Convergence almost Everywhere of Certain Partial Sums of Fourier Integrals

Suppose that R goes to infinity through a second-order lacunaryset. Let S_R denote the Rth spherical partial inverse Fourierintegral on R^d. Then S_Rf converges almost everywhere to f, providedthat f satisfies . 2000 Mathematics SubjectClassification 42B15, 42B25, 42B08.     

Auslander-Regular Algebras and Maximal Orders

Let R be an Auslander-regular, Cohen-Macaulay, Noetherian ringthat is stably free. Then, we prove that R is a domain and amaximal order in its division ring of fractions. In particular,this applies to the Sklyanin algebra S and shows that, whenS satisfies a polynomial identity, it is actually a finite moduleover its centre.     

Indecomposable flat cotorsion modules

An additive functor from the category of flat right R-modulesto the category of abelian groups is continuous if it is isomorphicto a functor of the form–_R M, where M is a left R-module.It is shown that for any simple subfunctor A of– M thereis a unique indecomposable flat cotorsion module U_R for whichA(U)0. It is also proved that every subfunctor of a continuousfunctor contains a simple subfunctor. This implies that everyflat right R-module may be purely embedded into a product ofindecomposable flat cotorsion modules. If CE(R) is the cotorsion envelope of R_R and S= End;_R CE(R),then a local ring monomorphism is constructed from R/J(R) toS/J(S). This local morphism of rings is used to associate asemiperfect ring to any semilocal ring. It also proved thatif R is a semilocal ring and M a simple left R-module, thenthe functor–_R M on the category of flat right R-modulesis uniform, and therefore contains a unique simple subfunctor.     

A Hilbert Basis Theorem for Quantum Groups

A noncommutative version of the Hilbert basis theorem is usedto show that certain R-symmetric algebras S_R(V) are Noetherian.This result applies in particular to the coordinate ring ofquantum matrices A_R(V) associated with an R-matrix R operatingon the tensor square of a vector space V, to show that, undera natural set of hypotheses on R, the algebra A_R(V) is Noetherianand its augmentation ideal has a polynormal set of generators.As a corollary we deduce that these properties hold for thegeneric quantized function algebras R_q[G] over any field ofcharacteristic zero, for G an arbitrary connected, simply connected,semisimple group over C. That R_q[G] is Noetherian recovers aresult due to Joseph [10], with a different proof.1991 MathematicsSubject Classification 17B37, 16P40.     

An Optimal Krasnosel'skii-Type Theorem for the Dimension of the Kernel of a Starshaped Set

It is shown for a closed connected nonconvex and locally compactsubset S of a real normed linear space that ker S = {conv S_z: z D reg S}, where reg S denotes the set of regular pointsof S, D is a relatively open subset of S containing the setlnc S of local nonconvexity points of S, and S_z = {s S : zis visible from s via S}. An analogous intersection formula,with the set sph S of spherical points of S in place of regS, is shown to hold for a closed connected nonconvex set S ina real Banach space which is uniformly convex and uniformlysmooth. A routine procedure then leads to a Krasnosel'skii-typecharacterization of the dimension of the kernel of a closedconnected nonconvex subset S of R^d with lnc S bounded. Thisimproves a recent result by removing the hypothesis of compactnessof S and settles an open problem.     

Stability and Accuracy of Semi-discretized Finite Difference Methods

The relationship between the accuracy and stability of semi-discretizedfinite-difference schemes for the advection equation u_t = u_xis analysed. Given the scheme of accuracy p we establish that,subject to stability, p min {r+s+R+S, 2(r+R+1), 2(s+S)}. This is done by using the theory of order stars. Furthermore,we find stable methods which attain this bound for various choicesof r, s, R and S.     

A Note on a Maximal Function Over Arbitrary Sets of Directions

Let M_S be the universal maximal operator over unit vectors ofarbitrary directions. This operator is not bounded in L²(R²).We consider a sequence of operators over sets of finite equidistributeddirections converging to M_S. We provide a new proof of N. Katz'sbound for such operators. As a corollary, we deduce that M_Sis bounded from some subsets of L² to L². These subsets arecomposed of positive functions whose Fourier transforms havea logarithmic decay or which are supported on a disc. 1991 MathematicsSubject Classification 42B25.     

