Boundary reducible handle additions on simple 3-manifolds

Let M be a simple manifold, and F be a component of δM of genus two. For a slope γ on F, we denote by M（γ） the manifold obtained by attaching a 2-handle to M along a regular neighborhood of γon F. In this paper, we shall prove that there is at most one separating slope γ on F such that M（γ） is δ-reducible.     

Pseudo-Anosov extensions and degree one maps between hyperbolic surface bundles

Let F′,F be any two closed orientable surfaces of genus g′ > g≥ 1, and f:F→ F be any pseudo-Anosov map. Then we can “extend” f to be a pseudo- Anosov map f′:F′→ F′ so that there is a fiber preserving degree one map M(F′,f′)→ M(F,f) between the hyperbolic surface bundles. Moreover the extension f′ can be chosen so that the surface bundles M(F′,f′) and M(F,f) have the same first Betti numbers. Y. Ni is partially supported by a Centennial fellowship of the Graduate School at Princeton University. S.C. Wang is partially supported by MSTC     

Maximality of analytic operator algebras

LetM be a σ-finite von Neumann algebra andα be an action ofR onM. LetH ^∞(α) be the associated analytic subalgebra; i.e.H ^∞(α)={X ∈M: sp_∞(X) [0, ∞]}. We prove that every σ-weakly closed subalgebra ofM that containsH ^∞(α) isH ^∞(γ) for some actionγ ofR onM. Also we show that (assumingZ(M)∩M ^α = Ci)H ^∞(α) is a maximal σ-weakly closed subalgebra ofM if and only if eitherH ^∞(α)={A ∈M: (I−F)xF=0} for some projectionF ∈M, or sp(α)=Γ(α).     

Incompleteness and minimality of complex exponential system

A necessary and sufficient condition is obtained for the incompleteness of a complex exponential system E(A,M)in C_α,where C_αis the weighted Banach space consisting of all complex continuous functions f on the real axis R with f(t)exp(-α(t))vanishing at infinity,in the uniform norm‖f‖_α=sup{|f(t)e~(-α(t))|:t∈R}with respect to the weightα(t).If the incompleteness holds, then the complex exponential system E(?)is minimal and each function in the closure of the linear span of complex exponential system E(?)can be extended to an entire function represented by a Taylor-Dirichlet series.     

The localization of 1-cohomology of transitive Lie algebroids

For a transitive Lie algebroid A on a connected manifold M and its representation on a vector bundle F, we define a morphism of cohomology groups rk: Hk(A,F) → Hk(Lx,Fx), called the localization map, where Lx is the adjoint algebra at x ∈ M. The main result in this paper is that if M is simply connected, or H (LX,FX) is trivial, then T is injective. This means that the Lie algebroid 1-cohomology is totally determined by the 1-cohomology of its adjoint Lie algebra in the above two cases.     

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex Euclidean space

We treat m-dimensional real submanifolds M of complex space forms ̿M when the maximal holomorphic tangent subspace is (m−1)-dimensional. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient space and in this paper we study the condition h(FX,Y)−h(X,FY) = g(FX,Y)η, η∊ T⊥(M), on the structure F and on the second fundamental form h of these submanifolds. Especially when the ambient space ̿M is a complex Euclidean space, we obtain a complete classification of submanifolds M which satisfy these conditions.Mathematics Subject Classifications (2000): 53C15, 53C40, 53B20.     

On universality of the Lerch zeta-function

It is known that the Lerch zeta-function L(λ, α, s) with transcendental parameter α is universal in the Voronin sense; i.e., every analytic function can be approximated by shifts L(λ, α, s + iτ) uniformly on compact subsets of some region. In this paper, the universality for some classes of composite functions F(L(λ, α, s)) is obtained. In particular, general theorems imply the universality of the functions sin(L(λ, α, s)) and sinh(L(λ, α, s)).     

Composition of fractional Orlicz maximal operators and A 1-weights on spaces of homogeneous type

For a Young function θ with 0 ≤α 〈 1, let Mα,θ be the fractional Orlicz maximal operator defined in the context of the spaces of homogeneous type （X, d, μ） by Mα,θf（x） = supx∈（B）α ｜｜f｜｜θ,B, where ｜｜f｜｜θ,B is the mean Luxemburg norm of f on a ball B. When α= 0 we simply denote it by Me. In this paper we prove that if Ф and ψare two Young functions, there exists a third Young function θ such that the composition Mα,ψ o MФ is pointwise equivalent to Mα,θ. As a consequence we prove that for some Young functions θ, if Mα,θf 〈∞a.e. and δ ∈（0,1） then （Mα,θf）δ is an A1-weight.     

