Constant angle surfaces in $ {\Bbb S}^2\times {\Bbb R} $

In this article we study surfaces in for which the unit normal makes a constant angle with the -direction. We give a complete classification for surfaces satisfying this simple geometric condition.     

Singular oscillatory integrals on {\mathbb{R}^n}

Let ${\mathcal{P}_{d,n}}Let P_d,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRⁿ{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? P_d,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that

supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      3. Natural Tiling, Lattice Tiling and Lebesgue Measure of Integral Self-Affine Tiles   总被引：1，自引：0，他引：1   Gabardo  Jean-Pierre; Yu  Xiaojiang 《Journal London Mathematical Society》2006,74(1):184-204 In the existing theory of self-affine tiles, one knows thatthe Lebesgue measure of any integral self-affine tile correspondingto a standard digit set must be a positive integer and everyintegral self-affine tile admits some lattice Zn as a translationtiling set of Rn. In this paper, we give algorithms to evaluatethe Lebesgue measure of any such integral self-affine tile Kand to determine all of the lattice tilings of Rn by K. Moreover,we also propose and determine algorithmically another type oftranslation tiling of Rn by K, which we call natural tiling.We also provide an algorithm to decide whether or not Lebesguemeasure of the set K (K+j), j–Zn, is strictly positive.      4. Covering and packing in ${\Bbb Z}^n$ and ${\Bbb R}^n$, (I)      Wolfgang M. Schmidt  David M. Tuller 《Monatshefte für Mathematik》2008,153(3):265-281 Given a finite subset of an additive group such as or , we are interested in efficient covering of by translates of , and efficient packing of translates of in . A set provides a covering if the translates with cover (i.e., their union is ), and the covering will be efficient if has small density in . On the other hand, a set will provide a packing if the translated sets with are mutually disjoint, and the packing is efficient if has large density. In the present part (I) we will derive some facts on these concepts when , and give estimates for the minimal covering densities and maximal packing densities of finite sets . In part (II) we will again deal with , and study the behaviour of such densities under linear transformations. In part (III) we will turn to . Authors’ address: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395, USA The first author was partially supported by NSF DMS 0074531.      5. Disk-Like Self-Affine Tiles in R 2      C. Bandt  Y. Wang 《Discrete and Computational Geometry》2001,26(4):591-601 We give simple necessary and sufficient conditions for self-affine tiles in R 2 to be homeomorphic to a disk. Received October 10, 2000, and in revised form February 16, 2001, and April 25, 2001. Online publication July 25, 2001.      6. On modified quaternionic analysis in $ {\Bbb R}^3 $      Sirkka-Liisa Eriksson-Bique  Heinz Leutwiler 《Archiv der Mathematik》1998,70(3):228-234       7. Evolution of hypersurfaces in $ {\Bbb R}^N$ by Gaussian curvature      Pierangelo Marcati  Manuela Molinari 《NoDEA : Nonlinear Differential Equations and Applications》1999,6(2):119-132 In this paper we first prove short time existence of a classical solution for the problem which describes the evolution by Gaussian curvature of a strictly convex hypersurface in . Then we give a proof of the existence of a viscosity solution for this problem in such a way as to define a generalized motion existing for each time. Received November 24, 1997      8. Topological Structures in Colombeau Algebras: Investigation of the Duals of {\cal G}_{{\rm c}}(\Omega), {\cal G}(\Omega) and {\cal G}_{{\cal S}}({\Bbb R}^n)      Claudia Garetto 《Monatshefte für Mathematik》2005,146(3):203-226       9. The Second Variational Formula for Laguerre Minimal Hypersurfaces