Schrödinger operators on manifolds, essential self-adjointness, and absence of eigenvalues

Suppose thatM ⁿ is a complete, noncompact, Riemannian manifold. If Δ denotes the Laplace operator ofM, one has associated Schrödinger operators ? Δ +V. Conditions onV are formulated, which ensures the essential self-adjointness of ? Δ +V. In particular, ifV ∈ Q_α,loc (M ⁿ), the local Stummel class, andV ≥ ? c outside of a compact set, then ? Δ +V is essentially self-adjoint on C ₀ ^∞ (M ⁿ). In addition, essential self-adjointness is proved for potentials which are strongly singular at a point. The absence of eigenvalues of ?Δ +V is also studied. This relies upon Rellich-type identities. The results on strongly singular potentials make use of a generalization of the classical uncertainty principle, inR ⁿ, to Riemannian manifolds with a pole.     

The rate of the normal approximation for jackknifingU-statistics

Let U_n be a U-statistic with symmetric kernel h(x,y) such that Eh(X_1,X_2)=θ and Var E[h(X_1,X_2)-θ|X_j]>0.Let f(x) be a function defined on R and f″ be bounded.If f(θ) is the parameterof interest,a natural estimator is f(U_n).It is known that the distribution function of z_n=(n~(1/2){Jf(U_n)-f(θ)})/(S_n~*) converges to the standard normal distribution Φ(x) as n→∞,where Jf(U_n) isthe jackknife estimator of f(U_n),and S_n~(*2) is the jackknife estimator of the asymptotic variance ofn~(1/2) Jf(U_n).It is of theoretical value to study the rate of the normal approximation of the statistic.In this paper,assuming the existence of fourth moment of h(X_1,X_2),we show that(?)|P{z_n≤x}-Φ(x)|=O(n~(-1/2)log n).This improves the earlier results of Cheng(1981).     

Splitting manifolds as M × R where M has a k-fold end structure

Let X be a locally finite simplicial complex of dimension n, n? 5, equipped with a k-fold end structure [4] and consider a piecewise linear (n + 1)-dimensional manifold M that is proper homotopy equivalent to X × R by F:M → X × R, where R is the set of real numbers. The question arises as to whether or not the manifold M can be split, i.e., written as M = N × R where N is a n-manifold and where there is a proper homotopy between F and (p₁ ° F₀) × id:N × R → X × R, preserving the natural (k+1)-fold end structure, where F₀ is F|_N and p₁ is the projection X × R → X. Of particular significance is the fact that X is noncompact. When the construction of such splittings is attempted, algebraic obstructions arise, which vanish if and only if the construction can be completed. This paper develops such an obstruction theory by utilizing methods of L.C. Siebenmann and the k-fold end structures of F. Waldhausen.     

Base manifolds for fibrations of projective irreducible symplectic manifolds

Given a projective irreducible symplectic manifold M of dimension 2n, a projective manifold X and a surjective holomorphic map f:M→X with connected fibers of positive dimension, we prove that X is biholomorphic to the projective space of dimension n. The proof is obtained by exploiting two geometric structures at general points of X: the affine structure arising from the action variables of the Lagrangian fibration f and the structure defined by the variety of minimal rational tangents on the Fano manifold X.     

Planar normal sections on the natural imbedding of a flag manifold

We study the setX[M] of planar normal sections on the natural imbedding of a flag manifoldM. We characterizeX[M] and show that it is a real algebraic submanifold of P ⁿ (n=dimM). From results of Chen and Ferus, it may be concluded thatX[M] measures how farM is from a symmetricR-space. We compute the Euler characteristic ofX[M] and its complexificationX _c [M]. Our main result shows, in particular, thatx(X[M]) andx(X _c [M]) depend only on dimM and not on the nature ofM itself.     

Monge-Ampère equations and moduli spaces of manifolds of circular type

A (bounded) manifold of circular type is a complex manifold M of dimension n admitting a (bounded) exhaustive real function u, defined on M minus a point xo, so that: (a) it is a smooth solution on M?{xo} to the Monge-Ampère equation n(ddcu)=0; (b) xo is a singular point for u of logarithmic type and eu extends smoothly on the blow up of M at xo; (c) ddc(eu)>0 at any point of M?{xo}. This class of manifolds naturally includes all smoothly bounded, strictly linearly convex domains and all smoothly bounded, strongly pseudoconvex circular domains of Cn.A set of modular parameters for bounded manifolds of circular type is considered. In particular, for each biholomorphic equivalence class of them it is proved the existence of an essentially unique manifold in normal form. It is also shown that the class of normalizing maps for an n-dimensional manifold M is a new holomorphic invariant with the following property: it is parameterized by the points of a finite dimensional real manifold of dimension n² when M is a (non-convex) circular domain while it is of dimension n²+2n when M is a strictly linearly convex domain. New characterizations of the circular domains and of the unit ball are also obtained.     

