Abstract. In this paper we prove the local existence and uniqueness of C 1+γ solutions of the Boussinesq equations with initial data v0, θ0 ∈ C 1+γ , ω0, ∇θ0 ∈ L q for 0 < γ < 1 and 1 < q < 2. We also obtain a blow-up criterion for this local solutions. More precisely we show that the gradient of the passive scalar θ controls the breakdown of C 1+γ solutions of the Boussinesq equations. §1. IntroductionThe interactive motion of a passive scalar (e.g. temperature) and atmosphere with an external potential force is modeled by the following Boussinesq equations:In (1) p denotes the scalar pressure of the fluid flow. It is suggested that these equations have strong resemblance with the 3-D Euler equations in many aspects (see e.g. [7]). In particular the problem of finite time blow-up of smooth solutions of the Boussinesq equations is outstanding as in the case of 3-D Euler equations.In [4], authors proved local existence of solutions of the Boussinesq equations in the Sobolev spaces H m (R 2 ), m > 2, and obtained a blow-up criterion of the smooth solutions. The proofs are similar to Kato's [6] and Beale-Kato-Majda's [2] respectively for the 3-D Euler equations.In this paper we extend the previous results to the case of Hölder continuous initial data. We first prove the unique local existence of the solutions of

Diffuse leptomeningeal disseminated glioneuronal tumor (DL-GNT) is a rare brain tumor that presents as a plaque-like subarachnoid tumor, commonly involving the basal cisterns and interhemispheric fissure of children but lacking intraparenchymal tumor. Histologically, the tumors are composed of sheets of monotonous rounded cells. Here, we report three cases of DL-GNTs, focusing on clinicopathologic features. Two patients were adult male, but one patient was child. The patients presented with seizures (n = 1) or headaches (n = 2). In all patients, radiography revealed characteristic leptomeningeal thickening and enhancement with minor superficial parenchymal lesions. All three cases were diffusely positive for both GFAP and synaptophysin, and scattered positive for OLIG2 and NeuN, but negative for IDH-1 (H09). Electron microscopic examination showed astrocytic and neuronal differentiation. The patient with the anaplastic tumor died due to aggressive progression of the tumor, but the remaining two patients were stable without tumor recurrence for 23 and 37 months. Thus, these findings suggest that DL-GNT can occur in both children and adult and both supra- and infra-tentorial leptomeninges. It has unique radiological and histopathological features and biological behavior. Further clinicopathological data with molecular genetic study are required for establishing DL-GNT as a unique entity.

The present study sought to evaluate the effect of resin shades on the degree of the polymerization. To this end, response variables affected by the degree of polymerization were examined in this study -namely, microhardness, polymerization shrinkage, and color change. Two commercial composite resins of four different shades were employed in this study: shades A3, A3.5, B3, and C3 of Z250 (Z2) and shades A3, A3.5, B3, and B4 of Solitaire 2 (S2). After light curing, the reflectance/absorbance, microhardness, polymerization shrinkage, and color change of the specimens were measured. On reflectance and absorbance, Z2 and S2 showed similar distribution curves regardless of the resin shade, with shade A3.5 of Z2 and shade A3 of S2 exhibiting the lowest/highest distributions. Similarly for attenuation coefficient and microhardness, the lowest/highest values were exhibited by shade A3.5 of Z2 and shade A3 of S2. On polymerization shrinkage, no statistically significant differences were observed among the different shades of Z2. Similarly for color change, Z2 specimens exhibited only a slight (ΔE*=0.5-0.9) color change after immersion in distilled water for 10 days, except for shades A3 and A3.5. Taken together, results of the present study suggested that the degree of polymerization of the tested composite resins was minimally affected by resin shade.

Tissue engineering scaffolds provide temporary mechanical support for tissue regeneration. To regenerate tissues more efficiently, an ideal structure of scaffolds should have appropriate porosity and pore structure. In this paper, we generate the Schwarz primitive P surface with various volume fractions using a phase-field model. The phase-field model enables us to design various surface-to-volume ratio structures with high porosity and mechanical properties. Comparing the Schwarz P surface's von Mises stress with that of triply periodic cylinders and cubes, we draw conclusions about the mechanical properties of the Schwarz P surface.

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