A Bayesian approach to the estimation of maps between Riemannian manifolds. II: Examples

Let gJ be a smooth compact oriented manifold without boundary, imbedded in a Euclidean space E ^s , and let γ be a smooth map of gJ into a Riemannian manifold Λ. An unknown state θ ∈ gJ is observed via X = θ + ɛξ, where ɛ > 0 is a small parameter and ξ is a white Gaussian noise. For a given smooth prior λ on gJ and smooth estimators g(X) of the map γ we have derived a second-order asymptotic expansion for the related Bayesian risk [3]. In this paper, we apply this technique to a variety of examples. The second part examines the first-order conditions for equality-constrained regression problems. The geometric tools that are utilized in [3] are naturally applicable to these regression problems.     

On the geometry of tangent hyperquadric bundles: CR and pseudoharmonic vector fields

We study the natural almost CR structure on the total space of a subbundle of hyperquadrics of the tangent bundle T(M) over a semi-Riemannian manifold (M, g) and show that if the Reeb vector ξ of an almost contact Riemannian manifold is a CR map then the natural almost CR structure on M is strictly pseudoconvex and a posteriori ξ is pseudohermitian. If in addition ξ is geodesic then it is a harmonic vector field. As an other application, we study pseudoharmonic vector fields on a compact strictly pseudoconvex CR manifold M, i.e. unit (with respect to the Webster metric associated with a fixed contact form on M) vector fields X ε H(M) whose horizontal lift X↑ to the canonical circle bundle S¹ → C(M) → M is a critical point of the Dirichlet energy functional associated to the Fefferman metric (a Lorentz metric on C(M)). We show that the Euler–Lagrange equations satisfied by X^↑ project on a nonlinear system of subelliptic PDEs on M. Mathematics Subject Classifications (2000): 53C50, 53C25, 32V20     

Iteration of closed geodesics in stationary Lorentzian manifolds

Following the lines of Bott in (Commun Pure Appl Math 9:171–206, 1956), we study the Morse index of the iterates of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic γ, we prove the existence of a locally constant integer valued map Λ_γ on the unit circle with the property that the Morse index of the iterated γ ^N is equal, up to a correction term ε_γ∈{0,1}, to the sum of the values of Λ_γ at the N-th roots of unity. The discontinuities of Λ_γ occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincaré map of γ. We discuss some applications of the theory.     

Weighted approximation of random functions

ＷＥＩＧＨＴＥＤＡＰＰＲＯＸＩＭＡＴＩＯＮＯＦＲＡＮＤＯＭＦＵＮＣＴＩＯＮＳＹＵＪＩＡＲＯＮＧＡｂｓｔｒａｃｔ：Ｌｅｔ（Ω,Ａ,Ｐ）ｂｅａｐｒｏｂａｂｉｌｉｔｙｓｐａｃｅ,Ｘ（ｔ,ω）ａｒａｎｄｏｍｆｕｎｃｔｉｏｎｃｏｎｔｉｎｕｏｕｓｉｎｐｒｏｂａｂ...     

Nearly flat almost monotone measures are big pieces of lipschitz graphs

A concentrated (ξ, m) almost monotone measure inR ⁿ is a Radon measure Φ satisfying the two following conditions: (1) Θ ^m (Φ,x)≥1 for every x ∈spt (Φ) and (2) for everyx ∈R ⁿ the ratioexp [ξ(r)]r^−mΦ(B(x,r)) is increasing as a function of r>0. Here ξ is an increasing function such thatlim _r→0-ξ(r)=0. We prove that there is a relatively open dense setReg (Φ) ∋spt (Φ) such that at each x∈Reg(Φ) the support of Φ has the following regularity property: given ε>0 and λ>0 there is an m dimensional spaceW ⊂R ⁿ and a λ-Lipschitz function f from x+W into x+W^‖ so that (100-ε)% ofspt(Φ) ∩B (x, r) coincides with the graph of f, at some scale r>0 depending on x, ε, and λ.     

