An Example on Sobolev Space Approximation

For each d2 we construct a connected open set R^d such that = int (clos()), and for each k 1 and each p [1, ), the subsetW^k, () fails to be dense in the Sobolev space W^{k, p}(), in thenorm of W^{k, p}(). 1991 Mathematics Subject Classification 46E35,46F05.     

A Relation between Hausdorff Dimension and a Condition on Time Sets for the Image by the Brownian Sheet to Possess Interior Points

We prove that if W_{N, d} is a Brownian sheet mapping to R^d and E is a set in (0, )^N of Hausdorff dimensiongreater than , then for almost every rotation about a point x and translation _x such that _x(E) (0, )^N, the set _x(E) is such that almost surely W(E) containsinterior points. The techniques are adapted from Kahane andRosen and generalize to higher dimensional time and range.     

A problem of Hirst on continued fractions with sequences of partial quotients

Let B denote an infinite sequence of positive integers b₁ <b₂ < ..., and let denote the exponent of convergence ofthe series _{n = 1} 1/b_n; that is, = inf {s 0 : _{n = 1} 1/b_n^s <}. Define E(B) = {x [0, 1]: a_n(x) B (n 1) and a_n(x) asn }. K. E. Hirst [Proc. Amer. Math. Soc. 38 (1973) 221–227]proved the inequality dim_H E(B) /2 and conjectured (see ibid.,p. 225 and [T. W. Cusick, Quart. J. Math. Oxford (2) 41 (1990)p. 278]) that equality holds. In this paper, we give a positiveanswer to this conjecture.     

Precise Spectral Asymptotics for Logistic Equations of Population Dynamics

The semilinear elliptic eigenvalue problem with superlinearpure power nonlinearity is considered. This problem is treatedfrom the standpoint of L²-theory and the precise asymptoticformula for the eigenvalue parameter = () as is established,where is the L²-norm of the solution u associated with . 2000Mathematics Subject Classification 35P30 (primary), 35J60 (secondary).     

A Comparison Estimate for the Heat Equation with an Application to the Heat Content of the S-Adic Von Koch Snowflake

Let D be an open set in Euclidean space R^m with boundary D,and let :D[0, ) be a bounded, measurable function. Let u:DDx[0,)[0, ) be the unique weak solution of the heat equation [formula] with initial condition [formula] and with inhomogeneous Dirichlet boundary condition [formula] Then u(x; t) represents the temperature at a point xD at timet if D has initial temperature 0, while the temperature at apoint xD is kept fixed at (x) for all t>0. We define thetotal heat content (or energy) in D at time t by [formula] In this paper we wish to examine the effect of imposing additionalcooling on some subset C on both u and E_D. 1991 MathematicsSubject Classification 35K05, 60J65, 28A80.     

Asymptotics of the m-Coefficient for Eigenvalue Problems with Eigenparameter in the Boundary Conditions

For Sturm-Liouville problems on [a, ) with a regular -dependentboundary condition at a, and the limit point case at , a techniqueof W. N. Everitt [1] is employed to obtain asymptotic formulaefor the associated m()-functions on rays and lines in the complex-plane. The method relies on asymptotic formulae for solutionsof the initial value problem for –u'+qu = u, as || ,which the author has given in [4]. For the case of the regularleft endpoint, the asymptotic formulae on vertical lines sufficeto provide a direct proof of the formula for the total variationof the associated spectral function, a question which the authorhad raised in [3; Remark 5.2].     

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.     

Cardinal Invariants and Eventually Different Functions

In this paper we study several kinds of maximal almost disjointfamilies. In the main result of this paper we show that forsuccessor cardinals , there is an unexpected connection betweeninvariants a_e(), b() and a certain cardinal invariant m_d(⁺)on ⁺. As a corollary we get for example the following result.For a successor cardinal , even assuming that ^< = and 2= ⁺, the following is not provable in Zermelo–Fraenkelset theory. There is a ⁺-cc poset which does not collapse andwhich forces a() = ⁺ < a_e() = ⁺⁺ = 2. We also apply the ideasfrom the proofs of these results to study a = a() and non(M).2000 Mathematics Subject Classification 03E17 (primary), 03E05(secondary).     

