Further characterization of ellipsoids

In this paper the following result is proved: Let K be a planar convex body and let γ be a differentiable closed convex curve. If for every point p in the interior of K the set of the midpoints, of all the chords through p, are forming a curve α(p) which is directly homothetic to γ, then K is an ellipse.     

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.     

Propagation of polarization in elastodynamics with residual stress and travel times

 We show that knowing the displacement-to-traction map associated to the equations of isotropic elastodynamics with residual stress we can determine the lens maps of compressional and shear waves. We derive several consequences of this for the inverse problem of determining the residual stress and the Lamé parameters from the displacement-to-traction map. Received: 6 December 2001 / Revised version: 29 October 2002 / Published online: 8 April 2003 RID="⋆" ID="⋆" The author thanks the Department of Mathematics at the University of Washington for its hospitality during his visit in fall 2000. RID="⋆⋆" ID="⋆⋆" Partly supported by NSF grant DMS-0070488 and a John Simon Guggenheim fellowship. The author also thanks MSRI for partial support and for providing a very stimulating environment during the inverse problems program in fall 2001.     

Some bendings of a long cylinder

Elementary tools are applied to describe piecewise-linear isometric embeddings of cylindrical surfaces in ℝ³. Let T² be a flat torus, let γ⊂T² be the shortest closed geodesic of length l_o, and let k be a fixed positive integer. We assume that if l is the length of any closed geodesic on T² which is homotopic neither to γ nor to any power of γ, then l>kl₀. It is shown how to embed T² in ℝ³ if k is sufficiently large. The same problem is solved for a flat skew torus T². It is also shown that if a knot of arbitrary type in ℝ³ is fixed and k is sufficiently large, then T² can be isometrically embedded in ℝ³ as a tube knotted according to the type of fixed knot. Bibliography; 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 66–83. Translated by S. Yu. Pilyugin.     

Free limits of forcing and more on Aronszajn trees

We prove that the Souslin Hypothesis does not imply “every Aron. (=Aronszajn) tree is special”. For this end we introduce variants of the notion “special Aron. tree”. We also introduce a limit of forcings bigger than the inverse limit, and prove it preserves properness and related notions not less than inverse limit, and the proof is easier in some respects. The result was announced in [9]. The author thanks Uri Avraham for detecting many errors.     

Completions for partially ordered semigroups

A standard completion γ assigns a closure system to each partially ordered set in such a way that the point closures are precisely the (order-theoretical) principal ideals. If S is a partially ordered semigroup such that all left and all right translations are γ-continuous (i.e., Y∈γS implies {x∈S:y·x∈Y}∈γS and {x∈S:x·y∈Y}∈γS for all y∈S), then S is called a γ-semigroup. If S is a γ-semigroup, then the completion γS is a complete residuated semigroup, and the canonical principal ideal embedding of S in γS is a semigroup homomorphism. We investigate the universal properties of γ-semigroup completions and find that under rather weak conditions on γ, the category of complete residuated semigroups is a reflective subcategory of the category of γ-semigroups. Our results apply, for example, to the Dedekind-MacNeille completion by cuts, but also to certain join-completions associated with so-called “subset systems”. Related facts are derived for conditional completions. A first draft of this paper by the second author, containing parts of Section 2, was received on August 9, 1985.     

On Hom-spaces of tame algebras

Let Λ be a finite dimensional algebra over an algebraically closed field k and Λ has tame representation type. In this paper, the structure of Hom-spaces of all pairs of indecomposable Λ-modules having dimension smaller than or equal to a fixed natural number is described, and their dimensions are calculated in terms of a finite number of finitely generated Λ-modules and generic Λ-modules. In particular, such spaces are essentially controlled by those of the corresponding generic modules. The first author thanks the support of project “43374F” of Fondo Sectorial SEP-Conacyt. Y. Zhang thanks the support of Important project 10331030 of Natural Science Foundation of China.     

On the renewal of a Wiener field on a plane with the use of its values on closed curves

Forw(u, v), (u, v)∉ γ (here,w(x, y), x≥0, y≥0, is a Wiener field and γ is a certain closed curve on a plane), we construct the best mean-square estimate on the basis of the values ofw(x, y) for (x, y)∈ γ. We also calculate the error of this estimate. Donetsk University, Donetsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 744–752, June, 1999.     

Polygons inscribed in a closed curve and a three-dimensional convex body

Here are samples of results obtained in the paper. Let γ be a centrally symmetric closed curve in ℝ ⁿ that does not contain its center of symmetry, O. Then γ is circumscribed about a square (with center O), as well as about a rhombus (also with center O) whose vertices split γ into parts of equal length. If n is odd, then there is a centrally symmetric equilateral 2n-link polyline inscribed in γ and lying in a hyperplane. Let K ⊂ ℝ³ be a convex body, and let x ∈ (0; 1). Then K is circumscribed about an affine-regular pentagonal prism P such that the ratio of the lateral edge l of P to the longest chord of K parallel to l is equal to x. Bibliography: 7 titles.     

