Homology of GLn over algebraically closed fields

In this paper we define higher pre-Bloch groups _n(F) of a fieldF. When the base field is algebraically closed, we study itsconnection to the homology of the general linear groups withcoefficients in /l , where l is a positive integer. As a resultof our investigation we give a necessary and sufficient conditionfor the natural map H_n(GL_n–1(F), /l ) H_n(GL_n(F), /l )to be bijective. We prove that this map is bijective for n4.We also demonstrate that a certain property of _n() is equivalentto the validity of the Friedlander–Milnor isomorphismconjecture for (n+1)th homology of GL_n().     

Von Neumann Eta-Invariants and C*-Algebra K-Theory

Let M be a smooth, compact, oriented, odd-dimensional Riemannianmanifold and let M be anormal covering of M. It is proved that the relative von Neumanneta-invariant ₍₂₎() of Cheeger and Gromov is a homotopy invariant when is torsion-free, discreteand the Baum–Connes assembly map µ_max:K₀(B) K₀(C*)is an isomorphism.     

Metrical Theorems on Prime Values of the Integer Parts of Real Sequences II

Let [ ] denote the integer part. Among other results in [3]we gave a complete solution to the following problem. PROBLEM. Given an increasing sequence a_n R⁺, n = 1, 2, ...,where a_n as n , are there infinitely many primes in the sequence[a_n] for almost all ?     

Contractions et Hyperdistributions a Spectre de Carleson

Let =(_n)_n1 be a log concave sequence such that lim inf_n+n/n^c>0for some c>0 and ((log _n)/n)_n1 is nonincreasing for some<1/2. We show that, if T is a contraction on the Hilbertspace with spectrum a Carleson set, and if ||T^–n||=O(_n)as n tends to + with _n11/(n log _n)=+, then T is unitary. Onthe other hand, if _n11/(n log _n)<+, then there exists a (non-unitary)contraction T on the Hilbert space such that the spectrum ofT is a Carleson set, ||T^–n||=O(_n) as n tends to +, andlim sup_n+||T^–n||=+.     

Identity Theorems for Functions of Bounded Characteristic

Suppose that f(z) is a meromorphic function of bounded characteristicin the unit disk :|z|<1. Then we shall say that f(z)N. Itfollows (for example from [3, Lemma 6.7, p. 174 and the following])that where h₁(z), h₂(z) are holomorphic in and have positive realpart there, while ₁(z), ₂(z) are Blaschke products, that is, where p is a positive integer or zero, 0<|a_j|<1, c isa constant and (1–|a_j|)<. We note in particular that, if c0, so that f(z)0, (1.1) so that f(z)=0 only at the points a_j. Suppose now that z_j isa sequence of distinct points in such that |z_j|1 as j and (1–|z_j|)=. (1.2) If f(z_j)=0 for each j and fN, then f(z)0. N. Danikas [1] has shown that the same conclusion obtains iff(z_j)0 sufficiently rapidly as j. Let _j, _j be sequences of positivenumbers such that _j< and _j as j. Danikas then defines and proves Theorem A.     

On Free Modules for Finite Subgroups of Algebraic Groups

We show that given an affine algebraic group G over a fieldK and a finite subgroup scheme H of G there exists a finitedimensional G-module V such that V|_H is free. The problem israised in the recent paper by Kuzucuglu and Zalesski [15] which containsa treatment of the special case in which K is the algebraicclosure of a finite field and H is reduced. Our treatment isdivided into two parts, according to whether K has zero or positivecharacteristic. The essence of the characteristic 0 case isa proof that, for given n, there exists a polynomial GL_n(Q)-moduleV of dimension , where the product is over all primes less than or equal to n+1, such thatV is free as a QH-module for every finite subgroup H of GL_n(Q).The module V is the tensor product of the exterior algebra *(E),on the natural GL_n(Q)-module E, and Steinberg modules St_p, onefor each prime not exceeding n+1. The Steinberg modules alsoplay the major role in the case in which K has characteristicp>0 and the key point in our treatment is to show that fora finite subgroup scheme H of a general linear group scheme(or universal Chevalley group scheme) G over K, the Steinbergmodule St_pⁿ for G is injective (and projective) on restrictionto H for n>>0. A curious consequence of this is that,despite the wild behaviour of the modular representation theoryof finite groups, one has the following. Let H be a finite groupand V a finite dimensional vector space. Then there exists a(well-understood) faithful rational representation GL(V)GL(W)such that, for each faithful representation : HGL(V), the compositeo: HGL(W) is free, in particular all representations o are equivalent.     

