Anzai Skew Products with Lebesgue Component of Infinite Multiplicity

Let f be a 1-periodic C¹-function whose Fourier coefficientssatisfy the condition _n|n|³|f(n|² < . For every R\Q andm Z\{0}, we consider the Anzai skew product T(x, y) = (x +, y + mx + f(x)) acting on the 2-torus. It is shown that T hasinfinite Lebesgue spectrum on the orthocomplement L²(dx) ofthe space of functions depending only on the first variable.This extends some earlier results of Kushnirenko, Choe, Lemaczyk,Rudolph, and the author. 1991 Mathematics Subject Classification28D05.     

On Extensions of Myers' Theorem

Let M be a compact Riemannian manifold, and let h be a smoothfunction on M. Let p^h(x) = inf||–₁(Ric_x(,)–2Hess(h_x(,)).Here Ric_x denotes the Ricci curvature at x and Hess(h) is theHessian of h. Then M has finite fundamental group if ^h–p^h<0. Here ^h =:+2L_h is the Bismut-Witten Laplacian. This leadsto a quick proof of recent results on extension of Myers' theoremto manifolds with mostly positive curvature. There is also asimilar result for noncompact manifolds.     

Sharp Distortion Estimate for Locally Schlicht Bloch Functions

Let B be the space of locally schlicht Bloch functions f whichare analytic in the unit disc with f(0) = f'(0) – 1 =0 satisfying 0 < |f'(z)|(1 – |z|²) 1. For each fixedz₀ we shall determine the shape of the set {logf'(z₀): fB},that is, we shall give the sharp distortion estimate for locallyschilcht Bloch functions.     

On Weighted Distortion in Conformal Mapping II

Some years ago, Blatter [1] gave a result of the form for any function f regular and univalentin D: |z| < 1, where is the hyperbolic distance betweenz₁ and z₂. Kim and Minda [5] pointed out that the multiplieron the right is incorrect. They say that Blatter's proof givesthe correct multiplier, but Blatter's formula for in termsof z₁, z₂ is wrong. Kim and Minda proved the generalized formula where D₁(f) = f'(z) (1 – |z|²),valid for p P with some P, . In each case there was an appropriate equality statement. Kimand Minda made the important and easily verified remark thatthese problems are linearly invariant in the sense that if theresult is proved for f, then it follows for , where U is a linear transformation of the planeonto itself and T is a linear transformation of D onto itself.This means that we need to prove such results only in an appropriatelynormalized context. 1991 Mathematics Subject Classification30C75, 30F30.     

An Upper Bound for the Hyperbolic Metric of a Convex Domain

In [1], Beardon introduced the Apollonian metric defined forany domain D in Rⁿ by This metric is Möbius invariant, and for simply connectedplane domains it satisfies the inequality _D2_D, where _D denotesthe hyperbolic distance in D, and so gives a lower bound onthe hyperbolic distance. Furthermore, it is shown in [1, Theorem6.1] that for convex plane domains, the Apollonian metric satisfies, and, by considering the example of the infinite strip {x + iy:|y|<1}, that the best possibleconstant in this inequality is at least . In this paper we makethe following improvements.     

Correction: on the Intersection Matrices of Curves lying on Irregular Algebraic Surfaces

Professor W. F. Hammond has kindly drawn my attention to a blunderin 4 of the above paper. He referred to the ( – 2r) xß submatrix D of the skew-symmetric matrix displayednear the top of page 181, of which it is asserted that it issquare and non-singular, and pointed out that, from the factthat the matrix of which D forms part is regular, it may onlybe deduced that the columns of D are linearly independent; thatis, it only follows that – 2r ß. The validity of the equation – 2r = ß is essentialto the succeeding argument and, fortunately, may be establishedby alternative means. Using the nomenclature of the paper, wehave on F the set ₁^*, ..., _2r^*, ₁^*, ..., _ß^* of independent3-cycles (independent because they cut independent 1-cycleson the curve C), which may be completed, to form a basis forsuch cycles on F, by a further set ₁', ..., _2q–2r–pof independent 3-cycles, each of which meets C in a cycle homologousto zero on C. The cycles ₁^*, ..., ^* are invariant cycles andare independent on F so that, if > 2r + ß, thereis a non-trivial linear combination ^* of these having zero intersectionon C with each of the cycles ₁^*, ..., _2r^*, ₁^*, ..., _ß^*.Thus we have. (^* ._k^*)c = 0 = (^* ._i^*)c i.e. (^* ._k^*) = 0 = (^* ._i^* on F (1 k 2r; 1 i ß). Furthermore, (_j . C) 0 on C and we have (^* ._j .C)_C = 0 i.e. (^* ._j) = 0 on F (1 j 2q – 2r – ß). It now follows that ^* 0 on F (for it has zero intersectionwith every member of a basic set of 3-cycles on F). But thiscondradicts the assumption that ^* is a non-trivial linear combinationof the independent cycles ₁^*, ...,^*; and hence < 2r + ß.     

