One-Dimensional p-Adic Subanalytic Sets

In this paper we extend two theorems from [2] on p-adic subanalyticsets, where p is a fixed prime number, Q_p is the field of p-adicnumbers and Z_p is the ring of p-adic integers. One of thesetheorems [2, 3.32] says that each subanalytic subset of Z_p issemialgebraic. This is extended here as follows.     

On Artin's Conjecture, I: Systems of Diagonal Forms

As a special case of a well-known conjecture of Artin, it isexpected that a system of R additive forms of degree k, say [formula] with integer coefficients a_ij, has a non-trivial solution inQ_p for all primes p whenever [formula] Here we adopt the convention that a solution of (1) is non-trivialif not all the x_i are 0. To date, this has been verified onlywhen R=1, by Davenport and Lewis [4], and for odd k when R=2,by Davenport and Lewis [7]. For larger values of R, and in particularwhen k is even, more severe conditions on N are required toassure the existence of p-adic solutions of (1) for all primesp. In another important contribution, Davenport and Lewis [6]showed that the conditions [formula] are sufficient. There have been a number of refinements of theseresults. Schmidt [13] obtained N>>R²k³ log k, and Low,Pitman and Wolff [10] improved the work of Davenport and Lewisby showing the weaker constraints [formula] to be sufficient for p-adic solubility of (1). A noticeable feature of these results is that for even k, onealways encounters a factor k³ log k, in spite of the expectedk² in (2). In this paper we show that one can reach the expectedorder of magnitude k². 1991 Mathematics Subject Classification11D72, 11D79.     

Minima of Trigonometric Polynomials

Let 0<n₁<n₂<...<n_N be N distinct integers, and leta₁, a₂, ..., a_N be complex numbers. We set [formula] and [formula] There are two well-known problems concerning the case a₁=a₂=...=a_N=1.1991 Mathematics Subject Classification 42A05.     

Estimates for Commutators of Orthogonal Fourier Series

In this paper we study weighted norm inequalities for the commutators[b, S_n] where b is a BMO function and S_n denotes the nth partialsum of the Fourier series relative to a system of orthogonalpolynomials on [–1, 1] with respect to general weights.Results about generalized Jacobi and Bessel Fourier series areobtained.     

Supplementary Note on 'Rational Invariants of Certain Orthogonal and Unitary Groups'

Given a field k and a finite group G acting on the rationalfunction field k(X₁, ..., X_n) as a group of k-automorphisms,an important Noether's problem asks whether the invariant subfield [forumal] is purely transcendental over k. 1991 Mathematics Subject Classification12F20, 20G40.     

Real closed fields with non-standard and standard analytic structure

We consider the ordered field which is the completion of thePuiseux series field over equipped with a ring of analyticfunctions on [–1, 1]ⁿ which contains the standard subanalyticfunctions as well as functions given by t-adically convergentpower series, thus combining the analytic structures of Denefand van den Dries [Ann. of Math. 128 (1988) 79–138] andLipshitz and Robinson [Bull. London Math. Soc. 38 (2006) 897–906].We prove quantifier elimination and o-minimality in the correspondinglanguage. We extend these constructions and results to rankn ordered fields _n (the maximal completions of iterated Puiseuxseries fields). We generalize the example of Hrushovski andPeterzil [J. Symbolic Logic 72 (2007) 119–122] of a sentencewhich is not true in any o-minimal expansion of (shown in [Bull.London Math. Soc. 38 (2006) 897–906] to be true in ano-minimal expansion of the Puiseux series field) to a towerof examples of sentences _n, true in _n, but not true in any o-minimalexpansion of any of the fields , ₁, ..., _n–1.     

