ℝ-Conformal invariants of curves

In this paper we describe the structure of the algebra of scalar differential invariants of curves in a plane with Euclidean or Minkowski metric with respect to ?-conformal transformations.     

Extreme norms on ℝ n

LetN ¹,N be thel ¹ andl norms on ⁿ . We denote by the set of all normsN on ⁿ such thatN NN ¹. The aim of the paper is to present a characterization of the extreme points of .This paper has been written while the author was a research fellow of the Alexander von Humboldt-Stiftung at the Mathematisches Institut der Eberhardt-Karls-Universität, Tübingen.     

Linear structures on the collections of minimal surfaces inℝ 3 andℝ 4

The collection of minimal herissons in ³ is endowed with a vector space structure. The existence of this structure is related to the fact that null curves inC ³ are described by a single map from the étalé space of the sheaf of germs of holomorphic sections of the line bundle of degree 2 over ₁ to C³, which islinear on stalks. There is an analogous construction for null curves inC ⁴. This gives a similar class of minimal surfaces in ⁴.     

Modified Mini finite element for the Stokes problem in ℝ2 or ℝ3

We analyze a modified version of the Mini finite element (or the Mini* finite element) for the Stokes problem in ℝ² or ℝ³. The cross‐grid element of order one in ℝ³ is also analyzed. The stability is verified with the aid of the macroelement technique introduced by Stenberg. Each of these methods converges with first order in h as the Mini element does. Numerical tests are given for the Mini* element in comparison with the Mini element when Ω is a unit square on ℝ². This revised version was published online in June 2006 with corrections to the Cover Date.     

On Borsuk's Problem in ℝ3

Mathematical Notes -     

Isotrope Kurventheorie aufS 2 × ℝ

In this note we define an isotropic metric on the threedimensional manifoldS ² × . This metric will allow an symmetric riemannian connection , wich will be used to do differential geometry on S² × . We develope theory of curves onS ² × and show some relations to the theory of curves of threedimensional isotropic spaceI ₃.     

Nonlinear Least Squares in ℝ N

Recent research and new paradigms in mathematics, engineering, and science assume nonlinear signal models of the form ?=∪ _i∈I V _i consisting of a union of subspaces V _i instead of a single subspace ?=V. These models have been used in sampling and reconstruction of signals with finite rate of innovation, the Generalized Principle Component Analysis and the subspace segmentation problem in computer vision, and problems related to sparsity, compressed sensing, and dictionary design. In this paper, we develop an algorithm that searches for the best nonlinear model of the form ?=∪ _i=1 ^l V _i ?? ^N that is optimally compatible with a set of observations ?={f ₁,…,f _m }?? ^N . When l=1 this becomes the classical least squares optimization. Thus, this problem is a nonlinear version of the least squares problem. We test our algorithm on synthetic data as well as images.     

Mixed finite elements in ℝ3

Summary We present here some new families of non conforming finite elements in ³. These two families of finite elements, built on tetrahedrons or on cubes are respectively conforming in the spacesH(curl) andH(div). We give some applications of these elements for the approximation of Maxwell's equations and equations of elasticity.First, we introduce some notations K is a tetrahedron or a cube, thevolume of which is - K is its boundary - f is a face ofK, thesurface of which is - a is an edge, the length of which is - L ² (K) is the usual Hilbert space of square integrable functions defined onK - H ^m (K) {L ²(K); L ²(K); ||m}, where =(₁, ₂, ₃) is a multi-index; ||=₁+₂+₃ - curlu u, (defined by using the distributional derivative) foru=(u ₁,u ₂,u ₃);u _iL ² (K) - H(curl) {u(L ² (K))³; curlu(L ² (K)) ³} - divu ·u - H(div) {u(L ² (K)) ³; divuL ² (K)} - D ^k u is thek-th differential operator associated tou, which is a (k+1)-multilinear operator acting on ³ - k is an index - _k is the linear space of polynomials, the degree of which is less or equal tok - _k is the group of all permutations of the set {1, 2, ...,k} - c orc will stand for any constant depending possibly on      

On Urysohn’s ℝ-Tree

In the short note of 1927, Urysohn constructed the metric space R that is nowhere locally separable. There is no publication with indications that R is a (noncomplete) ?-tree that has valency c at each point. The author in 1989, as well as Polterovich and Shnirelman in 1997, constructed ?-trees isometric to R unaware of the paper by Urysohn. In this paper the author considers various constructions of the ?-tree R and of the minimal complete ?-tree of valency c including R, as well as the characterizations of ?-trees, their properties, and connections with ultrametric spaces.

     

Heat kernel estimates for an operator with unbounded diffusion coefficients in ℝ and ℝ2

We prove heat kernel bounds for the operator $(1+|x|^{2})^{\frac{\alpha}{2}}\Delta$ in ? ^N , N=1,2. The equivalence between Nash and Sobolev inequalities and ultracontractivity will be often used.     

