Van Kampen’s embedding obstruction for discrete groups

We give a lower bound to the dimension of a contractible manifold on which a given group can act properly discontinuously. In particular, we show that the n-fold product of nonabelian free groups cannot act properly discontinuously on ℝ^{2 n −1}. Oblatum 19-I-2001 & 29-V-2002?Published online: 5 September 2002 RID="*" ID="*"All three authors gratefully acknowledge the support by the National Science Foundation.     

A primal all-integer algorithm based on irreducible solutions

 This paper introduces an exact primal augmentation algorithm for solving general linear integer programs. The algorithm iteratively substitutes one column in a tableau by other columns that correspond to irreducible solutions of certain linear diophantine inequalities. We prove that various versions of our algorithm are finite. It is a major concern in this paper to show how the subproblem of replacing a column can be accomplished effectively. An implementation of the presented algorithms is given. Computational results for a number of hard 0/1 integer programs from the MIPLIB demonstrate the practical power of the method. Received: April 23, 2001 / Accepted: May 2002 Published online: March 21, 2003 RID="*" ID="*" Supported by grants FKZ 0037KD0099 and FKZ 2495A/0028G of the Kultusministerium of Sachsen-Anhalt. RID="*" ID="*" Supported by grants FKZ 0037KD0099 and FKZ 2495A/0028G of the Kultusministerium of Sachsen-Anhalt. RID="*" ID="*" Supported by grants FKZ 0037KD0099 and FKZ 2495A/0028G of the Kultusministerium of Sachsen-Anhalt. RID="#" ID="#"Supported by a Gerhard-Hess-Preis and grant WE 1462 of the Deutsche Forschungsgemeinschaft, and by the European DONET program TMR ERB FMRX-CT98-0202. Mathematics Subject Classification (1991): 90C10     

Differentiable functions defined in closed sets. A problem of Whitney

In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed subset X of ℝ ⁿ is the restriction of a function of class 𝒞 ^p . A necessary and sufficient criterion was given in the case n=1 by Whitney, using limits of finite differences, and in the case p=1 by Glaeser (1958), using limits of secants. We introduce a necessary geometric criterion, for general n and p, involving limits of finite differences, that we conjecture is sufficient at least if X has a “tame topology”. We prove that, if X is a compact subanalytic set, then there exists q=q _X (p) such that the criterion of order q implies that f is 𝒞 ^p . The result gives a new approach to higher-order tangent bundles (or bundles of differential operators) on singular spaces. Oblatum 21-XI-2001 & 3-VII-2002?Published online: 8 November 2002 RID="*" ID="*"Research partially supported by the following grants: E.B. – NSERC OGP0009070, P.M. – NSERC OGP0008949 and the Killam Foundation, W.P. – KBN 5 PO3A 005 21.     

Submanifolds with constant Möbius scalar curvature in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

 Let M ^m be a m-dimensional submanifold in the n-dimensional unit sphere S ⁿ without umbilic point. Two basic invariants of M ^m under the M?bius transformation group of S ⁿ are a 1-form Φ called M?bius form and a symmetric (0,2) tensor A called Blaschke tensor. In this paper, we prove the following rigidity theorem: Let M ^m be a m-dimensional (m≥3) submanifold with vanishing M?bius form and with constant M?bius scalar curvature R in S ⁿ , denote the trace-free Blaschke tensor by . If , then either ||?||≡0 and M ^m is M?bius equivalent to a minimal submanifold with constant scalar curvature in S ⁿ ; or and M ^m is M?bius equivalent to in for some c≥0 and . Received: 15 May 2002 / Revised version: 3 February 2003 Published online: 19 May 2003 RID="*" ID="*" Partially supported by grants of CSC, NSFC and Outstanding Youth Foundation of Henan, China. RID="†" ID="†" Partially supported by the Alexander Humboldt von Stiftung and Zhongdian grant of NSFC. Mathematics Subject Classification (2000): Primary 53A30; Secondary 53B25     

Pathwidth of Planar and Line Graphs

 We prove that for every 2-connected planar graph the pathwidth of its geometric dual is less than the pathwidth of its line graph. This implies that pathwidth(H)≤ pathwidth(H ^*)+1 for every planar triangulation H and leads us to a conjecture that pathwidth(G)≤pathwidth(G ^*)+1 for every 2-connected graph G. Received: May 8, 2001 Final version received: March 26, 2002 RID="*" ID="*" I acknowledge support by EC contract IST-1999-14186, Project ALCOM-FT (Algorithms and Complexity - Future Technologies) and support by the RFBR grant N01-01-00235. Acknowledgments. I am grateful to Petr Golovach, Roland Opfer and anonymous referee for their useful comments and suggestions.     

