Propertyf −1(S) =g −1(S) for entire and meromorphicP-adic functions

LetW be an algebraically closed filed of characteristic zero, letK be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value, and letA(K) (resp. ℳ(K)) be the set of entire (resp. meromorphic) functions inK. For everyn≥7, we show that the setS _n(b) of zeros of the polynomialx ⁿ−b (b≠0) is such that, iff, g ∈W[x] or iff, g ∈A(K), satisfyf ⁻¹(S _n(b))=g ⁻¹(S _n(b)), thenf ⁿ=g ⁿ. For everyn≥14, we show thatS _n(b) is such that iff, g ∈W({tx}) or iff, g ∈ ℳ(K) satisfyf ⁻¹(S _n(b))=g ⁻¹(S _n(b)), then eitherf ⁿ=g ⁿ, orfg is a constant. Analogous properties are true for complex entire and meromorphic functions withn≥8 andn≥15, respectively. For everyn≥9, we show that the setY _n(c) of zeros of the polynomial , (withc≠0 and 1) is an ursim ofn points forW[x], and forA(K). For everyn≥16, we show thatY _n(c) is an ursim ofn points forW(x), and for ℳ(K). We follow a method based on thep-adic Nevanlinna Theory and use certain improvement of a lemma obtained by Frank and Reinders.     

The alternating semigroups: Generators and congruences

The idea of an even permutation has recently been generalized, via path notation, to one-one partial transformations (charts) in the symmetric inverse semigroupsC _n. The even charts form the alternating semigroupA _n ^c ⊂C _n . Generators ofA _n ^c are identified: It then follows that forn≥5,A _n ^c is the collection of restrictions of the even permutations (of rankn). Like theC _n case, the congruences ofA _n ^c form a chain. Part of this research was supported by a Mary Washington College faculty development grant.     

Dynamical Borel-Cantelli lemmas for gibbs measures

LetT: X→X be a deterministic dynamical system preserving a probability measure μ. A dynamical Borel-Cantelli lemma asserts that for certain sequences of subsetsA _n ⊃ X and μ-almost every pointx∈X the inclusionT ⁿ x∈A _n holds for infinitely manyn. We discuss here systems which are either symbolic (topological) Markov chain or Anosov diffeomorphisms preserving Gibbs measures. We find sufficient conditions on sequences of cylinders and rectangles, respectively, that ensure the dynamical Borel-Cantelli lemma. Partially supported by NSF grant DMS-9732728. Partially supported by NSF grant DMS-9704489.     

On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces

We prove that ifC is a bounded closed convex subset of a uniformly convex Banach space,T:C→C is a nonlinear contraction, andS _n =(I+T+…+T ⁿ⁻¹ )/n, then lim _n ‖S _n (x)−TS _n (x)‖=0 uniformly inx inC. T also satisfies an inequality analogous to Zarantonello’s Hilbert space inequality. which permits the study of the structure of the weak ω-limit set of an orbit. These results are valid forB-convex spaces if some additional condition is imposed on the mapping. Partially supported by NSF Grant MCS-7802305A01.     

Inversion and characterization of the hemispherical transform

Explicit inversion formulas are obtained for the hemispherical transform(FΜ)(x) = Μ{y ∃S ⁿ :x. y ≥ 0},x ∃S ⁿ, whereS ⁿ is thendimensional unit sphere in ℝⁿ⁺¹,n ≥ 2, and Μ is a finite Borel measure onS ⁿ. If Μ is absolutely continuous with respect to Lebesgue measuredy onS ⁿ, i.e.,dΜ(y) =f(y)dy, we write(F f)(x) = ∫ _{x.y> 0} f(y)dy and consider the following cases: (a)f ∃C ^∞(Sⁿ); (b)f ∃ L^p(S ⁿ), 1 ≤ p < ∞; and (c)f ∃C(Sⁿ). In the case (a), our inversion formulas involve a certain polynomial of the Laplace-Beltrami operator. In the remaining cases, the relevant wavelet transforms are employed. The range ofF is characterized and the action in the scale of Sobolev spacesL _p ^γ (Sⁿ) is studied. For zonalf ∃ L¹(S ²), the hemispherical transformF f was inverted explicitly by P. Funk (1916); we reproduce his argument in higher dimensions. Partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Deformation retracts and the Hochschild homology of polynomial rings

We assume given a ringA with unit, and a subcomplex of the reduced bar complex ofA. We assume that this subcomplex is a deformation retract of the whole complex and thus has homology equal to the Hochschild homology ofA, but it will typically be smaller and easier to calculate with. We use these to construct (accordingly small) deformation retracts for the reduced bar complexes ofA[t] andA[t,t ⁻¹]. WhenA is a Banach algebra, we also do this construction forC ^∞(S¹;A). Partially supported by N.S.F. Grant No. DMS 92-03398.     