Nilfactors of ℝ m and configurations in sets of positive upper density in ℝ m

We use ergodic theoretic tools to solve a classical problem in geometric Ramsey theory. LetE be a measurable subset of ℝ ^m , with . LetV = {0,v ₁,...,v _k} ⊂ ℝ^m. We show that fort large enough, we can find an isometric copy oftV arbitrarily close toE. This is a generalization of a theorem of Furstenberg, Katznelson and Weiss [FuKaW] showing a similar property form=k=2.     

Parabolic NTA domains in ℝ2

We show that each connected component of the boundary of a parabolic NTA domain in ?² is given by a graph. We then apply this observation to classify blowup solutions in ?² to a free boundary problem for caloric measure first considered by Hofmann, Lewis and Nyström [16 Hofmann, S., Lewis, J.L., Nyström, K. (2004). Caloric measure in parabolic flat domains. Duke Math. J. 122(2):281–346. MR 2053754 (2005e:35092)[Crossref], [Web of Science ®] , [Google Scholar]].     

Anisotropic Meshless Frames on ℝ n

We present a construction of anisotropic multiresolution and anisotropic wavelet frames based on multilevel ellipsoid covers (dilations) of ℝ ⁿ . The wavelets we construct are C ^∞ functions, can have any prescribed number of vanishing moments and fast decay with respect to the anisotropic quasi-distance induced by the cover. The dual wavelets are also C ^∞, with the same number of vanishing moments, but with only mild decay with respect to the quasi-distance. An alternative construction yields a meshless frame whose elements do not have vanishing moments, but do have fast anisotropic decay.     

Foliations and global injectivity in ℝ n

Let F: ? ⁿ → ? ⁿ be a polynomial local diffeomorphism and let S _F denote the set of not proper points of F. The Jelonek’s real Jacobian Conjecture states that if codim(S _F ) ≥ 2, then F is bijective. We prove a weak version of such conjecture establishing the sufficiency of a necessary condition for the bijectivity of F.     

Classification of Refinable Splines in ℝ d

A refinable spline in ℝ ^d is a compactly supported refinable function whose support can be decomposed into simplices such that the function is a polynomial on each simplex. The best-known refinable splines in ℝ ^d are the box splines. Refinable splines play a key role in many applications, such as numerical computation, approximation theory and computer-aided geometric design. Such functions have been classified in one dimension in Dai et al. (Appl. Comput. Harmon. Anal. 22(3), 374–381, 2007), Lawton et al. (Comput. Math. 3, 137–145, 1995). In higher dimensions Sun (J. Approx. Theory 86, 240–252, 1996) characterized those splines when the dilation matrices are of the form A=mI, where m∈ℤ and I is the identity matrix. For more general dilation matrices the problem becomes more complex. In this paper we give a complete classification of refinable splines in ℝ ^d for arbitrary dilation matrices A∈M _d (ℤ).     

Acute triangulations of polyhedra and ℝ N

We study the problem of acute triangulations of convex polyhedra and the space ? ⁿ . Here an acute triangulation is a triangulation into simplices whose dihedral angles are acute. We prove that acute triangulations of the n-cube do not exist for n≥4. Further, we prove that acute triangulations of the space ? ⁿ do not exist for n≥5. In the opposite direction, in ?³, we present a construction of an acute triangulation of the cube, the regular octahedron and a non-trivial acute triangulation of the regular tetrahedron. We also prove nonexistence of an acute triangulation of ?⁴ if all dihedral angles are bounded away from π/2.     

Hitting Simplices with Points in ℝ3

The so-called first selection lemma states the following: given any set P of n points in ℝ ^d , there exists a point in ℝ ^d contained in at least c _d n ^d+1−O(n ^d ) simplices spanned by P, where the constant c _d depends on d. We present improved bounds on the first selection lemma in ℝ³. In particular, we prove that c ₃≥0.00227, improving the previous best result of c ₃≥0.00162 by Wagner (On k-sets and applications. Ph.D. thesis, ETH Zurich, 2003). This makes progress, for the three-dimensional case, on the open problems of Bukh et al. (Stabbing simplices by points and flats. Discrete Comput. Geom., 2010) (where it is proven that c ₃≤1/4⁴≈0.00390) and Boros and Füredi (The number of triangles covering the center of an n-set. Geom. Dedic. 17(1):69–77, 1984) (where the two-dimensional case was settled).     

Degenerate homogeneous affine surfaces in ℝ3

We study degenerate homogeneous affine surfaces in ³. It is proved that such a surface is either an open part of a plane, a cylinder on an ellipse, parabola or hyperbola or of the surface given byxz – 1/2y ²=0.