Dense subgroups of <Emphasis Type="Italic">n</Emphasis>-dimensional Möbius groups

We will derive a new discreteness condition for n-dimensional M?bius subgroups as well as obtain some results concerning classification of such groups. We will also discuss dense subgroups of n-dimensional M?bius groups. The main result is that any dense group of an n-dimensional M?bius group contains a dense subgroup which is generated by at most n elements if . Received: 5 June 2001 / Published online: 24 February 2003 RID="*" ID="*" The research was partly supported by FNS of China, grant number 19801011     

Green functions and eigenfunction expansions for the square of the Laplace–Beltrami operator on plane domains

For a certain class of domains Ω⊂ℂ with smooth boundary and Δtilde;_Ω=w ²Δ the Laplace–Beltrami operator with respect to the Poincaré metric ds ²=w(z)^-2 dz dz on Ω, we (1) show that the Green function for the biharmonic operator Δtilde;_Ω ², with Dirichlet boundary data, is positive on Ω×Ω; and (2) obtain an eigenfunction expansion for the operator Δtilde;_Ω, which reduces to the ordinary non-Euclidean Fourier transform of Helgason for Ω=𝔻 (the unit disc). In both cases the proofs go via uniformization, and in (1) we obtain a Myrberg-like formula for the corresponding Green function. Finally, the latter formula as well as the eigenfunction expansion are worked out more explicitly in the simplest case of Ω an annulus, and a result is established concerning the convergence of the series ∑ _ω∈G (1-|ω0|²) ^s for G the covering group of the uniformization map of Ω and 0<s<1. Received: August 21, 2000?Published online: October 30, 2002 RID="*" ID="*"The first author was supported by GA AV CR grants no. A1019701 and A1019005.     

Non-amenable finitely presented torsion-by-cyclic groups

Abstract. – We construct a finitely presented non-amenable group without free non-cyclic subgroups thus providing a finitely presented counterexample to von Neumann’s problem. Our group is an extension of a group of finite exponent n ≫ 1 by a cyclic group, so it satisfies the identity [x,y] ⁿ = 1. Manuscrit reĉu le 8 février 2001. RID="*" ID="*"Both authors were supported in part by the NSF grant DMS 0072307. In addition, the research of the first author was supported in part by the Russian Fund for Basic Research 99-01-00894 and by the INTAS grant, the research of the second author was supported in part by the NSF grant DMS 9978802.     

Asymptotic characterization of standing waves and of the static limit for a class of wave equations of higher order with a variable coefficient and time-independent incitation

We consider the equation (?1)^m?^m (p?^mu) + ?u = ? in ?ⁿ × (0, ∞) for arbitrary positive integers m and n and under the assumptions p ? 1, ? ? C(?ⁿ) and p > 0. Even if the differential operator (?1)^m?^m (p?^mu) has no eigenvalues, the solution u(x,t) may increase as t → ∞ for 2m ≥ n. For this case, we derive necessary and sufficient conditions for the convergence of u(x,t) as t → ∞. Furthermore, we characterize the functions occurring in these conditions as solutions of the homogeneous static equation (?1)^m?^m (p?^mu) = 0, which satisfy appropriate asymptotic conditions at infinity. We also give an asymptotic characterization of the static limit.     

Approximation in Lp( ) from Spaces Spanned by the Perturbed Integer Translates of a Radial Function

The problem of approximating smooth L_p-functions from spaces spanned by the integer translates of a radially symmetric function φ is very well understood. In case the points of translation, Ξ, are scattered throughout ^d, the approximation problem is only well understood in the “stationary” setting. In this work, we provide lower bounds on the obtainable approximation orders in the “non-stationary” setting under the assumption that Ξ is a small perturbation of ^d. The functions which we can approximate belong to certain Besov spaces. Our results, which are similar in many respects to the known results for the case Ξ= ^d, apply specifically to the examples of the Gauss kernel and the generalized multiquadric.     

Rigidity and gamma convergence for solid‐solid phase transitions with SO(2) invariance

The singularly perturbed two‐well problem in the theory of solid‐solid phase transitions takes the form where u : Ω ? ?ⁿ → ?ⁿ is the deformation, and W vanishes for all matrices in K = SO(n)A ∪ SO(n)B. We focus on the case n = 2 and derive, by means of Gamma convergence, a sharp‐interface limit for I_ε. The proof is based on a rigidity estimate for low‐energy functions. Our rigidity argument also gives an optimal two‐well Liouville estimate: if ?u has a small BV norm (compared to the diameter of the domain), then, in the L¹ sense, either the distance of ?u from SO(2)A or the one from SO(2)B is controlled by the distance of ?u from K. This implies that the oscillation of ?u in weak L¹ is controlled by the L¹ norm of the distance of ?u to K. © 2006 Wiley Periodicals, Inc.     