Manifolds polarized by vector bundles

LetX be a complex projective manifold of dimension n and let ε be an ample vector bundle of rank r. Let also τ = τ (X,ε) = min {t ∈ ℝ : K_X + t det ε is nef} be the nef value of the pair (X, ε). In this paper we classify the pairs (X, ε) such that{ Mathematics Subject Classification (2000)14J60; 14J40; 14E30     

Lie jets and symmetries of prolongations of geometric objects

The Lie jet L _θ λ of a field of geometric objects λ on a smooth manifold M with respect to a field θ of Weil A-velocities is a generalization of the Lie derivative L _v λ of a field λ with respect to a vector field v. In this paper, Lie jets L _θ λ are applied to the study of A-smooth diffeomorphisms on a Weil bundle T ^A M of a smooth manifold M, which are symmetries of prolongations of geometric objects from M to T ^A M. It is shown that vanishing of a Lie jet L _θ λ is a necessary and sufficient condition for the prolongation λ ^A of a field of geometric objects λ to be invariant with respect to the transformation of the Weil bundle T ^A M induced by the field θ. The case of symmetries of prolongations of fields of geometric objects to the second-order tangent bundle T ² M are considered in more detail.     

Asymtotics of <Emphasis Type="Italic">M</Emphasis>-estmation in Non-linear Regression

Consider the standard non-linear regression model y _i = g(x _i , θ ₀)+ε _i , i = 1, ... ,n where g(x, θ) is a continuous function on a bounded closed region X × Θ, θ ₀ is the unknown parameter vector in Θ ⊂ R ^p , {x ₁, x ₂, ... , x _n } is a deterministic design of experiment and {ε₁, ε₂, ... , ε _n } is a sequence of independent random variables. This paper establishes the existences of M-estimates and the asymptotic uniform linearity of M-scores in a family of non-linear regression models when the errors are independent and identically distributed. This result is then used to obtain the asymptotic distribution of a class of M-estimators for a large class of non-linear regression models. At the same time, we point out that Theorem 2 of Wang (1995) (J. of Multivariate Analysis, vol. 54, pp. 227–238, Corrigenda. vol. 55, p. 350) is not correct. This research was supported by the Natural Science Foundation of China (Grant No. 19831010 and grant No. 39930160) and the Doctoral Foundation of China     

Global existence and uniform decay for the coupled Klein-Gordon-Schr?dinger equations

This paper is concerned to the existence, uniqueness and uniform decay for the solutions of the coupled Klein-Gordon-Schr?dinger damped equations where ω is a bounded domain of R ⁿ , n≤ 3, F : R ²→R is a C ¹-function; γ, β; θ are constants such that γ, β > 0 and 1 ≤ 2θ≤ 2. Received January 1999 – Accepted October 1999     

Lip Automorphism Germ and Lip Automorphism

Let X be a metric space, ε^n(X) be the standard trivial Lip n-bundle over X, and Φ be a Lip automorphism germ of ε^n(X). This paper proves that there is a Lip automorphism Φ‘ of ε^n(X) such that the germ of Φ‘ is Φ.     

On Vertices, focal curvatures and differential geometry of space curves

The focal curve of an immersed smooth curve γ : θ ↦ γ (θ), in Euclidean space ℝ^m+1, consists of the centres of its osculating hyperspheres. This curve may be parametrised in terms of the Frenet frame of γ (t, n₁, . . . , n_m), as C_γ (θ) = (γ +c₁n₁+ c₂n₂ + • • • + c_mn_m)(θ), where the coefficients c₁, . . . , c_m-1 are smooth functions that we call the focal curvatures of γ . We discovered a remarkable formula relating the Euclidean curvatures κ_i , i = 1, . . . ,m, of γ with its focal curvatures. We show that the focal curvatures satisfy a system of Frenet equations (not vectorial, but scalar!). We use the properties of the focal curvatures in order to give, for ℓ = 1, . . . ,m, necessary and sufficient conditions for the radius of the osculating ℓ-dimensional sphere to be critical. We also give necessary and sufficient conditions for a point of γ to be a vertex. Finally, we show explicitly the relations of the Frenet frame and the Euclidean curvatures of γ with the Frenet frame and the Euclidean curvatures of its focal curve C_γ.     

