Problemi variazionali conformi

Letf: (M,g)→(N,g′) be a differentiable map between the riemannian manifoldsM andN, M being compact.K. Uhlenbeck pointed out a functionalE ^m(f), related to the energy density off, that depends only on the conformal structure ofM. In this paper we prove thatE ^m(f) is stationary with respect to deformations of the riemannian metric ofM if and only iff is weakly conformal; in this casef provides a local minimum ofE ^m.     

Galerkin method and optimal error algorithms in hilbert spaces

Summary We consider operator equations of the formLu=f, whereL belongs to the class of linear, bounded (by a constantM) and coercive (with a constantm) operators from a Hilbert spaceV onto its dualV ^* andf belongs to a Hilbert spaceWV ^*. We study optimality of the Galerkin methodP _n ^* Lu _n =P _n ^* f, whereu _n V _n ,V _n is subspace ofV, P _n is the orthogonal projector ontoV _n andP _n ^* is dual toP _n . We show that the Galerkin method is quasi optimal independently of the choice of the subspaceV _n and the spaceW ifM>m. In the caseM=m, optimality of the method depends strongly on the choice ofV _n andW. Therefore we define a new algorithm which is always optimal (independently of the choice ofV _n andW and relations betweenM andm).     

On classification of immersions ofn-manifolds in (2n−1)-manifolds

Under the assumption of (f, M ⁿ ,N ²ⁿ⁻¹) being trivial, the classification of immersions homotopic tof: M ⁿ →N ²ⁿ⁻¹ is obtained in many cases. The triviality of (f, M ⁿ ,P ²ⁿ⁻¹) is proved for anyM ⁿ andf. LetM, N be differentiable manifolds of dimensionn and2n−1 respectively. For a mapf: M → N, denote byI[M, N] _f the set of regular homotopy classes of immersions homotopic tof. It has been proved in [1] that, whenn>1,I[M, N] _f is nonempty for anyf. In this paper we will determine the setI[M, N] _f in some cases. For example, ifN=P ²ⁿ⁻¹ or more generally, the lens spacesS _m ²ⁿ⁻¹ =Z _m /S ²ⁿ⁻¹,M is any orientablen-manifold or nonorientable butn≡0, 1, 3 mod 4, then, for anyf, theI[M, N] _f is determined completely. WhenN=R ²ⁿ⁻¹, the setI[M, N] of regular homotopy classes of all immersions has been enumerated by James and Thomas in [2] and McClendon in [3] forn>3. Applying our results toN=R ²ⁿ⁻¹ we obtain their results again, except for the casen≡2 mod 4 andM nonorientable. Whenn=3, McClendon's results cannot be used. Our results include the casesn=3,M orientable or not (for orientableM, I[M, R⁵] is known by Wu [4]).     

Three dimensional expansive diffeomorphisms with homoclinic points

LetM be a compact connected oriented three dimensional manifold andf:MM an expansive diffeomorphism such that (f)=M. Let us also assume that there is a hyperbolic periodic point with a homoclinic intersection. Thenf is conjugate to an Anosov isomorphism ofT ³. Moreover, we show that at a homoclinic point the stable and unstable manifolds of the hyperbolic periodic point are topologically transverse.     

On multilinear operators commuting with Lie derivatives

LetE ₁, ...,E _k andE be natural vector bundles defined over the categoryMf _m ⁺ of smooth orientedm-dimensional manifolds and orientation preserving local diffeomorphisms, withm2. LetM be an object ofMf _m ⁺ which is connected. We give a complete classification of all separately continuousk-linear operatorsD : _c(E ₁ M) × ... × _c(E _k M) (EM) defined on sections with compact supports, which commute whith Lie derivatives, i.e., which satisfy

Inequalities for the interpolation points in Chebyshev approximation by polynomials

Letf andg be approximated in the Chebyshev sense by polynomials of degree n and n–1, respectively. It is shown that if the sum and difference of the normalized (n+1)-st derivatives off andg do not change sign, then the interpolation points ofg separate those off. A corollary is that the zeros of the Chebyshev polynomialT _n separate the interpolation points off iff ⁽ⁿ⁺¹⁾ does not change sign. The sharpness of this result is demonstrated.     

