Conformal deformations of prescribing scalar curvature on Riemannian manifolds with negative curvature

We study the conformal deformation for prescribing scalar curvature function on Cartan-Hadamard manifoldM ⁿ (n≥3) with strongly negative curvature. By employing the supersubsolution method and a careful construction for the supersolution, we obtain the best possible asymptotic behavior for near infinity so that the problem of complete conformal deformation is solvable. In more general cases, we prove an asymptotic estimation on the solutions of the conformal scalar curvature equation. Project partially supported by the NNSF of China     

Martin boundary of unlimited covering surfaces

LetW be an open Riemann surface and ap-sheeted (1<p<∞) unlimited covering surface ofW. Denote by Δ¹ (resp., ) the minimal Martin boundary ofW (resp., ). For ζ ∈ Δ, let ζ be the (cardinal) number of the set of pionts which lie over ζ and the class of open connected subsetsM ofW such thatM∪{ζ} is a minimal fine neighborhood of ζ. Our main result is the following: , where is the number of components of π^-1 M and π is the projection of ontoW. Moreover, some applications of the above results are discussed whenW is the unit disc.     

A class of multipliers for $$\mathcal{W}^ \bot $$

Let denote the class of ergodic probability preserving transformations which are disjoint from every weakly mixing system. Let be the class of multipliers for , i.e. ergodic transformations whose all ergodic joinings with any element of are also in . Fix an ergodic rotationT, a mildly mixing actionS of a locally compact second countable groupG and an ergodic cocycle ϕ forT with values inG. The main result of the paper is a sufficient (and also necessary by [LeP] whenG is countable Abelian andS is Bernoullian) condition for the skew product build fromT, ϕ andS to be an element of . Moreover, the self-joinings of such extensions ofT are described with an application to study semisimple extensions of rotations. Dedicated to Hillel Furstenberg on the occasion of his retirement The first-named author was supported in part by CRDF, grant UM1-2546-KH-03. The second-named author was supported in part by KBN grant 1P03A 03826.     

Eigenvalues and power growth

LetX be a complex Banach space and letT be a bounded linear operator onX. Denote by σ _p (T) the point spectrum ofT and by the unit circle. We investigate how the growth of the sequence ‖T ⁿ ‖ is influenced by the size of the set (T) and by the geometry of the spaceX. We also prove analogous results forC ₀-semigroups(T _t )_t≥0. Research partially supported by grants from NSERC, FQRNT and the Canada research chairs program.     

Bounded point evaluations and cyclic polynomials on the spaces $$\hat F$$

LetF be aBK space withAK and denote the set of all formal power series with such that ε F for the sequence of coefficients of . We give a necessary and sufficient condition for a point to be a bounded point evaluation on , and for a polynomial to be cyclic in . As special cases, we obtain the results for the space ℓ _p (β) in [7]. Research of the authors supported under the research project #1232 of the Serbian Ministry of Sciences and Tecnology and, in the case of the second author, also by the DAAD foundation (German Academic Exchange Service), grant 911 103 102 8.     

On the word problem and the conjugacy problem for groups of the formF/V(R)

LetF be a free group with at most countable system of free generators, letR be its normal subgroup recursively enumerable with respect to , and let be a variety of groups that differs from and for which the corresponding verbal subgroupV of the free group of countable rank is recursive. It is proved that the word problem inF/V(R) is solvable if and only if this problem is solvable inF/R, and if , then there exists anR such, that the conjugacy problem inF/R is solvable, but this problem is unsolvable inF/V(R) for any Abelian variety (all algorithmic problems are regarded with respect to the images of under the corresponding natural epimorphisms). Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 3–9, January, 1997. Translated by M. I. Anokhin     

Eigenvalue estimate on a compact Riemann manifold

LetM be a compact Riemann manifold with the Ricci curvature ≽ - R(R = const. > 0) . Denote by d the diameter ofM. Then the first eigenvalue λ₁ ofM satisfies . Moreover if , then      

On residue complexes,dualizing sheaves and local cohomology modules

According to Grothendieck Duality Theory [RD], on each varietyV over a fieldk, there is a canonical complex of -modules, theresidue complex . These complexes satisfy (and are characterized by) functorial properties in the categoryV ofk-varieties. In [Ye] a complex is constructed explicitly (when the fieldk is perfect). The main result of this paper is that the two families of complexes, and , which carry certain additional data (such as trace maps…), are uniquely isomorphic. As a corollary we recover Lipman’s canonical dualizing sheaf of [Li], and we obtain formulas for residues of local cohomology classes of differential forms.     