A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges

The solution of the Stokes problem in three-dimensional domainswith edges has anisotropic singular behaviour which is treatednumerically by using anisotropic finite element meshes. Thevelocity is approximated by Crouzeix–Raviart (nonconformingP₁ ) elements and the pressure by piecewise constants. Thismethod is stable for general meshes (without minimal or maximalangle condition). Denoting by N_e the number of elements in themesh, the interpolation and consistency errors are of the optimalorder h N_e^–1/3 which is proved for tensor product meshes.As a by-product, we analyse also nonconforming prismatic elementswith P₁ [oplus ] span {x₃²} as the local space for the velocitywhere x₃ is the direction of the edge.     

Projective Prime Ideals and Localisation in PI-Rings

The results here generalise [2, Proposition 4.3] and [9, Theorem5.11]. We shall prove the following. THEOREM A. Let R be a Noetherian PI-ring. Let P be a non-idempotentprime ideal of R such that P_R is projective. Then P is leftlocalisable and R_P is a prime principal left and right idealring. We also have the following theorem. THEOREM B. Let R be a Noetherian PI-ring. Let M be a non-idempotentmaximal ideal of R such that M_R is projective. Then M has theleft AR-property and M contains a right regular element of R.     

On Simultaneous Diagonal Inequalities

Let F₁, ..., F_t be diagonal forms of degree k with real coefficientsin s variables, and let be a positive real number. The solubilityof the system of inequalities |F₁(x)|<,...,|F_t(x)|< in integers x₁, ..., x_s has been considered by a number of authorsover the last quarter-century, starting with the work of Cook[9] and Pitman [13] on the case t = 2. More recently, Brüdernand Cook [8] have shown that the above system is soluble providedthat s is sufficiently large in terms of k and t and that theforms F₁, ..., F_t satisfy certain additional conditions. Whathas not yet been considered is the possibility of allowing theforms F₁, ..., F_t to have different degrees. However, with therecent work of Wooley [18,20] on the corresponding problem forequations, the study of such systems has become a feasible prospect.In this paper we take a first step in that direction by studyingthe analogue of the system considered in [18] and [20]. Let₁, ..., _s and µ₁, ..., µ_s be real numbers such thatfor each i either _i or µ_i is nonzero. We define the forms and consider the solubility of the system of inequalities in rational integers x₁, ..., x_s. Although the methods developedby Wooley [19] hold some promise for studying more general systems,we do not pursue this in the present paper. We devote most ofour effort to proving the following theorem.     

Asymptotically Sharp Bounds in the Hardy-Littlewood Inequalities on Mean Values of Analytic Functions

Let f be analytic in the unit disc, and let it belong to theHardy space H^p, equipped with the usual norm ||f||_p. It is knownfrom the work of Hardy and Littlewood that for q > _p, theconstants [formula] with the usual extension to the case where q = , have C(p,q)< . The authors prove that [formula] and [formula] 2000 Mathematics Subject Classification 30D55, 30A10.     

The Singularities of Quantum Groups

If C[G] C[H] is an extension of Hopf domains of degree d, thenH G is an étale map. Equivalently, the variety X^{C}[H]of d-dimensional C[H]-modules compatible with the trace mapof the extension, is a smooth GL_d-variety with quotient G. Ifwe replace C[H] by a non-commutative Hopf algebra H, we constructsimilarly a GL^d-variety and quotient map : X_H G. The smoothlocus of H over C[G] is the set of points g G such that X_His smooth along {^-1}(g). We relate this set to the separabilitylocus of H over C[G] as well as to the (ordinary) smooth locusof the commutative extension C[G] Z where Z is the centre ofH. In particular, we prove that the smooth locus coincides withthe separability locus whenever H is a reflexive Azumaya algebra.This implies that the quantum function algebras O(G) and quantisedenveloping algebras U{g} are as singular as possible. 1991 MathematicsSubject Classification: 16W30, 16R30.     