Conditional limit theorems for exponential families and finite versions of de Finetti's theorem

Consider an exponential familyP _λ which is maximal, smooth, and has uniformly bounded standardized fourth moments. Consider a sequenceX ₁,X ₂,... of i.i.d. random variables with parameter λ. LetQ _nsk be the law ofX ₁,...,X _k given thatS _n=X ₁+...+X _n=s. Choose λ so thatE _λ(X ₁)=s/n. Ifk andn→∞ butk/n→0, then $$\parallel Q_{nsk} - P_\lambda ^k \parallel = \gamma \frac{k}{n} + o\left( {\frac{k}{n}} \right)$$ where γ=1/2E{|1?Z ²|} andZ isN(0,1). The error term is uniform ins, the value ofS _n. Similar results are given fork/n→θ and for mixtures of theP _λ ^k . Versions of de Finetti's theorem follow.     

On the dimension of ordered sets with the 2-cutset property

An ordered setP is said to have 2-cutset property if, for every elementx ofP, there is a setS of elements ofP which are noncomparable tox, with |S|?2, such that every maximal chain inP meets {x}∪S. We consider the following question: Does there exist ordered sets with the 2-cutset property which have arbitrarily large dimension? We answer the question in the negative by establishing the following two results.Theorem: There are positive integersc andd such that every ordered setP with the 2-cutset property can be represented asP=X∪Y, whereX is an ordinal sum of intervals ofP having dimension ?d, andY is a subset ofP having width ?c. Corollary: There is a positive integern such that every ordered set with the 2-cutset property has dimension ?n.     

Alcuni risultati sugli ideali massimali di anelli di polinomi

Riassunto In questo lavoro viene studiata la cardinalità minima dei sistemi di generatori di ideali massimaliM dell'anello di polinomiR[X ₁, …,X _n] (R dominio d'integrità) tali cheM⊃R=0. In particolare è dimostrato che sen≥2 edR è unS-dominio di dimensione 1, ogni siffatto ideale massimale può essere generato dan elementi.

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Summary This paper is concerned with estimates of the minimal number of generators for maximal idealsM in the polynomial ringR[X ₁, …,X _n] (R an integral domain) such thatM⊃R=0. In particular it is proved that ifn≥2 andR is a 1-dimensionalS-domain, every such maximal ideal can be generated byn elements.

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Lavoro eseguito nell'ambito dell'attività del G.N.S.A.G.A. del C.N.R.     

Upper bounds of small Eigenvalues of the Dirac operator and isometric immersions

We show that a topologically determined number of eigenvalues of the Dirac operatorD of a closed Riemannian spin manifoldM of even dimensionn can be bounded by the data of an isometric immersion ofM into the Euclidian spaceR ^N . From this we obtain similar bounds of the eigenvalues ofD in terms of the scalar curvature ofM ifM admits a minimal immersion intoS ^N or,ifM is complex, a holomorphic isometric immersion intoPC ^N .     

Invertible linear maps preserving {1}-inverses of matrices over PID

Let R be a PID,chR = 2,n > 1, M_n(R) be then xn full matrix algebra over R.f denotes any invertible linear map preserving {1}-inverses from M_n(R) to itself. In this paper, we have proven thatf is an invertible linear map on M_n(R) preserving {1}-inverses if and only iff satisfies any one of the following two conditions: (i) there exists a matrixP ? GL _n(R) such thatf(A) =PAP ^?1 for allA ? M _n(R), (ii) there exists a matrixP ? GL _n(R) such thatf(A) =PA ^t P^?1 forA ? M _n(R).     

Approximation of convex bodies by polytopes with uniformly bounded valences

Given any convex bodyK in Euclideann-spaceR ⁿ and any number ?>0, does there always exist a polytopeP(K, ?)?R ⁿ such that the number of vertices of a facet ofP and the number of facets meeting in a common vertex are bounded by a constant depending on the dimensiond only and such that the Hausdorff-distance ? (K, P) ofK andP is less than ?? This question of Ewald posed at the Durham symposium in 1975 is answered in the affirmative.     