Zur Topologie der Dixmier-Abbildung

Summary  LetG be the coadjoint group of a finite-dimensional complex Lie algebrag. Forg solvable, the Dixmier-map is known to be a homeomorphism of the orbit space /G onto the space χ of primitive ideals in the enveloping algebra U(G) [6,15]. For , the Dixmier-map is known to be a bijection (and in general not a homeomorphism) with the space χ^l of all completely prime primitive ideals [7, 16]. Here we derive from a result ofW. SOERGEL [18], that this map issheet- wise a homeomorphism onto the image. Here a sheet is a maximal irreducible subset consisting of orbits of a fixed dimension; obviouslyg decomposes into finitely many sheets [3]. The results of this paper hold more generally forg semisimple, if one restricts to a sheet of polarizable orbits, where a Dixmier-map can be defined. Relative to a fixed polarization (a parabolic subalgebra)p ⊂g let I be the annihilator of the generic module induced from p. The „relative enveloping algebra“ ) has been studied e.g. bySOERGEL [19, 18]. Its center Z is described here by a relative Harish-Chandra isomorphism of the normalization with a suitable ring of group invariants (3.2). We study here the extension ofU by . We suggest that this very mild central extension ofU generates good properties and is very suitable for the study of the Dixmier-map (cf.4.3,5.6). In particular, we conjecture in case : Every minimal primitive ideal of is generated by a maximal ideal of the center. This would generalize for a well known theorem ofM. Duflo (casep Borel, where ). AsJ. Dixmier communicated in a letter, the main result here is exactly what he had hoped for when he first introduced a notion of sheets many years ago.

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Added in proof: This conjecture will be proved in a subsequent paper.     

The Artin-Rees property and homology

The Artin-Rees property for a finitely generated nilpotent groupG is used to prove that for any finitely generatedG-moduleM, where is the completion ofM with respect to the augmentation ideal ofZ[G]. Applications to topology are given. Partially supported by NSF Grant GP 33960X.     

Reconstruction of a plane convex body from the curvature of its boundary

Let denote the class of plane convex bodies having a width functionw, wherew′ is absolutely continuous. It is proved that a body in is determined (up to translation) by the radius of curvature function of its boundary. This result is then used for a characterization of the extreme (indecomposable) bodies in and for a density theorem for Reuleaux polygons in . The content of this paper is a revised version of a part of the Master of Science thesis written by the author under the supervision of Professor Micha A. Perles at the Hebrew University of Jerusalem and submitted in October, 1971.     

A relative cohomological invariant for group pairs

We define a cohomological invariantE(G, S, M) whereG is a group,S is a non empty family of (not necessarily distinct) subgroups of infinite index inG andM is a -module ( is the field of two elements). In this paper we are interested in the special case where the family of subgroups consists of just one subgroup, andM is the -module . The invariant will be denoted byE(G, S). We study the relations of this invariant with other endse(G), e(G, S) ande(G,S)), and some results are obtained in the case whereG andS have certain properties of duality.     

Nonparametric estimation of Matusita's measure of affinity between absolutely continuous distributions

LetF andG be two distribution functions defined on the same probability space which are absolutely continuous with respect to the Lebesgue measure with probability densitiesf andg, respectively. Matusita [3] defines a measure of the closeness, affinity, betweenF andG as: . Based on two independent samples fromF andG we propose to estimate ρ by , where and are taken to be the kernel estimates off(x) andg(x), respectively, as given by Parzen [5]. In this note sufficient conditions are given such that (i) asx→∞ and (ii) with probability one, asn→∞. Research supported in part by the National Research Council of Canada and by McMaster University Science and Engineering Research Board. The author is presently with the Department of Mathematical Sciences, Memphis State University, Memphis, Tennessee 38152.     

