On some operators inc 0

Theorem. Let S be a bounded Suslin set in the plane. Then there is a bounded linear operator T in c_o, whose point spectrum σ _e (T)=S.     

n-connectedness in pure 2-complexes

A graph is a 1-dimensional simplicial complex. In this work we study an interpretation of “n-connectedness” for 2-dimensional simplicial complexes. We prove a 2-dimensional analogue of a theorem by Whitney for graphs: Theorem (A Whitney type theorem for pure 2-complexes).Let G be a pure 2-complex with no end-triangles. Then G is n-connected if and only if the valence of e is at least n for every interior edge e of G, and there does not exist a juncture set J of less than n edges of G. Examples ofn-connected pure 2-complexes are then given, and some consequences are proved.     

Gorenstein differential graded algebras

We propose a definition of Gorenstein Differential Graded Algebra. In order to give examples, we introduce the technical notion of Gorenstein morphism. This enables us to prove the following: Theorem:Let A be a noetherian local commutative ring, let L be a bounded complex of finitely generated projective A-modules which is not homotopy equivalent to zero, and let ɛ=Hom _A (L, L)be the endomorphism Differential Graded Algebra of L. Then ɛ is a Gorenstein Differential Graded Algebra if and only if A is a Gorenstein ring. Theorem:Let A be a noetherian local commutative ring with a sequence of elements a=(a ₁,…,a _n )in the maximal ideal, and let K(a)be the Koszul complex of a.Then K(a)is a Gorenstein Differential Graded Algebra if and only if A is a Gorenstein ring. Theorem:Let A be a noetherian local commutative ring containing a field k, and let X be a simply connected topological space with dim _k H*(X;k)<∞,which has poincaré duality over k. Let C*(X;A)be the singular cochain Differential Graded Algebra of X with coefficients in A. Then C*(X; A)is a Gorenstein Differential Graded Algebra if and only if A is a Gorenstein ring. The second of these theorems is a generalization of a result by Avramov and Golod from [4].     

A sum formula related to ellipsoids with applications to lattice point theory

In this article we consider finite and infinitep-dimensional sums over functionsf, where the argument off is represented by a positive definite quadratic form. We develop a sum formula like theEuler-Maclaurin orPoisson sum formula. Applications to exponential sums and lattice point problems are given.     

Regularity properties of functional equations and inequalities

Summary By a well-known theorem of Lebesgue and Fréchet every measurable additive real function is continuous. This result was improved by Ostrowski who showed that a (Jensen-) convex real function must be continuous if it is bounded above on a set of positive Lebesgue measure. Recently, R. Trautner provided a short and elegant proof of the Lebesgue—Fréchet theorem based on a representation theorem for sequences on the real line.We consider here a locally compact topological groupX with some Haar measure. Then the following generalizes Trautner's theorem: Theorem.Let M be a measurable subset of X of positive finite Haar measure. Then there is a neighbourhood W of the identity e such that for each sequence (z _n )in W there is a subsequence (z _nk )and points y and x _k in M with z _nk =x _k ·y ^–1 for k . Using this theorem we obtain the following extensions of the theorems of Lebesgue and Fréchet and of Ostrowski. Theorem.Let R and T be topological spaces. Suppose that R has a countable base and that X is metrizable. If g: X R and H: R × X T are mappings where g is measurable on a set M of positive finite Haar measure and H is continuous in its first variable, then any solution f: X T of f(x · y) = H(g)(x), y) for x, yX is continuous. Theorem.Let G: X × X be a mapping. If there is a subset M of X of positive finite Haar measure such that for each yX the mapping x G(x, y) is bounded above on M, then any solution f: x of f(x · y) G(x, y) for x, yX is locally bounded above. We also prove category analogues of the above results and obtain similar results for general binary mappings in place of the group operation in the argument off.     

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 the orientable genus of graphs embedded in the klein bottle

Let G be a graph embedded in the Klein bottle with “representativity” at least four. We give a formula for the orientable genus of G, which also implies a polynomially bounded algorithm. The formula is in terms of the number of times certain closed curves on the Klein bottle intersect the graph. In particular, it shows that a cut-and-paste technique for re-embedding graphs is the best possible.     

