On the solutions of the translation equation in rings of formal power series

We study in this paper solutions of the translation equation in rings of formal power series K[X] where K ∈R, C (so called one-parameter groups or flows), and even, more generally, homomorphisms Ф from an abelian group (G, +) into the group Г(K) of invertible power series in K[X]. This problem can equivalently be formulated as the question of constructing homomorphisms Ф from (G, +) into the differential group Г¹∞ describing the chain rules of higher order of C∞ functions with fixed point 0. In this paper we present the general form of these homomorphisms Ф : G → Г(K) (or L¹∞),Ф = (f_n ⁿ≤1,forwhich f₁ = l, f₂ = ... = f_p+l =0,f_p+2 ≠ 0 for fixed, but arbitrary p ≤ 0 (see Theorem 5, Corollary 6 and Theorem 6). This representation uses a sequence (w _n ^p )_n≥p+2 of universal polynomials in f_p+2 and a sequence of parameters, which determines the individual one-parameter group. Instead of (w _n ^p )_n≥p+2 we may also use another sequence (L _n ^p )_n≥p+2 of universal polynomials, and we describe the connection between these forms of the solutions.     

Overcompleteness of Sequences of Reproducing Kernels in Model Spaces

We give necessary conditions and sufficient conditions for sequences of reproducing kernels (k_Θ(·, λ_n))_{n ≥ 1} to be overcomplete in a given model space K_Θ^p where Θ is an inner function in H^∞, p ∈ (1, ∞), and where (λ_n)_{n ≥ 1} is an infinite sequence of pairwise distinct points of Under certain conditions on Θ we obtain an exact characterization of overcompleteness. As a consequence we are able to describe the overcomplete exponential systems in L² (0, a).     

A lattice in more than two Kac-Moody groups is arithmetic

Let Γ < G ₁ × … × G _n be an irreducible lattice in a product of infinite irreducible complete Kac-Moody groups of simply laced type over finite fields. We show that if n ≥ 3, then each G _i is a simple algebraic group over a local field and Γ is an S-arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n ≥ 2: either Γ is an S-arithmetic (hence linear) group, or Γ is not residually finite. In that case, it is even virtually simple when the ground field is large enough. More general CAT(0) groups are also considered throughout.     

A symmetry of sphere map implies its chaos*

A well-known example, given by Shub, shows that for any |d| ≥ 2 there is a self-map of the sphere Sⁿ, n ≥ 2, of degree d for which the set of non-wandering points consists of two points. It is natural to ask which additional assumptions guarantee an infinite number of periodic points of such a map. In this paper we show that if a continuous map f : Sⁿ → Sⁿ commutes with a free homeomorphism g : Sⁿ → Sⁿ of a finite order, then f has infinitely many minimal periods, and consequently infinitely many periodic points. In other words the assumption of the symmetry of f originates a kind of chaos. We also give an estimate of the number of periodic points. *Research supported by KBN grant nr 2 P03A 045 22.     

Simple groups of finite morley rank and Tits buildings

Theorem A:If ℬ is an infinite Moufang polygon of finite Morley rank, then ℬ is either the projective plane, the symplectic quadrangle, or the split Cayley hexagon over some algebraically closed field. In particular, ℬ is an algebraic polygon. It follows that any infinite simple group of finite Morley rank with a spherical MoufangBN-pair of Tits rank 2 is eitherPSL ₃(K),PSp ₄(K) orG ₂(K) for some algebraically closed fieldK. Spherical irreducible buildings of Tits rank ≥ 3 are uniquely determined by their rank 2 residues (i.e. polygons). Using Theorem A we show Theorem B:If G is an infinite simple group of finite Morley rank with a spherical Moufang BN-pair of Tits rank ≥ 2, then G is (interpretably) isomorphic to a simple algebraic group over an algebraically closed field. Theorem C:Let K be an infinite field, and let G(K) denote the group of K-rational points of an isotropic adjoint absolutely simple K-algebraic group G of K-rank ≥ 2. Then G(K) has finite Morley rank if and only if the field K is algebraically closed. We also obtain a result aboutBN-pairs in splitK-algebraic groups: such aBN-pair always contains the root groups. Furthermore, we give a proof that the sets of points, lines and flags of any ℵ₁-categorical polygon have Morley degree 1. Partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany). Supported by the Minerva Foundation (Germany). Research Director at the Fund for Scientific Research-Flanders (Belgium).     

