On the chromatic numbers of spheres in ℝ n

Let χ(S _r ^n?1 )) be the minimum number of colours needed to colour the points of a sphere S _r ^n?1 of radius $r \geqslant \tfrac{1} {2}$ in ? ⁿ so that any two points at the distance 1 apart receive different colours. In 1981 P. Erd?s conjectured that χ(S _r ^n?1 )→∞ for all $r \geqslant \tfrac{1} {2}$ . This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ(S _r ^n?1 ) ≥ n. In the same paper, Lovász claimed that if $r < \sqrt {\frac{n} {{2n + 2}}}$ , then χ(S _r ^n?1 ) ≤ n+1, and he conjectured that χ(S _r ^n?1 ) grows exponentially, provided $r \geqslant \sqrt {\frac{n} {{2n + 2}}}$ . In this paper, we show that Lovász’ claim is wrong and his conjecture is true: actually we prove that the quantity χ(S _r ^n?1 ) grows exponentially for any $r > \tfrac{1} {2}$ .     

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.     

On the representation of biharmonic functions with singularities in ℝ n

Biharmonic functions are defined on Euclidean spaces, Riemannian manifolds, infinite trees, and more generally on abstract harmonic spaces. In this note, we consider biharmonic functions b defined on annular sets Ω \ K and obtain Laurent-type decompositions for b in the Euclidean spaces and in infinite trees. Particular importance is given to the investigation when b extends as a distribution on Ω.     

Estimates of perturbed Oseen semigroups and their applications to the Navier-Stokes system in ℝ n

For perturbed Oseen semigroups in ? ⁿ , we establish their power L _p ? L _q estimates. These estimates are used to prove the existence of small global solutions to perturbed nonlinear Oseen systems and also of estimates of their L _p -norms as t → ∞.     

On the structure of incoming and outgoing subspaces for a wave equations in ℝ n

We investigate the structure of incoming and outgoing subspaces in the Lax-Phillips scheme for the classic wave equation in ℝ ⁿ . Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 5, pp. 708–712, May, 1999.     

Rigid Sets of Dimension n – 1 in ℝ n

We give conditions allowing an intrinsic isometry on a dense subset to be extended to an isometry of the whole set. This enables us to find examples of (n-1)-dimensional sets rigid in ⁿ .     

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.     

About the Cohomology of the Lie Superalgebra of Vector Fields on ℝ n|n

In this article, we compute the first space of cohomology of Vect (? ^n|n ), the Lie superalgebra of vector fields on the supermanifold ? ^n|n with coefficients in 𝒻 (? ^n|n ), the space of smooth functions on ? ^n|n . We give a super analog of the cohomologies of vector fields that where studied for instance by Fuchs [2 Fuchs , D. B. ( 1986 ). Cohomology of Infinite Dimensional Lie Algebras . New York : Consultants Bureau . [Google Scholar]]. This work allows us to classify the deformations of the action of Vect(? ^n|n ) on 𝒻 (? ^n|n ).     

Set of ambiguous points for functions in ℝ n

It is known that an arbitrary function in the open unit disk can have at most countable set of ambiguous points. Point ζ on the unit circle is an ambiguous point of a function if there exist two Jordan arcs, lying in the unit ball, except the endpoint ζ, such that cluster sets of function along these arcs are disjoint. We investigate whether it is possible to modify the notion of ambiguous point to keep the analogous result true for functions defined in the n-dimensional Euclidean unit ball.     

Cube-tilings of ℝ n and nonlinear codesand nonlinear codes

Families of nonlattice tilings of ℝ ⁿ by unit cubes are constructed. These tilings are specializations of certain families of nonlinear codes overGF(2). These cube-tilings provide building blocks for the construction of cube-tilings such that no two cubes have a high-dimensional face in common. We construct cube-tilings of ℝ ⁿ such that no two cubes have a common face of dimension exceeding .     

Existence and symmetry for elliptic equations in ℝn with arbitrary growth in the gradient

We define the Polish space R of non-degenerate rank-1 systems. Each non-degenerate rank-1 system can be viewed as a measure-preserving transformation of an atomless, σ-finite measure space and as a homeomorphism of a Cantor space. We completely characterize when two non-degenerate rank-1 systems are topologically isomorphic. We also analyze the complexity of the topological isomorphism relation on R, showing that it is \({F_\sigma }\) as a subset of R× R and bi-reducible to E₀. We also explicitly describe when a non-degenerate rank-1 system is topologically isomorphic to its inverse.     

On the relationship between differentiability and absolute continuity of measures on ℝn

Summary We give an elementary proof of the fact that a finite Borel measure on ⁿ is absolutely continuous with a C ¹ density if and only if it has directional derivatives which are continuous almost everywhere. The Radon-Nikodym derivative of a differentiable measure is given in terms of the directional derivatives.     

An explicit counter-example for the shephard problem of convex bodies in ℝ n

Comparing the volume of the projection body of a double cone and that of the projection body of a ball, we give an explicit counter-example for the Shephard problem of convex bodies in Rⁿ (n ≥ 3) and an affirmative answer to the question of Zhang.     

Hypersurfaces in ℝn and critical points in their external region

In this paper we study the hypersurfaces given as connected compact regular fibers of a differentiable map , in the cases in which has finitely many nondegenerate critical points in the unbounded component of .     

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.     

The stability of fixed points of discrete dynamical systems in the space conv ℝn

Conditions for the asymptotic Lyapunov stability of the fixed points of discrete dynamical systems in the space conv ?ⁿ are established.