Beardon's Diophantine Equations and Non-Free Mobius Groups

Let µ be a real number. The Möbius group G_µis the matrix group generated by It is known that G_µ is free if |µ| 2 (see [1])or if µ is transcendental (see [3, 8]). Moreover, thereis a set of irrational algebraic numbers µ which is densein (–2, 2) and for which G_µ is non-free [2, p. 528].We may assume that µ > 0, and in this paper we considerrational µ in (0, 2). The following problem is difficult. Let G^nf denote the set of all rational numbers µ in (0,2) for which G_µ is non-free. In 1969 Lyndon and Ullman[8] proved that G^nf contains the elements of the forms p/(p²+ 1) and 1/(p + 1), where p = 1, 2, ..., and that if µ₀ G^nf, then µ₀/p G^nf for p = 1, 2, .... In 1993 Beardon[2] studied problem (P) by means of the words of the form A^rB^s A^t and A^r B^s A^t B^u A^v, and he obtained a sufficient conditionfor solvability of (P), included implicitly in [2, pp. 530–531],by means of the following Diophantine equations: 1991 Mathematics SubjectClassification 20E05, 20H20, 11D09.     

On time regularity and related conditions for power-bounded operators

Let T be a bounded linear operator in a complex Banach space.Our main result gives various characterizations of the condition:T is power-bounded and an estimate ||(I – T)Tⁿ || cn^–1/2 holds for all positive integers n. In particular, this conditionholds if and only if T = β S + (1 – β)I, forsome β (0, 1) and some power-bounded operator S; or ifand only if T is power-bounded and the discrete semigroup (Tⁿ)is dominated by the continuous semigroup (e^{– t(I –T)})_{t 0} in a natural sense. As a consequence of our main results,for 1/2 < 1 we characterize the condition that T is power-boundedand ||(I – T)Tⁿ || c n^– for all n, in terms ofestimates on the semigroup e^{–t(I – T)}.     

Kato Class Potentials for Higher Order Elliptic Operators

Our goal in this paper is to determine conditions on a potentialV which ensure that an operator such as H:=(–)^m+V (1) acting on L²(R^N) defines a semigroup in L^p(R^N) for various valuesof p including p=1. The operator is defined as a quadratic formsum. That is, we put for (all integrals are on R^N and are with respect to Lebesgue measure), and note thatthe closure of the form is non-negative and has domain equalto the Sobolev space W^m,2. We then assume that the potentialhas quadratic form bound less than 1 with respect to Q₀, anddefine This form is closed and is associated with a semibounded self-adjointoperator H in L² (see [17, p. 348; 5, Theorem 4.23]). One canthen ask whether the semigroup e^–Ht defined on L² fort0 is extendable to a strongly continuous one-parameter semigroupon L^p for other values of p, and if so whether one can describethe domain and spectrum of its generator.     

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.     

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].     

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.     

A New Lower Bound for the L1 Mean of the Exponential Sum with the MoBius Function

The L¹ means of various exponential sums with arithmeticallyinteresting coefficients have been investigated in many recentpapers. For example, Balog and Perelli proved in [1] that for a suitable positive number c.The method of proving the lower bound in [1] is rather flexibleand can work well with many multiplicative functions in placeof µ(n), the Möbius function, whose Dirichlet serieshave a suitable expression by the Riemann -function. In this short note we improve on the above lower bound. 1991Mathematics Subject Classification 11L03, 42A70.     

Multiple Positive Solutions of Semilinear Differential Equations with Singularities

The existence of positive solutions of a second order differentialequation of the form z'+g(t)f(z)=0 (1.1) with the separated boundary conditions: z(0) – ßz'(0)= 0 and z(1)+z'(1) = 0 has proved to be important in physicsand applied mathematics. For example, the Thomas–Fermiequation, where f = z^3/2 and g = t^–1/2 (see [12, 13, 24]),so g has a singularity at 0, was developed in studies of atomicstructures (see for example, [24]) and atomic calculations [6].The separated boundary conditions are obtained from the usualThomas–Fermi boundary conditions by a change of variableand a normalization (see [22, 24]). The generalized Emden–Fowlerequation, where f = z^p, p > 0 and g is continuous (see [24,28]) arises in the fields of gas dynamics, nuclear physics,chemically reacting systems [28] and in the study of multipoletoroidal plasmas [4]. In most of these applications, the physicalinterest lies in the existence and uniqueness of positive solutions.     

Linear Independence of Hecke Operators in the Homology of X0(N)

In Merel's recent proof [7] of the uniform boundedness conjecturefor the torsion of elliptic curves over number fields, a keystep is to show that for sufficiently large primes N, the Heckeoperators T₁, T₂, ..., T_D are linearly independent in theiractions on the cycle e from 0 to i in H₁(X₀(N) (C), Q). In particular,he shows independence when max(D⁸, 400D⁴) < N/(log N)⁴. Inthis paper we use analytic techniques to show that one can chooseD considerably larger than this, provided that N is large.     

