POLYNOMIAL NUMERICAL INDEX FOR SOME COMPLEX VECTOR-VALUED FUNCTION SPACES

We study the relation between the polynomial numerical indicesof a complex vector-valued function space and the ones of itsrange space. It is proved that the spaces C(K, X) and L(µ,X) have the same polynomial numerical index as the complex Banachspace X for every compact Hausdorff space K and every -finitemeasure µ, which does not hold any more in the real case.We give an example of a complex Banach space X such that, forevery k 2, the polynomial numerical index of order k of X isthe greatest possible, namely 1, while the one of X** is theleast possible, namely k^k/(1–k). We also give new examplesof Banach spaces with the polynomial Daugavet property, namelyL(µ, X) when µ is atomless, and C_w(K, X), C_w*(K,X*) when K is perfect.     

Lorentz Spaces Of Vector-Valued Measures

Given a non-atomic, finite and complete measure space (,,µ)and a Banach space X, the modulus of continuity for a vectormeasure F is defined as the function _F(t) = sup_µ(E)t |F|(E)and the space V^p,q(X) of vector measures such that t^–1/p'_F(t) L^q((0,µ()],dt/t) is introduced. It is shown thatV^p,q(X) contains isometrically L^p,q(X) and that L^p,q(X) = V^p,q(X)if and only if X has the Radon–Nikodym property. It isalso proved that V^p,q(X) coincides with the space of cone absolutelysumming operators from L^p',q^' into X and the duality V^p,q(X^*)=(Lp^',q^'(X))^*where 1/p+1/p^'= 1/q+1/q^' = 1. Finally, V^p,q(X) is identifiedwith the interpolation space obtained by the real method (V¹(X),V(X))_1/p^',q. Spaces where the variation of F is replaced bythe semivariation are also considered.     

Circle Actions and Morse Theory on Quaternion-Kahler Manifolds

In this paper we study quaternion-Kähler manifolds endowedwith an isometric S¹-action. We consider the corresponding momentmap µ and prove that the only compact quaternion-Kählermanifold with positive scalar curvature which admits an isometriccircle action free on µ^–1(0) is the quaternionicprojective space HPⁿ.     

Non-Complex Symplectic 4-Manifolds with Formula

Simply connected closed symplectic 4-manifolds with and K² = 0 are investigated. As aresult, it is confirmed that most of homotopy elliptic surfaces{E(1)_k|K is a fibred knot in S³} constructed by R. Fintusheland R. Stern in Invent. Math. 134 (1998) 363–400 are simplyconnected closed minimal symplectic 4-manifolds that do notadmit a complex structure. 2000 Mathematics Subject Classification57R17, 57R57 (primary), 14J26 (secondary).     

Bounded-Norm Matrix-Inverse Mappings

This paper introduces the concept of bounded-norm matrix-inversemappings, i.e. mappings µ : R^mxnR^nxm such that, for allnonzero mxn matrices A, the matrix µ(A) is a generalizedinverse of A and ||µ(A)||> k/s(A), where K < 0 isa constant and s(A) is the nonzero singular value of A havingsmallest absolute value. It is shown how the definition of suchmappings is motivated by the need to ensure finite terminationof the inner-iterations of generalized elimination methods forthe solution of nonlinearly constrained optimization problems.The main result of the paper is that the mapping defined byµ(A) = A^b is a bounded-norm matrix-inverse mapping, providedthat the basic inverse A^b is calculated using Gaussian eliminationwith complete pivoting. The concept of bounded-norm matrix-inversemappings is then extended to that of boundednorm least-squaresmatrix-inverse mappings. It is proved that the mapping definedby µ(A) = A^ß is a bounded-norm least-squaresmatrix-inverse mapping, provided that the basic least-squaresinverse A^ß is calculated using the QR decompositionwith column pivoting.     

