Torus manifolds with non-abelian symmetries

Let (G) be a connected compact non-abelian Lie-group and (T) a maximal torus of (G). A torus manifold with (G)-action is defined to be a smooth connected closed oriented manifold of dimension (2\dim T) with an almost effective action of (G) such that (M^T\neq \emptyset). We show that if there is a torus manifold (M) with (G)-action then the action of a finite covering group of (G) factors through (\tilde{G}=\prod SU(l_i+1)\times\prod SO(2l_i+1)\times \prod SO(2l_i)\times T^{l_0}). The action of (\tilde{G}) on (M) restricts to an action of (\tilde{G}'=\prod SU(l_i+1)\times\prod SO(2l_i+1)\times \prod U(l_i)\times T^{l_0}) which has the same orbits as the (\tilde{G})-action. We define invariants of torus manifolds with (G)-action which determine their (\tilde{G}')-equivariant diffeomorphism type. We call these invariants admissible 5-tuples. A simply connected torus manifold with (G)-action is determined by its admissible 5-tuple up to (\tilde{G})-equivariant diffeomorphism. Furthermore we prove that all admissible 5-tuples may be realised by torus manifolds with (\tilde{G}")-action where (\tilde{G}") is a finite covering group of (\tilde{G}').Comment: 56 pages; a mistake in section 6 corrected; accepted for publication in Trans. Am. Math. So

Periodic Sequences modulo mm

We give a few remarks on the periodic sequence $a_n=\binom{n}{x}~(mod~m)$ where $x,m,n\in \mathbb{N}$, which is periodic with minimal length of the period being $$\ell(m,x)={\displaystyle\prod^w_{i=1}p^{\lfloor\log_{p_i}x\rfloor+b_i}_i}=m{\displaystyle\prod^w_{i=1}p^{\lfloor\log_{p_i}x\rfloor}_i}$$ where $m=\prod^w_{i=1}p^{b_i}_i$. We prove certain interesting properties of $\ell(m,x)$ and derive a few other results and congruences.Comment: 7 pages, preprint. Comments are welcom

Enumerating contingency tables via random permanents

Given m positive integers R=(r_i), n positive integers C=(c_j) such that sum r_i = sum c_j =N, and mn non-negative weights W=(w_{ij}), we consider the total weight T=T(R, C; W) of non-negative integer matrices (contingency tables) D=(d_{ij}) with the row sums r_i, column sums c_j, and the weight of D equal to prod w_{ij}^{d_{ij}}. We present a randomized algorithm of a polynomial in N complexity which computes a number T'=T'(R,C; W) such that T' < T < alpha(R, C) T' where alpha(R,C) = min{prod r_i! r_i^{-r_i}, prod c_j! c_j^{-c_j}} N^N/N!. In many cases, ln T' provides an asymptotically accurate estimate of ln T. The idea of the algorithm is to express T as the expectation of the permanent of an N x N random matrix with exponentially distributed entries and approximate the expectation by the integral T' of an efficiently computable log-concave function on R^{mn}. Applications to counting integer flows in graphs are also discussed.Comment: 19 pages, bounds are sharpened, references are adde