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Let$E$ç$S$^{1}be a set with Minkowski dimension$d(E)1$. We consider the Hardy-Littlewood maximal function, the Hilbert transform and the maximal Hilbert transform along the directions of$E$. The main result of this paper shows that these operators are bounded on$L$_{$rad$}^{$p$}(R^{2}) for$p>1+d(E)$and unbounded when$p<1+d(E)$. We also give some end-point results.

In 1979, G. Parisi predicted a variational formula for the thermodynamic limit of the free energy in the Sherrington-Kirkpatrick model and described the role played by its minimizer. This formula was verified in the seminal work of Talagrand and later generalized to the mixed p-spin models by Panchenko. In this paper, we prove that the minimizer in Parisi's formula is unique at any temperature and external field by establishing the strict convexity of the Parisi functional.

IN 1970 Lawson [22] proved that two embedded closed diffeomorphic minimal surfaces in the unit three-dimensional sphere S3 in lRJ are ambiently isotopic in S3. Lawson proved this theorem by first proving that an embedded orientable closed minimal surface of genus y in a closed orientable Riemannian three-manifold Mwith positive Ricci curvature disconnects M-into two genus-8 handlebodics. A result of Frankel [7] was used to prove this. Lawson then applied a deep result of Waldhauscn [33] that states that decompositions of SJ into two genus-cl handlebodies arc unique up to ambient isotopy. More prcciscly. Waldhauscns uniqucncss thcorcm states that whenever a closed surface of genus y in S separates SJ into handlcbodics, then the embedding of the surface is as simple as possible; in other words, the surface is obtained from a two-sphere Sz c Sby adding handles in an unknotted manner.

Japan’s lunar probe satellite “Kaguya”, China’s lunar probe satellite “Chang’ e-1 lunar probe” left the earth on September 14th, 2007 and October 24th, 2007, began the exploration to the moon. The launching of the China’s lunar satellite is the proud to the global Chinese. However, at the same time as we are proud of the event, we must admit that the pictures published which were sent back by Japanese are clearer and more beautiful than Chinese. While we are saying that the Japan has a better technology, maybe we also need to say that the Japanese can hold better chance than Chinese when taking the pictures. The following two pictures were sent back by the satellite. Picture one is sent back by Japan’s satellite, Picture two is sent back by China’s. It can be seen that there is a long way for China if we want to become stronger at science technology. We also hope that we can have some contribution in the prosperity of China now and in the future.