐dimethyl‐2‐benzopyrylium p‐toluenesulfonate 31 is considered as the intermediate for the production of 12‐15. Overall yields of isochromenes 12‐15 were 26‐6% starting from 2‐methylresorcinol for seven steps.