On numbers badly approximable by dyadic rationals

We consider a problem originating both from circle coverings and badly approximable numbers in the case of dyadic diophantine approximation. For the unit circle we give an elementary proof that the set {x ∈ : 2 ⁿ x ≥ c (mod 1) n ≥ 0} is a fractal set whose Hausdorff dimension depends continuously on c and is constant on intervals which form a set of Lebesgue measure 1. Hence it has a fractal graph. We completely characterize the intervals where the dimension remains unchanged. As a consequence we can describe the graph of c ↦ dim _H {x ∈ [0; 1]: x − m/2 ⁿ < c/2 ⁿ (mod 1) finitely often}.     

The logic of pseudo-S-integers

Let {n _i } be a sequence of natural numbers and let {p _i } be a listing of rational primes. Then an abelian groupG={x ∈ √| ord _pi x ≥ −n _i } is called a group of pseudo-integers. We investigate the logical properties of such groups of pseudo-integers and the counterparts of such groups in global fields in the case the number of primes allowed to appear in the denominator is infinite. We show that, while the addition problem of any recursive group of pseudo-integers is decidable, the Diophantine problem for some recursive groups of pseudo-integers with infinite number of primes allowed in the denominator, is not decidable. More precisely, there exist recursive groups of pseudo-integers, where infinite number of primes are allowed to appear in the denominator, such that there is no uniform algorithm to decide whether a polynomial equation over ℤ in several variables has solutions in the group. This result is obtained by giving a Diophantine definition of ℤ over these groups. The proof is based on the strong Hasse norm principal. The research for this paper has been partially supported by NSA grant MDA904-96-1-0019.     

Propriétés diophantiennes du développement en cotangente continue de Lehmer

Diophantine Properties of Lehmer’s Continued Cotangent Developments. This article deals with an algorithm devised by Lehmer which enables us to write any real positive number as the sum of an alternating series of cotangents of integers n _ν, ν ≥ 0, in a unique way. We continue the work begun by Lehmer and continued by Shallit: amongst other things, we give explicitly the link between the rational approximations of a given real number coming from this algorithm and the usual convergents of the same real number and we produce a quasi-optimal bound for the growth of the sequence (n _ν)_{ν ≥ 0} associated to an algebraic number. We also determine the regular continued fractions of an exceptional class of continued cotangent developments, which enables us to produce optimal irrationality measures of these expansions.     

On permutations of order dividing a given integer

We give a detailed analysis of the proportion of elements in the symmetric group on n points whose order divides m, for n sufficiently large and m≥n with m=O(n).      

p-Groupes abéliens de type (p,…,p) et disques ouverts p-adiques

Let k be an algebraically closed field of characteristic p>0, W(k) its ring of Witt vectors and R a complete discrete valuation ring dominating W(k). Consider finite groups G≃ (ℤ/pℤ) ⁿ , p≥ 2, n≥1. In a former paper we showed that a given realization of such a G as a group of k-automorphisms of k[[z]] must satisfy some conditions in order to have a lifting as a group of R-automorphisms of R[[Z]]. In this note, we give for every G (all p≥ 2, n>1) a realization as an automorphism group of k[[z]] which ca be lifted as a group of R-automorphisms of R[[Z]] for suitable R. Received: 22 December 1998     

On Buchsbaum bundles on quadric hypersurfaces

Let E be an indecomposable rank two vector bundle on the projective space ℙ ⁿ , n ≥ 3, over an algebraically closed field of characteristic zero. It is well known that E is arithmetically Buchsbaum if and only if n = 3 and E is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface Q _n ⊂ ℙ ⁿ⁺¹, n ≥ 3. We give in fact a full classification and prove that n must be at most 5. As to k-Buchsbaum rank two vector bundles on Q ₃, k ≥ 2, we prove two boundedness results.     

Cyclic branched coverings of lens spaces

Some infinite family is constructed of orientable three-dimensional closed manifoldsM _n (p, q), where n ≥ 2, p ≥ 3, 0 < q < p, and (p, q) = 1, such that M _n (p, q) is an n-fold cyclic covering of the lens space L(p, q) branched over a two-component link.     