Separation by a codimension-1 map with a normal crossing point

Letf:M ^n–1N ⁿ be an immersion with normal crossings of a closed orientable (n–1)-manifold into an orientablen-manifold. We show, under a certain homological condition, that iff has a multiple point of multiplicitym, then the number of connected components ofN–f(M) is greater than or equal tom+1, generalizing results of Biasi and Romero Fuster (Illinois J. Math. 36 (1992), 500–504) and Biasi, Motta and Saeki (Topology Appl. 52 (1993), 81–87). In fact, this result holds more generally for every codimension-1 continuous map with a normal crossing point of multiplicitym. We also give various geometrical applications of this theorem, among which is an application to the topology of generic space curves.     

Monotonie bei den Quadraturverfahren von Gauß und Newton-Cotes

Summary LetQ _n be the quadrature rule of Gauss or Newton-Cotes withn abscissas. It is proven here, thatf ⁽²ⁿ⁾0 impliesQ _n ^G [f]Q _m ^G [f] (for allm>n) andQ _2n–1 ^NC [f]Q _2n ^NC [f]Q _2n+1 ^NC [f]. It follows that the sequenceQ _n[f] (n=1, 2, ...) is monotone, if all derivatives off are positive.

     

Limiting equations in asymptotic stability problems for nonautonomous differential equations

Summary By limiting equations, we prove some asymptotic stability theorems for the origin ofR ⁿ with respect to the solutions of a differential equation , also when the functionf is not defined forx=0. Further we examine similar problems concerning the asymptotic stability of a setM ofR ⁿ that can be unbounded.

---

Riassunto Mediante le equazioni limiti, si dimostrano alcuni teoremi di stabilità asintotica per l'origine diR ⁿ rispetto alle soluzioni di un'equazione differenziale , anche quando la funzionef non è definita perx=0. Vengono inoltre esaminati analoghi problemi relativi alla stabilità asintotica di un insiemeM diR ⁿ anche non limitato.

---

---

Work performed under the auspices of the Italian Council of Research (G.N.F.M. del C.N.R.).     

Tighter Bounds on the Solution of a Divide-and-Conquer Maximin Recurrence

LetM(n) be defined by the recurrencewherefis an arbitrary nondecreasing function andM(1) is given. The recurrenceM(n) is a divide-and-conquer maximin recurrence, which occurs in a variety of problems in the analysis of algorithms. In this paper, a new upper bound onM(n) is first derived. The derived bound is smaller than the one proposed previously by Li and Reingold. It is at most two times the exact solution ofM(n). Using the bound, we further show thatM(n) ≤ 2E(n), whereE(n) is defined by the recurrenceE(n) = E(⌊n/2⌋) + E(⌈n/2⌉) + f(⌊n/2⌋). From this result, we can conclude that a divide-and-conquer algorithm whose time complexity is expressed asM(n) is as efficient as a divide-and-conquer algorithm whose time complexity is expressed asE(n).     

Semistability and finite maps

Let ${f : Y \longrightarrow M}Let f : Y ? M{f : Y \longrightarrow M} be a surjective holomorphic map between compact connected K?hler manifolds such that each fiber of f is a finite subset of Y. Let ω be a K?hler form on M. Using a criterion of Demailly and Paun (Ann. Math. 159 (2004), 1247–1274) it follows that the form f*ω represents a K?hler class. Using this we prove that for any semistable sheaf E ? M{E\, \longrightarrow\,M} , the pullback f*E is also semistable. Furthermore, f*E is shown to be polystable provided E is reflexive and polystable. These results remain valid for principal bundles on M and also for Higgs G-sheaves.     

Straffe Untermannigfaltigkeiten in konvexen Hyperflächen

It will be proved that a tight substantial embedding ofS ^m×Sⁿ intoE ^m+n+2 whose image lies in a strictly convex hypersurface is projectively equivalent to the productC ₁×C ₂E ^m+1×E ^m+1=E ^m+n+2 of two convex hypersurfacesC ₁ undC ₂.     

Necessary and sufficient conditions for the convergence of integrated and mean-integratedr-th order error of histogram density estimates

Suppose thatX _l ,..., X _n are samples drawn from a population with density functionf andf _n (x)=f _n (x;X _l ,..., X _n is an estimate off(x), Denote bym _nr =|f _n (x)–f(n)| ^r dx andM _nr =E(m _nr) the Integratedr-th Order Error and Mean Integratedr-th Order Error off _n for somer1 (whenr=2,they are the familiar and widely studied ISE and MISE), In this paper the same necessary and sufficient condition for and a.s. is obtained whenf _n (x) is the ordinary histogram estimator.The Project supported by National Natural Science Foundation of China.     