On Galois isomorphisms between ideals in extensions of local fields

LetL/K be a totally ramified, finite abelian extension of local fields, let and be the valuation rings, and letG be the Galois group. We consider the powers of the maximal ideal of as modules over the group ring . We show that, ifG has orderp ^m (withp the residue field characteristic), ifG is not cyclic (or ifG has orderp), and if a certain mild hypothesis on the ramification ofL/K holds, then and are isomorphic iffr≡r′ modp ^m . We also give a generalisation of this result to certain extensions not ofp-power degree, and show that, in the casep=2, the hypotheses thatG is abelian and not cyclic can be removed.     

Limit theorems for the maximal eigenvalues of the mean-field Hamiltonian with random potential

Let , be the mean-field Hamiltonian with and random i.i.d. potential ξ_V. We prove limit theorems for the extreme eigenvalues of as |V|→∞. The limiting distributions are the same as for the corresponding extremes of ξ_V only if either (i) ξ_V is undbounded and , or (ii) ξ_V is bounded with “sharp” peaks and . Localization properties for the corresponding eigenfunctions are also studied. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 147–168, April–June, 1999.     

Strong monotonicity of operator functions

LetA, B be bounded selfadjoint operators on a Hilbert space. We will give a formula to get the maximum subspace such that is invariant forA andB, and . We will use this to show strong monotonicity or strong convexity of operator functions. We will see that when 0≤A≤B, andB−A is of finite rank,A ^t ≤B ^t for somet>1 if and only if the null space ofB−A is invariant forA.     

Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices

Let and be a perturbed eigenpair of a diagonalisable matrixA. The problem is to bound the error in and . We present one absolute perturbation bound and two relative perturbation bounds. The absolute perturbation bound is an extension of Davis and Kahan's sin θ Theorem from Hermitian to diagonalisable matrices. The two relative perturbation bounds assume that and are an exact eigenpair of a perturbed matrixD ₁ AD ₂ , whereD ₁ andD ₂ are non-singular, butD ₁ AD ₂ is not necessarily diagonalisable. We derive a bound on the relative error in and a sin θ theorem based on a relative eigenvalue separation. The perturbation bounds contain both the deviation ofD ₁ andD ₂ from similarity and the deviation ofD ₂ from identity. This work was partially supported by NSF grant CCR-9400921.     

Eine Verallgemeinerung derCW-Komplexe

In the definition ofCW-complexes, the one-point spaceP, respectively the spaceP∪* with basepoint *, play the roll of the only “building-stone”. Let be a family of compact spaces. Then the definition of a generalizedCW-complex over is obtained from the definition of aCW-complex by replacingP by the spaces of and formation of the mapping cone by a slightly modified construction. LetCW * denote the category of all pointed spaces which have the homotopy type of a generalizedCW-complex over . If , thenCW * is the category of all pointedCW-spaces.CW * is closed under the formation of direct sums and of mapping cones, cylinders and tori, and is formally characterized as the smallest such subcategory of Top * containing the spaces W∪*, . Following the methods of E. H. Brown, it is proved, that any half exact homotopy functor onCW * is representable, and any cohomology theory onCW is naturally equivalent to the cohomology theory of an Ω-spectrum; for example, the singular cohomo logy is representable onCW for any family of compact spaces.      

The spectrum of compact hypersurface in sphere

Let M be a compact minimal hypersurface of sphere Sn 1(1). Let (M) be H (r)-torus of sphere Sn 1 (1).Assume they have the same constant mean curvature H, the result in [1] is that ifSpec0(M, g) =Spec0((M), g),then for 3≤ n ≤ 6, r2≤n-1/n or n ≥ 6, r2 ≥ n-1, then M is isometric to (M). We improved the result and prove that: if Spec0(M,g) =Spec0((M),g), then M is isometric to (M). Generally, if Specp(M,g) =Specp((M),g), here p is fixed and satisfies that n(n - 1) ≠ 6p(n - p), then M is isometric to (M).     

Compactness of minimal closed invariant sets of actions of unipotent groups

Let G be a connected Lie group, let Γ be a lattice in G, and let be a unipotent subgroup of G. It is proved that, for the natural action of on G/Γ, every minimal closed -invariant subset is compact. Dedicated to Professor Jacques Tits on the occasion of his sixtieth birthday