Submanifolds with constant Möbius scalar curvature in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

 Let M ^m be a m-dimensional submanifold in the n-dimensional unit sphere S ⁿ without umbilic point. Two basic invariants of M ^m under the M?bius transformation group of S ⁿ are a 1-form Φ called M?bius form and a symmetric (0,2) tensor A called Blaschke tensor. In this paper, we prove the following rigidity theorem: Let M ^m be a m-dimensional (m≥3) submanifold with vanishing M?bius form and with constant M?bius scalar curvature R in S ⁿ , denote the trace-free Blaschke tensor by . If , then either ||?||≡0 and M ^m is M?bius equivalent to a minimal submanifold with constant scalar curvature in S ⁿ ; or and M ^m is M?bius equivalent to in for some c≥0 and . Received: 15 May 2002 / Revised version: 3 February 2003 Published online: 19 May 2003 RID="*" ID="*" Partially supported by grants of CSC, NSFC and Outstanding Youth Foundation of Henan, China. RID="†" ID="†" Partially supported by the Alexander Humboldt von Stiftung and Zhongdian grant of NSFC. Mathematics Subject Classification (2000): Primary 53A30; Secondary 53B25     

Infinite simple C*-algebras and Reduced cross products of abelian C*-algebras and free groups

Summary Let Γ=〈g ₁〉*〈g ₂〉*...*〈g _n 〉*... be a free product of cyclic groups with generators {g _i }, andC _r ^* (Γ,℘ _Λ) be the C*-algebra generated by the reduced group C*-algebraC _r ^* Γ and a set of projectionsP _gL associated with a subset Λ of {g _i }. We prove the following: (1)C _r ^* (Γ,℘ _Λ) is *-isomorphic to the reduced cross product for certain Hausdorff compact spaceX _Λ constructed from Γ and its boundary ∂Γ. (2)C _r ^* (Γ,℘ _Λ) is either a purely infinite, simple C*-algebra or an extension of a purely infinite, simple C*-altebra, depending on the pair (Γ, Λ). (3)C _r ^* (Г,℘ _Λ) is nuclear if and only if the subgroup Γ_Λ generated by {g _i }/Λ is amenable. Partially supported by RMC grant 45/290/603 from the University of Newcastle Partially supported by NSF grant DMS-9225076 and a Taft travel grant from the University of Cincinnati     

Sausages are good packings

LetB ^d be thed-dimensional unit ball and, for an integern, letC _n ={x ¹,...,x ⁿ } be a packing set forB ^d , i.e.,|x ⁱ −x ^j |≥2, 1≤i<j≤n. We show that for every a dimensiond(ρ) exists such that, ford≥d(ρ),V(conv(C _n )+ρB ^d )≥V(conv(S _n )+ρB ^d ), whereS _n is a “sausage” arrangement ofn balls, holds. This gives considerable improvement to Fejes Tóth's “sausage” conjecture in high dimensions. Further, we prove that, for every convex bodyK and ρ<1/32d ⁻²,V(conv(C _n )+ρK)≥V(conv(S _n )+ρK), whereC _n is a packing set with respect toK andS _n is a minimal “sausage” arrangement ofK, holds.     

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.     

Schouten curvature functions on locally conformally flat Riemannian manifolds

Consider a compact Riemannian manifold (M, g) with metric g and dimension n ≥ 3. The Schouten tensor A _g associated with g is a symmetric (0, 2)-tensor field describing the non-conformally-invariant part of the curvature tensor of g. In this paper, we consider the elementary symmetric functions {σ _k (A _g ), 1 ≤ k ≤ n} of the eigenvalues of A _g with respect to g; we call σ _k (A _g ) the k-th Schouten curvature function. We give an isometric classification for compact locally conformally flat manifolds which satisfy the conditions: A _g is semi-positive definite and σ _k (A _g ) is a nonzero constant for some k ∈ {2, ... , n}. If k = 2, we obtain a classification result under the weaker conditions that σ₂(A _g ) is a non-negative constant and (M ⁿ , g) has nonnegative Ricci curvature. The corresponding result for the case k = 1 is well known. We also give an isometric classification for complete locally conformally flat manifolds with constant scalar curvature and non-negative Ricci curvature. Udo Simon: Partially supported by Chinese-German cooperation projects, DFG PI 158/4-4 and PI 158/4-5, and NSFC.     

Holonomy on poisson manifolds and the modular class

(1) It is shown that ifc is real-valued measurable then the Maharam type of (c, P(c),σ) is 2 ^c . This answers a question of D. Fremlin [Fr, (P2f)]. (2) A different construction of a model with a real-valued measurable cardinal is given from that of R. Solovay [So]. This answers a question of D. Fremlin [Fr, (P1)]. (3) The forcing with aκ-complete ideal over a setX, |X| ≥κ cannot be isomorphic to Random × Cohen or Cohen × Random. The result forX=κ was proved in [Gi-Sh1] but, as was pointed out to us by M. Burke, the application of it in [Gi-Sh2] requires dealing with anyX. The application is: ifA _n is a set of reals forn<ω then for some pairwise disjointB _n (forn<ω) we haveB _n ⊆A _n but they have the same outer Lebesgue measure. Partially supported by the Israeli Basic Research Fund. Publ. Number 582.     

