The occupation time of Brownian motion in a ball

LetW be a Wiener process of dimensiond3, starting from 0, and letX(t) be the total time spent byW in the ball centered at 0 with radiust. We give an affirmative answer to a conjecture of Taylor and Tricot⁽¹⁶⁾ on the tail distribution ofX(t). Lévy's lower functions ofX(t) are characterized by an integral test.     

The binding number of a graph and its triangle

In this paper we prove the following conjecture of Woodall^[1]: if bind (G)3/2, thenG contains a triangle. Moreover we also prove that if bind (G)3/2, then each vertex is contained in a 4-cycle, each edge is contained in a 5-cycle whenV(G)11, and there exists a 6-cycle inG.     

Synthesis and reactions of 2-(alkylthio)-4,4-dimenthyl-1,3-thiazole-5(4H)-thiones

Six 2-(alkylthio)-substituted 4,4-dimethyl-1,3-thiazole-5(4H)-thiones were synthesized according to a new method. The reactions of these compounds with allyl- and benzyllithium reagents, 1,3-dipoles, and dimethyl acetylenedicarboxylate proceeded in a similar manner to 2-alkyl-substituted analogues, while methyllithium reacted in a different way yielding trithio-orthoester derivatives.