Weierstrass preparation of quadratic differentials

Special cases or variants of the following play an important role in the asymptotic analysis of ordinary differential equations with turning points. Theorem.Let a(t, x) be a smooth complex-valued function germ at the origin in C × R ^m which is holomorphic in t. Suppose that a(t, 0) does not vanish identically. Then there is a smooth change of variable t = g(s, x), holomorphic in s, such that a(t, x)dt ² =P(s, x)ds ² where P is a monic polynomial in s.     

Extending partial isomorphisms for the small index property of many ω-categorical structures

Theorem:Let A be a finite K _m -free graph, p ₁ , …, p _n partial isomorphisms on A. Then there exists a finite extension B, which is also a K _m -free graph, and automorphisms f _i of B extending the p _i . A paper by Hodges, Hodkinson, Lascar and Shelah shows how this theorem can be used to prove the small index property for the generic countable graph of this class. The same method also works for a certain class of continuum many non-isomorphic ω-categorical countable digraphs and more generally for structures in an arbitrary finite relational language, which are built in a similar fashion. Hrushovski proved this theorem for the class of all finite graphs [Hr]; the proof presented here stems from his proof. Supported by EC-grant ERBCHBGCT 920013.     

Order-two continuous Hausdorff images of compact ordinals

Theorem. LetK be a Hausdorff space. The following conditions are equivalent: (a)K is homeomorphic to a compact scattered ordered space; (b)K is an order-two image of a compact ordinal.     

Another logarithmic functional equation

Summary. Let f : ]0,￥[? \Bbb R f :\,]0,\infty[\to \Bbb R be a real valued function on the set of positive reals. The functional equations¶¶f(x + y) - f(x) - f(y) = f(x^-1 + y^-1) f(x + y) - f(x) - f(y) = f(x^{-1} + y^{-1}) ¶and¶f(xy) = f(x) + f(y) f(xy) = f(x) + f(y) ¶are equivalent to each other.     

Minimum ideal triangulations of hyperbolic 3-manifolds

Let σ(n) be the minimum number of ideal hyperbolic tetrahedra necessary to construct a finite volumen-cusped hyperbolic 3-manifold, orientable or not. Let σ_or(n) be the corresponding number when we restrict ourselves to orientable manifolds. The correct values of σ(n) and σ_or(n) and the corresponding manifolds are given forn=1,2,3,4 and 5. We then show that 2n−1≤σ(n)≤σ_or(n)≤4n−4 forn≥5 and that σ_or(n)≥2n for alln. Both authors were supported by NSF Grants DMS-8711495, DMS-8802266 and Williams College Research Funds.     

The category of a map and the grade of a module

Letf: Y → X be a continuous map between connectedCW complexes. The homologyH _* (F) of the homotopy fibre is then a module over the loop space homologyH _* (ΩX). Theorem:If H _* (F; R) and H _* (ΩX; R) are R-free (R a principal ideal domain) then for some H _* (ΩX; R)-projective module P=P_>0 and for some m ≤ cat f: Some applications are also given. Research partially supported by a NATO travel grant. Research partially supported by an NSERC operating grant. U.A. au CNRS 751.     

On the sup-norm of Maass cusp forms of large level

We establish upper bounds for the sup-norm of Hecke-Maass eigenforms on arithmetic surfaces. In a first part, the case of open modular surfaces is studied. Let f{f} be an Hecke–Maass cuspidal newform of square-free level N{N} and bounded Laplace eigenvalue. Recently, V. Blomer and R. Holowinsky [Invent. Math., 179 (3)] provide a non-trivial bound when f{f} is non-exceptional. Our approach is different in that we rely on the geometric side of the trace formula. The improved bound ||f||_￥ << N^-1/23 ||f||₂{||f||_\infty \ll N^{-1/23} ||f||_2} is established. In a second part, we show that a corresponding result holds true for compact arithmetic surfaces and with a better exponent 1/12. The proof requires an estimate for the number of lattice points in a certain annulus domain. A key input is that a congruence subgroup (multiplicative group) is included in an order (ring). This structure enables us to introduce a diophantine argument.     

