Singular Integrals and Maximal Functions Associated to Surfaces of Revolution

We obtain L^p estimates for singular integrals and maximal functionsassociated to hypersurfaces in Rⁿ⁺¹, n 2, which are obtainedby rotating a curve around one of the coordinate axes.     

On an Operator Theory Approach to the Corona Problem

This paper deals with an operator theory approach to the coronaconjecture for H(Dⁿ). Treil gave a counter-example to this conjecturein the case where n = 1 for operator-valued functions; thusone might hope to use this to disprove the corona conjecturefor H(Dⁿ) (for n 2). This paper shows that this natural approachtowards a negative answer fails. On the other hand, the secondresult here shows that ‘commutant lifting’ cannotbe true for more than two contractions for any constant. Thisobstructs a natural attempted proof of the corona conjecturefor H(Dⁿ) (for n 2) by our previous result.     

The quaternion group as a subgroup of the sphere braid groups

Let n 3. We prove that the quaternion group of order 8 is realisedas a subgroup of the sphere braid group B_n(²) if and only ifn is even. If n is divisible by 4, then the commutator subgroupof B_n(²) contains such a subgroup. Further, for all n 3, B_n(²)contains a subgroup isomorphic to the dicyclic group of order4n.     

Distance Entre Puissances D'une Unite Approchee Bornee

Let A be a Banach algebra and let p and q be two positive integers.We show that if A has a left bounded sequential approximate identity (e_n)_n1such that lim, inf_n+|e^p_n-e^{p+q}_n| (p/p+q)^p/qq/p+q} then A has a left-bounded sequential identity (f_n)_{n1} such thatf²_{n = fn} for n1. A simple example shows that the constant (p/p+q)^{p/q q/p+q} is best possible. This result is based on some algebraic or integral formulaewhich associate an idempotent to elements of a Banach algebrasatisfying some inequalities involving polynomials or entirefunctions.     

Deformation Theory and The Computation of Zeta Functions

We present a new approach to the problem of computing the zetafunction of a hypersurface over a finite field. For a hypersurfacedefined by a polynomial of degree d in n variables over thefield of q elements, one desires an algorithm whose runningtime is a polynomial function of dⁿ log(q). (Here we assumed 2, for otherwise the problem is easy.) The case n = 1 isrelated to univariate polynomial factorisation and is comparativelystraightforward. When n = 2 one is counting points on curves,and the method of Schoof and Pila yields a complexity of , where the function C_d depends exponentiallyon d. For arbitrary n, the theorem of the author and Wan givesa complexity which is a polynomial function of (pdⁿ log(q))ⁿ,where p is the characteristic of the field. A complexity estimateof this form can also be achieved for smooth hypersurfaces usingthe method of Kedlaya, although this has only been worked outin full for curves. The new approach we present should yielda complexity which is a small polynomial function of pdⁿ log(q).In this paper, we work this out in full for Artin–Schreierhypersurfaces defined by equations of the form Z^p – Z= f, where the polynomial f has a diagonal leading form. Themethod utilises a relative p-adic cohomology theory for familiesof hypersurfaces, due in essence to Dwork. As a corollary ofour main theorem, we obtain the following curious result. Letf be a multivariate polynomial with integer coefficients whoseleading form is diagonal. There exists an explicit deterministicalgorithm which takes as input a prime p, outputs the numberof solutions to the congruence equation f = 0 op, and runs in bit operations, for any >0. This improves upon the elementary estimate of bit operations, where n is the number of variables,which can be achieved using Berlekamp's root counting algorithm.2000 Mathematics Subject Classification 11Y99, 11M38, 11T99.     

A Classification of Horizontally Homothetic Submersions from Space Forms of Nonnegative Curvature

In this paper, it is proved that for n 2, any horizontallyhomothetic submersion : Rⁿ⁺¹ (Nⁿ, h) is a Riemannian submersionup to a homothety. It is also shown that if : Sⁿ⁺¹ (Nⁿ, h)is a horizontally homothetic submersion, then n = 2m, (Nⁿ, h)is isometric to CP^m and, up to a homothety, is a standard Hopffibration S^2m+1 CP^m. 2000 Mathematics Subject Classification53C20, 53C12.     

Chaotic Homeomorphisms of Rn, Lifted from Torus Homeomorphisms

We establish the existence of self-homeomorphisms of Rⁿ, n 2, which are chaotic in the sense of Devaney, preserve volumeand are spatially periodic. Moreover, we show that in the spaceof volume-preserving homeomorphisms of the n-torus with meanrotation zero, those with chaotic lifts to Rⁿ are dense, withrespect to the uniform topology. An application is given forfixed points of 2-dimensional torus homeomorphisms (Conley–Zehnder–FranksTheorem). 1991 Mathematics Subject Classification 54H20.     

Matheron's Conjecture for the Covariogram Problem

The covariogram of a convex body K provides the volumes of theintersections of K with all its possible translates. Matheronconjectured in 1986 that this information determines K amongall convex bodies, up to certain known ambiguities. It is provedthat this is the case if K R² is not C¹, or it is not strictlyconvex, or its boundary contains two arbitrarily small C² openportions ‘on opposite sides’. Examples are alsoconstructed that show that this conjecture is false in Rⁿ forany n 4.     

Existence of dicritical singularities of Levi-flat hypersurfaces and holomorphic foliations

We study holomorphic foliations tangent to singular real-analytic Levi-flat hypersurfaces in compact complex manifolds of complex dimension two. We give some hypotheses to guarantee the existence of dicritical singularities of these objects. As consequence, we give some applications to holomorphic foliations tangent to real-analytic Levi-flat hypersurfaces with singularities in \(\mathbb {P}^2\).     

