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
2021115151 2146489966 1724455341 2156463490
We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals of polynomial functions on the orthogonal group O(d) and the symplectic group Sp(d). We obtain an exact character expansion and the asymptotic behavior for large d. Thus we can show the asymptotic freeness of Haar-distributed orthogonal and symplectic random matrices, as well as the convergence of integrals of the Itzykson–Zuber type. We study compositions of the integer @math in which the first part, successive differences, and the last part are constrained to lie in prescribed sets @math , respectively. A simple condition on @math guarantees that the generating function @math has only a simple pole on its circle of convergence and this at @math , a function independent of @math and @math . Thus the number of compositions is asymptotic to @math for a suitable constant @math . We prove a multivariate central and local limit theorem and apply it to various statistics of random locally restricted compositions of @math , such as number of parts, numbers of parts of given sizes, and number of rises. The first and last parts are shown to have limiting distributions and to be asymptotically independent. If @math has only finitely many positive elements @math , or finitely many negative elements @math , then the largest part and number of distinct part sizes are almost surely @math . On the other hand, when both @math and @math have a positive asymptotic lower "local log-density", we prove that the largest part and number of distinct part sizes are almost surely @math , and we give sufficient conditions for the largest part to be almost surely asymptotic to @math . Let @math be an i.i.d. sequence of random variables in @math , @math . We show that, for any function @math , under regularity conditions, [n^ 1 2 (n^ -1 i=1 ^n (X_i) f ^(X_i) - (x) ,dx ) P 0, ] where @math is the classical kernel estimator of the density of @math . This result is striking because it speeds up traditional rates, in root @math , derived from the central limit theorem when @math . Although this paper highlights some applications, we mainly address theoretical issues related to the later result. We derive upper bounds for the rate of convergence in probability. These bounds depend on the regularity of the functions @math and @math , the dimension @math and the bandwidth of the kernel estimator @math . Moreover, they are shown to be accurate since they are used as renormalizing sequences in two central limit theorems each reflecting different degrees of smoothness of @math . As an application to regression modelling with random design, we provide the asymptotic normality of the estimation of the linear functionals of a regression function. As a consequence of the above result, the asymptotic variance does not depend on the regression function. Finally, we debate the choice of the bandwidth for integral approximation and we highlight the good behavior of our procedure through simulations. We treat the use of more complex higher degree polynomial curves and surfaces of degree higher than 2, which have many desirable properties for object recognition and position estimation, and attack the instability problem arising in their use with partial and noisy data. The scenario discussed in this paper is one where we have a set of objects that are modeled as implicit polynomial functions, or a set of representations of classes of objects with each object in a class modeled as an implicit polynomial function, stored in the database. Then, given partial data from one of the objects, we want to recognize the object (or the object class) or collect more data in order to get better parameter estimates for more reliable recognition. Two problems arising in this scenario are discussed: 1) the problem of recognizing these polynomials by comparing them in terms of their coefficients; and 2) the problem of where to collect data so as to improve the parameter estimates as quickly as possible. We use an asymptotic Bayesian approximation for solving the two problems. The intrinsic dimensionality of polynomials and the use of the Mahalanobis distance are discussed.