Nearly <Emphasis Type="Italic">H</Emphasis>-closed spaces

For a subset M of a topological space X, the θ-closure {ie626-01} M is defined as the set of all x ∈ X such that any closed neighborhood of x intersects M. A Urysohn space X is said to be U-closed if, whenever X ∪ {ξ} is a Urysohn space obtained from X by adding one point ξ, the point ξ is isolated in X ∪ {ξ}. The θ-closure operator is applied to study compactness-type properties of (weakly) U-and H-closed and closed-hereditarily U-and H-closed spaces. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 48, General Topology, 2007.     

Hypothesis testing for the common mean of two normal distributions in the presence of an indifference zone

Summary LetX ₁,...,X _m andY _t,...,Y be independent, random samples from populations which are N(θ,σ _x ² ) and N(θ,σ _y ² ), respectively, with all parameters unknown. In testingH ₀:θ=0 againstH ₁:θ≠0, thet-test based upon either sample is known to be admissible in the two-sample setting. If, however, one testsH ₀ againstH ₁:|θ|≧ε>0, with ε arbitrary, our main results show: (i) the construction of a test which is better than the particulart-test chosen, (ii) eacht-test is admissible under the invariance principle with respect to the group of scale changes, and (iii) there does not exist a test which simultaneously is better than botht-tests.     

Injectivity of differentiable maps ℝ2 → ℝ2 at infinity

The main result given in Theorem 1.1 is a condition for a map X, defined on the complement of a disk D in ℝ² with values in ℝ², to be extended to a topological embedding of ℝ², not necessarily surjective. The map X is supposed to be just differentiable with the condition that, for some ε > 0, at each point the eigenvalues of the differential do not belong to the real interval (-ε,∞). The extension is obtained by restricting X to the complement of some larger disc. The result has important connections with the property of asymptotic stability at infinity for differentiable vector fields.     

Density estimation for a class of stationary nonlinear processes

Let {X _t ;t∈ℤ be a strictly stationary nonlinear process of the formX _t =ε _t +∑ _r=1 ^∞ W _rt , whereW _rt can be written as a functiong _r (ε _t−1,...ε _t-r-q ), {ε _t ;t∈ℤ is a sequence of independent and identically distributed (i.i.d.) random variables withE|ε₁| ^g < ∞ for some γ>0 andq≥0 is fixed integer. Under certain mild regularity conditions ofg _r and {ε _t } we then show thatX ₁ has a density functionf and that the standard kernel type estimator baded on a realization {X ₁,...,X _n } from {X _t } is, asymptotically, normal and converges a.s. tof(x) asn→∞. The research of this author was partially carried out while he was a research scholar, on a sabbatical leave, at the Department of Statistics and Probability, Michigan State University.     

Preservation of the injectivity of the Gauss map for general projections

Let X be a smooth irreducible quasi-projective variety of dimension n in P ^N with N ≥ 2n + 2. Let γ be its Gauss map, let be the embedding obtained from the general projection in P ^N and let γ′ be its Gauss map. We say that the general projection preserves the injectivity of the Gauss map if γ(Q) ≠ γ(Q′) implies γ′(Q) ≠ γ′ (Q′). We prove that this property holds in the following cases: N≥ 3n + 1; N ≥ 3n with n ≥ 2; N ≥ 3n−1 with n ≥ 4 and X does not contain a linear (n−1)-space. In case N = 3n−1 and X does contain a linear (n−1)-space (such smooth varieties exist) then the general projection does not preserver the injectivity of the Gauss map. This shows that there does not exist a straightforward kind of Bertini theorem for properties related to the Gauss map. The author is affiliated with the University at Leuven as a research fellow. This paper belongs to the FWO-project G.0318.06.     