Examples of Harmonic Mappings Obtained by Reduction

Our main result is to prove the existence of an equivariantharmonic map from CP^m # (endowed with some non-homogeneous metric) into S^2m, for all m 2. Weuse reduction techniques and the direct method of the calculusof variations.     

Convex Hypersurfaces with Prescribed Curvature and Boundary in Riemannian Manifolds

In this article a priori estimates at the boundary for the second fundamental form of n-dimensional convex hypersurfaces M with prescribed curvature quotient Sn (M)/Sl (M) in Riemannian manifolds are derived. A consequence of these estimates and other known results is an existence theorem for such hypersurfaces, which is a generalization of a recent result of Ivochkina and Tomi to the Riemannian case.     

On a Comparison Principle and the Critical Fujita Exponents for Solutions of Semilinear Parabolic Inequalities

The paper examines blow-up phenomena for the inequality u_t–Lu–|u|^q–1u_t–L–||^q–1 (*) in the half-space x Rⁿ, n 1, where L is a linear second-order partial differential operatorin divergence form. The paper studies weak solutions of (*) that belong only locallyto the corresponding Sobolev spaces in the half-space x Rⁿ. It also requires no conditionsfor the behavior of solutions of (*) on the hyperplane t = 0. The existence of critical blow-up exponents is obtained forsolutions of (*) as a special case of a comparison principlefor the corresponding solutions of (*). For example, the well-knownFujita result is a consequence of the comparison principle. The approach developed in the paper is directly applicable tothe study of analogous problems involving nonlinear differentialoperators. Its elliptic analogue has been recently developedby the authors.     

The Coherence of One-Relator Groups with Torsion and the Hanna Neumann Conjecture

A conjecture is proposed, bounding the number of cycles withlabel Wⁿ in a labeled directed graph. Some partial results towardsthis conjecture are established. As a consequence, it is provedthat a₁, a₂,...|Wⁿ is coherent for n 4. Furthermore, it iscoherent for n 2, provided that the strengthened Hanna Neumannconjecture holds. 2000 Mathematics Subject Classification 20F06,05C38.     

On Hardy, BMO and Lipschitz spaces of invariantly harmonic functions in the unit ball

The invariantly harmonic functions in the unit ball Bⁿ in Cⁿare those annihilated by the Bergman Laplacian . The Poisson-Szegökernel P(z,) solves the Dirichlet problem for : if f C(Sⁿ),the Poisson-Szegö transform of f, where d is the normalized Lebesgue measure on Sⁿ,is the unique invariantly harmonic function u in Bⁿ, continuousup to the boundary, such that u=f on Sⁿ. The Poisson-Szegötransform establishes, loosely speaking, a one-to-one correspondencebetween function theory in Sⁿ and invariantly harmonic functiontheory in Bⁿ. When n 2, it is natural to consider on Sⁿ functionspaces related to its natural non-isotropic metric, for theseare the spaces arising from complex analysis. In the paper,different characterizations of such spaces of smooth functionsare given in terms of their invariantly harmonic extensions,using maximal functions and area integrals, as in the correspondingEuclidean theory. Particular attention is given to characterizationin terms of purely radial or purely tangential derivatives.The smoothness is measured in two different scales: that ofSobolev spaces and that of Lipschitz spaces, including BMO andBesov spaces. 1991 Mathematics Subject Classification: 32A35,32A37, 32M15, 42B25.     

Hypersurfaces in a Unit Sphere Sn+1(1) with Constant Scalar Curvature

The paper considers n-dimensional hypersurfaces with constantscalar curvature of a unit sphere S^n–1(1). The hypersurfaceS^k(c₁)xS^n–k(c₂) in a unit sphere Sⁿ⁺¹(1) is characterized,and it is shown that there exist many compact hypersurfaceswith constant scalar curvature in a unit sphere Sⁿ⁺¹(1) whichare not congruent to each other in it. In particular, it isproved that if M is an n-dimensional (n > 3) complete locallyconformally flat hypersurface with constant scalar curvaturen(n–1)r in a unit sphere Sⁿ⁺¹(1), then r > 1–2/n,and (1) when r (n–2)/(n–1), if then M is isometric to S¹xS^n–1(c),where S is the squared norm of the second fundamental form ofM; (2) there are no complete hypersurfaces in Sⁿ⁺¹(1) with constantscalar curvature n(n–1)r and with two distinct principalcurvatures, one of which is simple, such that r = (n–2)/(n–1)and      

Dirichlet Series and Curious infinite Products

By number-theoretical methods we give a result generalizingthose of J. O. Shallit, one particular case of which reads asfollows: ...=2^–;, where a(n) is equal to 1 if the sum of the digits of n in basetwo is even, and – 1 if this sum is odd. Moreover we provea conjecture of Shallit concerning a way of approximating 2^–     

Graphs with many r-cliques have large complete r-partite subgraphs

Let r2 and c>0. Every graph on n vertices with at least cn^rcliques on r vertices contains a complete r-partite subgraphwith r–1 parts of size c^rlog n and one part of size greaterthan n^1–cr–1. This result implies a quantitativeform of the Erdös–Stone theorem.     

Equivalent birational embeddings

Let X be a projective variety of dimension r over an algebraicallyclosed field. It is proven that two birational embeddings ofX in ⁿ with n r + 2 are equivalent up to Cremona transformationsof ⁿ.     

The eigenvalues of limits of radial Toeplitz operators

Let A² be the Bergman space on the unit disk. A bounded operatorS on A² is called radial if Szⁿ = _n zⁿ for all n 0, where _nis a bounded sequence of complex numbers. We characterize theeigenvalues of radial operators that belong to the Toeplitzalgebra.