On the number of affine equivalence classes of spherical tube hypersurfaces

We consider Levi non-degenerate tube hypersurfaces in \mathbbCⁿ⁺¹{\mathbb{C}^{n+1}} that are (k, n − k)-spherical, i.e. locally CR-equivalent to the hyperquadric with Levi form of signature (k, n − k), with n ≤ 2k. We show that the number of affine equivalence classes of such hypersurfaces is infinite (in fact, uncountable) in the following cases: (i) k = n − 2, n ≥ 7; (ii) k = n − 3, n ≥ 7; (iii) k ≤ n − 4. For all other values of k and n, except for k = 3, n = 6, the number of affine classes was known to be finite. The exceptional case k = 3, n = 6 has been recently resolved by Fels and Kaup who gave an example of a family of (3, 3)-spherical tube hypersurfaces that contains uncountably many pairwise affinely non-equivalent elements. In this paper we deal with the Fels–Kaup example by different methods. We give a direct proof of the sphericity of the hypersurfaces in the Fels–Kaup family, and use the j-invariant to show that this family indeed contains an uncountable subfamily of pairwise affinely non-equivalent hypersurfaces.     

On Beauville structures on the groups S n and A n

It is known that the symmetric group S _n , for n ≥ 5, and the alternating group A _n , for large n, admit a Beauville structure. In this paper we prove that A _n admits a Beauville (resp. strongly real Beauville) structure if and only if n ≥ 6 (resp n ≥ 7). We also show that S _n admits a strongly real Beauville structure for n ≥ 5.     

Complementary Regions of Knot and Link Diagrams

An increasing sequence of integers is said to be universal for knots and links if every knot and link has a reduced projection on the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. In this paper, it is proved that the following infinite sequences are each universal for knots and links: (3, 5, 7, . . .), (2, n, n + 1, n + 2, . . .) for each n ≥ 3, (3, n, n + 1, n + 2, . . .) for each n ≥ 4. Moreover, the finite sequences (2, 4, 5) and (3, 4, n) for each n ≥ 5 are universal for all knots and links. It is also shown that every knot has a projection with exactly two odd-sided faces, which can be taken to be triangles, and every link of n components has a projection with at most n odd-sided faces if n is even and n + 1 odd-sided faces if n is odd.     

Depth of functions of the <Emphasis Type="Italic">k</Emphasis>-valued logic in infinite bases

The realization of functions of the k-valued logic by circuits is considered over an arbitrary infinite complete basis B. The Shannon function D _B (n) of the circuit depth over B is examined (for any positive integer n the value D _B (n) is the minimal depth sufficient to realize every function of the k-valued logic of n variables by a circuit over B). It is shown that for each fixed k ≥ 2 and for any infinite complete basis B either there exists a constant α ≥ 1 such that D _B (n) = α for all sufficiently large n, or there exist constants β (β > 0), γ, δ such that βlog₂ n ≤ D _B (n) ≤ γlog₂ n + δ for all n.     

On closeness of the sum of n subspaces of a Hilbert space

We give necessary and sufficient conditions for the sum of subspaces H ₁,…, H _n , n ≥ 2, of a Hilbert space H to be a subspace and present various properties of the n-tuples of subspaces with closed sum.     

Homomorphisms of direct powers of algebras

We give an example of n-coconnected algebra that is not (n+1)-coconnected, for any n ≥ 2. Received June 8, 2005; accepted in final form August 28, 2005.     

Chaotic Tensor Product Semigroups

Suppose {G₁(t)}_{t ≥ 0} and {G₂(t)_{t ≥ 0} be two semigroups on an infinite dimensional separable reflexive Banach space X. In this paper we give sufficient conditions for tensor product semigroup G(t): X → G₂(t)X G₁(t) to become chaotic in L with the strong operator topology and chaotic in the ideal of compact operators on X with the norm operator topology.     

Comparison theorems for the volume of a complex submanifold of a Kaehler manifold

LetM be a Kaehler manifold of real dimension 2n with holomorphic sectional curvatureK _H≥4λ and antiholomorphic Ricci curvatureρ _A≥(2n−2)λ, andP is a complex hypersurface. We give a bound for the quotient (volume ofP)/(volume ofM) and prove that this bound is attained if and only ifP=C P ⁿ⁻¹(λ) andM=C P ⁿ(λ). Moreover, we give some results on the volume of of tubes aboutP inM. Work partially supported by a DGICYT Grant No. PS87-0115-CO3-01.