Best Approximation in an Asymmetrically Weighted L1 Measure

Let f(x) be a given, real-valued, continuous function definedon an interval [a,b]of the real line. Given a set of m real-valued,continuous functions _j(x) defined on [a,b], a linear approximatingfunction can be formed with any real setA = {a₁, a₂,..., a_m}. We present results for determining A sothat F(A, x) is a best approximation to(x) when the measureof goodness of approximation is a weighted sum of |F(A, x)–f(x)|,the weights being positive constants, w, when F(A, x) f(x)and w2 otherwise (when w, = w2 = 1, the measure is the L₁, norm).The results are derived from a linear programming formulationof the problem. In particular, we give a theorem which shows when such bestapproximations interpolate the function at fixed ordinates whichare independent of f(x). We show how the fixed points can becalculated and we present numerical results to indicate thatthe theorem is quite robust.     

Regularity of Rees Algebras

Let B = k[x₁, ..., x_n] be a polynomial ring over a field k,and let A be a quotient ring of B by a homogeneous ideal J.Let m denote the maximal graded ideal of A. Then the Rees algebraR = A[m t] also has a presentation as a quotient ring of thepolynomial ring k[x₁, ..., x_n, y₁, ..., y_n] by a homogeneousideal J^*. For instance, if A = k[x₁, ..., x_n], then Rk[x₁,...,x_n,y₁,...,y_n]/(x_iy_j–x_jy_i|i, j=1,...,n). In this paper we want to compare the homological propertiesof the homogeneous ideals J and J^*.     

The Singular Homology of the Hawaiian Earring

The singular homology groups of compact CW-complexes are finitelygenerated, but the groups of compact metric spaces in generalare very easy to generate infinitely and our understanding ofthese groups is far from complete even for the following compactsubset of the plane, called the Hawaiian earring: Griffiths [11] gave a presentation of the fundamental groupof H and the proof was completed by Morgan and Morrison [15].The same group is presented as the free -product of integers Z in [4, Appendix]. Hence the firstintegral singular homology group H₁(H) is the abelianizationof the group . These results have been generalized to non-metrizable counterparts H_I of H(see Section 3). In Section 2 we prove that H₁(X) is torsion-free and H_i(X) =0 for each one-dimensional normal space X and for each i 2.The result for i 2 is a slight generalization of [2, Theorem5]. In Section 3 we provide an explicit presentation of H₁(H)and also H₁(H_I) by using results of [4]. Throughout this paper, a continuum means a compact connectedmetric space and all maps are assumed to be continuous. Allhomology groups have the integers Z as the coefficients. Thebouquet with n circles is denoted by B_n. The base point (0, 0) of B_n is denoted by o forsimplicity.     

Betti Numbers of Semialgebraic and Sub-Pfaffian Sets

Let X be a subset in [–1,1]^n₀R^n₀ defined by the formula X={x₀|Q₁x₁Q₂x₂...Qx ((x₀,x₁,...x)X)}, where Q_i{ }, Q_i Q_i+1, x_i [–1, 1]^n_i, and X may be eitheran open or a closed set in [–1,1]^n₀+...+n, being the differencebetween a finite CW-complex and its subcomplex. An upper boundon each Betti number of X is expressed via a sum of Betti numbersof some sets defined by quantifier-free formulae involving X. In important particular cases of semialgebraic and semi-Pfaffiansets defined by quantifier-free formulae with polynomials andPfaffian functions respectively, upper bounds on Betti numbersof X are well known. The results allow to extend the boundsto sets defined with quantifiers, in particular to sub-Pfaffiansets.     

A Bound on the Presentation Rank of a Finite Group

Let G be a finite group, and let I_G be the augmentation idealof ZG. We denote by d(G) the minimum number of generators forthe group G, and by d(I_G) the minimum number of elements ofI_G needed to generate I_G as a G-module. The connection betweend(G) and d(I_G) is given by the following result due to Roggenkamp]14]: where pr(G) is a non-negative integer, called the presentationrank of G, whose definition comes from the study of relationmodules (see [4] for more details). 1991 Mathematics SubjectClassification 20D20.     