Generalisation of the Bogomolov-Miyaoka-Yau Inequality to Singular Surfaces

The paper considers pairs (X, B) where X is a normal projectivesurface over C, and B is a Q-divisor whose coefficients are1 or 1–1/m for some natural number m. A log canonicalsingularity on such a pair is a quotient by a finite or infinitegroup, so if (X, B) has log canonical singularities, the orbifoldEuler number e_orb(X, B) can be defined. The main result is aBogomolov-Miyaoka-Yau-type inequality which implies that if(X, B) has log canonical singularities and (X, K_X + B) 0 then(K_X+B)² 3e_orb(X, B). The actual inequality proved is somewhatstronger and it also implies all the previously published versionsof the Bogomolov-Miyaoka-Yau inequality. The proof involvesthe Log Minimal Model Program, Q-sheaves when K_X+B is nef, anda study of the changes in the two sides of the inequality undera contraction. The paper also contains a further generalisationwhere the coefficients of B can be arbitrary rational numbersin [0, 1], a different condition is imposed on the singularitiesand K_X+B is required to be nef. Some applications of the inequalitiesare also given, for example, estimating the number of singularitiesor certain kinds of configurations of curves on surfaces. 1991Mathematics Subject Classification: 14J17, 14J60, 14C17.     

The Derivation Problem for Group Algebras of Connected Locally Compact Groups

The derivation problem for a locally compact group G is to decidewhether for each derivation D from L¹(G) into L¹(G) there isa bounded measure µM(G) with D(a) = aµ–µa(a L¹(G)). In this paper we obtain an affirmative answer forthe case of connected groups. To explain the contents of thispaper we give an equivalent formulation of the problem. Supposethat the group G acts as a group of homeomorphisms of the locallycompact space X. Related to this there is an action of G onM(X). A bounded crossed homomorphism from G to M(X) is a map with bounded range and satisfying (gh) = g(h)+(g) (g, h G).The problem for bounded crossed homomorphisms is to decide iffor each such there is an element µ of M(X) with (g)= gµ– µ (g G). The derivation problem isequivalent to this bounded crossed homomorphism problem forthe special case X = G where G acts on X by conjugation (togetherwith some mild continuity hypotheses about the map :GM(X) whichare often automatically satisfied). The bounded crossed homomorphismproblem always has a positive solution if G is amenable anda closely related calculation shows that in solving the boundedcrossed homomorphism problem we need only solve it for functions which are zero on H where H is a given amenable subgroup ofG. It can happen that this condition of being zero on H forces to be zero even when H is a comparatively small subgroup ofG. If h is an element of G such that ‘hⁿx ’ asn for all x X then for any two measures µ and , forlarge values of n, µ and hⁿ have little overlap so ||µ+ hⁿ|| ||µ|| + ||||. Thus if H is the subgroup generatedby h, for any g G .     

Eigenvectors of Order-Preserving Linear Operators

Suppose that K is a closed, total cone in a real Banach spaceX, that A:XX is a bounded linear operator which maps K intoitself, and that A' denotes the Banach space adjoint of A. Assumethat r, the spectral radius of A, is positive, and that thereexist x₀0 and m1 with A^m(x₀)=r^mx₀ (or, more generally, thatthere exist x₀(–K) and m1 with A^m(x₀)r^mx₀). If, in addition,A satisfies some hypotheses of a type used in mean ergodic theorems,it is proved that there exist uK–{0} and K'–{0}with A(u)=ru, A'()=r and (u)>0. The support boundary of Kis used to discuss the algebraic simplicity of the eigenvaluer. The relation of the support boundary to H. Schaefer's ideasof quasi-interior elements of K and irreducible operators Ais treated, and it is noted that, if dim(X)>1, then thereexists an xK–{0} which is not a quasi-interior point.The motivation for the results is recent work of Toland, whoconsidered the case in which X is a Hilbert space and A is self-adjoint;the theorems in the paper generalize several of Toland's propositions.     

On the L1 Mean of the Exponential Sum Formed with the Mobius Function

In this paper we study the L¹ mean (1) of the exponential sum M()=_nXµ(n)e(n), where µ(n)is the Möbius function and e(x)=e^2ix. From the Cauchy–Schwarzinequality and Parseval's identity, we have , (2) and it is an interesting problem to investigate whether (2)reflects the true order of magnitude of (1).     