The lattice of flats and its underlying flag matroid polytope

LetM be a matroid andF the collection of all linear orderings of bases ofM, orflags ofM. We define the flag matroid polytope Δ(F). We determine when two vertices of Δ(F) are adjacent, and provide a bijection between maximal chains in the lattice of flats ofM and certain maximal faces of Δ(F). Supported in part by NSA grant MDA904-95-1-1056.     

Regression of polynomial statistics on the sample mean and natural exponential families

A polynomial Q = Q(X ₁, …, X _n ) of degree m in independent identically distributed random variables with distribution function F is an unbiased estimator of a functional q(α ₁(F), …, α _m (F)), where q(u ₁, …, u _m ) is a polynomial in u ₁, …, u _m and α _j (F) is the jth moment of F (assuming the necessary moment of F exists). It is shown that the relation E(Q | X ₁ + … + X n) = 0 holds if and only if q(α ₁(θ), …, α _m (θ)) ≡ 0, where α _j (θ) is the jth moment of the natural exponential family generated by F. This result, based on the fact that X ₁ + … + X n is a complete sufficient statistic for a parameter θ in a sample from a natural exponential family of distributions F _θ(x) = ∫_−∞ ^x e ^θu−k(θ) dF(u), explains why the distributions appearing as solutions of regression problems are the same as solutions of problems for natural exponential families though, at the first glance, the latter seem unrelated to the former.     

K-Theory of pseudodifferential operators with semi-periodic symbols

Let 𝔄 denote the C^*-algebra of bounded operators on L ² ℝ generated by: (i) all multiplications a(M) by functions a∈C[ − ∞, + ∞], (ii) all multiplications by 2π-periodic continuous functions, and (iii) all operator of the form F ⁻¹ b(M)F, where F denotes the Fourier transform and b∈C[ − ∞, + ∞]. A given A ∈ 𝔄 is a Fredholm operator if and only if σ(A) and γ(A) are invertible, where σ denotes the continuous extension of the usual principal symbol, while γ denotes an operator-valued “boundary principal symbol” (the “boundary” here consists of two copies of the circle, one at each end of the real line). We give two proofs of the fact that K ₀(𝔄) is isomorphic to ℤ and that K ₁(𝔄) is isomorphic to ℤ ⊕ ℤ . We do it first by computing the connecting mappings in the six-term exact sequence associated to σ. For the second proof, we show that the image of γ is isomorphic to the direct sum of two copies of the crossed product , where α denotes the translation-by-one automorphism. Its K-theory can be computed using the Pimsner–Voiculescu exact sequence, and that information suffices for the analysis of the standard cyclic exact sequence associated to γ. Received: February 2006     

Trigonometric approximation in reflexive Orlicz spaces

The Lipschitz classes Lip(α, M) , 0 α≤ 1 are defined for Orlicz space generated by the Young function M, and the degree of approximation by matrix transforms of f ∈ Lip (α, M) is estimated by n-α .     

Classification of framed links in 3-manifolds

We present a short and complete proof of the following Pontryagin theorem, whose original proof was complicated and has never been published in detail. Let M be a connected oriented closed smooth 3-manifold, L ₁(M) be the set of framed links in M up to a framed cobordism, and deg: L ₁(M) → H ₁(M; ℤ) be the map taking a framed link to its homology class. Then for each α ∈ H ₁(M; ℤ) there is a one-to-one correspondence between the set deg⁻¹ α and the group ℤ_2d(α), where d(α) is the divisibility of the projection of α to the free part of H ₁(M; ℤ).     

Transformation formulas associated with integrals involving the Riemann Ξ-function

Using residue calculus and the theory of Mellin transforms, we evaluate integrals of a certain type involving the Riemann Ξ-function, which give transformation formulas of the form F(z, α) = F(z, β), where αβ = 1. This gives a unified approach for generating certain modular transformation formulas, including a famous formula of Ramanujan and Guinand.     

