Second Order Parallel Tensors on P-Sasakian Manifolds

ＳｅｃｏｎｄＯｒｄｅｒＰａｒａｌｌｅｌＴｅｎｓｏｒｓｏｎＰ－ＳａｓａｋｉａｎＭａｎｉｆｏｌｄｓ￥ＤｅｂａｓｉｓｈＴａｒａｆｄａｒａｎｄＵ．Ｃ．Ｄｅ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＵｎｉｖｅｒｓｉｔｙｏｆＫａｌｙａｎｉ，Ｋａｌｙａｎｉ...     

Sub－product Clases of l－Groups

Ｓｕｂ┐ｐｒｏｄｕｃｔＣｌａｓｅｓｏｆｌ┐ＧｒｏｕｐｓＴｏｎｇＤａｏｒｏｎｇ（仝道荣）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｅｈａｉＵｎｉｖｅｒｓｉｔｙ，Ｎａｎｊｉｎｇ，２１００２４）ＡｂｓｔｒａｃｔＡｃｌａｓｓｏｆｌ－ｇｒｏｕｐｓｉｓ...     

Torsion Classes and Torsion Value Selectors of hl-groups

ＴｏｒｓｉｏｎＣｌａｓｅｓａｎｄＴｏｒｓｉｏｎＶａｌｕｅＳｅｌｅｃｔｏｒｓｏｆｈｌ┐ｇｒｏｕｐｓＴｏｎｇＤａｏｒｏｎｇ（仝道荣）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｅｈａｉＵｎｉｖｅｒｓｉｔｙ，Ｎａｎｊｉｎｇ，２１００２４）Ｍｉｃｈｅ...     

Heegaard分解的重新分块

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　ＬｅｔＭｂｅａｃｏｍｐａｃｔ 3 ｍａｎｉｆｏｌｄ .ＩｆｔｈｅｒｅｉｓａｐｒｏｐｅｒｌｙｅｍｂｅｄｄｅｄｃｌｏｓｅｄｓｕｒｆａｃｅＳｉｎＭｗｈｉｃｈｓｅｐａｒａｔｅｓＭｉｎｔｏｔｗｏｃｏｍｐｒｅｓｓｉｏｎｂｏｄｉｅｓＨ1ａｎｄＨ2 ,ｔｈｅｎＭｃａｎｂｅｗｒｉｔｔｅｎａｓＭ =Ｈ1∪ＳＨ2 .ＴｈｉｓｓｔｒｕｃｔｕｒｅｏｎＭｉｓｃａｌｌｅｄａＨｅｅｇａａｒｄｓｐｌｉｔｔｉｎｇｏｆＭａｎｄＳｉｓａｓｐｌｉｔｔｉｎｇｓｕｒｆａｃｅ .Ｈ1∪ＳＨ2 ｉｓｓａｉｄｔｏｂｅｒｅｄｕｃｉｂｌｅ…     

The Limit Distribution of Some PP CM Statistics with Enough High Dimensions and Large Sample Size

ＴｈｅＬｉｍｉｔＤｉｓｔｒｉｂｕｔｉｏｎｏｆＳｏｍｅＰＰＣＭＳｔａｔｉｓｔｉｃｓｗｉｔｈＥｎｏｕｇｈＨｉｇｈＤｉｍｅｎｓｉｏｎｓａｎｄＬａｒｇｅＳａｍｐｌｅＳｉｚｅＣｈｅｎｇＰｉｎｇ（成平）（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｓＳｃｉｅｎｃｅ，Ａｃａ...     

The Limit Distribution of PPCramer-von Mises Normality Test Statistics for High Dimension and Large Sample Size

ＴｈｅＬｉｍｉｔＤｉｓｔｒｉｂｕｔｉｏｎｏｆＰＰＣｒａｍ︺ｒ┐ｖｏｎＭｉｓｅｓＮｏｒｍａｌｉｔｙＴｅｓｔＳｔａｔｉｓｔｉｃｓｆｏｒＨｉｇｈＤｉｍｅｎｓｉｏｎａｎｄＬａｒｇｅＳａｍｐｌｅＳｉｚｅＬｉｕＹｉｘｉｎｇ（刘义兴）ａｎｄＣｈｅｎｇＰｉｎｇ（成平...     