Simple sheaves along dihedral Lagrangians

We prove the unicity of a complex of sheavesF whose microsupport is carried by a “dihedral” Lagrangian Λ ofT ^* X (X=a real manifold) and which is simple with a prescribed shift at a regular point of Λ. Our method consists in reducing Λ, by a real contact transformation, to the conormal bundle to aC ¹-hypersuface, and then in using [K-S 1, Prop. 6.2.1] in the variant of [D'A-Z 1]. This is similar to [Z 2] but more general, since complex contact transformations and calculations of shifts are not required. We then consider the case of a complex manifoldX, and obtain some vanishing theorems for the complex of “microfunctions along Λ” similar to those of [A-G], [A-H], [K-S 1] (cf. also [D'A-Z 3 5], [Z 2]).     

Symmetric block bases of sequences with large average growth

We show that if 0<ε≦1, 1≦p<2 andx ₁, …,x _n is a sequence of unit vectors in a normed spaceX such thatE ‖∑ _l ⁿ ε_i x _l‖≧n ^1/p, then one can find a block basisy ₁, …,y _m ofx ₁, …,x _n which is (1+ε)-symmetric and has cardinality at leastγn ^2/p-1(logn)⁻¹, where γ depends on ε only. Two examples are given which show that this bound is close to being best possible. The first is a sequencex ₁, …,x _n satisfying the above conditions with no 2-symmetric block basis of cardinality exceeding 2n ^2/p-1. This sequence is not linearly independent. The second example is a sequence which satisfies a lowerp-estimate but which has no 2-symmetric block basis of cardinality exceedingCn ^2/p-1(logn)^4/3, whereC is an absolute constant. This applies when 1≦p≦3/2. Finally, we obtain improvements of the lower bound when the spaceX containing the sequence satisfies certain type-condition. These results extend results of Amir and Milman in [1] and [2]. We include an appendix giving a simple counterexample to a question about norm-attaining operators.     

A necessary and sufficient condition for a lower bound for fourth-order pseudodifferential operators

We give necessary and sufficient conditions for the lower bound {fx55-01} to hold for any compact setK ⊂X, an open set ofR ⁿ , andP =P* ∃ ψ _phg ⁴ (X) with p(x, ξ) ~ q ₂ ² + p₃ + p₂ + ..., q₂ beingtransversally elliptic with respect to the characteristic manifold Σ =q ₂ ^-1 (0).     

Multiparametric estimating equations

Summary Let {p(x, θ): θ∈Θ} be a family of densities where θ=(θ₁,θ₂), being θ₁ ∈ Θ₁ ak-dimensional parameter of interest, θ₂ ∈ Θ₂ a nuisance parameter and Θ=Θ₁×Θ₂. To estimate θ₁, vector estimating equations g(x,θ₁)=(g₁(x,θ₁),...,g_k(x,θ₁))=0 are considered. The standardized form of g(x,θ₁) is defined as g_s=(E_θ(∂g/∂θ′₁))⁻¹g. Then, within the classG ₁ of unbiased equations (i.e. satisfying E_θ(g)=0 (θ∈Θ)), an equationg ^*=0 is said to be optimum if the covariance matrices ofg _s andg _s ^* are such that is non-negative definite for allg∈ G ₁ and θ∈Θ. Sufficient conditions for optimality are discussed and, in particular, conditions for the optimality of the maximum conditional likelihood equation are analyzed. Special attention is given to non-regular cases. In addition, measures of the information about θ₁ contained in an estimating equation are presented and a Rao-Blackwell theorem is given. CIENES