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.     

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).     

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 Exact Hausdorff Measure Function of the Level Sets of Multi-parameter Symmetric Stable Process

In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued stable processes. We show that the exact Hausdorff measure function of level sets of N-parameter Rd-valued symmetric stable processes of index α is Ф（r） = r^N-d/α （log log l/r）d/α when Nα 〉 d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general （N, d, α） strictly stable processes.     

On 2-almost contact tachibana manifolds

LeM be a (2m+2)-dimensional Riemannian manifold with two structure vector fieldsξ _r (r=2m+1, 2m+2) and letη ^r =ξ _r ^b be their corresponding covectors (or Pfaffians). These vector fields define onM a 2-almost contact structure. If the 2-formϕ=η ^2m+1∧η ^2m+2 is harmonic, then, following S. Tachibana [12],M is a Tachibana manifold and in this caseM is covered with 2 families of minimal submanifolds tangent toD ^⊥={ξ r} and its complementary orthogonal distributionD ^⊤. On such a manifold a canonical eigenfunction α of the Laplacian is associated. Since the corresponding eingenvalue is negative,M cannot be compact. Any horizontal vector fieldX orthogonal to α^# is a skew-symmetric Killing vector field (see [6]). Next, we assume that the Tachibana manifoldM under consideration is endowed with a framedf-structure defined by an endomorphism ϕ of the tangent bundleTM. Infinitesimal automorphisms of the symplectic form Ω _ϕ are obtained.     

Regularity of discs attached to a submanifold ofC n

Letp be an analytic disc attached to a generating CR-submanifoldM of C ⁿ . It is proved that some recently introduced conditions onp andM which imply that the family of all smallC ^α holomorphic perturbations ofp alongM is a Banach submanifold of (A^α(D))ⁿ are equivalent. These conditions are given in terms of the partial indices of the discp attached toM and “holomorphic sections” of the conormal bundle ofM along p(∂D). Also, a sufficient geometric conditionon p andM is given so that the family of all smallC ^α holomorphic perturbationsof p alongM, fixed at some boundary point, is a Banach submanifold of (A ^α (D))ⁿ.     

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 .     

On the Kauffman skein modules

 Let k be a subring of the field of rational functions in α, s which contains α ^±1 ,s ^±1 . Let M be a compact oriented 3-manifold, and let K(M) denote the Kauffman skein module of M over k. Then K(M) is the k-module freely generated by isotopy classes of framed links in M modulo the Kauffman skein relations. In the case of , the field of rational functions in α, s, we give a basis for the Kauffman skein module of the solid torus and a basis for the relative Kauffman skein module of the solid torus with two points on the boundary. We then show that K(S ¹ × S ² is freely generated by the empty link, i.e., . Received: 20 October 2001 / Revised version: 20 March 2002     

The distance between two separating,reducing slopes is at most 4

Let M be a simple 3-manifold such that one component of ∂M, say F, has genus at least two. For a slope α on F, we denote by M(α) the manifold obtained by attaching a 2-handle to M along a regular neighborhood of α on F. If M(α) is reducible, then α is called a reducing slope. In this paper, we shall prove that the distance between two separating, reducing slopes on F is at most 4. This work is supported by NSFC (10625102).     

On the first pontrjagin class of homotopy complex projective spaces

Let M ²ⁿ be a closed smooth manifold homotopy equivalent to the complex projective space ℂP(n). It is known that the first Pontrjagin class p ₁(M) of M ²ⁿ has the form (n+1+24α(M))u ² for some integer α(M) where u is a generator of H ²(M; ℤ). We prove that α(M) is even when n is even but not divisible by 64.     

Sudakov-type minoration for gaussian chaos processes

Consider a setA of symmetricn×n matricesa=(a _i,j) _i,j≤n . Consider an independent sequence (g _i) _i≤n of standard normal random variables, and letM=Esup_a∈A|Σ_i,j⪯na_i,jg_ig_j|. Denote byN ₂(A, α) (resp.N _t(A, α)) the smallest number of balls of radiusα for thel ₂ norm ofR ^{n 2} (resp. the operator norm) needed to coverA. Then for a universal constantK we haveα(logN ₂(A, α))^1/4≤KM. This inequality is best possible. We also show that forδ≥0, there exists a constantK(δ) such thatα(logN _t≤K(δ)M. Work partially supported by an N.S.F. grant.     