解变分不等式问题的混合方法

By using Fukushima‘s differentiable merit function,Taji,Fukushima and Ibaraki have given a globally convergent modified Newton method for the strongly monotone variational inequality problem and proved their method to be quadratically convergent under certain assumptions in 1993. In this paper a hybrid method for the variational inequality problem under the assumptions that the mapping F is continuously differentiable and its Jacobian matrix F(x) is positive definite for all x∈S rather than strongly monotone and that the set S is nonempty, polyhedral,closed and convex is proposed. Armijo-type line search and trust region strategies as well as Fukushima‘s differentiable merit function are incorporated into the method. It is then shown that the method is well defined and globally convergent and that,under the same assumptions as those of Taji et al. ,the method reduces to the basic Newton method and hence the rate of convergence is quadratic. Computational experiences show the efficiency of the proposed method.     

A new method for smoothing surfaces and computing hermite interpolants

Let Ω be a rectangular bounded domain of a plane equipped with a rectangular partition Δ. Assume a piecewise bivariate function that is differentiable up to order (k,l) except at the knots of Δ, where it is less differentiable. In this paper, we introduce a new method for smoothing the above function at the knots. More precisely, we describe algorithms allowing one to transform it into another function that will be differentiable up to order (k,l) in the whole domain Ω. Then, as an application of this method, we give a recursive computation of tensor product Hermite spline interpolants. To illustrate our results, some numerical examples are presented. AMS subject classification (2000)  41A05, 41A15, 65D05, 65D07, 65D10     

On the best approximation in the mean of continuous functions by polynomials,which are majorized by a given function

We discuss the problem of the best approximation in the mean of functions, which are continuous on a segment, by the polynomials in a differentiable Chebyshev system that do not exceed a given differentiable function. We prove that the extremal polynomial is unique and develop its characteristic properties.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 365–374, April, 1972.The author thanks A. L. Garkavi for posing the problem.     

A universal mapping characterization of the completion by cuts

Closed maps of lattices are defined to be those satisfying the condition that the inverse image of closed ideals are closed ideas. Residuated maps are closed and closed maps are complete-join homomorphisms. The natural embeddingj of a lattice into its completion by cuts is a closed map. For every closed map ϕ from a latticeL into a complete latticeM, there exists a unique closed map φ* from the completion by cutsL intoM such that φ*j=φ. This characterizes the completion by cuts. Presented by R. P. Dilworth.     

Large sample properties of the mle and mcle for the natural parameter of a truncated exponential family

Summary Consider a truncated exponential family of absolutely continuous distributions with natural parameter θ and truncation parameter γ. Strong consistency and asymptotic normality are shown to hold for the maximum likelihood and maximum conditional likelihood estimates of θ with γ unknown. Moreover, these two estimates are also shown to have the same limiting distribution, coinciding with that of the maximum likelihood estimate for θ when γ is assumed to be known.     

Teichmüller spaces and holomorphic motions

The subject of holomorphic motions over the open unit disc has found important applications in complex dynamics. In this paper, we study holomorphic motions over more general parameter spaces. The Teichmüller space of a closed subset of the Reimann sphere is shown to be a universal parameter space for holomorphic motions of the set over a simply connected complex Banach manifold. As a consequence, we prove a generalization of the “Harmonic γ-Lemma” of Bers and Royden. We also study some other applications.     

On the global convergence of path-following methods to determine all solutions to a system of nonlinear equations

In this paper we prove a general theorem stating a sufficient condition for the inverse image of a point under a continuously differentiable map from ℝ ⁿ to ℝ ^k to be connected. This result is applied to the trajectories generated by the Newton flow. Several examples demonstrate the applicability of the results to nontrivial problems. This work was supported by the Deutsche Forschungsgemeinschaft.     

Constrained open mapping theorem with applications

We prove a generalization of Graves? Open Mapping Theorem for a class of mappings which can be approximated at a reference point by a set-valued one having particular properties. The nonlinear mapping is restricted to a closed convex subset of a Banach space. We apply the results to derive necessary and sufficient conditions ensuring the existence of a differentiable selection for the inverse mapping. A slight generalization of a sufficient condition by J. Klamka on so-called constrained exact local controllability of nonlinear and semi-linear dynamic systems is also proved.     

Uniform quotient mappings of the plane with non-discrete point inverses

We construct uniform quotient mappings of the plane which have points with non-discrete inverse and good modulus of continuity. In particular, we find a monotone uniform quotient mappingf: ℝ² → ℝ² having a point with nontrivial inverse such that lim _r→0[Ω(r)/r ^γ]=0 for anyγ<1/2, where Ω is the modulus of continuity off. Supported by DFG grant Hi 584/2-1.     

Directional derivatives of difference functions of nonsmooth max-functions

In this paper, we give an upper estimate for the Clarke-Rockafellar directional derivatives of a function of the form f - g, where f, g are max-functions defined by locally Lipschitz but not necessarily differentiable functions on a closed convex set in a Euclidean space. As an application, we give a sufficient condition for f - g to have an error bound.     

Singularities of o‐minimal Peano derivatives

The notion of Peano differentiability generalizes the differentiability in the usual sense to higher order. Peano differentiable functions have derivatives which are sometimes differentiable or continuous or not even locally bounded. We give a complete characterisation of the sets in which Peano differentiable functions which are definable in an o‐minimal expansion of a real closed field are continuously differentiable. Thereby, we also distinguish between several kinds of discontinuities (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)