Stability and Asymptotic Stability of Functional-Differential Equations

We investigate asymptotic behaviour of solutions of the functional-differentialequation where f and g arelocally Lipschitz functions, C is a continuous matrix and thesmooth lag function obeys 0 (t) t for t 0. We transformthe equation into a delay equation with an infinity of delaysand use a theorem of Söderlind to derive sufficient conditionsfor stability and for asymptotic stability in the case lim_t (t) = . The situation is qualitatively different when lim_t (t) = * < and we outline stability conditions for thatcase by employing direct techniques.     

Coherent Products in a Finite Group along a Linear Ordering

Let C = (C, ) be a linear ordering, E a subset of {(x, y):x< y in C} whose transitive closure is the linear orderingC, and let :E G be a map from E to a finite group G = (G, •).We showed with M. Pouzet that, when C is countable, there isF E whose transitive closure is still C, and such that (p) = (x_o, x₁)•(x₁, x₂)•....•(x_{n– 1}, x_n) G depends only upon the extremities x₀, x_n ofp, where p = (x_o, x₁...,x_n) (with 1 n < ) is a finite sequencefor which (x_i, x_{i + 1}) F for all i < n. Here, we show thatthis property does not hold if C is the real line, but is stilltrue if C does not embed an ₁-dense linear ordering, or evena 2-dense linear ordering when Martin's Axiom holds (it followsin particular that it is independent of ZFC for linear orderingsof size ). On the other hand, we prove that this property isalways valid if E = {(x,y):x < y in C}, regardless of anyother condition on C.     

Sandwiching C0-Semigroups

Let T = {T(t)}_t0 be a C₀-semigroup on a Banach space X. Thefollowing results are proved. (i) If X is separable, there exist separable Hilbert spacesX₀ and X₁, continuous dense embeddings j₀:X₀ X and j₁:X X₁,and C₀-semigroups T₀ and T₁ on X₀ and X₁ respectively, suchthat j₀ T₀(t) = T(t) j₀ and T₁(t) j₁ = j₁ T(t) for all t 0. (ii) If T is -reflexive, there exist reflexive Banach spacesX₀ and X₁ , continuous dense embeddings j:D(A²) X₀, j₀:X₀ X, j₁:X X₁, and C₀-semigroups T₀ and T₁ on X₀ and X₁ respectively,such that T₀(t) j = j T(t), j₀ T₀(t) = T(t) j₀ and T(t) j₁ = j₁ T(t) for all t 0, and such that (A₀) = (A) = (A₁),where A_k is the generator of T_k, k = 0, Ø, 1.     

Herman Rings and Arnold Disks

For (,a) C* x C, let f_,a be the rational map defined by f_,a(z)= z² (az+1)/(z+a). If R/Z is a Brjuno number, we let D bethe set of parameters (,a) such that f_,a has a fixed Hermanring with rotation number (we consider that (e²ⁱ,0) D). Resultsobtained by McMullen and Sullivan imply that, for any g D, theconnected component of D(C* x (C/{0,1})) that contains g isisomorphic to a punctured disk. We show that there is a holomorphic injection F:DD such thatF(0) = (e²ⁱ ,0) and , where r is the conformal radius at 0 of the Siegel disk of the quadraticpolynomial z e²ⁱ z(1+z). As a consequence, we show that for a (0,1/3), if f_l,a has afixed Herman ring with rotation number and if m_a is the modulusof the Herman ring, then, as a0, we have e m_a=(r/a) + O(a). We finally explain how to adapt the results to the complex standardfamily z e^(a/2)(z-1/z).     