Beste Quadraturformeln für Integrale mit einer Gewichtsfunktion

Let be a fixed matrix with elements that are 0 or 1 and letX be a fixed set ofm+1 different knots. The problem is to find necessary and sufficient conditions for (E, X) to guarantee the existence of a quadrature formula with a remainder term of type for any choice of a weight functionw(t) and satisfyingR(f)=0 forf a polynomial of degree at mostn–1. The result generalizes the corresponding result ofI. J. Schoenberg for the special case of quasi-Lagrange-matricesE. —in case of the existence ofR it is possible to calculate the best quadrature formulaR ^* in the sense ofSard by integrating splines of degree 2n–1. But ifE contains onlyn ones it is sufficient to integrate polynomials of degreen–1.     

Homotopies for computation of fixed points on unbounded regions

Givenf: R ⁿ R ^n* with some conditions, our aim is to compute a fixed pointx f(x) off; hereR ⁿ isn-dimensional Euclidean space andR ^n* is the collection of nonempty subsets ofR ⁿ . A typical application of the algorithm can be motivated as follows: Beginning with the constant mapf ₀:R ⁿ {0} R ⁿ and its fixed pointx ₀ = 0, we deformf _t ast tof f and follow the pathx _t of fixed points off _t . Cluster points of thex _t 's ast are fixed points off. This research was supported in part by Army Research Office-Durham Contract DAHC-04-71-C-0041 and by National Science Foundation Grant GK-5695.     

Convex polynomial and spline approximation in C[−1, 1]

We prove that a convex functionf C[–1, 1] can be approximated by convex polynomialsp _n of degreen at the rate of ₃(f, 1/n). We show this by proving that the error in approximatingf by C² convex cubic splines withn knots is bounded by ₃(f, 1/n) and that such a spline approximant has anL third derivative which is bounded by n³₃(f, 1/n). Also we prove that iff C²[–1, 1], then it is approximable at the rate ofn ^–2 (f, 1/n) and the two estimates yield the desired result.Communicated by Ronald A. DeVore.     

Cohomology of compact hyperkähler manifolds and its applications

We announce the structure theorem for theH ²(M)-generated part of cohomology of a compact hyperkähler manifold. This computation uses an action of the Lie algebra so(4,n–2) wheren=dimH ²(M) on the total cohomology space ofM. We also prove that every two points of the connected component of the moduli space of holomorphically symplectic manifolds can be connected with so-called twistor lines — projective lines holomorphically embedded in the moduli space and corresponding to the hyperkähler structures. This has interesting implications for the geometry of compact hyperkähler manifolds and of holomorphic vector bundles over such manifolds.     

On the degree of complex rational approximation to real functions

Iff∈C[?1, 1] is real-valued, letE _R ^mn (f) andE _C ^mn (f) be the errors in best approximation tof in the supremum norm by rational functions of type (m, n) with real and complex coefficients, respectively. We show that form≥n?1≥0 $$\gamma _{mn} = \inf \{ {{E_{mn}^C (f)} \mathord{\left/ {\vphantom {{E_{mn}^C (f)} {E_{mn}^R (f)}}} \right. \kern-\nulldelimiterspace} {E_{mn}^R (f)}}:f \in C[ - 1,1]\} = \tfrac{1}{2}.$$      

Isometric immersions with homothetical Gauss map

Let f:M be an isometric immersion of an m-dimensional Riemannian manifold M into the n-dimensional Euclidean space. Its Gauss map g:MG _m ( ⁿ ) into the Grassmannian G _m ( ⁿ ) is defined by assigning to every point of M its tangent space, considered as a vector subspace of ⁿ . The third fundamental form b of f is the pull-back of the canonical Riemannian metric on G _m ( ⁿ ) via g. In this article we derive a complete classification of all those f (with flat normal bundle) for which the Gauss map g is homothetical; i.e. b is a constant multiple of the Riemannian metric on M. Using these results we furthermore classify all those f (with flat normal bundle) for which the third fundamental form b is parallel w.r.t. the Levi-Civita connection on M.     

Note on the degree of concavity of the determinant on sets of positive definite quadratic forms

An upper bound is determined for the for whichD(f) is a concave function off, wheref ranges over Minkowski-reduced positive definite quadratic forms inn variables with diagonal coefficients unity andD(f) denotes the determinant off In answer to a question of C E Nelson, it is shown thatD(f) is not concave in the casen 4