We introduce a space of stable meromorphic differentials with poles of prescribed orders and define its tautological cohomology ring. This space, just as the space of holomorphic differentials, is stratified according to the set of multiplicities of zeros of the differential. The main goal of this paper is to compute the Poincar 'e-dual cohomology classes of all strata. We prove that all these classes are tautological and give an algorithm to compute them. In a second part of the paper we study the Picard group of the strata. We use the tools introduced in the first part to deduce several relations in these Picard groups.
We consider the two following functions: Both @math and @math are polynomials for large values of @math (this result is due to Aaron Pixton, see @cite_32 and @cite_1 ). We denote by @math and @math the asymptotic polynomials. The two following conjectures have been proposed.
@cite_1 @cite_32
2472026787 2276886773
Let @math denote a random symmetric @math matrix whose upper diagonal entries are independent and identically distributed Bernoulli random variables (which take values @math and @math with probability @math each). It is widely conjectured that @math is singular with probability at most @math . On the other hand, the best known upper bound on the singularity probability of @math , due to Vershynin (2011), is @math , for some unspecified small constant @math . This improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the singularity probability decays faster than any polynomial. In this paper, improving on all previous results, we show that the probability of singularity of @math is at most @math for all sufficiently large @math . The proof utilizes and extends a novel combinatorial approach to discrete random matrix theory, which has been recently introduced by the authors together with Luh and Samotij. In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models. More specifically, we define a large class of graph polynomials @math and show that if @math and there is a disk @math centered at zero in the complex plane such that @math does not vanish on @math for all bounded degree graphs @math , then for each @math in the interior of @math there exists a deterministic polynomial-time approximation algorithm for evaluating @math at @math . This gives an explicit connection between absence of zeros of graph polynomials and the existence of efficient approximation algorithms, allowing us to show new relationships between well-known conjectures. Our work builds on a recent line of work initiated b... We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . We derive a new upper bound on the diameter of a polyhedron @math P = x ? R n : A x ≤ b , where @math A ? Z m × n . The bound is polynomial in @math n and the largest absolute value of a sub-determinant of @math A , denoted by @math Δ . More precisely, we show that the diameter of @math P is bounded by @math O ( Δ 2 n 4 log n Δ ) . If @math P is bounded, then we show that the diameter of @math P is at most @math O ( Δ 2 n 3.5 log n Δ ) . For the special case in which @math A is a totally unimodular matrix, the bounds are @math O ( n 4 log n ) and @math O ( n 3.5 log n ) respectively. This improves over the previous best bound of @math O ( m 16 n 3 ( log m n ) 3 ) due to Dyer and Frieze (Math Program 64:1---16, 1994).