Intertwining Maps from Certain Group Algebras

In [17, 18, 19], we began to investigate the continuity propertiesof homomorphisms from (non-abelian) group algebras. Alreadyin [19], we worked with general intertwining maps [3, 12]. Thesemaps not only provide a unified approach to both homomorphismsand derivations, but also have some significance in their ownright in connection with the cohomology comparison problem [4]. The present paper is a continuation of [17, 18, 19]; this timewe focus on groups which are connected or factorizable in thesense of [26]. In [26], G. A. Willis showed that if G is a connectedor factorizable, locally compact group, then every derivationfrom L¹(G) into a Banach L¹(G)-module is automatically continuous.For general intertwining maps from L¹(G), this conclusion isfalse: if G is connected and, for some nN, has an infinite numberof inequivalent, n-dimensional, irreducible unitary representations,then there is a discontinuous homomorphism from L¹(G into aBanach algebra by [18, Theorem 2.2] (provided that the continuumhypothesis is assumed). Hence, for an arbitrary intertwiningmap from L¹(G), the best we can reasonably hope for is a resultasserting the continuity of on a ‘large’, preferablydense subspace of L¹(G). Even if the target space of is a Banachmodule (which implies that the continuity ideal I() of is closed),it is not a priori evident that is automatically continuous:the proofs of the automatic continuity theorems in [26] relyon the fact that we can always confine ourselves to restrictionsto L¹(G) of derivations from M(G) [25, Lemmas 3.1 and 3.4].It is not clear if this strategy still works for an arbitraryintertwining map from L¹(G) into a Banach L¹(G)-module.     

Oscillation Properties of a Semi-discrete Approximation to the Beam Equation with a Second Order Term

The solution of the equation w(x)u_tt+[p(x)u_xx]_xx–[p(x)u_x]_x=0, 0< x < L, t > 0, where it is assumed that w, p,and q are positive on the interval [0, L], is approximated bythe method of straight lines. The resulting approximation isa linear system of differential equations with coefficient matrixS. The matrix S is studied under a variety of boundary conditionswhich result in a conservative system. In all cases the matrixS is shown to be similar to an oscillation matrix.     

On Fusion in Unipotent Blocks

The context of this note is as follows. One considers a connectedreductive group G and a Frobenius endomorphism F: G G definingG over a finite field of order q. One denotes by G^F the associated(finite) group of fixed points. Let l be a prime not dividing q. We are interested in the l-blocksof the finite group G^F. Such a block is called unipotent ifthere is a unipotent character (see, for instance, [6, Definition12.1]) among its representations in characteristic zero. Roughlyspeaking, it is believed that the study of arbitrary blocksof G^F might be reduced to unipotent blocks (see [2, Théorème2.3], [5, Remark 3.6]). In view of certain conjectures aboutblocks (see, for instance, [9]), it would be interesting tofurther reduce the study of unipotent blocks to the study ofprincipal blocks (blocks containing the trivial character).Our Theorem 7 is a step in that direction: we show that thelocal structure of any unipotent block of G^F is very close tothat of a principal block of a group of related type (notionof ‘control of fusion’, see [13, 49]). 1991 MathematicsSubject Classification 20Cxx.     

Scattering Matrix for Manifolds with Conical Ends

Let M be a manifold with conical ends. (For precise definitionssee the next section; we only mention here that the cross-sectionK can have a nonempty boundary.) We study the scattering forthe Laplace operator on M. The first question that we are interestedin is the structure of the absolute scattering matrix S(s).If M is a compact perturbation of Rⁿ, then it is well-knownthat S(s) is a smooth perturbation of the antipodal map on asphere, that is, S(s)f(·)=f(–·) (mod C) On the other hand, if M is a manifold with a scattering metric(see [8] for the exact definition), it has been proved in [9]that S(s) is a Fourier integral operator on K, of order 0, associatedto the canonical diffeomorphism given by the geodesic flow atdistance . In our case it is possible to prove that S(s) isin fact equal to the wave operator at a time t = plus C terms.See Theorem 3.1 for the precise formulation. This result isnot too difficult and is obtained using only the separationof variables and the asymptotics of the Bessel functions. Our second result is deeper and concerns the scattering phasep(s) (the logarithm of the determinant of the (relative) scatteringmatrix).     

RIGIDITY OF PERIODIC DIFFEOMORPHISMS OF HOMOTOPY K3 SURFACES

In this paper, we show that homotopy K3 surfaces do not admita periodic diffeomorphism of odd prime order 3 acting triviallyon cohomology. This gives a negative answer for period 3 toProblem 4.124 in Kirby's problem list. In addition, we givean obstruction in terms of the rationality and the sign of thespin numbers to the non-existence of a periodic diffeomorphismof odd prime order acting trivially on cohomology of homotopyK3 surfaces. The main strategy is to calculate the Seiberg–Witteninvariant for the trivial spin^c structure in the presence ofsuch a Z_p-symmetry in two ways: (1) the new interpretation ofthe Seiberg–Witten invariants of Furuta and Fang, and(2) the theorem of Morgan and Szabó on the Seiberg–Witteninvariant of homotopy K3 surfaces for the trivial Spin^c structure.As a consequence, we derive a contradiction for any periodicdiffeomorphism of prime order 3 acting trivially on cohomologyof homotopy K3 surfaces.     

