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
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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. We investigate the asymptotic number of elements of size @math in a particular class of closed lambda-terms (so-called @math -terms) which are related to axiom systems of combinatory logic. By deriving a differential equation for the generating function of the counting sequence we obtain a recurrence relation which can be solved asymptotically. We derive differential equations for the generating functions of the counting sequences of other more general classes of terms as well: the class of @math -terms and that of closed lambda-terms. Using elementary arguments we obtain upper and lower estimates for the number of closed lambda-terms of size @math . Moreover, a recurrence relation is derived which allows an efficient computation of the counting sequence. @math -terms are discussed briefly. We explicitly compute the diverging factor in the large genus asymptotics of the Weil–Petersson volumes of the moduli spaces of n-pointed complex algebraic curves. Modulo a universal multiplicative constant we prove the existence of a complete asymptotic expansion of the Weil–Petersson volumes in the inverse powers of the genus with coefficients that are polynomials in n. This is done by analyzing various recursions for the more general intersection numbers of tautological classes, whose large genus asymptotic behavior is also extensively studied. Abstract We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and k -Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and k - K -Schur functions – Schubert representatives for the K -theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules. We establish the asymptotic expansion of certain integrals of ψ classes on moduli spaces of curves ℳ¯g,n, when either the g or n goes to infinity. Our main tools are cut-join type recursion formulae from the Witten–Kontsevich theorem, as well as asymptotics of solutions to the first Painleve equation. We also raise a conjecture on large genus asymptotics for n-point functions of ψ classes and partially verify the positivity of coefficients in generalized Mirzakhani’s formula of higher Weil–Petersson volumes. A function @math is @math -resilient if all its Fourier coefficients of degree at most @math are zero, i.e., @math is uncorrelated with all low-degree parities. We study the notion of @math @math of Boolean functions, where we say that @math is @math -approximately @math -resilient if @math is @math -close to a @math -valued @math -resilient function in @math distance. We show that approximate resilience essentially characterizes the complexity of agnostic learning of a concept class @math over the uniform distribution. Roughly speaking, if all functions in a class @math are far from being @math -resilient then @math can be learned agnostically in time @math and conversely, if @math contains a function close to being @math -resilient then agnostic learning of @math in the statistical query (SQ) framework of Kearns has complexity of at least @math . This characterization is based on the duality between @math approximation by degree- @math polynomials and approximate @math -resilience that we establish. In particular, it implies that @math approximation by low-degree polynomials, known to be sufficient for agnostic learning over product distributions, is in fact necessary. Focusing on monotone Boolean functions, we exhibit the existence of near-optimal @math -approximately @math -resilient monotone functions for all @math . Prior to our work, it was conceivable even that every monotone function is @math -far from any @math -resilient function. Furthermore, we construct simple, explicit monotone functions based on @math and @math that are close to highly resilient functions. Our constructions are based on a fairly general resilience analysis and amplification. These structural results, together with the characterization, imply nearly optimal lower bounds for agnostic learning of monotone juntas. We study the asymptotic-capacity-achieving score function that was recently proposed by for bias-based traitor tracing codes. For the bias function, we choose the Dirichlet distribution with a cutoff. Using Bernstein's inequality and Bennett's inequality, we upper bound the false-positive and false-negative error probabilities. From these bounds we derive sufficient conditions for the scheme parameters. We solve these conditions in the limit of large coalition size c0 and obtain asymptotic solutions for the cutoff, the sufficient code length, and the corresponding accusation threshold. We find that the code length converges to its asymptote approximately as c0?1 2 @math , which is faster than the c0?1 3 @math of Tardos' score function. MSC:94B60 We study the distribution in short intervals of certain arithmetic functions, including the von Mangoldt function and the M "obius function, on polynomials over a finite field @math . Using the Grothendieck-Lefschetz trace formula, we reinterpret each moment of these distributions as a point-counting problem on a highly singular complete intersection variety. We compute part of the @math -adic cohomology of these varieties, yielding an asymptotic bound on each moment for fixed degree @math in the limit as @math . Let @math be a compact Riemannian submanifold of @math of dimension @math and let @math be a sample of i.i.d. points in @math with uniform distribution. We study the random operators @math where @math is the Gaussian kernel and @math as @math Such operators can be viewed as graph laplacians (for a weighted graph with vertices at data points) and they have been used in the machine learning literature to approximate the Laplace-Beltrami operator of @math @math (divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations @math for smooth functions @math both pointwise and uniformly in @math and @math (here @math and @math is the Riemannian volume measure). In particular, we show that for any class @math of three times differentiable functions on @math with uniformly bounded