Average Linking Numbers of Closed Orbits of Hyperbolic Flows

The existence of an asymptotic average of linking numbers ofperiodic orbits of a basic set of an Axiom A flow in S³ or R³is proved.     

Finiteness Properties of Certain Metabelian Arithmetic Groups in the Function Field Case

Consider the group scheme where R is an arbitrary commutative ring with 1 0 and a unitx R* acts on R by multiplication. We will study the finiteness properties of subgroups of G(O_S)where O_S is an S-arithmetic subring of a global function field.The subgroups we are interested in are of the form where Q is a subgroup of O_S*. The finiteness propertiesof these metabelian groups can be expressed in terms of the-invariant due to R. Bieri and R. Strebel. Theorem A. Let S be a finite set of places of a global functionfield (regarded as normalized discrete valuations) and O_S thecorresponding S-arithmetic ring. Let Q be a subgroup of O_S*.Then Q is finitely generated and for all integers n 1 the followingare equivalent:

(1) O_S Q is of type FP_n;

(2) O_S is n-tameas a ZQ-module;

(3) each p S restricts to a non-trivial homomorphism and the set is n-tame.

If these conditions hold for at least one n 1 then the identity holds.} Theorem B. Let r denote the rank of Q. Then the followinghold:

(1) the group O_S Q is not of type FP_r+1};

(2) if Qhas maximum rank r = |S| –1 then the group O_S Q is oftype FP_r.

In particular, is of type FP_{|S| –1} but not of type FP_|S|. 1991 Mathematics SubjectClassification: 20E08, 20F16, 20G30, 52A20.     

An Euler-Type Version of the Local Steiner Formula for Convex Bodies

The purpose of this note is to establish a new version of thelocal Steiner formula and to give an application to convex bodiesof constant width. This variant of the Steiner formula generalizesresults of Hann [3] and Hug [6], who use much less elementarytechniques than the methods of this paper. In fact, Hann askedfor a simpler proof of these results [4, Problem 2, p. 900].We remark that our formula can be considered as a Euclideananalogue of a spherical result proved in [2, p. 46], and thatour method can also be applied in hyperbolic space. For some remarks on related formulas in certain two-dimensionalMinkowski spaces, see Hann [5, p. 363]. For further information about the notions used below, we referto Schneider's book [9]. Let Kⁿ be the set of all convex bodiesin Euclidean space Rⁿ, that is, the set of all compact, convex,non-empty subsets of Rⁿ. Let S^n–1 be the unit sphere.For KKⁿ, let NorK be the set of all support elements of K, thatis, the pairs (x, u)RⁿxS^n–1 such that x is a boundarypoint of K and u is an outer unit normal vector of K at thepoint x. The support measures (or generalized curvature measures)of K, denoted by ₀(K.), ..., _n–1(K.), are the unique Borelmeasures on RⁿxS^n–1 that are concentrated on NorK andsatisfy [formula] for all integrable functions f:RⁿR; here denotes the Lebesguemeasure on Rⁿ. Equation (1), which is a consequence and a slightgeneralization of Theorem 4.2.1 in Schneider [9], is calledthe local Steiner formula. Our main result is the following.1991 Mathematics Subject Classification 52A20, 52A38, 52A55.     

Cocompact Spherical-Euclidean Spaceform Groups of Infinite VCD

We exhibit closed manifolds M covered by S^2n–1 x R^k forall n 2 and for sufficiently large k, with fundamental groupsof infinite virtual cohomological dimension. These examplesare based on results of Raghunathan on lattices in covers ofspin and symplectic groups, and address a problem first raisedby Wall.     

On the number of homotopy types of fibres of a definable map

In this paper we prove a single exponential upper bound on thenumber of possible homotopy types of the fibres of a Pfaffianmap in terms of the format of its graph. In particular, we showthat if a semi-algebraic set SR^m+n, where R is a real closedfield, is defined by a Boolean formula with s polynomials ofdegree less than d, and : R^m+nRⁿ is the projection on a subspace,then the number of different homotopy types of fibres of doesnot exceed s^2(m+1)n(2^m nd)^O(nm). As applications of our mainresults we prove single exponential bounds on the number ofhomotopy types of semi-algebraic sets defined by fewnomials,and by polynomials with bounded additive complexity. We alsoprove single exponential upper bounds on the radii of ballsguaranteeing local contractibility for semi-algebraic sets definedby polynomials with integer coefficients.     