Chern Classes in Bivariant Theory Definitions of Brasselet and of Sabbah

ＣｈｅｒｎＣｌａｓｓｅｓｉｎＢｉｖａｒｉａｎｔＴｈｅｏｒｙＤｅｆｉｎｉｔｉｏｎｓｏｆＢｒａｓｓｅｌｅｔａｎｄｏｆＳａｂｂａｈ￥ＪｉａｎｙｉＺＨＯＵ（ＳｉＧｍＡＣＮＲＳ，ＣＩＲＭＬｕｍｉｎｙＣａｓｅ９１６，Ｆ－１３２８８ＭａｒｓｅｉｌｌｅＣｅｄｅｘ９；...     

Variational Inclusions for Fuzzy Mappings

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＳｉｄｄｉｑｉａｎｄＡｎｓａｒｉ［４］ｄｅｖｅｌｏｐｅｄｉｔｅｒａｔｉｖｅａｌｇｏｒｉｔｈｍｓｆｏｒｆｉｎｄｉｎｇａｐｐｒｏｘｉｍａｔｅｓｏｌｕｔｉｏｎｔｏｎｅｗｃｌａｓｓｅｓｏｆｑｕａｓｉｖａｒｉａｔｉｏｎａｌｉｎｅｑｕａｌｉｔｉｅｓｉｎＨｉｌｂｅｒｔｓｐａｃｅｓ，ＨａｓｓｏｕｎｉａｎｄＭｏｕｄａｆｉ［１］ｅｘｔｅｎｄｔｈｅｍａｉｎｉｄｅａｓｏｆｐａｐｅｒ［４］ｔｏｍｏｒｅｇｅｎｅｒａｌｃａｓｅｓ，ｗｈｉｃｈｃｏｎｓｉｄｅｒｅｄａｃｌａｓｓｏｆｖａｒｉａｔｉｏｎ…     

Some Characterizations of a Class of M-matrices

ＳｏｍｅＣｈａｒａｃｔｅｒｉｚａｔｉｏｎｓｏｆａＣｌａｓｓｏｆ　Ｍ－ｍａｔｒｉｃｅｓＬｉＷｅｎ（黎稳）ａｎｄＺｈａｎｇＭｏｕｃｈｅｎｇ（张谋成）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ,ＳｏｕｔｈＣｈｉｎａＮｏｒｍａｌＵｖｉｖｅｒｓｉｔｙ,Ｇ...     

The Structure of NBe—rpp Semigroups

§ 1.ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＭａｉｎＲｅｓｕｌｔｓ　ＡｓｅｍｉｇｒｏｕｐＳｉｓｃａｌｌｅｄａｎｒｐｐｓｅｍｉｇｒｏｕｐｉｆａｌｌｐｒｉｎｃｉｐａｌｒｉｇｈｔｉｄｅａｌｓａＳ1(ａ∈Ｓ)ｏｆＳ ,ｒｅｇａｒｄｅｄａｓａｎＳ1 ｓｙｓｔｅｍ ,ａｒｅｐｒｏｊｅｃｔｉｖｅ (ｓｅｅ,[1 ]ａｎｄ [2 ] ) .ＡｓｅｍｉｇｒｏｕｐＳｉｓａｎｒｐｐｓｅｍｉｇｒｏｕｐｉｆａｎｄｏｎｌｙｉｆｆｏｒａｌｌａ∈Ｓ ,ｔｈｅｓｅｔＭａ ={ｅ∈Ｅ Ｓ1ａ Ｓｅａｎｄ ( ｘ ,ｙ∈Ｓ1)ａｘ=ａｙ ｅｘ=ｅｙ}ｉｓｎｏｎｅｍｐｔｙ ,ｗｈｅｒｅＥ …     