Periodic Solutions of Lienard Equations with Asymmetric Nonlinearities at Resonance

The existence of 2-periodic solutions of the second-order differentialequation where a, b satisfy and p(t)=p(t+2),t R, is examined. Assume that limits lim_x±F(x)=F(±)(F(x)=) and lim_x±g(x)=g(±)exist and are finite. It is proved that the equation has atleast one 2-periodic solution provided that the zeros of thefunction ₁ are simple and the zeros of the functions ₁, ₂ aredifferent and the signs of ₂ at the zeros of ₁ in [0,2/n) donot change or change more than two times, where ₁ and ₂ aredefined as follows: Moreover, it is also proved that the given equation has at leastone 2-periodic solution provided that the following conditionshold: with 1 p < q 2.     

Normal Elements of C*-Algebras of Real Rank Zero without Finite-Spectrum Approximants

We investigate inductive limits of Toeplitz-type C*-algebras.One example, which has real-rank zero, is the middle term ofan exact sequence where is a Bunce-Deddens algebra and I is AF. Using Berg's technique,we produce a normal element N that is not the limit of finite-spectrumnormals. Moreover, this is an example of a normal element inan inductive limit that is not the limit of normal elementsof the approximating subalgebras. A second example is an embedding of C() ( the closed disk) into , where is a simple AF algebra and is the Toeplitz algebra.Let _n, for n 2, be the CW complex obtained as the quotientof by an n-fold identification of the boundary. (So ₂ = RP².)Regarding C(_n) as a subalgebra of C(), we find nontrivial embeddingsof C(_n) into type I inductive limits. From this, we producea *-homomorphism, for n odd, C₀(_n\{pt}) _{n + 1}, that inducesan isomorphism on K-theory. More generally, for X a connectedCW complex minus a point, and for n odd, we show that the map is a split surjection.     

The Convergence of a Class of Quasimonotone Reaction-Diffusion Systems

It is proved that every solution of the Neumann initial-boundaryproblem converges to some equilibrium, if the system satisfies (i) F_i/u_j 0 for all 1 i j n, (ii) F(u * g(s)) h(s) * F(u) wheneveru and 0 s 1, where x *y = (x₁y₁, ..., x_ny_n) and g, h : [0, 1] [0, 1]ⁿ are continuousfunctions satisfying g_i(0) = h_i(0) = 0, g_i(1) = h_i(1) = 1, 0< g_i(s); h_i(s) < 1 for all s (0, 1) and i = 1, 2, ...,n, and (iii) the solution of the corresponding ordinary differentialequation system is bounded in . We also study the convergence of the solution of the Lotka–Volterrasystem where r_i > 0, 0, and a_ij 0 for i j.     

Regularity Conditions and Bernoulli Properties of Equilibrium States and g-Measures

When T : X X is a one-sided topologically mixing subshift offinite type and : X R is a continuous function, one can definethe Ruelle operator L : C(X) C(X) on the space C(X) of real-valuedcontinuous functions on X. The dual operator always has a probability measure as an eigenvectorcorresponding to a positive eigenvalue ( = with > 0). Necessary and sufficient conditionson such an eigenmeasure are obtained for to belong to twoimportant spaces of functions, W(X, T) and Bow (X, T). For example, Bow(X, T) if and only if is a measure with a certain approximateproduct structure. This is used to apply results of Bradleyto show that the natural extension of the unique equilibriumstate µ of Bow(X, T) has the weak Bernoulli propertyand hence is measure-theoretically isomorphic to a Bernoullishift. It is also shown that the unique equilibrium state ofa two-sided Bowen function has the weak Bernoulli property.The characterizations mentioned above are used in the case ofg-measures to obtain results on the ‘reverse’ ofa g-measure.     

