Uniqueness Theorems for CR Functions

Let M be a CR manifold embedded in ?^s of arbitrary codimension. M is called generic if the complex hull of the tangent space in all points of M is the whole ?^s. M is minimal (in sense of Tumanov) in p ? M if there does not exist any CR submanifold of M passing through p with the same CR dimension as M but of smaller dimension. Let M be generic and minimal in some point p ? M and N be a generic submanifold of M passing through p. We prove that a continuous CR function on M vanishes identically in some neigbourhood of p if its restriction to N either vanishes in p faster then some function with non-integrable logarithm or it vanishes on a subset of N of positive measure.     

A strange example on Property Np

 We exhibit an example of a line bundle M on a smooth complex projective variety Y s.t. M satisfies Property N ₁₀ , the 10-module of a minimal resolution of the ideal of the embedding of Y by M is nonzero and M ² does not satisfy Property N ₁₀ . Thus this is a completely convincing example showing that surprisingly it is not true that if a line bundle M satisfies Property N _p then any power of M satisfies Property N _p . We recall that in [Ru] we proved the following statement: if M is a line bundle on a smooth complex projective variety and M satisfies Property N _p then M ^s satisfies Property N _p if s≥p. Received: 5 March 2001     

Quotient Fields of a Model of IΔ0 + Ω1

In [4] the authors studied the residue field of a model M of IΔ₀ + Ω₁ for the principal ideal generated by a prime p. One of the main results is that M/<p> has a unique extension of each finite degree. In this paper we are interested in understanding the structure of any quotient field of M, i.e. we will study the quotient M/I for I a maximal ideal of M. We prove that any quotient field of M satisfies the property of having a unique extension of each finite degree. We will use some of Cherlin's ideas from [3], where he studies the ideal theory of non standard algebraic integers.     

The Levi classes generated by nilpotent groups

Given an arbitrary class M of groups, denote by L(M) the class of all groups G in which the normal closure of every element belongs to M. Consider the quasivariety _q F _p generated by the relatively free group in the class of nilpotent groups of length at most 2 with the commutant of exponent p (where p is an odd prime). We describe the Levi class that is generated by qF _p.     

The L p spectrum of Riemannian products

We prove that the L ^p spectrum of a Riemannian product M ₁×M ₂ coincides with the set theoretic sum of the L ^p spectra of M ₁ and M ₂ . Received: 13 June 2007     

Schur multiplicators of finite <Emphasis Type="Italic">p</Emphasis>-groups with fixed coclass

We investigate the Schur multiplicators M(G) of p-groups G using coclass theory. For p > 2 we show that there are at most finitely many p-groups G of coclass r with |M(G)| ≤ s for every r and s. We observe that this is not true for p = 2 by constructing infinite series of 2-groups G with coclass r and |M(G)| = 1. We investigate the Schur multiplicators of the 2-groups of coclass r further.     

Transitivity and full transitivity for p-local modules

A p-local module M is called (fully) transitive if for all x,y ? Mx,y\in M with U_M(x) = U_M(y) ( U_M(x)\leqq U_M(y)U_M(x)\leqq U_M(y)) there exists an automorphism (endomorphism) of M which maps x onto y. In this paper we examine the relationship of these two notions in the case of p-local modules. We show that a module M is fully transitive if and only if M?MM\oplus M is transitive in the case where the divisible part of M/tMM/tM has rank at most one. Moreover, we show that for the same class of modules transitivity implies full transitivity if p > 2. This extends theorems of Files, Goldsmith and of Kaplansky for torsion p-local modules.     

Eigenvalue estimate for a weighted p-Laplacian on compact manifolds with boundary

Let (M ⁿ , g) be a compact Riemannian manifold with convex boundary, let dμ = e ^h(x) dV (x) be a weighted measure on M, and let Δ_μ,p be the corresponding weighted p-Laplacian on M. We obtain a lower bound for the first nonzero Neumann eigenvalue of Δ_μ,p .     

