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Using a dataset that recorded a large number of investment transactions in China from 1991 to 2018, we examine the statistical properties of the Chinese venture capital (VC) syndication network. Our main findings are as follows. First, the number of investment transactions sharply increased after 2014. Second, more than half of the VC firms are located in Beijing, Shanghai, and Shenzhen. Third, the degree distribution becomes roughly straight on a log–log plot. Fourth, the hypothesis that the degree distribution follows a power-law distribution is not rejected for 2015 and 2016. Fifth, better connected VC firms increase their connectivity faster, which suggests the existence of preferential attachment.

Let $V$ be an $n$ dimensional vector space over an algebraic closure of a finite field $F_q$ and put $G = GL(V)$. For a positive integer $r$, we consider the variety $X_{uni} = G_{uni} \times V^{r-1}$, on which $G$ acts diagonally. $X_{uni}$ is the "unipotent part" of the enhanced variety of level $r$. $X_{uni}$ is partitioned into finitely many pieces $X_{\lambda}$ labelled by $r$-partitions $\lambda$ of $n$, and we consider the intersection cohomology $IC_{\lambda}$ associated to $X_{\lambda}$. In this paper, we show that the Frobenius trace functions (over $F_q$) associated to those $IC_{\lambda}$ satisfy certain orthogonality relations, which are very close to the equations characterizing the Kostka functions indexed by (a pair of) $r$-partitions. Using this we show, in some special cases, that the Kostka functions can be described in terms of those intersection cohomology, which is a (partial) generalization of the known results for the case $r = 1, 2$., Comment: v2: 52 pages. Proposition 6.3 was revised, and some examples were added in section 7. Final version, to appear in Transformation Groups

The modified Ariki-Koike algebra is a variation of the original Ariki-Koike algebra over an integral domain R. When R is a rational function field over the independent parameters, But for general R, is not isomorphic to , and has a simpler structure than . In this paper, we construct a cellular basis of which has a similar property as the cellular basis of introduced by Dipper-James-Mathas. By comparing these two cellular bases, we obtain some estimate on the decomposition numbers of in terms of the decomposition numbers of . We also prove the integral form of the Schur-Weyl reciprocity between a certain quantum algebra U q and on the tensor space

Letp be an odd prime and\(\mathbb{F}_p \) the finite field withp elements. In the present paper we shall investigate the number of points of certain quadratic hypersurfaces in the vector space\(\mathbb{F}_p^n \) and derive explicit formulas for them. In addition, we shall show that the class number of the real quadratic field\(\mathbb{Q}(\sqrt p )\) (wherep≡1 (mod 4)) over the field ℚ of rational numbers can be expressed by means of these formulas.

We determine the constants appearing in the Lusztig conjecture in the case of unipotent characters of classical groupsG subject to the condition that the characteristic is odd and the center ofG is connected. This enables us to compute the character values of unipotent characters of such finite classical groups.