Oriented matroids and complete-graph embeddings on surfaces

We provide a link between topological graph theory and pseudoline arrangements from the theory of oriented matroids. We investigate and generalize a function f that assigns to each simple pseudoline arrangement with an even number of elements a pair of complete-graph embeddings on a surface. Each element of the pair keeps the information of the oriented matroid we started with. We call a simple pseudoline arrangement triangular, when the cells in the cell decomposition of the projective plane are 2-colorable and when one color class of cells consists of triangles only. Precisely for triangular pseudoline arrangements, one element of the image pair of f is a triangular complete-graph embedding on a surface. We obtain all triangular complete-graph embeddings on surfaces this way, when we extend the definition of triangular complete pseudoline arrangements in a natural way to that of triangular curve arrangements on surfaces in which each pair of curves has a point in common where they cross. Thus Ringel's results on the triangular complete-graph embeddings can be interpreted as results on curve arrangements on surfaces. Furthermore, we establish the relationship between 2-colorable curve arrangements and Petrie dual maps. A data structure, called intersection pattern is provided for the study of curve arrangements on surfaces. Finally we show that an orientable surface of genus g admits a complete curve arrangement with at most 2g+1 curves in contrast to the non-orientable surface where the number of curves is not bounded.     

On stability of the Cauchy equation on semigroups

Summary We say that Hyers's theorem holds for the class of all complex-valued functions defined on a semigroup (S, +) (not necessarily commutative) if for anyf:S such that the set {f(x + y) – f(x) – f(y): x, y S} is bounded, there exists an additive functiona:S for which the functionf – a is bounded.Recently L. Székelyhidi (C. R. Math. Rep. Acad. Sci. Canada8 (1986) has proved that the validity of Hyers's theorem for the class of complex-valued functions onS implies its validity for functions mappingS into a semi-reflexive locally convex linear topological spaceX. We improve this result by assuming sequential completeness of the spaceX instead of its semi-reflexiveness. Our assumption onX is essentially weaker than that of Székelyhidi. Theorem.Suppose that Hyers's theorem holds for the class of all complex-valued functions on a semigroup (S, +) and let X be a sequentially complete locally convex linear topological (Hausdorff) space. If F: S X is a function for which the mapping (x, y) F(x + y) – F(x) – F(y) is bounded, then there exists an additive function A : S X such that F — A is bounded.     

Julia sets of permutable Holomorphic maps

Let f and g be two permutable transcendental holomorphic maps in the plane. We shall discuss the dynamical properties of f, g and f o g and prove, among other things, that if either f has no wandering domains or f is of bounded type, then the Julia sets of f and f(g) coincide. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

The Thurston boundary of Teichmüller space and complex of curves

Let S be a closed orientable surface with genus g?2. For a sequence σi in the Teichmüller space of S, which converges to a projective measured lamination [λ] in the Thurston boundary, we obtain a relation between λ and the geometric limit of pants decompositions whose lengths are uniformly bounded by a Bers constant L. We also show that this bounded pants decomposition is related to the Gromov boundary of complex of curves.     

Boundedness and convergence of martingales in rearrangement-invariant function spaces

Let (W, F, P)(\Omega, \cal F, P) be a complete nonatomic probability space. We shall give a characterization of rearrangement-invariant spaces X over W\Omega with the property that every martingale f = (f_n)_{n \geqq 0}f = (f_n)_{n \geqq 0} bounded in X converges with respect to the norm topology of X. Using the results, we shall consider the summability of martingales by Toeplitz matrices.     

A generalized moment problem

Let {λ _n} (n≧0) satisfy (1.1) we are considering the following problems: What are the necessary and sufficient conditions on a sequence {μ_n} (n≧0) in order that it should possess the representation (1.2) wherea(t) is of bounded variation or the representation (1.3) wheref(t) ∈L _M[0, 1] orf(t) is essentially bounded.