-functionals and multivariate Bernstein polynomials

This paper estimates upper and lower bounds for the approximation rates of iterated Boolean sums of multivariate Bernstein polynomials. Both direct and inverse inequalities for the approximation rate are established in terms of a certain K-functional. From these estimates, one can also determine the class of functions yielding optimal approximations to the iterated Boolean sums.     

Limit Theorems for Sums of Heavy-tailed Variables with Random Dependent Weights

Let be independent and identically distributed random variables with heavy-tailed distributions. Consider a sequence of random weights , independent of and focus on the weighted sums , where μ involves a suitable centering. We establish sufficient conditions for these weighted sums to converge to non-trivial limit processes, as n→∞, when appropriately normalized. The convergence holds, for example, if is strictly stationary, dependent, and W ₁ has lighter tails than U ₁. In particular, the weights W _j s can be strongly dependent. The limit processes are scale mixtures of stable Lévy motions. We establish weak convergence in the Skorohod J ₁-topology. We also consider multivariate weights and show that they converge weakly in the strong Skorohod M ₁-topology. The M ₁-topology, while weaker than the J ₁-topology, is strong enough for the supremum and infimum functionals to be continuous. This research was partially supported by a fellowship of the Horace H. Rackham School of Graduate Studies at the University of Michigan and the NSF Grants BCS-0318209 and DMS-0505747 at Boston University.     

A Supplement to the Baum-Katz-Spitzer Complete Convergence Theorem

Let {X, Xn; n≥ 1} be a sequence of i.i.d. Banach space valued random variables and let {an; n ≥ 1} be a sequence of positive constants such that an↑∞ and 1〈 lim inf n→∞ a2n/an≤lim sup n→∞ a2n/an〈∞ Set Sn=∑i=1^n Xi,n≥1.In this paper we prove that ∑n≥1 1/n P（｜｜Sn｜｜≥εan）〈∞ for all ε〉0 if and only if lim n→∞ Sn/an=0 a.s. This result generalizes the Baum-Katz-Spitzer complete convergence theorem. Combining our result and a corollary of Einmahl and Li, we solve a conjecture posed by Gut.     

基于核心素养的“递进式活动导学”初探——“机械能”教学案例

从三维目标走向核心素养,是学科教学高度、深度和内涵的提升,是学科教学对人的真正的回归.基于核心素养的“递进式活动导学”,是以核心素养为主导,以“递进式活动”为抓手,将教学目标细化为行为目标并在课堂教学中落实.文章以“上教版”八年级物理第四章第3节“机械能”第一课时为例,从教学目标细化和“递进式活动”的设计两个方面,讲述基于核心素养的“递进式活动导学”的教学思路,以期对初中物理核心素养在课堂教学中的落实有所帮助.     

Prophet inequalities for dependent random variables

A prophet inequality with constant a satisfying K-log K=2 is obtained for a class dependent random variables in the full information case, under the same assumptions the prophet constant 4 is obtained in the partial information case. Applications to sums of independent random variables are given     

Explicit multiplicative relations between Gauss sums

H. Hasse conjectured that all multiplicative relations between Gauss sums essentially follow from the Davenport-Hasse product formula and the norm relation for Gauss sums. While this is known to be false, very few counterexamples, now known as sign ambiguities, have been given. Here, we provide an explicit product formula giving an infinite class of new sign ambiguities and resolve the ambiguous sign in terms of the order of the ideal class of quadratic primes.     

On the Mean Value of the Dedekind Sum with the Weight of Hurwitz Zeta-Function

The main purpose of this paper is to use the mean value theorem of the Dirichlet L-functions to study the asymptotic behavior of the Dedekind sums with a weight of Hurwitz zeta-function and to give an interesting asymptotic formula.     

Involutions Fixing Products of Projective Spaces

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Invariance principles for non-isotropic long memory random fields

We prove that when a random field with bounded spectral density satisfies a Donsker type theorem, its dilated and properly normalised spectral field admits a weak limit. We apply this result to establish the convergence of partial sums for random fields obtained by filtering a white noise. In particular, we prove the convergence of partial sums for strongly-dependent fields whose memory does not satisfy the regularity conditions previously met in the literature.