Limits Along Parallel Lines and the Classical Fine Topology

The fine topology on Rⁿ (n2) is the coarsest topology for whichall superharmonic functions on Rⁿ are continuous. We refer toDoob [11, 1.XI] for its basic properties and its relationshipto the notion of thinness. This paper presents several theoremsrelating the fine topology to limits of functions along parallellines. (Results of this nature for the minimal fine topologyhave been given by Doob – see [10, Theorem 3.1] or [11,1.XII.23] – and the second author [15].) In particular,we will establish improvements and generalizations of resultsof Lusin and Privalov [18], Evans [12], Rudin [20], Bagemihland Seidel [6], Schneider [21], Berman [7], and Armitage andNelson [4], and will also solve a problem posed by the latterauthors. An early version of our first result is due to Evans [12, p.234], who proved that, if u is a superharmonic function on R³,then there is a set ER²x{0}, of two-dimensional measure 0, suchthat u(x, y,·) is continuous on R whenever (x, y, 0)E.We denote a typical point of Rⁿ by X=(X' x), where X'R^n–1and xR. Let :RⁿR^n–1x{0} denote the projection map givenby (X', x) = (X', 0). For any function f:Rⁿ[–, +] andpoint X we define the vertical and fine cluster sets of f atX respectively by C_V(f;X)={l[–, +]: there is a sequence (t_m) of numbersin R\{x} such that t_mx and f(X', t_m)l}| and C_F(f;X)={l[–, +]: for each neighbourhood N of l in [–,+], the set f^–1(N) is non-thin at X}. Sets which are open in the fine topology will be called finelyopen, and functions which are continuous with respect to thefine topology will be called finely continuous. Corollary 1(ii)below is an improvement of Evans' result.     

Essential norms and localization moduli of Sobolev embeddings for general domains

We introduce new measures of non-compactness for the embeddingoperator E_p,q():L_p¹() L_q() and describe their relations withthe essential norm of E_{p, q}(), ‘local’ isoperimetricand isocapacitary constants. An explicit formula for the essentialnorm of E_{p, q}() is obtained for domains with a power cusp onthe boundary and bounded C¹ domains. The Neumann problem fora particular Schrödinger operator is discussed on domainswith a power cusp.     

On Borel Sets in Function Spaces with the Weak Topology

It is proved that the duality map ,:(, weak)x(()*, weak*)R isnot Borel. More generally, the evaluation e:(C)(K),x KR, e(f,x) = f(x), is not Borel for any function space C(K) on a compactF-space. It is also shown that a non-coincidence of norm-Boreland weak-Borel sets in a function space does not imply thatthe duality map is non-Borel.     

A Strong Law for the Largest Nearest-Neighbour Link between Random Points

Suppose that X₁, X₂, X₃, ... are independent random points inR^d with common density f, having compact support with smoothboundary , with f| continuous. Let Rⁿ_{i, k} denote the distancefrom X_i to its kth nearest neighbour amongst the first n points,and let M_{n, k} = max_in Rⁿ_{i, k}. Let denote the volume of theunit ball. Then as n , , almost surely If instead the points lie in a compact smooth d-dimensionalRiemannian manifold K, then nM^d_{n, k}/log n (min_Kf)^–1,almost surely.     

Optimal Cardinals for Metrizable Barrelled Spaces

We seek the smallest or largest cardinals for which certainbasic results hold, as did Mazur when he proved that c is thesmallest infinite-dimensionality for a Fréchet space.As with Mazur, we make no axiomatic assumptions outside theusual ZFC model. We discover three instances in which the optimalcardinal is the dominating number and three in which it isthe bounding number b, apparently giving the first locally convexspace characterizations of these venerable and easily describedcardinals. Here are two samples: it is known that for any non-normablemetrizable locally convex space E, the minimal size _b(E) fora fundamental system of bounded sets must satisfy _{1 b}(E) c;we prove that _b(E) = . Again, it is known that if E is a non-normablemetrizable barrelled space of minimal dimension, then ₁ dim(E) c; we prove that dim(E) = b. The most important individualresult is the reconstruction of Tweddle's space without useof the Continuum Hypothesis (₁ = c). The reconstruction is vitalin the characterizations of b and in subsequent papers answeringopen questions about countable enlargements.