Necessary and Sufficient Conditions for Exponential Stability and Ultimate Boundedness of Systems Governed by Stochastic Partial Differential Equations

Consider the following infinite dimensional stochastic evolutionequation over some Hilbert space H with norm |·|: It is proved that under certain mild assumptions, the strongsolution X_t(x₀)VHV*, t 0, is mean square exponentially stableif and only if there exists a Lyapunov functional (·,·):HxR₊R¹ which satisfies the following conditions: (i)c₁|x|²–k₁e^–µ₁t(x,t)c₂|x|²+k₂+k₂e^–µ₂t; (ii) L(x,t)–c₃(x,t)+k₃e^–µ₃t, xV, t0; where L is the infinitesimal generator of the Markov processX_t and c_i, k_i, µ_i, i = 1, 2, 3, are positive constants.As a by-product, the characterization of exponential ultimateboundedness of the strong solution is established as the nulldecay rates (that is, µ_i = 0) are considered.     

Complete Manifolds with Sectional Curvature Bounded Below and Large Volume Growth

This paper provides a proof that an n-dimensional complete openRiemannian manifold M with sectional curvature K_M –1is diffeomorphic to a Euclidean n-space Rⁿ if the volume growthof geodesic balls in M is close to that of the balls in an n-dimensionalhyperbolic space Hⁿ(–1) of sectional curvature –1.     

The Milnor number of a function on a space curve germ

Given a finite function germ f:(X, 0) (, 0) on a reduced spacecurve singularity (X, 0), we show that µ(f) = µ(X,0) + deg(f) – 1. Here, µ(f) and µ(X, 0) denotethe Milnor numbers of the function and the curve, respectively,and deg(f) is the degree of f. We use this formula to obtainseveral consequences related to the topological triviality andWhitney equisingularity of families of curves and families offunctions on curves.     

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.     

On regularity of a quasiconformal mapping at a point of maximal stretching

Local behaviour of a K-quasiconformal mapping f at a point z₀of maximal stretching is studied. A sufficient condition forthe existence of the finite limit lim(f(z) – f(z₀))/(z– z₀)|z – z₀|^1/K–1 as z z₀, and a criterionfor z₀ to be a point of maximal stretching are given.     

Sharp Geometric Poincare Inequalities for Vector Fields and Non-Doubling Measures

We derive Sobolev–Poincaré inequalities that estimatethe L^q(d µ) norm of a function on a metric ball when µis an arbitrary Borel measure. The estimate is in terms of theL¹(d ) norm on the ball of a vector field gradient of the function,where d dx is a power of a fractional maximal function of µ.We show that the estimates are sharp in several senses, andwe derive isoperimetric inequalities as corollaries. 1991 MathematicsSubject Classification: 46E35, 42B25.     

Periodicity in Group Cohomology and Complete Resolutions

A group G is said to have periodic cohomology with period qafter k steps, if the functors Hⁱ(G, –) and H^i+q(G, –)are naturally equivalent for all i > k. Mislin and the authorhave conjectured that periodicity in cohomology after some stepsis the algebraic characterization of those groups G that admita finite-dimensional, free G-CW-complex, homotopy equivalentto a sphere. This conjecture was proved by Adem and Smith underthe extra hypothesis that the periodicity isomorphisms are givenby the cup product with an element in H^q(G,Z). It is expectedthat the periodicity isomorphisms will always be given by thecup product with an element in H^q(G,Z); this paper shows thatthis is the case if and only if the group G admits a completeresolution and its complete cohomology is calculated via completeresolutions. It is also shown that having the periodicity isomorphismsgiven by the cup product with an element in H^q(G,Z) is equivalentto silp G being finite, where silp G is the supremum of theinjective lengths of the projective ZG-modules. 2000 MathematicsSubject Classification 20J05, 57S25.     