Approximation of analytic by Borel sets and definable countable chain conditions

LetI be a σ-ideal on a Polish space such that each set fromI is contained in a Borel set fromI. We say thatI fails to fulfil theΣ ₁ ¹ countable chain condition if there is aΣ ₁ ¹ equivalence relation with uncountably many equivalence classes none of which is inI. Assuming definable determinacy, we show that if the family of Borel sets fromI is definable in the codes of Borel sets, then eachΣ ₁ ¹ set is equal to a Borel set modulo a set fromI iffI fulfils theΣ ₁ ¹ countable chain condition. Further we characterize the σ-idealsI generated by closed sets that satisfy the countable chain condition or, equivalently in this case, the approximation property forΣ ₁ ¹ sets mentioned above. It turns out that they are exactly of the formMGR(F)={A : ∀F ∈ F A ∩F is meager inF} for a countable family F of closed sets. In particular, we verify partially a conjecture of Kunen by showing that the σ-ideal of meager sets is the unique σ-ideal onR, or any Polish group, generated by closed sets which is invariant under translations and satisfies the countable chain condition. Research partially supported by NSF grant DMS-9317509.     

On the bounded version of Hilbert's tenth problem

 The paper establishes lower bounds on the provability of 𝒟=NP and the MRDP theorem in weak fragments of arithmetic. The theory I ⁵ E ₁ is shown to be unable to prove 𝒟=NP. This non-provability result is used to show that I ⁵ E ₁ cannot prove the MRDP theorem. On the other hand it is shown that I ¹ E ₁ proves 𝒟 contains all predicates of the form (∀i≤|b|)P(i,x)^Q(i,x) where ^ is =, <, or ≤, and I ⁰ E ₁ proves 𝒟 contains all predicates of the form (∀i≤b)P(i,x)=Q(i,x). Here P and Q are polynomials. A conjecture is made that 𝒟 contains NLOGTIME. However, it is shown that this conjecture would not be sufficient to imply 𝒟=N P. Weak reductions to equality are then considered as a way of showing 𝒟=NP. It is shown that the bit-wise less than predicate, ≤₂, and equality are both co-NLOGTIME complete under FDLOGTIME reductions. This is used to show that if the FDLOGTIME functions are definable in 𝒟 then 𝒟=N P. Received: 13 July 2001 / Revised version: 9 April 2002 / Published online: 19 December 2002 Key words or phrases: Bounded Arithmetic – Bounded Diophantine Complexity     

Radial rearrangement, harmonic measures and extensions of Beurling's shove theorem

LetI be a union of finitely many closed intervals in [−1, 0). LetI ^↞ be a single interval of the form [−1, −a] chosen to have the same logarithmic length asI. LetD be the unit disc. Then, Beurling [8] has shown that the harmonic measure of the circle ∂D at the origin in the slit discD/I is increased ifI is replaced byI ^↞. We prove a number of cognate results and extensions. For instance, we show that Beurling's result remains true if the intervals inI are not just one-dimensional, but if they in fact constitute polar rectangles centred on the negative real axis and having some fixed constant angular width. In doing this, we obtain a new proof of Beurling's result. We also discuss a conjecture of Matheson and Pruss [25] and some other open problems. Much of the present paper has been adapted from Chapter IV of the author's doctoral dissertation. The research was partially supported by Professor J. J. F. Fournier's NSERC Grant #4822.     

Distribution of Primitive A-Roots of Composite Moduli Ⅱ

Abstract We improve estimates for the distribution of primitive λ-roots of a composite modulus q yielding an asymptotic formula for the number of primitive λ-roots in any interval I of length ∣I∣ ≫ q ^1/2+∈. Similar results are obtained for the distribution of ordered pairs (x, x ⁻¹) with x a primitive λ-root, and for the number of primitive λ-roots satisfying inequalities such as |x − x ⁻¹| ≤ B. (Dedicated to Professor Wang Yuan on the occasion of his 75th birthday) *Project supported by the National Natural Science Foundation of China (No.19625102) and the 973 Project of the Ministry of Science and Technology of China.     