On （α,β）-Metrics of Scalar Flag Curvature with Constant S-curvature

In this paper, we study （α,β）-metrics of scalar flag curvature on a manifold M of dimension n （n 〉 3）. Suppose that an （α,β）-metric F is not a Finsler metric of Randers type, that is, F ≠k1 V√α^2 ＋ k2β^2 ＋ k3β, where k1 〉 0, k2 and k3 are scalar functions on M. We prove that F is of scalar flag curvature and of vanishing S-curvature if metric. In this case, F is a locally Minkowski and only if the flag curvature K = 0 and F is a Berwald metric.     

Maximal sets in α-recursion theory

Let α be an admissible ordinal, and leta ^* be the Σ₁-projectum ofa. Call an α-r.e. setM maximal if α→M is unbounded and for every α→r.e. setA, eitherA∩(α-M) or (α-A)∩(α-M) is bounded. Call and α-r.e. setM amaximal subset of α^* if α^*−M is undounded and for any α-r.e. setA, eitherA∩(α^*-M) or (⇌^*-A)∩(α^*-M) is unbounded in α^*. Sufficient conditions are given both for the existence of maximal sets, and for the existence of maximal subset of α^*. Necessary conditions for the existence of maximal sets are also given. In particular, if α ≧ ℵ ^L then it is shown that maximal sets do not exist. Research partially supported by NSF Grant GP-34088 X. Some of the results in this paper have been taken from the second author’s Ph. D. Thesis, written under the supervision of Gerald Sacks.     

On Asymptotic Behavior for Singularities of the Powers of Mean Curvature Flow

Let M^n be a smooth, compact manifold without boundary, and F0 ： M^n→ R^n＋1 a smooth immersion which is convex. The one-parameter families F（·, t） ： M^n× [0, T） → R^n＋1 of hypersurfaces Mt^n= F（·,t）（M^n） satisfy an initial value problem dF/dt （·,t） = -H^k（· ,t）v（· ,t）, F（· ,0） = F0（· ）, where H is the mean curvature and u（·,t） is the outer unit normal at F（·, t）, such that -Hu = H is the mean curvature vector, and k 〉 0 is a constant. This problem is called H^k-fiow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type Ⅰ and Type Ⅱ. It is proved that for Type Ⅰ singularity, the limiting hypersurface satisfies an elliptic equation; for Type Ⅱ singularity, the limiting hypersurface must be a translating soliton.     

Incompressibility and normal minimal surfaces

In this paper we describe a procedure for refining the given triangulation of a 3-manifold that scales the PL-metric according to a given weight function while creating no new normal surfaces. Let F be a closed orientable incompressible surface in an irreducible 3-manifold M. Then in every triangulation τ of M, F is isotopic to a τ-normal surface F(τ) that is of minimal PL-area (in the isotopy class of F). Using the above scaling refinement we prove the converse. If for every triangulation τ of M, F is isotopic to a τ-normal surface F(τ) that is of minimal PL-area, then we shall show that F is incompressible. Hence we get a characterisation of incompressibility of a surface in terms of existence of a minimal PL-area normal surface.     

Penultimate limiting forms in extreme value theory

Summary Let {X _n}_n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM _n=max (X ₁,…,X _n), suitably normalized with attraction coefficients {α_n}_n≧1(α_n>0) and {b _n}_n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s which better approximate the d.f. of(M _n−b_n)/a_n thanG(x), for moderaten. We thus consider differences of the formF ⁿ(a_nx+b_n)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф _α(x)=exp (−x^−α), x>0] or a type III [Ψ _α(x)= exp (−(−x)^α), x<0] d.f. of largest values, much closer toF ⁿ(a_nx+b_n) than the ultimate itself.     

Stability of homogeneity almost everywhere

We consider approximately ϕ-homogeneous mappings almost everywhere, that is functions F such that the difference F(αx) − ϕ(α)F(x) is in some sense bounded almost everywhere in a product space. We will prove, under some assumptions, that then either we have some kind of boundedness of ϕ and F, or there exist a homomorphism and a -homogeneous function , which are almost everywhere equal to ϕ and F, respectively. From this result we derive the superstability effect for the multiplicativity almost everywhere.