Some Remarks on p-Adic Analytic Groups

A theorem of Maranda [1, Section 30] states that if F is a finitegroup, p is a prime and p^e exactly divides |F|, then a Z_pF-latticeM is determined up to isomorphism by its finite quotient M/p^e+1M.If M is a free Z_p-module of rank d, this is equivalent to sayingthat representations of F in GL_d(Z_p) are determined up to equivalenceby their images modulo p^e+1. 1991 Mathematics Subject Classification20E18, 22E20.     

On Irregularities of Distribution II

Let a=(a₁, a₂, a₃, ...) be an arbitrary infinite sequence inU=[0, 1). Let Van der Corput [5] conjectured that d(a, n) (n=1, 2, ...) isunbounded, and this was proved in 1945 by van Aardenne-Ehrenfest[1]. Later she refined this [2], obtaining for infinitely many n. Here and later c₁, c₂, ... denote positiveabsolute constants. In 1954, Roth [8] showed that the quantity is closely related to the discrepancy of a suitable point setin U².     

Euler Characteristics of Coxeter Groups, PL-Triangulations of Closed Manifolds, and Cohomology of Subgroups of Artin Groups

The motivation for the theory of Euler characteristics of groups,which was introduced by C. T. C. Wall [21], was topology, butit has interesting connections to other branches of mathematicssuch as group theory and number theory. This paper investigatesEuler characteristics of Coxeter groups and their applications.In his paper [20], J.-P. Serre obtained several fundamentalresults concerning the Euler characteristics of Coxeter groups.In particular, he obtained a recursive formula for the Eulercharacteristic of a Coxeter group, as well as its relation tothe Poincaré series (see 3). Later, I. M. Chiswell obtainedin [10] a formula expressing the Euler characteristic of a Coxetergroup in terms of orders of finite parabolic subgroups (Theorem1). These formulae enable us to compute Euler characteristicsof arbitrary Coxeter groups. On the other hand, the Euler characteristics of Coxeter groupsW happen to be intimately related to their associated complexesF_W, which are defined by means of the posets of nontrivial parabolicsubgroups of finite order (see 2.1 for the precise definition).In particular, it follows from the recent result of M. W. Davis[13] that if F_W is a product of a simplex and a generalizedhomology 2n-sphere, then the Euler characteristic of W is zero(Corollary 3.1). The first objective of this paper is to generalizethe previously mentioned result to the case when F_W is a PL-triangulationof a closed 2n-manifold which is not necessarily a homology2n-sphere. In other words (as given below in Theorem 3), ifW is a Coxeter group such that F_W is a PL-triangulation of aclosed 2n-manifold, then the Euler characteristic of W is equalto 1–(F_W)/2.     

DESINGULARIZATIONS OF CALABI-YAU 3-FOLDS WITH CONICAL SINGULARITIES. II. THE OBSTRUCTED CASE

This is the second of two papers studying Calabi–Yau 3-foldswith conical singularities and their desingularizations. Inour first paper [Y.-M. Chan, Quart. J. Math. 57 (2006), 151–181]we constructed the desingularization of the conically singularmanifold M₀ by gluing an asymptotically conical (AC) Calabi–Yau3-fold Y into M₀ at the singular point, thus obtaining a 1-parameterfamily of compact, non-singular Calabi–Yau 3-folds M_tfor small t > 0. During the gluing process one may encountera kind of cohomological obstruction to defining a 3-form _t onM_t which interpolates between the 3-form ₀ on M₀ and the scaled3-form t³ _Y on Y if the rate at which the AC Calabi–Yau3-fold Y converges to the Calabi–Yau cone is equal to– 3. The first paper [3] studied the simpler case <–3 where there is no obstruction. This paper extends theresult in the first one by considering a more complicated situtationwhen = –3. Assuming the existence of singular Calabi–Yaumetrics on compact complex 3-folds with ordinary double points,our result in this paper can be applied to repairing such kindsof singularities, which is an analytic version of Friedman'sresult giving necessary and sufficient conditions for smoothingordinary double points.