Splitting off tori and the evaluation subgroup of the fundamental group

Given a spaceX what is the largest torusT ⁿ such thatX is homotopy equivalent toY×T ⁿ We find the answer depends on a simple property of the evaluation subgroup of the fundamental group,G ₁(X). As corollaries we have the Splitting theorem of Conner and Raymond and the fact that the dimension ofX must be greater than the rank ofG ₁(X).     

k-torsionless modules with finite Gorenstein dimension

Let R be a commutative Noetherian ring. It is shown that the finitely generated R-module M with finite Gorenstein dimension is reflexive if and only if M _p is reflexive for p ∈ Spec(R) with depth(R _p) ? 1, and $G - {\dim _{{R_p}}}$ (M _p) ? depth(R _p) ? 2 for p ∈ Spec(R) with depth(R _p) ? 2. This gives a generalization of Serre and Samuel’s results on reflexive modules over a regular local ring and a generalization of a recent result due to Belshoff. In addition, for n ? 2 we give a characterization of n-Gorenstein rings via Gorenstein dimension of the dual of modules. Finally it is shown that every R-module has a k-torsionless cover provided R is a k-Gorenstein ring.     

Bounding the number of bases of a matroid

Letb(M) andc(M), respectively, be the number of bases and circuits of a matroidM. For any given minor closed class? of matroids, the following two questions, are investigated in this paper. (1) When is there a polynomial functionp(x) such thatb(M)≤p(c(m)|E(M)|) for every matroidM in?? (2) When is there a polynomial functionp(x) such thatb(M)≤p(|E(M)|) for every matroidM in?? Let us denote byM _Mn the direct sum ofn copies ofU _1,2. We prove that the answer to the first question is affirmative if and only if someM _Mn is not in?. Furthermore, if all the members of? are representable over a fixed finite field, then we prove that the answer to the second question is affirmative if and only if, also, someM _Mn is not in?.     

Degeneracy of holomorphic curves in surfaces

LetX be a complex projective algebraic manifold of dimension 2 and let D₁, ..., D_u be distinct irreducible divisors onX such that no three of them share a common point. Let\(f:{\mathbb{C}} \to X\backslash ( \cup _{1 \leqslant i \leqslant u} D_i )\) be a holomorphic map. Assume thatu ? 4 and that there exist positive integers n₁, ... ,n_u,c such that n_in_J D _i.D_j) =c for all pairsi,j. Thenf is algebraically degenerate, i.e. its image is contained in an algebraic curve onX.     

A problem related to Foulkes’s conjecture

In this paper we study a class of symmetric matricesT indexed by positive integers m≥ n≥2 and defined as follows: for any positive integersp andq let ?_p,q be the set of partitions ofU = {1,2,3, ...,pq} into p blocks each of sizeq. Letm ≥n ≥ 2 be positive integers. By atransversal of α = A₁/A₂/.../A_n ∈ ?_n,m we mean a partitionß = B₁/B₂/.../B_m ∈ ? _m,n such that ‖A _i ∩B _j = 1 for every i= 1,2, ...,n and everyj = 1,2, ...,m. LetM be the zero-one matrix with rows indexed by the elements of ?_n,m and columns indexed by the elements of ?_m,n such that M_αß = 1 iffß is a transversal of α. We are interested in finding the eigenvalues and eigenspaces of the symmetric matrixT = MM^t. The nonsingularity ofT implies Foulkes’s Conjecture (for these values of m andn). In the casen = 2 we completely determine the eigenvalues and eigenspaces of T and in so doing demonstrate the non-singularity ofT. Forn = 3 we develop a fast algorithm for computing the eigenvalues ofT, and give numerical results in the cases m = 3,4, 5, 6.     

On subharmonic functions dominated by certain functions

Given two kinds of functionsf(X) andh(y) defined on them-dimensional Euclidean spaceR ^m (m≧1) and the set of positive real numbers respectively, we give an estimation of growth of subharmonic functionsu(P) defined onR ^m+n (n≧1) such that $$u(P) \leqq f\left( X \right)h\left( {\left\| Y \right\|} \right)$$ for anyP=(X, Y),X ∈R ^m, Y ∈R ⁿ, where ‖Y ‖ denotes the usual norm ofY. Using an obtained result, we give a sharpened form of an ordinary Phragmén-Lindelöf theorem with respect to the generalized cylinderD ×R ⁿ, with a bounded domainD inR ^m.     

On an Ulam's problem

The existence and the uniqueness (with respect to a filtration-equivalence) of a vector flowX on ? ⁿ ,n≥3, such that:

1. X has not any stationary points on ? ⁿ ;
2. all orbits ofX are bounded;
3. there exists a filtration forX are proved in the present note.