New central polynomials for the matrix algebra

Forn≥3 we find a central polynomial of degree (n−1)²+4 for then×n matrix algebra over a field of characteristic 0. Forn=3,4 our polynomial coincides with the known central polynomials of minimal degree and forn>4 the result gives new central polynomials. Until now, forn>4 the minimal degree of the known central polynomials wasn ². Partially supported by the Alexander von Humboldt Foundation in Germany and by Grant MM2/91 of the Ministry of Education and Science in Bulgaria.     

On weighted positivity and the Wiener regularity of a boundary point for the fractional Laplacian

A sufficient condition for the Wiener regularity of a boundary point with respect to the operator (− Δ)^μ inR ⁿ ,n≥1, is obtained, for μ∈(0,1/2n)/(1,1/2n−1). This extends some results for the polyharmonic operator obtained by Maz'ya and Maz'ya-Donchev. As in the polyharmonic case, the proof is based on a weighted positivity property of (− Δ)^μ, where the weight is a fundamental solution of this operator. It is shown that this property holds for μ as above while there is an interval [A _n , 1/2n−A _n ], whereA _n →1, asn→∞, with μ-values for which the property does not hold. This interval is non-empty forn≥8.     

Comparison theorems for the volume of a complex submanifold of a Kaehler manifold

LetM be a Kaehler manifold of real dimension 2n with holomorphic sectional curvatureK _H≥4λ and antiholomorphic Ricci curvatureρ _A≥(2n−2)λ, andP is a complex hypersurface. We give a bound for the quotient (volume ofP)/(volume ofM) and prove that this bound is attained if and only ifP=C P ⁿ⁻¹(λ) andM=C P ⁿ(λ). Moreover, we give some results on the volume of of tubes aboutP inM. Work partially supported by a DGICYT Grant No. PS87-0115-CO3-01.     

A note on smooth toral reductions of spheres

In this paper we show that there exist mod 2 obstructions to the smoothness of 3-Sasakian reductions of spheres. Specifically, ifS is a smooth 3-Sasakian manifold obtained by reduction of the 3-Sasakian sphereS ⁴ⁿ⁻¹ by a torus, and if the second Betti numberb ₂(S)≥2 then dimS=7, 11, 15, whereas, ifb ₂ (S)≥5 then dimS=7. We also show that the above bounds are sharp, in that we construct explicit examples of 3-Sasakian manifolds in the cases not excluded by these bounds. During the preparation of this work the authors were partially supported by an NSF grant. This article was processed by the author using the L^AT_EX style file from Springer-Verlag.     

Le relazioni irreduttibili fra gľinvarianti d’una forma qualunque Dľottavo ordine

Let C ⊂ P ³ be a smooth and connected projective curve of genus g ≥ 2. Here in characteristic ≠ 2 we prove that its generic plane section is in uniform position and that its numerical character is connected. Partially supported by MIUR and GNSAGA of INdAM (Italy)     

On graded quotient modules of mapping class groups of surfaces

Let Γ _{g, n} be the mapping class group of a compact Riemann surface of genusg withn points preserved (2−2g−n<0,g≥1,n≥0). The Torelli subgroup of Γ _{g, n} has a natural weight filtration {Γ_{g, n}(m)} _m≥1. Each graded quotient gr ^m Γ _{g, n} ⊗ ℚ (m≥1) is a finite dimensional vector space over ℚ on which the group Sp(2g, ℚ)×S _n naturally acts. In this paper, we have determined the Sp(2g, ℚ)×S _n module structure of gr ^m Γ _{g, n} ⊗ ℚ for 1≤m≤3. This includes a verification of an expectation by S. Morita. Also, for generalm, we have identified a certain Sp(2g, ℚ)-irreducible component of gr ^m Γ _{g, n} ⊗ ℚ by constructing explicitly elements in these modules.     

A symmetry of sphere map implies its chaos*

A well-known example, given by Shub, shows that for any |d| ≥ 2 there is a self-map of the sphere Sⁿ, n ≥ 2, of degree d for which the set of non-wandering points consists of two points. It is natural to ask which additional assumptions guarantee an infinite number of periodic points of such a map. In this paper we show that if a continuous map f : Sⁿ → Sⁿ commutes with a free homeomorphism g : Sⁿ → Sⁿ of a finite order, then f has infinitely many minimal periods, and consequently infinitely many periodic points. In other words the assumption of the symmetry of f originates a kind of chaos. We also give an estimate of the number of periodic points. *Research supported by KBN grant nr 2 P03A 045 22.     

Dynamics on Ahlfors quasi-circles

The celebrated theory of Denjoy introduced a topological invariant distinguishingC ¹ andC ² diffeomorphisms of the circle. AC ² diffeomorphism of the circle cannot have an infinite minimal set other than the circle itself. However, this is possible forC ¹ diffeomorphisms. In dimension two there is a related invariant distinguishingC ² andC ³ diffeomorphisms. Partially supported by NSF grant No. MCS-83202062.