A New Proof of a Theorem of Suslin

In this paper we compute the group H₂(SL₂(F)), for F an infinite field. In particular, using some techniques from homological algebra developed by Hutchinson [Hutchinson, K: K-Theory 4 (1990), 181–200], we give a new proof of the following theorem obtained by [Su2]: The group H₂(SL₂, (F)) is the fiber product of λ_*:K₂(F)→ I²(F)/I³(F) and σ: I²(F) → I²(F)/I³(F) where λ_* and σ map onto I²(F)/I³(F). (Received: February 2003)     

Sur l’annulation du deuxième foncteur de (co)homologie d’André-Quillen

For a given idealI of a commutative ringA, B=A/I, the vanishing of the second André-Quillen (co)homology functorH ₂ (A, B, δ) is characterized in terms of the canonical homomorphism α:S(I)→R(I) from the symmetric algebra of the idealI onto its Rees algebra. This is done by introducing a Koszul complex that characterizes commutative graded algebras which are symmetric algebras.

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This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.     

On the weak non-defectivity of veronese embeddings of projective spaces

Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<( _n ^n+x ). Here we prove that the order x Veronese embedding ofP ⁿ is not weakly (k−1)-defective, i.e. for a general S⊃P ⁿ such that #(S) = k+1 the projective space | I _2S (x)| of all degree t hypersurfaces ofP ⁿ singular at each point of S has dimension ( _n ^/n+x )−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I _2S (x)| has an ordinary double point at each P∈ S and Sing (F)=S. The author was partially supported by MIUR and GNSAGA of INdAM (Italy).     

Symbolic powers,Serre conditions and Cohen-Macaulay Rees algebras

Summary LetR be a Cohen-Macaulay ring andI an unmixed ideal of heightg which is generically a complete intersection and satisfiesI ⁽ⁿ⁾=Iⁿ for alln≥1. Under what conditions will the Rees algebra be Cohen-Macaulay or have good depth? A series of partial answers to this question is given, relating the Serre condition (S _r ) of the associated graded ring to the depth of the Rees algebra. A useful device in arguments of this nature is the canonical module of the Rees algebra. By making use of the technique of the fundamental divisor, it is shown that the canonical module has the expected form: ω _R[It] ≅(t(1−t) ^g−2). The third author was partially supported by the NSF This article was processed by the author using theLaTex style filecljour1 from Springer-Verlag.     

Closures of finitely generated ideals in Hardy spaces

LetH ^∞ be the algebra of bounded analytic functions in the unit diskD. LetI=I(f ₁,...,f _N) be the ideal generated byf ₁,...,f _N∈H ^∞ andJ=J(f ₁,...,f _N) the ideal of the functionsf∈H ^∞ for which there exists a constantC=C(f) such that |f(z)|≤C(|f ₁ (z)|+...;+|f _N (z)|),z∈D. It is clear that , but an example due to J. Bourgain shows thatJ is not, in general, in the norm closure ofI. Our first result asserts thatJ is included in the norm closure ofI ifI contains a Carleson-Newman Blaschke product, or equivalently, if there existss>0 such that

Gradient methods for multiple state optimal design problems

We optimise a distribution of two isotropic materials α I and β I (α < β) occupying the given body in R ^d . The optimality is described by an integral functional (cost) depending on temperatures u ₁, . . . , u _m of the body obtained for different source terms f ₁, . . . ,f _m with homogeneous Dirichlet boundary conditions. The relaxation of this optimal design problem with multiple state equations is needed, introducing the notion of composite materials as fine mixtures of different phases, mathematically described by the homogenisation theory. The necessary conditions of optimality are derived via the Gateaux derivative of the cost functional. Unfortunately, there could exist points in which necessary conditions of optimality do not give any information on the optimal design. In the case m < d we show that there exists an optimal design which is a rank-m sequential laminate with matrix material α I almost everywhere on Ω. Contrary to the optimality criteria method, which is commonly used for the numerical solution of optimal design problems (although it does not rely on a firm theory of convergence), this result enables us to effectively use classical gradient methods for minimising the cost functional.