The Symmetrized Bidisc and Lempert's Theorem

Let G C² be the open symmetrized bidisc, namely G = {(₁ + ₂,₁₂) : |₁| < 1, |₂| < 1}. In this paper, a proof is giventhat G is not biholomorphic to any convex domain in C². By combiningthis result with earlier work of Agler and Young, the authorshows that G is a bounded domain on which the Carathéodorydistance and the Kobayashi distance coincide, but which is notbiholomorphic to a convex set. 2000 Mathematics Subject Classification32F45 (primary), 15A18 (secondary).     

Cluster Sets of Harmonic Functions at the Boundary of a Half-Space

The purpose of this paper is to answer some questions posedby Doob [2] in 1965 concerning the boundary cluster sets ofharmonic and superharmonic functions on the half-space D givenby D = R^n–1 x (0, + ), where n 2. Let f: D [–,+] and let Z D. Following Doob, we write B_Z (respectively C_Z)for the non-tangential (respectively minimal fine) cluster setof f at Z. Thus l B_Z if and only if there is a sequence (X_m)of points in D which approaches Z non-tangentially and satisfiesf(X_m) l. Also, l C_Z if and only if there is a subset E ofD which is not minimally thin at Z with respect to D, and whichsatisfies f(X) l as X Z along E. (We refer to the book byDoob [3, 1.XII] for an account of the minimal fine topology.In particular, the latter equivalence may be found in [3, 1.XII.16].)If f is superharmonic on D, then (see [2, 6]) both sets B_Z andC_Z are subintervals of [–, +]. Let denote (n –1)-dimensional measure on D. The following results are due toDoob [2, Theorem 6.1 and p. 123]. 1991 Mathematics Subject Classification31B25.     

On the Behaviour of the First Eigenfunction of the p-Laplacian Near its Critical Points

In this paper, the behaviour of the positive eigenfunction of in u_| = 0, p > 1, isstudied near its critical points. Under some convexity and symmetryassumptions on , is seen to have a unique critical point atx = 0; also, the behaviour of both and is determined nearby.Positive solutions u to some general problems –_pu = f(u)in , u_| = 0, are also considered, with some convexity restrictionson u. 2000 Mathematics Subject Classification 35B05 (primary),35J65, 35J70 (secondary).     

Exceptional Functions and Normality

Yang proved in [10] that if f and f^(k) have no fix-points forevery fF, where F is a family of meromorphic functions in adomain G and k a fixed integer, then F is normal in G. In thispaper we prove normality for families F for which every fF omits₁ and f^(k) omits ₂, where ₁ and ₂ are analytic functions with. 1991 Mathematics SubjectClassification 30D35, 30D45.     

Lower Bounds for the Lp-Norm in Terms of the Mellin Transform

Given a measurable function f on (0, ) with Mellin transformF(s), let |f|_p denote the L^p-norm of f with respect to the measuredx/x. We prove that under certain assumptions, for instanceif f is real and non-negative and F() converges for in an openinterval and F() 0, then wherec_p (2e)^–1. We derive similar inequalities for complex-valuedf, for the L^p-norm of the derivative of f, and for the supremumof real-valued f and of its derivative. The lower bounds areeminently applicable when f is a convolution product.     