in {\mathbb {R}^n}      Yuping Song 《Results in Mathematics》2013,63(3-4):985-998 Li and Wang (Manuscr Math 122(1):73–95, 2007) presented Laguerre geometry for hypersurfaces in ${\mathbb{R}^{n}}$ and calculated the first variational formula of the Laguerre functional by using Laguerre invariants. In this paper we present the second variational formula for Laguerre minimal hypersurfaces. As an application of this variational formula we give the standard examples of Laguerre minimal hypersurfaces in ${\mathbb{R}^{n}}$ and show that they are stable Laguerre minimal hypersurfaces. Using this second variational formula we can prove that a surface with vanishing mean curvature in ${\mathbb{R}^{3}_{0}}$ is Laguerre equivalent to a stable Laguerre minimal surface in ${\mathbb{R}^{3}}$ under the Laguerre embedding. This example of stable Laguerre minimal surface in ${\mathbb{R}^{3}}$ is different from the one Palmer gave in (Rend Mat Appl 19(2):281–293, 1999).      10. Stability and Computation of Topological Invariants of Solids in ${\Bbb R}^n$      Frederic Chazal  Andre Lieutier 《Discrete and Computational Geometry》2007,37(4):601-617 In this work one proves that under quite general assumptions one can deduce the topology of a bounded open set in from an approximation of it. For this, one introduces the weak feature size (wfs) that extends for nonsmooth objects the notion of local feature size. Our results apply to open sets with positive wfs. This class includes subanalytic open sets which cover many cases encountered in practical applications. The proofs are based upon the study of distance functions to closed sets and their critical points. The notion of critical point is the same as the one used in riemannian geometry [22], [9], [20] and nonsmooth analysis [10]. As an application, one gives a way to compute the homology groups of open sets from noisy samples of points on their boundary.      11. Hypoelliptic Dunkl Convolution Equations in the Space of Distributions on {\Bbb R}^d      Khalifa Trimèche 《Journal of Fourier Analysis and Applications》2006,12(5):517-542 In this article we define and study the Dunkl convolution product and the Dunkl transform on spaces of distributions on By using the main results obtained, we study the hypoelliptic Dunkl convolution equations in the space of distributions.      12. A dual algorithm for the minimum covering weighted ball problem in {\mathbb{R}^n}      P. M. Dearing  Andrea M. Smith 《Journal of Global Optimization》2013,55(2):261-278 The nonlinear convex programming problem of finding the minimum covering weighted ball of a given finite set of points in ${\mathbb{R}^n}$ is solved by generating a finite sequence of subsets of the points and by finding the minimum covering weighted ball of each subset in the sequence until all points are covered. Each subset has at most n + 1 points and is affinely independent. The radii of the covering weighted balls are strictly increasing. The minimum covering weighted ball of each subset is found by using a directional search along either a ray or a circular arc, starting at the solution to the previous subset. The step size is computed explicitly at each iteration.      13. Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on $ {Bbb R}^n $      Enzo Mitidieri  Stanislav I. Pohozaev 《Journal of Evolution Equations》2001,1(2):189-220 No Abstract        14. Radial symmetry results for systems of integral equations on {\Omega\subset {\mathbb{R}}^n}      Xiaotao Huang  Dongsheng Li  Lihe Wang 《manuscripta mathematica》2012,137(3-4):317-330 The main purpose of this paper is to investigate positive solutions of the following system of integral equations with weight Riesz potential on Ω: $$\left\{\begin{array}{ll} u(x)=\int\limits_{\Omega}{|x|^{-\alpha}|x-y|^{-\mu}|y|^{-\beta}}{u^a(y)v^b(y)}dy,\\ v(x)=\int\limits_{\Omega}{|x|^{-\gamma}|x-y|^{-\nu}|y|^{-\kappa}}{u^c(y)v^d(y)}dy,\\ \end{array} \right.