Finite jet determination of local analytic CR automorphisms and their parametrization by 2-jets in the finite type case

We show that germs of local real-analytic CR automorphisms of a real-analytic hypersurface M in $\mathbb{C}$² at a point p M are uniquelydetermined by their jets of some finite order at p if and only if M is not Levi-flat near p. This seems to be the first necessary and sufficient result on finite jet determination and the first result of this kind in the infinite type case.If M is of finite type at p, we prove a stronger assertion: the local real-analytic CR automorphisms of M fixing p are analytically parametrized (and hence uniquely determined) by their 2-jets at p. This result is optimal since the automorphisms of the unit sphere are not determined by their 1-jets at a point of the sphere. The finite type condition is necessary since otherwise the needed jet order can be arbitrarily high [Kow1,2], [Z2]. Moreover, we show, by an example, that determination by 2-jets fails for finite type hypersurfaces already in $\mathbb{C)$³.We also give an application to the dynamics of germs of local biholomorphisms of $\mathbb{C)$².     

Approximate identities on some homogeneous Banach spaces

We study convergence of approximate identities on some complete semi-normed or normed spaces of locally L ^p functions where translations are isometries, namely Marcinkiewicz spaces M^p{\mathcal{M}^{p}} and Stepanoff spaces S^p{\mathcal{S}^p}, 1 ≤ p < ∞, as well as others where translations are not isometric but bounded (the bounded p-mean spaces M ^p ) or even unbounded (M^p₀{M^{p}_{0}}). We construct a function f that belongs to these spaces and has the property that all approximate identities f_e * f{\phi_\varepsilon * f} converge to f pointwise but they never converge in norm.     

The Hilbert Null-cone on Tuples of Matrices and Bilinear Forms

We describe the null-cone of the representation of G on M ^p , where either G = SL(W) × SL(V) and M = Hom(V,W) (linear maps), or G = SL(V) and M is one of the representations S ²(V ^*) (symmetric bilinear forms), Λ²(V ^*) (skew bilinear forms), or (arbitrary bilinear forms). Here V and W are vector spaces over an algebraically closed field K of characteristic zero and M ^p is the direct sum of p of copies of M. More specifically, we explicitly determine the irreducible components of the null-cone on M ^p . Results of Kraft and Wallach predict that their number stabilises at a certain value of p, and we determine this value. We also answer the question of when the null-cone in M ^p is defined by the polarisations of the invariants on M; typically, this is only the case if either dim V or p is small. A fundamental tool in our proofs is the Hilbert–Mumford criterion for nilpotency (also known as unstability).     

The structure of -tensorial cochains of differential operators

Let M be a manifold. Let F = C^∞(M, R). Then the associative algebra of differential operators on is a two-sided -module. We prove that there is a natural isomorphism between the -tensorial Hochschild p-cochains of and the jets, taken on the diagonal, of smooth functions on the Cartesian product of p + 1 copies of M. There is an induced isomorphism of the corresponding associative differential graded algebras. The normalised -tensorial p-cochains correspond isomorphically to jets of those above functions which vanish on all the contiguous subdiagonals x_{j + 1} = X_j, j = 0,…, p − 1 of M^{(p + 1)}. This isomorphism may offer a useful alternative view of infinite-order jets of functions of several variables, taken on the diagonal as cochains of .     

Pseudo-Riemannian manifolds modelled on symmetric spaces

We construct irreducible pseudo-Riemannian manifolds (M, g) of arbitrary signature (p, q) with the same curvature tensor as a pseudo-Riemannian symmetric space which is a direct product of a two-dimensional Riemannian space form M ₂(c) and a pseudo-Euclidean space with the signature (p, q − 2), or (p − 2, q), respectively.     

Covering homotopy 3-spheres

We prove the following theorem: for any closed orientable 3-manifoldM and any homotopy 3-sphere Σ, there exists a simple 3-fold branched coveringp:M→Σ. We also propose the conjecture that, for any primitive branched coveringp:M→N between orientable 3-manifolds,g(M)≥g(N), whereg denotes the Heegaard genus. By the above mentioned result, the genus 0 case of such conjecture is equivalent to the Poincaré conjecture.     