Base Sizes and Regular Orbits for Coprime Affine Permutation Groups

Let G be a permutation group on a finite set . A sequence B=(₁,..., _b) of points in is called a base if its pointwise stabilizerin G is the identity. Bases are of fundamental importance incomputational algorithms for permutation groups. For both practicaland theoretical reasons, one is interested in the minimal basesize for (G, ), For a nonredundant base B, the elementary inequality2^|B||G|||^|B| holds; in particular, |B|log|G|/log||. In the casewhen G is primitive on , Pyber [8, p. 207] has conjectured thatthe minimal base size is less than Clog|G|/log|| for some (large)universal constant C. It appears that the hardest case of Pyber's conjecture is thatof primitive affine groups. Let H=GV be a primitive affine group;here the point stabilizer G acts faithfully and irreduciblyon the elementary abelian regular normal subgroup V of H, andwe may assume that =V. For positive integers m, let mV denotethe direct sum of m copies of V. If (v₁, ..., v_m)mV belongsto a regular G-orbit, then (0, v₁, ..., v_m) is a base for theprimitive affine group H. Conversely, a base (₁, ..., _b) forH which contains 0V= gives rise to a regular G-orbit on (b–1)V. Thus Pyber's conjecture for affine groups can be viewed asa regular orbit problem for G-modules, and it is therefore aspecial case of an important problem in group representationtheory. For a related result on regular orbits for quasisimplegroups, see [4, Theorem 6].     

Packing, Tiling, Orthogonality and Completeness

Let R^d be an open set of measure 1. An open set DR^d is calleda ‘tight orthogonal packing region’ for if D–Ddoes not intersect the zeros of the Fourier transform of theindicator function of , and D has measure 1. Suppose that isa discrete subset of R^d. The main contribution of this paperis a new way of proving the following result: D tiles R^d whentranslated at the locations if and only if the set of exponentialsE = {exp 2i, x: } is an orthonormal basis for L²(). (This resulthas been proved by different methods by Lagarias, Reeds andWang [9] and, in the case of being the cube, by Iosevich andPedersen [3]. When is the unit cube in R^d, it is a tight orthogonalpacking region of itself.) In our approach, orthogonality ofE is viewed as a statement about ‘packing’ R^d withtranslates of a certain non-negative function and, additionally,we have completeness of E in L²() if and only if the above-mentionedpacking is in fact a tiling. We then formulate the tiling conditionin Fourier analytic language, and use this to prove our result.2000 Mathematics Subject Classification 52C22, 42B99, 11K70.     

The Representation of Some Integers as a Subset Sum

Let A N. The cardinality (the sum of the elements) of A willbe denoted by |A| ((A)). Let m N and p be a prime. Let A {1, 2,...,p}. We prove thefollowing results. If |A| [(p+m–2)/m]+m, then for every integer x such that0 x p – 1, there is B A such that |B| = m and (B) x mod p. Moreover, the bound is attained. If |A| [(p+m–2)/m]+m!, then there is B A such that |B| 0 mod m and (B) = (m – 1)!p. If |A| [(p + 1)/3]+29, then for every even integer x such that4p s x p(p + 170)/48, there is S A such that x = (S). In particular,for every even integer a 2 such that p 192a – 170, thereare an integer j 0 and S A such that (S) = a^j+1.     

Discontinuous Homomorphisms from Non-Commutative Banach Algebras

In the 1970s, a question of Kaplansky about discontinuous homomorphismsfrom certain commutative Banach algebras was resolved. Let Abe the commutative C*-algebra C(), where is an infinite compactspace. Then, if the continuum hypothesis (CH) be assumed, thereis a discontinuous homomorphism from C() into a Banach algebra[2, 7]. In fact, let A be a commutative Banach algebra. Then(with (CH)) there is a discontinuous homomorphism from A intoa Banach algebra whenever the character space _A of A is infinite[3, Theorem 3] and also whenever there is a non-maximal, primeideal P in A such that |A/P|=2^₀ [4, 8]. (It is an open questionwhether or not every infinite-dimensional, commutative Banachalgebra A satisfies this latter condition.) 1991 MathematicsSubject Classification 46H40.     