$$ where a, b, c, d, α, β, γ, κ, μ, ν are constants. With the method of moving planes, we establish the symmetry of both the solution and the bounded C 1 domain Ω if u and v are constants on ?Ω. Moreover, we also give a symmetry result for systems of integral equations with weight Riesz potential on exterior domains.      15. Linear Fractional Transformation Methods in $ {\shadC}^n $      《复变函数与椭圆型方程》2012,57(1):69-81 We consider the generalization of linear fractional transformations of the plane to $ {\shadC}^n $ . Analogs of the one-variable theory are developed including fixed point sets and points of symmetry. The domains in $ {\shadC}^n $ that are images of the ball under these transformations are found. Finally, we see some examples where classical fixed point results follow from this theory in a natural way.      16. On the moving frame of a conformal map from 2-disk into {\mathbb{R}^n}      Yuxiang Li  Yong Luo  Hongyan Tang 《Calculus of Variations and Partial Differential Equations》2013,46(1-2):31-37 Let f be a conformal map from the 2-disk into ${\mathbb{R}^n}$ . We prove that the image f(B) have a normal tangent vector basis (e 1, e 2) with ${\|d(e_{1}, e_{2})\|_{L^2(B)} \leq C\|A\|_{L^2(B)}}$ when the total Gauss curvature ${\int_B |K_{f}| d\mu_f < 2\pi}$ .      17. A Single Cell in an Arrangement of Convex Polyhedra in ${\Bbb R}^3$      Esther Ezra  Micha Sharir 《Discrete and Computational Geometry》2007,37(1):21-41 We show that the combinatorial complexity of a single cell in an arrangement of k convex polyhedra in 3-space having n facets in total is , for any , thus settling a conjecture of Aronov et al. We then extend our analysis and show that the overall complexity of the zone of a low-degree algebraic surface, or of the boundary of an arbitrary convex set, in an arrangement of k convex polyhedra in 3-space with n facets in total, is also , for any . Finally, we present a deterministic algorithm that constructs a single cell in an arrangement of this kind, in time , for any .      18. Existence and bifurcation of positive solutions of a semilinear elliptic problem on {\Bbb R}^N      Josè L. Gámez 《NoDEA : Nonlinear Differential Equations and Applications》1997,4(3):341-357 Existence of positive solutions of a semilinear elliptic problem on , arising from population genetics is obtained. We use an approximation procedure, by accumulation of branches of solutions of the problem stated on balls of sufficiently large radia. Received January 19, 1996      19. Set-theoretic complete intersection monomial curves in $${\mathbb{P}^n}$$      Tran?Hoai?Ngoc?NhanEmail author 《Archiv der Mathematik》2012,99(1):37-41 In this paper, we give a sufficient numerical criterion for a monomial curve in a projective space to be a set-theoretic complete intersection. Our main result generalizes a similar statement proven by Keum for monomial curves in three-dimensional projective space. We also prove that there are infinitely many set-theoretic complete intersection monomial curves in the projective n?space for any suitably chosen n ? 1 integers. In particular, for any positive integers p, q, where gcd(p, q) = 1, the monomial curve defined by p, q, r is a set-theoretic complete intersection for every \({r \geq pq( q - 1)}\).      20. Sharp higher-order Sobolev inequalities in the hyperbolic space {\mathbb{H}^n}      Genqian Liu 《Calculus of Variations and Partial Differential Equations》2013,47(3-4):567-588 In this paper, we obtain the sharp k-th order Sobolev inequalities in the hyperbolic space ${\mathbb{H}^n}$ for all k = 1, 2, 3, . . . . This gives an answer to an open question raised by Aubin in [Aubin, Princeton University Press, Princeton (1982), pp. 176–177] for ${W^{k,2}(\mathbb{H}^n)}$ with k > 1. In addition, we prove that the associated Sobolev constants are optimal.