Compactification and maximal diameter theorem for noncompact manifolds with radial curvature bounded below

Let M be a complete open n-manifold with a base point p, at which the radial sectional curvature along every minimizing geodesic emanating from p is bounded below by the radial curvature function of a model surface. We discuss the maximal diameter theorem for the compactification of M by attaching the ideal boundary. Under certain conditions we prove that p becomes a pole and that M is isometric to the n-model. Received: 24 September 2000; in final form: 21 November 2001 / Published online: 17 June 2002 Dedicated to Professor Su Bu-Chin on the occasion of his one hundredth birthday The work of the first author was partially supported by the Grant-in-Aid for Scientific Research, No. 12440021 and for Exploratory Research, No. 13874012     

On m-Accretivity of Perturbed Bochner Laplacian in L p Spaces on Riemannian Manifolds

We consider a differential expression ${H=\nabla^*\nabla+V}We consider a differential expression H=?^*?+V{H=\nabla^*\nabla+V}, where ?{\nabla} is a Hermitian connection on a Hermitian vector bundle E over a manifold of bounded geometry (M, g) with metric g, and V is a locally integrable section of the bundle of endomorphisms of E. We give a sufficient condition for H to have an m-accretive realization in the space L ^p (E), where 1 < p <  +∞. We study the same problem for the operator Δ _M  + V in L ^p (M), where 1 < p < ∞, Δ _M is the scalar Laplacian on a complete Riemannian manifold M, and V is a locally integrable function on M.     

On a class of finite rings

In [7], Corbas determined all finite rings in which the product of any two zero-divisors is zero, and showed that they are of two types, one of characteristic p ²and the other of characteristic p².

The purpose of this paper is to address the problem of the classification of finite rings such that.

(i)the set of all zero-divisors form an ideal M.

(ii)M ³=(0); and.

(iii)M ³≠(0).

Because of (i), these rings are called completely primary and we shall call a finite completely primary ring R which satisfies conditions (i), (ii) and (iii), a ring with property(T). These rings are of three types, namely, of characteristic p p ² and p ³. The characteristic p ² case is subdivided into cases in which p?M ² p?ann(M)?M ² and p?M ?ann(M), where ann(M) denotes the two-sided annihilator of where M in R.     

Component group of thep-new subvariety ofJ 0(M p )

For an abelian varietyA over ℚ _p , the special fibre in the Néron model ofA over ℤ _p is the extension of a finite group scheme over ℤ _p , called the group of connected components, by the connected component of identity. WhenA is the Jacobian variety of an algebraic curve, its component group has been calculated in many cases. We determine in this paper the component group of thep-new subvariety ofJ ₀(M _p ), forM>1 a positive integer andp≥5 a prime not dividingM. Such a subvariety is not the Jacobian of any obvious curve, but it is not clear if it can ever be realised as the Jacobian of a curve.     

M ‐elliptic pseudo‐differential operators on L p (ℝn )

We construct the minimal and maximal extensions in L ^p (?ⁿ ), 1 < p < ∞, for M ‐elliptic pseudo‐differential operators initiated by Garello and Morando. We prove that they are equal and determine the domains of the minimal, and hence maximal, extensions of M ‐elliptic pseudo‐differential operators. For M ‐elliptic pseudodifferential operators with constant coefficients, the spectra and essential spectra are computed. An application to quantization is given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Universal theories for rigid soluble groups

A group is said to be p-rigid, where p is a natural number, if it has a normal series of the form G = G ₁ > G ₂ > … > G _p  > G _p+1 = 1, whose quotients G _i /G _i+1 are Abelian and are torsion free when treated as \mathbbZ \mathbb{Z} [G/G _i ]-modules. Examples of rigid groups are free soluble groups. We point out a recursive system of universal axioms distinguishing p-rigid groups in the class of p-soluble groups. It is proved that if F is a free p-soluble group, G is an arbitrary p-rigid group, and W is an iterated wreath product of p infinite cyclic groups, then ∀-theories for these groups satisfy the inclusions A(F) ê A(G) ê A(W) \mathcal{A}(F) \supseteq \mathcal{A}(G) \supseteq \mathcal{A}(W) . We construct an ∃-axiom distinguishing among p-rigid groups those that are universally equivalent to W. An arbitrary p-rigid group embeds in a divisible decomposed p-rigid group M = M(α₁,…, α _p ). The latter group factors into a semidirect product of Abelian groups A ₁ A ₂…A _p , in which case every quotient M _i /M _i+1 of its rigid series is isomorphic to A _i and is a divisible module of rank α_i over a ring \mathbbZ \mathbb{Z} [M/M _i ]. We specify a recursive system of axioms distinguishing among M-groups those that are Muniversally equivalent to M. As a consequence, it is stated that the universal theory of M with constants in M is decidable. By contrast, the universal theory of W with constants is undecidable.