Extremal Properties of Contraction Semigroups on Hilbert and Banach Spaces

Let f be a unit vector and T = {T(t) = e^tA: t 0} be a (C₀)contraction semigroup generated by A on a complex Hilbert spaceX. If |T(t)f,f| 1 as t then f is an eigenvector of A correspondingto a purely imaginary eigenvalue. If one allows X to be a Banachspace, the same situation can be considered by replacing T(t)f,fby (T(t)f) where is a unit vector in X^* dual to f. If |(T(t)f)| 1, as t , is f an eigenvector of A? The answer is sometimesyes and sometimes no.     

Optimal Stopping in the L log L-Inequality of Hardy and Littlewood

Let B = (B_t)_t0 be standard Brownian motion started at zero.We prove for all c > 1and all stopping times for B satisfying E(^r) < for somer > 1/2. This inequality is sharp, and equality is attainedat the stopping time whereu_* = 1 + 1/e^c(c – 1) and = (c – 1)/c for c >1, with X_t = |B_t| and S_t = max_0rt|B_r|. Likewise, we prove for all c > 1 and all stopping times for B satisfying E(^r < for some r > 1/2. This inequalityis sharp, and equality is attained at the stopping time where v_* = c/e(c – 1) and =(c – 1)/c for c > 1. These results contain and refinethe results on the L log L-inequality of Gilat [6] which areobtained by analytic methods. The method of proof used hereis probabilistic and is based upon solving the optimal stoppingproblem with the payoff whereF(x) equals either xlog⁺ x or x log x. This optimal stoppingproblem has some new interesting features, but in essence issolved by applying the principle of smooth fit and the maximalityprinciple. The results extend to the case when B starts at anygiven point (as well as to all non-negative submartingales).1991 Mathematics Subject Classification 60G40, 60J65, 60E15.     

A Necessary and Sufficient Condition for a Linear Differential System to be Strongly Monotone

In order to present the results of this note, we begin withsome definitions. Consider a differential system [formula] where IR is an open interval, and f(t, x), (t, x)IxRⁿ, is acontinuous vector function with continuous first derivativesf_r/x_s, r, s=1, 2, ..., n. Let D_xf(t, x), (t, x)IxRⁿ, denote the Jacobi matrix of f(t,x), with respect to the variables x₁, ..., x_n. Let x(t, t₀,x₀), tI(t₀, x₀) denote the maximal solution of the system (1)through the point (t₀, x₀)IxRⁿ. For two vectors x, yRⁿ, we use the notations x>y and x>>yaccording to the following definitions: [formula] An nxn matrix A=(a_rs) is called reducible if n2 and there existsa partition [formula] (p1, q1, p+q=n) such that [formula] The matrix A is called irreducible if n=1, or if n2 and A isnot reducible. The system (1) is called strongly monotone if for any t₀I, x₁,x₂Rⁿ [formula] holds for all t>t₀ as long as both solutions x(t, t₀, x_i),i=1, 2, are defined. The system is called cooperative if forall (t, x)IxRⁿ the off-diagonal elements of the nxn matrix D_xf(t,x) are nonnegative. 1991 Mathematics Subject Classification34A30, 34C99.     