derivatives @math as soon as @math and also prove asymptotic normality of @math (functional CLT) for a fixed @math and uniformly in @math In this paper we derive precise tail-area approximations for the sum of an arbitrary finite number of independent heavy-tailed random variables. In order to achieve second-order asymptotics, a mild regularity condition is imposed on the class of distribution functions with regularly varying tails. @PARASPLIT Higher-order asymptotics are also obtained when considering a semiparametric subclass of distribution functions with regularly varying tails. These semiparametric subclasses are shown to be closed under convolutions and a convolution algebra is constructed to evaluate the parameters of a convolution from the parameters of the constituent distributions in the convolution. A Maple code is presented which does this task. We consider the problem of approximating a Boolean functionf∶ 0,1 n → 0,1 by the sign of an integer polynomialp of degreek. For us, a polynomialp(x) predicts the value off(x) if, wheneverp(x)≥0,f(x)=1, and wheneverp(x)<0,f(x)=0. A low-degree polynomialp is a good approximator forf if it predictsf at almost all points. Given a positive integerk, and a Boolean functionf, we ask, “how good is the best degreek approximation tof?” We introduce a new lower bound technique which applies to any Boolean function. We show that the lower bound technique yields tight bounds in the casef is parity. Minsky and Papert [10] proved that a perceptron cannot compute parity; our bounds indicate exactly how well a perceptron canapproximate it. As a consequence, we are able to give the first correct proof that, for a random oracleA, PP A is properly contained in PSPACE A . We are also able to prove the old AC0 exponential-size lower bounds in a new way. This allows us to prove the new result that an AC0 circuit with one majority gate cannot approximate parity. Our proof depends only on basic properties of integer polynomials. In this paper we consider the large genus asymptotics for two classes of Siegel-Veech constants associated with an arbitrary connected stratum @math of Abelian differentials. The first is the saddle connection Siegel-Veech constant @math counting saddle connections between two distinct, fixed zeros of prescribed orders @math and @math , and the second is the area Siegel-Veech constant @math counting maximal cylinders weighted by area. By combining a combinatorial analysis of explicit formulas of Eskin-Masur-Zorich that express these constants in terms of Masur-Veech strata volumes, with a recent result for the large genus asymptotics of these volumes, we show that @math and @math , both as @math tends to @math . The former result confirms a prediction of Zorich and the latter confirms one of Eskin-Zorich in the case of connected strata. This paper makes two contributions to the areas of anti-concentration and non-asymptotic random matrix theory. First, we study the counting problem in inverse Littlewood-Offord theory for general random variables: for random variables @math which are i.i.d. copies of a random variable @math (satisfying some mild hypotheses), how many integer vectors @math in a prescribed box have large @math ? Building on recent work of Ferber, Jain, Luh, and Samotij (who treated the case when @math is a Rademacher random variable), we provide significantly better bounds for this problem than those obtained using the inverse Littlewood-Offord theorems of Tao and Vu, and Nguyen and Vu. Next, we study the non-asymptotic behavior of the least singular value @math of a random @math square matrix @math with i.i.d. entries, which are only assumed to be centered with variance @math . As an application of our counting theorem, and utilizing and developing a recent work by the author, we show that for all @math , @math , where @math is an absolute constant, thereby providing a considerably simpler proof of an approximate version of an essentially optimal result due to Rebrova and Tikhomirov. The determinantal complexity of a polynomial f is defined here as the minimal size of a matrix M with affine entries such that f = det M. This function gives a minoration of the more traditional size of an arithmetical formula. Consider the polynomial “permanent” perm d of a d × d matrix with entries X i,j . A conjecture in complexity theory says that the determinantal complexity (dc) of perm d should not be polynomial in d. In this article we prove that dc(permd)≥d2 2, improving the previously known minoration, 2d. We also begin a systematic study of the function dc, and compute it for the homogeneous polynomials of degree 2. We study the asymptotic behaviour of the nodal length of random @math -spherical harmonics @math of high degree @math , i.e. the length of their zero set @math . It is found that the nodal lengths are asymptotically equivalent, in the @math -sense, to the "sample trispectrum", i.e., the integral of @math , the fourth-order Hermite polynomial of the values of @math . A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit. Abstract Let P ( x ) be a function from GF (2 n ) to GF (2). P ( x ) is called “bent” if all Fourier coefficients of (−1) P(x) are ±1. The polynomial degree of a bent function P ( x ) is studied, as are the properties of the Fourier transform of (−1) P(x) , and a connection with Hadamard matrices. Recently Bogdanov and Viola (FOCS 2007) and Lovett (ECCC-07) constructed pseudorandom generators that fool degree k polynomials over F2 for an arbitrary constant k. We show that such generators can also be used to fool branching programs of width 2 and polynomial length that read k bits of inputs at a time. This model generalizes polynomials of degree k over F2 and includes some other interesting classes of functions, for instance k-DNF. The constructions of Bogdanov and Viola and Lovett consist of adding a constant number of independent copies of a generator that fools linear functions (an -biased set). It is natural to ask, in light of our first result, whether such generators can fool branching programs of width larger than 2. Our second result is a lower bound showing