Bihermitian Structures on Complex Surfaces

Bihermitian complex surfaces are oriented conformal four-manifoldsadmitting two independent compatible complex structures. Non-anti-self-dualbihermitian structures on R⁴ and the four-dimensional torusT⁴ have recently been discovered by P. Kobak. We show that anoriented compact 4-manifold, admitting a non-anti-self-dualbihermitian structure, is a torus or K3 surface in the stronglybihermitian case (when the two complex structures are independentat each point) or, otherwise, must be obtained from the complexprojective plane or a minimal ruled surface of genus less than2 by blowing up points along some anti-canonical divisor (butthe actual existence of bihermitian structures in the lattercase is still an open question). The paper includes a generalmethod for constructing non-anti-self-dual bihermitian structureson tori, K3 surfaces and S¹ x S³. Further properties of compactbihermitian surfaces are also investigated. 1991 MathematicsSubject Classification: 53C12, 53C55, 32J15.     

Shapiro's Cyclic Sum

Shapiro's cyclic sum is defined by , If K is the cone in Rⁿ of points withnon-negative coordinates, it is shown that the minimum of Ein K is a fixed point of T², where T is the non-linear operatordefined by (Tx)_i = x_n–i+1/(x_n–i+2 + x_n–i+3)²for i = 1,2,...,n. It is conjectured that Tx = S^kx, where Sis the shift operator in Rⁿ, and a proof is given under someadditional hypotheses. One of the consequences is a simple proofthat at the minimum point, a_i(x) = a_n–i+1–k(x) fori = 1,2,...,n.     

Stable Jacobson Radicals and Semiprime Smash Products

We prove that if H is a finite-dimensional semisimple Hopf algebraacting on a PI-algebra R of characteristic 0, and R is eitheraffine or algebraic over k, then the Jacobson radical of R isH-stable. Under the same hypotheses, we show that the smashproduct algebra R#H is semiprimitive provided that R is H-semiprime.More generally we show that the ‘finite’ Jacobsonradical is H-stable, and that R#H is semiprimitive providedthat R is H-semiprimitive and all irreducible representationsof R are finite-dimensional. We also consider R#H when R isan FCR-algebra. Finally, we prove a general relationship betweenstability of the radical and semiprimeness of R#H; in particularif for a given H, any action of H stabilizes the Jacobson radical,then also any action of H stabilizes the prime radical. 2000Mathematics Subject Classification 16W30, 16N20, 16R99, 16S40.     

On 4-manifolds with zero-entropy metrics

We sharpen a recent result of Paternain and Petean by showingthat a closed 4-manifold which admits a Riemannian metric withzero topological entropy and infinite fundamental group eitheris aspherical or has a finite covering space homeomorphic toS³ x S¹ or S² x S¹ x S¹.     

A Criterion for Finite Topological Determinacy of Map-Germs

Let f, g: (Rⁿ, 0) (R^p, 0) be two C map-germs. Then f and gare C⁰-equivalent if there exist homeomorphism-germs h and lof (Rⁿ, 0) and (R^p, 0) respectively such that g = l f h^–1.Let k be a positive integer. A germ f is k-C⁰-determined ifevery germ g with j^k g(0) = j^k f(0) is C⁰-equivalent to f. Moreover,we say that f is finitely topologically determined if f is k-C⁰-determinedfor some finite k. We prove a theorem giving a sufficient conditionfor a germ to be finitely topologically determined. We explainthis condition below. Let N and P be two C manifolds. Consider the jet bundle J^k(N,P) with fiber J^k(n, p). Let z in J^k(n, p) and let f be suchthat z = j^kf(0). Define Whether (f) < k depends only on z, not on f. We can thereforedefine the set Let W^k(N, P) be the subbundle of J^k(N, P) with fiber W^k(n, p).Mather has constructed a finite Whitney (b)-regular stratificationS^k(n, p) of J^k(n, p) – W^k(n, p) such that all strata aresemialgebraic and K-invariant, having the property that if S^k(N,P) denotes the corresponding stratification of J^k(N, P) –W^k(N, P) and f C(N, P) is a C map such that j^kf is multitransverseto S^k(N, P), j^kf(N) W^k(N, P) = and N is compact (or f is proper),then f is topologically stable. For a map-germ f: (Rⁿ, 0) (R^p, 0), we define a certain ojasiewiczinequality. The inequality implies that there exists a representativef: U R^p such that j^kf(U – 0) W^k (Rⁿ, R^p = and suchthat j^kf is multitransverse to S^k (Rⁿ, R^p) at any finite setof points S U – 0. Moreover, the inequality controlsthe rate j^kf becomes non-transverse as we approach 0. We showthat if f satisfies this inequality, then f is finitely topologicallydetermined. 1991 Mathematics Subject Classification: 58C27.     

Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

We prove a nearly optimal bound on the number of stable homotopytypes occurring in a k-parameter semi-algebraic family of setsin R, each defined in terms of m quadratic inequalities. Ourbound is exponential in k and m, but polynomial in . More precisely,we prove the following. Let R be a real closed field and let = {P₁, ... , P_m} R[Y₁, ... ,Y,X₁, ... ,X_k], with deg_Y(P_i) 2, deg_X(P_i) d, 1 i m. Let S R^+k be a semi-algebraic set,defined by a Boolean formula without negations, with atoms ofthe form P 0, P 0, P . Let : R^+k R^k be the projection onthe last k coordinates. Then the number of stable homotopy typesamongst the fibers S_x = ^–1(x) S is bounded by (2^mkd)^O(mk).     

Dualizing Complex and the Canonical Element Conjecture II

In this paper we continue our study of the Canonical ElementConjecture (henceforth C.E.C.) via the dualizing complex. Throughoutthe work (A, m, k) will denote a noetherian complete local ringA of dimension n, m its maximal ideal and k=A/m. Since A iscomplete, we can find a complete local Gorenstein ring (R, m_R,k) (complete intersection) such that dim R=dim A and A=R/I.Let denote the canonical module of A, that is, =Hom_R (A, R),which may be identified with the annihilator of I in R, an idealof R. When A is a domain, we change notation and denote I byP; in this case P is a height 0 prime ideal of R.     

Correction: the Embedding of Radical Rings in Simple Radical Rings

As G. M. Bergman has pointed out, in the proof of the lemmaon p. 187, we cannot conclude that $$\stackrel{\&macr;}{S}$$is universal in the sense stated. However, the proof can becompleted as follows: Any element of $$\stackrel{\&macr;}{S}$$can be obtained as the first component of the solution u ofa system (A–I)u+a = 0, (1) where A S_n, a ⁿS and A–I has an inverse over L. SinceS is generated by R and k{s}, A can (by the last part of Lemma3.2 of [1]) be taken to be linear in these arguments, say A= A₀ + sA1, where A₀ R_n, A₀ R_n, A₁ K_n. Multiplying by (I–sA₁)^–1,we reduce this equation to the form (S_vB_v–I)u+a=0, (2) with the same solution u as before, where B_v R_n, s_v k{s}¹and a ⁿS. Now consider the retraction S k{s} (3) obtained by mapping R 0. If we denote its effect by x x^*,then (2) goes over into an equation –I.v + a^* 0, (4) which clearly has a unique solution v in k{s}; therefore theretraction (3) can be extended to a homomorphism $$\stackrel{\&macr;}{S}$$ k{s}, again denoted by x x^*, provided we can show that u₁^*does not depend on the equation (1) used to define it. Thisamounts to showing that if an equation (1), or equivalently(2), has the solution u₁ = 0, then after retraction we get v₁= 0 in (4), i.e. a₁^* = 0. We shall use induction on n; if u₁= 0 in (2), then by leaving out the first row and column ofthe matrix on the left of (2), we have an equation for u₂,...,u_n and by the induction hypothesis, their values after retractionare uniquely determined. Now from (2) we have where B = (b_ij^v). Applying ^* and observing that b_ij^vR, we seethat a₁ ^* = 0, as we wished to show. The proof still appliesfor n = 1, so we have a well-defined mapping $$\stackrel{\&macr;}{S}$$ k{s}, which is a homomorphism. Now the proof of the lemma canbe completed as before.