On surfaces with zero vanishing cycles

We show that using an idea from a paper by Van de Ven one may obtain a simple proof of Zak’s classification of smooth projective surfaces with zero vanishing cycles. This method of proof allows one to extend Zak’s theorem to the case of finite characteristic.     

Verlinde algebras and the intersection form on vanishing cycles

We prove Zuber's conjecture [Z] establishing connections of the fusion rules of the su( N )_k WZW model of conformal field theory and the intersection form on vanishing cycles of the associated fusion potential.     

Gersten’s conjecture for some complexes of vanishing cycles

A result of Bloch-Ogus is extended by a proof which may apply top-adic vanishing cycles and theories satisfying the same formalism.     

几类微分方程解的迭代增长级数与零点迭代收敛指数

本文研究了几类微分方程解的迭代增长级及零点迭代收敛指数．     

Iterated joining means

Aequationes mathematicae - Using a simple dynamical system generated by means M and N which are considered on adjacent intervals, we show how to find their joints, that is means extending both M...     

Iterated Kleene computability

Dedicated to the memory of A. I. Mal'tsev.     

Iterated monoidal categories

We develop a notion of an n-fold monoidal category and show that it corresponds in a precise way to the notion of an n-fold loop space. Specifically, the group completion of the nerve of such a category is an n-fold loop space, and free n-fold monoidal categories give rise to a finite simplicial operad of the same homotopy type as the classical little cubes operad used to parametrize the higher H-space structure of an n-fold loop space. We also show directly that this operad has the same homotopy type as the n-th Smith filtration of the Barratt-Eccles operad and the n-th filtration of Berger's complete graph operad. Moreover, this operad contains an equivalent preoperad which gives rise to Milgram's small model for when n=2 and is very closely related to Milgram's model of for n>2.     

Iterated conditional expectations

Consider the probability space ([0,1),B,λ), where B is the Borel σ-algebra on [0,1) and λ the Lebesgue measure. Let f=₁[0,1/2) and g=₁[1/2,1). Then for any ε>0 there exists a finite sequence of sub-σ-algebras Gj⊂B(j=1,…,N), such that putting f₀=f and fj=E(fj−1|Gj), j=1,…,N, we have ‖fN−g_‖∞<ε; here E(⋅|Gj) denotes the operator of conditional expectation given σ-algebra Gj. This is a particular case of a surprising result by Cherny and Grigoriev (2007) [1] in which f and g are arbitrary equidistributed bounded random variables on a nonatomic probability space. The proof given in Cherny and Grigoriev (2007) [1] is very complicated. The purpose of this note is to give a straightforward analytic proof of the above mentioned result, motivated by a simple geometric idea, and then show that the general result is implied by its special case.     

迭代Brown运动的一个Chung型重对数律

X及Y分别为Rd1及Rd2中的相互独立的标准Brown运动,满足X（0）=Y（0）=0．定义,称为一个迭代Brown运动．本文给出了关于Zd1,d2的一个Chung型重对数律．     

Iterated sumsets and setpartitions

The Ramanujan Journal - Let $$G\cong {\mathbb {Z}}/m_1{\mathbb {Z}}\times \cdots \times {\mathbb {Z}}/m_r{\mathbb {Z}}$$ be a finite abelian group with $$m_1\mid \cdots \mid m_r=\exp (G)$$ . The...     

Nonstationary Iterated Tikhonov Regularization

A convergence rate is established for nonstationary iterated Tikhonov regularization, applied to ill-posed problems involving closed, densely defined linear operators, under general conditions on the iteration parameters. It is also shown that an order-optimal accuracy is attained when a certain a posteriori stopping rule is used to determine the iteration number.     

Iterated Galerkin versus Iterated Collocation for Integral Equations of the Second Kind

We consider the numerical solution of one-dimensional Fredholmintegral equations of the second kind by the Galerkin and collocationmethods and their iterated variants, using spline bases. Inparticular, we state and prove new superconvergence resultsfor the iterated solutions, under general smoothness requirementson the kernel and solution. We find that the smoothness requirementsfor the iterated collocation method are more stringent thanthose for the iterated Galerkin method, and show by examplethat these more stringent smoothness conditions are in a certainsense necessary. In the light of these results the Galerkinand collocation schernes are compared.     

Iterated relative recursive enumerability

A result of Soare and Stob asserts that for any non-recursive r.e. setC, there exists a r.e.[C] setA such thatAC is not of r.e. degree. A setY is called [of]m-REA (m-REA[C] [degree] iff it is [Turing equivalent to] the result of applyingm-many iterated hops to the empty set (toC), where a hop is any function of the formXX W _e ^X . The cited result is the special casem=0,n=1 of our Theorem. Form=0,1, and any (m+1)-REA setC, ifC is not ofm-REA degree, then for alln there exists an-r.e.[C] setA such thatA C is not of (m+n)-REA degree. We conjecture that this holds also form2.German speakers should not be unduely influenced by the acronym for this titlePartially supported by an NSF Postdoctoral Fellowship and the US Army Research Office through the Mathematical Sciences Institute of Cornell University