Hypersurface Singularities with 2-Dimensional Critical Locus

Consider an analytic germ f:(C^m, 0)(C, 0) (m3) whose criticallocus is a 2-dimensional complete intersection with an isolatedsingularity (icis). We prove that the homotopy type of the Milnorfiber of f is a bouquet of spheres, provided that the extendedcodimension of the germ f is finite. This result generalizesthe cases when the dimension of the critical locus is zero [8],respectively one [12]. Notice that if the critical locus isnot an icis, then the Milnor fiber, in general, is not homotopicallyequivalent to a wedge of spheres. For example, the Milnor fiberof the germ f:(C⁴, 0)(C, 0), defined by f(x₁, x₂, x₃, x₄) =x₁x₂x₃x₄ has the homotopy type of S¹xS¹xS¹. On the other hand,the finiteness of the extended codimension seems to be the rightgeneralization of the isolated singularity condition; see forexample [9–12, 17, 18]. In the last few years different types of ‘bouquet theorems’have appeared. Some of them deal with germs f:(X, x)(C, 0) wheref defines an isolated singularity. In some cases, similarlyto the Milnor case [8], F has the homotopy type of a bouquetof (dim X–1)-spheres, for example when X is an icis [2],or X is a complete intersection [5]. Moreover, in [13] Siersmaproved that F has a bouquet decomposition FF₀Sⁿ...Sⁿ (whereF₀ is the complex link of (X, x)), provided that both (X, x)and f have an isolated singularity. Actually, Siersma conjecturedand Tibr proved [16] a more general bouquet theorem for thecase when (X, x) is a stratified space and f defines an isolatedsingularity (in the sense of the stratified spaces). In thiscase F_iF_i, where the F_i are repeated suspensions of complexlinks of strata of X. (If (X, x) has the ‘Milnor property’,then the result has been proved by Lê; for details see[6].) In our situation, the space-germ (X, x) is smooth, but f hasbig singular locus. Surprisingly, for dim Sing f^–1(0)2,the Milnor fiber is again a bouquet (actually, a bouquet ofspheres, maybe of different dimensions). This result is in thespirit of Siersma's paper [12], where dim Sing f^–1(0)= 1. In that case, there is only a rather small topologicalobstruction for the Milnor fiber to be homotopically equivalentto a bouquet of spheres (as explained in Corollary 2.4). Inthe present paper, we attack the dim Sing f^–1(0) = 2 case.In our investigation some results of Zaharia are crucial [17,18].     

Tjurina and Milnor Numbers of Matrix Singularities

To gain understanding of the deformations of determinants andPfaffians resulting from deformations of matrices, the deformationtheory of composites f F with isolated singularities is studied,where f : YC is a function with (possibly non-isolated) singularityand F : XY is a map into the domain of f, and F only is deformed.The corresponding T¹(F) is identified as (something like) thecohomology of a derived functor, and a canonical long exactsequence is constructed from which it follows that = µ(f F) – ß₀ + ß₁, where is the length of T¹(F) and ß_i is the lengthof Tor_i^O_Y(O_Y/J_f, O_X). This explains numerical coincidences observedin lists of simple matrix singularities due to Bruce, Tari,Goryunov, Zakalyukin and Haslinger. When f has Cohen–Macaulaysingular locus (for example when f is the determinant function),relations between and the rank of the vanishing homology ofthe zero locus of f F are obtained.     

A Mod Two Analogue of a Conjecture of Cooke

The mod two cohomology of the three connective covering of S³has the form F₂[X_2n] E(Sq¹X_2n) where x_2n is in degree 2n and n = 2. If F denotes the homotopytheoretic fibre of the map S³ B²S¹ of degree 2, then the mod2 cohomology of F is also of the same form for n = 1. Notice(cf. Section 7 of the present paper) that the existence of spaceswhose cohomology has this form for high values of n would immediatelyprovide Arf invariant elements in the stable stem. Hence, itis worthwhile to determine for what values of n the above algebracan be realized as the mod2 cohomology of some space. The purposeof this paper is to construct a further example of a space withsuch a cohomology algebra for n = 4 and to show that no othervalues of n are admissible. More precisely, we prove the following.