Natural Tiling, Lattice Tiling and Lebesgue Measure of Integral Self-Affine Tiles

In the existing theory of self-affine tiles, one knows thatthe Lebesgue measure of any integral self-affine tile correspondingto a standard digit set must be a positive integer and everyintegral self-affine tile admits some lattice Zⁿ as a translationtiling set of Rⁿ. In this paper, we give algorithms to evaluatethe Lebesgue measure of any such integral self-affine tile Kand to determine all of the lattice tilings of Rⁿ by K. Moreover,we also propose and determine algorithmically another type oftranslation tiling of Rⁿ by K, which we call natural tiling.We also provide an algorithm to decide whether or not Lebesguemeasure of the set K (K+j), j–Zⁿ, is strictly positive.     

Covering and packing in ${\Bbb Z}^n$ and ${\Bbb R}^n$, (I)

Given a finite subset of an additive group such as or , we are interested in efficient covering of by translates of , and efficient packing of translates of in . A set provides a covering if the translates with cover (i.e., their union is ), and the covering will be efficient if has small density in . On the other hand, a set will provide a packing if the translated sets with are mutually disjoint, and the packing is efficient if has large density. In the present part (I) we will derive some facts on these concepts when , and give estimates for the minimal covering densities and maximal packing densities of finite sets . In part (II) we will again deal with , and study the behaviour of such densities under linear transformations. In part (III) we will turn to . Authors’ address: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395, USA The first author was partially supported by NSF DMS 0074531.     

Disk-Like Self-Affine Tiles in R 2

We give simple necessary and sufficient conditions for self-affine tiles in R ² to be homeomorphic to a disk. Received October 10, 2000, and in revised form February 16, 2001, and April 25, 2001. Online publication July 25, 2001.     

Evolution of hypersurfaces in $ {\Bbb R}^N$ by Gaussian curvature

In this paper we first prove short time existence of a classical solution for the problem which describes the evolution by Gaussian curvature of a strictly convex hypersurface in . Then we give a proof of the existence of a viscosity solution for this problem in such a way as to define a generalized motion existing for each time. Received November 24, 1997     

The Second Variational Formula for Laguerre Minimal Hypersurfaces in {\mathbb {R}^n}

Li and Wang (Manuscr Math 122(1):73–95, 2007) presented Laguerre geometry for hypersurfaces in ${\mathbb{R}^{n}}$ and calculated the first variational formula of the Laguerre functional by using Laguerre invariants. In this paper we present the second variational formula for Laguerre minimal hypersurfaces. As an application of this variational formula we give the standard examples of Laguerre minimal hypersurfaces in ${\mathbb{R}^{n}}$ and show that they are stable Laguerre minimal hypersurfaces. Using this second variational formula we can prove that a surface with vanishing mean curvature in ${\mathbb{R}^{3}_{0}}$ is Laguerre equivalent to a stable Laguerre minimal surface in ${\mathbb{R}^{3}}$ under the Laguerre embedding. This example of stable Laguerre minimal surface in ${\mathbb{R}^{3}}$ is different from the one Palmer gave in (Rend Mat Appl 19(2):281–293, 1999).     

Stability and Computation of Topological Invariants of Solids in ${\Bbb R}^n$

In this work one proves that under quite general assumptions one can deduce the topology of a bounded open set in from an approximation of it. For this, one introduces the weak feature size (wfs) that extends for nonsmooth objects the notion of local feature size. Our results apply to open sets with positive wfs. This class includes subanalytic open sets which cover many cases encountered in practical applications. The proofs are based upon the study of distance functions to closed sets and their critical points. The notion of critical point is the same as the one used in riemannian geometry [22], [9], [20] and nonsmooth analysis [10]. As an application, one gives a way to compute the homology groups of open sets from noisy samples of points on their boundary.     

Hypoelliptic Dunkl Convolution Equations in the Space of Distributions on {\Bbb R}^d

In this article we define and study the Dunkl convolution product and the Dunkl transform on spaces of distributions on By using the main results obtained, we study the hypoelliptic Dunkl convolution equations in the space of distributions.     

A dual algorithm for the minimum covering weighted ball problem in {\mathbb{R}^n}

The nonlinear convex programming problem of finding the minimum covering weighted ball of a given finite set of points in ${\mathbb{R}^n}$ is solved by generating a finite sequence of subsets of the points and by finding the minimum covering weighted ball of each subset in the sequence until all points are covered. Each subset has at most n + 1 points and is affinely independent. The radii of the covering weighted balls are strictly increasing. The minimum covering weighted ball of each subset is found by using a directional search along either a ray or a circular arc, starting at the solution to the previous subset. The step size is computed explicitly at each iteration.     

Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on $ {Bbb R}^n $

No Abstract       

Radial symmetry results for systems of integral equations on {\Omega\subset {\mathbb{R}}^n}

The main purpose of this paper is to investigate positive solutions of the following system of integral equations with weight Riesz potential on Ω: $$\left\{\begin{array}{ll} u(x)=\int\limits_{\Omega}{|x|^{-\alpha}|x-y|^{-\mu}|y|^{-\beta}}{u^a(y)v^b(y)}dy,\\ v(x)=\int\limits_{\Omega}{|x|^{-\gamma}|x-y|^{-\nu}|y|^{-\kappa}}{u^c(y)v^d(y)}dy,\\ \end{array} \right.$$ where a, b, c, d, α, β, γ, κ, μ, ν are constants. With the method of moving planes, we establish the symmetry of both the solution and the bounded C ¹ domain Ω if u and v are constants on ?Ω. Moreover, we also give a symmetry result for systems of integral equations with weight Riesz potential on exterior domains.     

Linear Fractional Transformation Methods in $ {\shadC}^n $

We consider the generalization of linear fractional transformations of the plane to $ {\shadC}^n $ . Analogs of the one-variable theory are developed including fixed point sets and points of symmetry. The domains in $ {\shadC}^n $ that are images of the ball under these transformations are found. Finally, we see some examples where classical fixed point results follow from this theory in a natural way.     

On the moving frame of a conformal map from 2-disk into {\mathbb{R}^n}

Let f be a conformal map from the 2-disk into ${\mathbb{R}^n}$ . We prove that the image f(B) have a normal tangent vector basis (e ₁, e ₂) with ${\|d(e_{1}, e_{2})\|_{L^2(B)} \leq C\|A\|_{L^2(B)}}$ when the total Gauss curvature ${\int_B |K_{f}| d\mu_f < 2\pi}$ .     

A Single Cell in an Arrangement of Convex Polyhedra in ${\Bbb R}^3$

We show that the combinatorial complexity of a single cell in an arrangement of k convex polyhedra in 3-space having n facets in total is , for any , thus settling a conjecture of Aronov et al. We then extend our analysis and show that the overall complexity of the zone of a low-degree algebraic surface, or of the boundary of an arbitrary convex set, in an arrangement of k convex polyhedra in 3-space with n facets in total, is also , for any . Finally, we present a deterministic algorithm that constructs a single cell in an arrangement of this kind, in time , for any .     

Existence and bifurcation of positive solutions of a semilinear elliptic problem on {\Bbb R}^N

Existence of positive solutions of a semilinear elliptic problem on , arising from population genetics is obtained. We use an approximation procedure, by accumulation of branches of solutions of the problem stated on balls of sufficiently large radia. Received January 19, 1996     

Set-theoretic complete intersection monomial curves in $${\mathbb{P}^n}$$

In this paper, we give a sufficient numerical criterion for a monomial curve in a projective space to be a set-theoretic complete intersection. Our main result generalizes a similar statement proven by Keum for monomial curves in three-dimensional projective space. We also prove that there are infinitely many set-theoretic complete intersection monomial curves in the projective n?space for any suitably chosen n ? 1 integers. In particular, for any positive integers p, q, where gcd(p, q) = 1, the monomial curve defined by p, q, r is a set-theoretic complete intersection for every \({r \geq pq( q - 1)}\).     

Sharp higher-order Sobolev inequalities in the hyperbolic space {\mathbb{H}^n}

In this paper, we obtain the sharp k-th order Sobolev inequalities in the hyperbolic space ${\mathbb{H}^n}$ for all k = 1, 2, 3, . . . . This gives an answer to an open question raised by Aubin in [Aubin, Princeton University Press, Princeton (1982), pp. 176–177] for ${W^{k,2}(\mathbb{H}^n)}$ with k > 1. In addition, we prove that the associated Sobolev constants are optimal.