A uniqueness condition for the polyharmonic equation in free space

Consider the polyharmonic wave equation ?u + (? Δ)^mu = f in ?ⁿ × (0, ∞) with time-independent right-hand side. We study the asymptotic behaviour of u ( x , t) as t → ∞ and show that u( x , t) either converges or increases with order t^α or In t as t → ∞. In the first case we study the limit $ u_0 \left({\bf x} \right) \colone \mathop {\lim }\limits_{t \to \infty } \,u\left({{\bf x},t} \right) $ and give a uniqueness condition that characterizes u₀ among the solutions of the polyharmonic equation ( ? Δ)^mu = f in ?ⁿ. Furthermore we prove in the case 2m ? n that the polyharmonic equation has a solution satisfying the uniqueness condition if and only if f is orthogonal to certain solutions of the homogeneous polyharmonic equation.     

Half-Linear Eigenvalue Problems

We consider the eigenvalue problem for t ? [0, b], where aⁿ = |a|ⁿ sgna, a ? ?, λ ? ?, the constants μ, v are real such that 0 ≤ μ < n and derive asymptotic estimates for solutions of the differential equation in the definite case q(t)> 0 which corresponds to the well-known WKB-approximation in the linear case n = 1, μ = 0. In the second part we investigate the asymptotic distribution of the eigenvalues in the general case of two -point boundary conditions and refine these results for the so called separated boundary conditions.     

On the convergence of the Weiszfeld algorithm

In this work we analyze the paper “Brimberg, J. (1995): The Fermat-Weber location problem revisited. Mathematical Programming 71, 71–76” which claims to close the question on the conjecture posed by Chandrasekaran and Tamir in 1989 on the convergence of the Weiszfeld algorithm. Some counterexamples are shown to the proofs showed in Brimberg’s paper. Received: January 1999 / Accepted: December 2001?Published online April 12, 2002 RID="*" ID="*"Partially supported by PB/11/FS/97 of Fundación Séneca of the Comunidad Autónoma de la Región de Murcia RID="**" ID="**"Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica (I+I+D), project TIC2000-1750-C06-06 RID="*" RID="**"     

On the well‐posedness of the Cauchy problem for an MHD system in Besov spaces

This paper is devoted to the study of the Cauchy problem of incompressible magneto‐hydrodynamics system in the framework of Besov spaces. In the case of spatial dimension n?3, we establish the global well‐posedness of the Cauchy problem of an incompressible magneto‐hydrodynamics system for small data and the local one for large data in the Besov space ? (?ⁿ), 1?p<∞ and 1?r?∞. Meanwhile, we also prove the weak–strong uniqueness of solutions with data in ? (?ⁿ)∩L²(?ⁿ) for n/2p+2/r>1. In the case of n=2, we establish the global well‐posedness of solutions for large initial data in homogeneous Besov space ? (?²) for 2<p<∞ and 1?r<∞. Copyright © 2008 John Wiley & Sons, Ltd.     

Skew loops and quadric surfaces

A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in R ³ with a point of positive curvature and no skew loops are the quadrics. In particular: Ellipsoids are the only closed surfaces without skew loops. Our efforts also yield results about skew loops on cylinders and positively curved surfaces. Received: January 7, 2002 RID="*" ID="*"The first author was partially supported by the NSF grant DMS-0204190.     

Intersecting Curves in the Plane

 We prove that for every family of n pairwise intersecting simple closed planar curves in general position, at least (4/5)n ²−O(n) points lie on more than one curve. This improves the previous lower bound of (3/4)n ²−O(n) due to Richter and Thomassen. Received: March 29, 2000 Final version received: August 30, 2001 RID="*" ID="*" Research supported in part by NSF grant DMS-9970325 Acknowledgments. I thank Bruce Richter for informing me about this problem, Gelasio Salazar for reading a preliminary version of the paper, and a Referee for useful comments. Current Address: Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399, USA. e-mail: mubayi@microsoft.com 1991 Mathematics Subject Classification. 05C35, 52C10     

Resonance phenomena for a class of partial differential equations of higher order in cylindrical waveguides

We consider a domain Ω in ?ⁿ of the form Ω = ?^l × Ω′ with bounded Ω′ ? ?^n?l. In Ω we study the Dirichlet initial and boundary value problem for the equation ? u + [(? ? ?… ? ?)^m + (? ? ?… ? ?)^m]u = fe^?iωt. We show that resonances can occur if 2m ≥ l. In particular, the amplitude of u may increase like t^α (α rational, 0<α<1) or like in t as t∞∞. Furthermore, we prove that the limiting amplitude principle holds in the remaining cases.     

Frames in the bargmann space of entire functions

We look at the decomposition of arbitrary f in L²( R ) in terms of the family of functions φ_mn(x) = π^?1/4exp{ ? 1/2imnab + i max ? 1/2(x ? nb)²}, with a, b > 0. We derive bounds and explicit formulas for the minimal expansion coefficients in the case where ab = 2π/N, N an integer ≧ 2. Transported to the Hilbert space F of entire functions introduced by V. Bargmann, these results are expressed as inequalities of the form We conjecture that these inequalities remain true for all a, b such that ab < 2π.