Trivial centralizers for codimension-one attractors

We show that if is a codimension-one hyperbolic attractor fora C^r diffeomorphism f, where 2 r , and f is not Anosov, thenthere is a neighborhood of f in Diff^r(M) and an open and denseset of such that any g has a trivial centralizer on thebasin of attraction for .     

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.     

Embedding lp, {infty} into Lp, {infty}[0, 1] Complementably

We show that l^p, embeds into L^p, [0, 1] as a complemented sublattice.     

Inequalities for Oscillating p-Functions

Upper bounds on [ ) are derived for those p-functions such thatp() = m is minimum, and p' (+) = q(l-m).     

On the Reflexivity of the Space {pi}p(E, F) of p-Absolutely Summing Operators, 1 <= p < + {infty}

We give a representation of the dual of the space _p(E,F) ofp-absolutely summing operators (1 p < + ) under certainconditions on E and F. One deduces that the space _p(E, F), 1 p < + , is reflexive if and only if E and F are reflexive.We improve results of Gordon, Lewis, Retherford and Saphar.     

Existence of Periodic Solutions in Nonlinear Asymmetric Oscillations

The existence of periodic solutions for the nonlinear asymmetricoscillator x' + x⁺ – rßx^– = h(t),(' =d/dt (is discussed, where , rß are positive constantssatisfying for some positive integer n N and h(t) L (0,2) is 2-periodic with x^±= max {±x,0}. 2000 Mathematics Subject Classification34C10, 34C25.     

A reverse Denjoy theorem

Suppose that C₁ and C₂ are two simple curves joining 0 to ,non-intersecting in the finite plane except at 0 and enclosinga domain D which is such that, for all large r, has measure at most 2, where 0 < < .Suppose also that u is a non-constant subharmonic function inthe plane such that u(z) = B(|z|, u) for all large z C₁ C₂.Let A_D(r, u) = inf { u(z):z D and | z | = r }. It is shownthat if A_D(r, u) = O(1) (or A_D(r, u) = o(B(r, u))), then lim_r B(r, u)/r^/2 > 0 (or lim_r log B(r, u)/log r /2).     

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).     

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.     

Heat Kernel Estimates and LP Spectral Independence of Elliptic Operators

Let be an open subset of R^d, and let T_p for p[1, ) be consistentC₀-semigroups given by kernels that satisfy an upper heat kernelestimate. Denoting their generators by A_p, we show that thespectrum (A_p) is independent of p[1, ). We also treat the caseof weighted L^p-spaces for weights that satisfy a subexponentialgrowth condition. An example shows that independence of thespectrum may fail for an exponential weight. 1991 MathematicsSubject Classification 47D06, 47A10, 35P05.     

Asymptotic Interpolating Sequences in Uniform Algebras

T. Hosokawa, K. Izuchi and D. Zheng recently introduced theconcept of asymptotic interpolating sequences (of type 1) inthe unit disk for H(D). It is shown that these sequences coincidewith sequences that are interpolating for the algebra QA. Alsoa characterization is given of the interpolating sequences oftype 1 for H(D), and asymptotic interpolating sequences in thespectrum of H(D) are studied. The existence of asymptotic interpolatingsequences of type 1 for H() on arbitrary domains is verified.It is shown that any asymptotic interpolating sequence in auniform algebra eventually is interpolating.     

Approximation of Ground State Eigenvalues and Eigenfunctions of Dirichlet Laplacians

Let be a bounded connected open set in R^N, N 2, and let –0be the Dirichlet Laplacian defined in L²(). Let > 0 be thesmallest eigenvalue of –, and let > 0 be its correspondingeigenfunction, normalized by ||||₂ = 1. For sufficiently small>0 we let R() be a connected open subset of satisfying Let – 0 be the Dirichlet Laplacian on R(), and let >0and >0 be its ground state eigenvalue and ground state eigenfunction,respectively, normalized by ||||₂=1. For functions f definedon , we let Sf denote the restriction of f to R(). For functionsg defined on R(), we let Tg be the extension of g to satisfying 1991 Mathematics SubjectClassification 47F05.