Totally Real Minimal Surfaces with Non-Circular Ellipse of Curvature in the Nearly Kahler S6

In [2] we discussed almost complex curves in the nearly KählerS⁶. These are surfaces with constant Kähler angle 0 or and, as a consequence of this, are also minimal and have circularellipse of curvature. We also considered minimal immersionswith constant Kähler angle not equal to 0 or , but withellipse of curvature a circle. We showed that these are linearlyfull in a totally geodesic S⁵ in S⁶ and that (in the simplyconnected case) each belongs to the S¹-family of horizontallifts of a totally real (non-totally geodesic) minimal surfacein CP². Indeed, every element of such an S¹-family has constantKähler angle and in each family all constant Kählerangles occur. In particular, every minimal immersion with constantKähler angle and ellipse of curvature a circle is obtainedby rotating an almost complex curve which is linearly full ina totally geodesic S⁵.     

On the Discreteness and Convergence in n-Dimensional Mobius Groups

Throughout this paper, we adopt the same notations as in [1,6, 8] such as the Möbius group M(Rⁿ), the Clifford algebraC_n–1, the Clifford matrix group SL(2, _n), the Cliffordnorm of ||A||=(|a|²+|b|²+|c|²+|d|²) (1) and the Clifford metric of SL(2, _n) or of the Möbius groupM(Rⁿ) d(A₁,A₂)=||A₁–A₂||(|a₁–a₂|²+|b₁–b₂|²+|c₁–c₂|²+|d₁–d₂|²)(2) where |·| is the norm of a Clifford number and represents f_i M(), i = 1,2, and so on. In addition, we adopt some notions in [6, 12]:the elementary group, the uniformly bounded torsion, and soon. For example, the definition of the uniformly bounded torsionis as follows.     

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.     

Multistep approximation of linear sectorial evolution equations

This paper concerns the linear multistep approximation of alinear sectorial evolution equation u_t = Au on a complex Banachspace X. Given a strictly A()-stable q-step method of orderp whose stability region includes a sectorial region containingthe spectrum of the operator A, the corresponding evolutionsemigroup for the method is Cⁿ(hA), n 0, defined on X^q, whereC(z) L (C^q) denotes the one-step map associated with the method.It is shown that for appropriately chosen V, Y: C C^q, basedon the principal right and left eigenvectors of C(z), Cⁿ(hA)approximates the semigroup V(hA)e^nhAY^H(hA) with optimal orderp.     

On the Analytic Order-Preserving Discrete-Time Dynamical Systems in Rn with every Fixed Point Stable

This paper studies the asymptotic behaviour of an analytic order-preservingdiscrete-time dynamical system in Rⁿ, which is usually generatedby a periodic cooperative system. The author proves that forsuch a dynamical system, if every fixed point is Liapunov stableand every positive semi-orbit has compact closure, then everypositive semi-orbit converges. This result does not requirethe assumption ‘strongly’ and gives an affirmativeanswer to the conjecture proposed by the author in [17] forthe analytic case.     

An extension of the recursively enumerable Turing degrees

Consider the countable semilattice _T consisting of the recursivelyenumerable Turing degrees. Although _T is known to be structurallyrich, a major source of frustration is that no specific, naturaldegrees in _T have been discovered, except the bottom and topdegrees, 0 and 0'. In order to overcome this difficulty, weembed _T into a larger degree structure which is better behaved.Namely, consider the countable distributive lattice _w consistingof the weak degrees (also known as Muchnik degrees) of massproblems associated with non-empty ₀¹ subsets of 2. It is knownthat _w contains a bottom degree 0 and a top degree 1 and isstructurally rich. Moreover, _w contains many specific, naturaldegrees other than 0 and 1. In particular, we show that in _wone has 0 < d < r₁ < f(r₂, 1) < 1. Here, d is theweak degree of the diagonally non-recursive functions, and r_nis the weak degree of the n-random reals. It is known that r₁can be characterized as the maximum weak degree of a ₀¹ subsetof 2 of positive measure. We now show thatf(r₂, 1) can be characterizedas the maximum weak degree of a ₀¹ subset of 2, the Turing upwardclosure of which is of positive measure. We exhibit a naturalembedding of _T into _w which is one-to-one, preserves the semilatticestructure of _T, carries 0 to 0, and carries 0' to 1. Identifying_T with its image in _w, we show that all of the degrees in _Texcept 0 and 1 are incomparable with the specific degrees d,r₁, andf(r₂, 1) in _w.