Correction: Abelian Varieties defined over their Fields of Moduli, I

Bull London Math. Soc, 4 (1972), 370–372. The proof of the theorem contains an error. Before giving acorrect proof, we state two lemmas. LEMMA 1. Let K/k be a cyclic Galois extension of degree m, let generate Gal (K/k), and let (A, I, ) be defined over K. Supposethat there exists an isomorphism :(A,I,) (A, I, ) over K suchthat v^m–1 ... = 1, where v is the canonical isomorphism(A^m, I^m, ^m) (A, I, ). Then (A, I, ) has a model over k, whichbecomes isomorphic to (A, I, ) over K. Proof. This follows easily from [7], as is essentially explainedon p. 371. LEMMA 2. Let G be an abelian pro-finite group and let : G Q/Z be a continuous character of G whose image has order p.Then either: (a) there exist subgroups G' and H of G such that H is cyclicof order p^m for some m, (G') = 0, and G = G' x H, or (b) for any m > 0 there exists a continuous character m ofG such that p^m _m = . Proof. If (b) is false for a given m, then there exists an element G, of order p^r for some r m, such that () ¦ 0. (Considerthe sequence dual to 0 Ker (p^m) G ^pm G). There exists an opensubgroup G_o of G such that (G₀) = 0 and has order p^r in G/G₀.Choose H to be the subgroup of G generated by , and then aneasy application to G/G₀ of the theory of finite abelian groupsshows the existence of G' (note that () ¦ 0 implies that is not a p-th. power in G). We now prove the theorem. The proof is correct up to the statement(iv) (except that (i) should read: F' k₁ F'ab). To removea minor ambiguity in the proof of (iv), choose to be an elementof Gal (F'_ab/k₂) whose image $$\stackrel{\&macr;}{\sigma}$$ in Gal (k₁/k₂) generates this last group. The error occursin the statement that the canonical map v : A^P A acts on pointsby sending a^p a; it, of course, sends a a. The proof is correct, however, in the case that it is possibleto choose so that ^p = 1 (in Gal (F'/k₂)). By applying Lemma 2 to G = Gal (F'_ab/k₂) and the map G Gal(k₁/k₂) one sees that only the following two cases have to beconsidered. (a) It is possible to choose so that ^pm = 1, for some m, andG = G' x H where G' acts trivially on k₁ and H is generatedby . (b) For any m > 0 there exists a field K, F'ab K k₁ k₂is a cyclic Galois extension of degree p^m. In the first case, we let K F'_ab be the fixed field of G'.Then (A, I, ), regarded as being defined over K, has a modelover k₂. Indeed, if m = 1, then this was observed above, butwhen m > 1 the same argument applies. In the second case, let : (A, I, ) (A$$\stackrel{\&macr;}{\sigma}$$, I$$\stackrel{\&macr;}{\sigma }$$, $$\stackrel{\&macr;}{\sigma}$$) be an isomorphism defined over k₁ and let v ... ^p–1 = µ(R). If is replaced by for some Aut_k1((A, I, )) then is replacedby ^P. Thus, as µ(R) is finite, we may assume that ^pm–1= 1 for some m. Choose K, as in (b), to be of degree p^m overk₂. Let _m be a generator of Gal (K/k₂) whose restriction tok₁ is $$\stackrel{\&macr;}{\sigma }$$. Then : (A, I, ) (A^{$$\stackrel{\&macr;}{\sigma }$$}, I^{$$\stackrel{\&macr;}{\sigma}$$}, ^{$$\stackrel{\&macr;}{\sigma }$$} = (A^{$$\stackrel{\&macr;}{\sigma}$$m}, I^{$$\stackrel{\&macr;}{\sigma }$$m}, ^{$$\stackrel{\&macr;}{\sigma}$$m} is an isomorphism defined over K and v ^mpm–1, ... ^m =^pm–1 = 1, and so, by) Lemma 1, (A, I, ) has a model overk₂ which becomes isomorphic to (A, I, over K. The proof may now be completed as before. Addendum: Professor Shimura has pointed out to me that the claimon lines 25 and 26 of p. 371, viz that µ(R) is a puresubgroup of _R^*t, does not hold for all rings R. Thus this condition,which appears to be essential for the validity of the theorem,should be included in the hypotheses. It holds, for example,if µ(R) is a direct summand of µ(F).     

A remark on the representation of Zolotarev polynomials

Zolotarev polynomials are the polynomials that have minimaldeviation from zero on [–1, 1] with respect to the norm||xⁿ – x^n–1 + a_n–2 x^n–2 + ... + a₁x+ a_n|| for given and for all a_k . This note complements the paper of F. Pehersforfer [J. LondonMath. Soc. (1) 74 (2006) 143–153] with exact (not asymptotic)construction of the Zolotarev polynomials with respect to thenorm L₁ for || < 1 and with respect to the norm L₂ for || 1 in the form of Bernstein–Szegö orthogonal polynomials.For all in L₁ and L₂ norms, the Zolotarev polynomials satisfyexactly (not asymptotically) the triple recurrence relationof the Chebyshev polynomials.