Strikte Konvexität für Variationsprobleme auf demn-dimensionalen Torus

We consider a variational problem with an integrandF:R ⁿ ×R×R ⁿ →R that isZ-periodic in the firstn+1 variables and satisfies certain growth-conditions. By a recent result of Moser, there exist for every α∈R ⁿ minimal solutionsu:R ⁿ →R minimising ƒF(x, u(x), u _x (x)) dx with respect to compactly supported variations ofu and such that sup |u(x)-αx|<∞. Given such a minimal solutionu we define the average action (whereB _r is ther-ball around 0∈R ⁿ ) and show thatM(α) is indeed independent of the minimal solutionu satisfying sup |u(x)-αx|<∞. We prove that this average actionM(α) is strictly convex in α.      

Completeness of the system {z^λn} in L^2[B,α0]

Let B be an unbounded domain located outside an angle domain with vertex at the origin, A ={λn}（n = 1,2,...） be a sequence of complex numbers satisfying sup ｜ arg（λn）｜ 〈 α 〈 π/2 and denote by M（∧） = {z^λ, λ ∈ ∧} the corresponding system of functions z^λ（λ∈∧）. Let α0（z） be a weight function defined on B. We obtain a completeness theorem for the system M（∧） in the Hilbert space L^2 [B, α0].     

An algorithm for finding planar surfaces in three-manifolds

By a slope in the boundary ∂M of a 3-manifold, we mean the isotopy class α of a finite set of disjoint simple closed curves in ∂M that are nontrivial and pairwise nonparallel. In this paper, we construct an algorithm to decide whether or not a given orientable 3-manifold M contains an essential planar surface whose boundary has a given slope α. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 197–202, 2005.     

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).     

Stable outer conjugacy and strong Morita equivalence of group actions on pro-C *-algebras

We show that two continuous inverse limit actions α and β of a locally compact group G on two pro-C ^*-algebras A and B are stably outer conjugate if and only if there is a full Hilbert A-module E and a continuous action u of G on E such that E and E ^*(the dual module of E) are countably generated in M(E)(the multiplier module of E), respectively M(E ^*) and the pair (E, u) implements a strong Morita equivalence between α and β. This is a generalization of a result of F. Combes [Proc. London Math. Soc. 49(1984), 289–306].      

On the geometry of tangent hyperquadric bundles: CR and pseudoharmonic vector fields

We study the natural almost CR structure on the total space of a subbundle of hyperquadrics of the tangent bundle T(M) over a semi-Riemannian manifold (M, g) and show that if the Reeb vector ξ of an almost contact Riemannian manifold is a CR map then the natural almost CR structure on M is strictly pseudoconvex and a posteriori ξ is pseudohermitian. If in addition ξ is geodesic then it is a harmonic vector field. As an other application, we study pseudoharmonic vector fields on a compact strictly pseudoconvex CR manifold M, i.e. unit (with respect to the Webster metric associated with a fixed contact form on M) vector fields X ε H(M) whose horizontal lift X↑ to the canonical circle bundle S¹ → C(M) → M is a critical point of the Dirichlet energy functional associated to the Fefferman metric (a Lorentz metric on C(M)). We show that the Euler–Lagrange equations satisfied by X^↑ project on a nonlinear system of subelliptic PDEs on M. Mathematics Subject Classifications (2000): 53C50, 53C25, 32V20     

Higher order commutators for a class of rough operators

In this paper we study the (L ^p (u ^p ),L ^q (v ^q )) boundedness of the higher order commutatorsT _Ω,α,b ^m andM _Ω,α,b ^m formed by the fractional integral operatorT _Ω,α , the fractional maximal operatorM _Ω,α , and a functionb(x) in BMO(v), respectively. Our results imporve and extend the corresponding results obtained by Segovia and Torrea in 1993 [9]. The project was supported by NNSF of China.     

ESTIMATE FOR DISTANCE-COEFFICIENT OF MATRICES

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Sojourn time tails in the single server queue with heavy-tailed service times

We consider the GI/GI/1 queue with regularly varying service requirement distribution of index −α. It is well known that, in the M/G/1 FCFS queue, the sojourn time distribution is also regularly varying, of index 1−α, whereas in the case of LCFS or Processor Sharing, the sojourn time distribution is regularly varying of index −α. That raises the question whether there exist service disciplines that give rise to a regularly varying sojourn time distribution with any index −γ∈[−α,1−α]. In this paper that question is answered affirmatively.