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Let p n : C → C p_n:\mathbb {C} \rightarrow \mathbb {C} be a random complex polynomial whose roots are sampled i.i.d. from a radial distribution 2 π r u ( r ) d r 2\pi r u(r) dr in the complex plane. A natural question is how the distribution of roots evolves under repeated (say n / 2 − n/2- times) differentiation of the polynomial. We conjecture a mean-field expansion for the evolution of ψ ( s ) = u ( s ) s \psi (s) = u(s) s : ∂ ψ ∂ t = ∂ ∂ x ( ( 1 x ∫ 0 x ψ ( s ) d s ) − 1 ψ ( x ) ) . \begin{equation*} \frac {\partial \psi }{\partial t} = \frac {\partial }{\partial x} \left ( \left ( \frac {1}{x} \int _{0}^{x} \psi (s) ds \right )^{-1} \psi (x) \right ). \end{equation*} The evolution of ψ ( s ) ≡ 1 \psi (s) \equiv 1 corresponds to the evolution of random Taylor polynomials p n ( z ) = ∑ k = 0 n γ k z k k ! where γ k ∼ N C ( 0 , 1 ) . \begin{equation*} p_n(z) = \sum _{k=0}^{n}{ \gamma _k \frac {z^k}{k!}} \quad \text {where} \quad \gamma _k \sim \mathcal {N}_{\mathbb {C}}(0,1). \end{equation*} We discuss some numerical examples suggesting that this particular solution may be stable. We prove that the solution is linearly stable. The linear stability analysis reduces to the classical Hardy integral inequality. Many open problems are discussed.

Given any n-tuple of complex numbers, one can canonically define a polynomial of degree n+1 that has the entries of this n-tuple as its critical points. In 2002, Beardon, Carne, and Ng studied a map $\theta\colon \mathbb{C}^n\to \mathbb{C}^n$ which outputs the critical values of the canonical polynomial constructed from the input, and they proved that this map is onto. Along the way, they showed that $\theta$ is a local homeomorphism whenever the entries of the input are distinct and nonzero, and, implicitly, they produced a polynomial expression for the Jacobian determinant of $\theta$. In this article we extend and generalize both the local homeomorphism result and the elegant determinant identity to analogous situations where the critical points occur with multiplicities. This involves stratifying $\mathbb{C}^n$ according to which coordinates are equal and generalizing $\theta$ to a similar map $\mathbb{C}^\ell \to \mathbb{C}^\ell$ where $\ell$ is the number of distinct critical points. The more complicated determinant identity that we establish is closely connected to the multinomial identity known as Dyson's conjecture., Comment: 13 pages

A Banach space operator T T is completely hereditarily normaloid, T ∈ C H N T\in \mathcal {CHN} , if either every part, and (also) T p − 1 T_p^{-1} for every invertible part T p T_p , of T T is normaloid or if for every complex number λ \lambda every part of T − λ I T-\lambda I is normaloid. Sufficient conditions for the perturbation T + A T+A of T ∈ C H N T\in \mathcal {CHN} by an algebraic operator A A to satisfy Weyl’s theorem are proved. Our sufficient conditions lead us to the conclusion that the conjugate operator ( T + A ) ∗ (T+A)^* satisfies a a -Weyl’s theorem.

In this paper, we first introduce the concept of single elements in a module. A systematic study of single elements in the AlgL-module U is initiated, where £ is a completely distributive subspace lattice on a Hilbert space H. Furthermore, as an application of single elements, we study module isomorphisms between norm closed AlgN-modules, where N is a nest, and obtain the following result: Suppose that U, V are norm closed AlgN-modules and that Φ: U → V is a module isomorphism. Then U = V and there exists a non-zero complex number A such that Φ(T) = λT, VT ∈ U.

We prove an extension of Hardy’s classical characterization of real Gaussians of the form e − π α x 2 e^{-\pi \alpha x^2} , α > 0 \alpha >0 , to the case of complex Gaussians in which α \alpha is a complex number with positive real part. Such functions represent rotations in the complex plane of real Gaussians. A condition on the rate of decay of analytic extensions of a function f f and its Fourier transform f ^ \widehat f along some pair of lines in the complex plane is shown to imply that f f is a complex Gaussian.

A Seifert matrix is a square integral matrix V V satisfying det ( V − V T ) = ± 1. \begin{equation*}\det (V - V^T) =\pm 1. \end{equation*} To such a matrix and unit complex number ω \omega there corresponds a signature, σ ω ( V ) = sign ( ( 1 − ω ) V + ( 1 − ω ¯ ) V T ) . \begin{equation*}\sigma _\omega (V) = \mbox {sign}( (1 - \omega )V + (1 - \bar {\omega })V^T). \end{equation*} Let S S denote the set of unit complex numbers with positive imaginary part. We show that { σ ω } ω ∈ S \{\sigma _\omega \}_ { \omega \in S } is linearly independent, viewed as a set of functions on the set of all Seifert matrices. If V V is metabolic, then σ ω ( V ) = 0 \sigma _\omega (V) = 0 unless ω \omega is a root of the Alexander polynomial, Δ V ( t ) = det ( V − t V T ) \Delta _V(t) = \det (V - tV^T) . Let A A denote the set of all unit roots of all Alexander polynomials with positive imaginary part. We show that { σ ω } ω ∈ A \{\sigma _\omega \}_ { \omega \in A } is linearly independent when viewed as a set of functions on the set of all metabolic Seifert matrices. To each knot K ⊂ S 3 K \subset S^3 one can associate a Seifert matrix V K V_K , and σ ω ( V K ) \sigma _\omega (V_K) induces a knot invariant. Topological applications of our results include a proof that the set of functions { σ ω } ω ∈ S \{\sigma _\omega \}_ { \omega \in S } is linearly independent on the set of all knots and that the set of two–sided averaged signature functions, { σ ω ∗ } ω ∈ S \{\sigma ^*_\omega \}_ { \omega \in S } , forms a linearly independent set of homomorphisms on the knot concordance group. Also, if ν ∈ S \nu \in S is the root of some Alexander polynomial, then there is a slice knot K K whose signature function σ ω ( K ) \sigma _\omega (K) is nontrivial only at ω = ν \omega = \nu and ω = ν ¯ \omega = \overline {\nu } . We demonstrate that the results extend to the higher-dimensional setting.

Let ir: X -* T be a deformation of a normal Gorenstein surface singularity over the complex number field C. We assume that T is a neighborhood of the origin of C. Then we prove that 7r admits a simultaneous log-canonical model if and only if an invariant -Pt * Pt of each fiber Xt is constant.

In this paper we answer a question of Curto and Fialkow: there exists a quadratically hyponormal weighted shift which is not positively quadratically hyponormal. Let Rbe a complex Hilbert space and let ?(R) be the algebra of bounded linear operators on R-. An operator T E ?J(H) is said to be normal if T*T = TT*, hyponormal if T*T > TT*, and subnormal if T = N ,H where N is normal on some Hilbert space IC D Rt. If T is subnormal, then T is also hyponormal. Recall that given a bounded sequence of positive numbers a: ao, a,... (called weights), the (unilateral) weighted shift W, associated with a is the operator on P2(Z+) defined by Wen := anen+1 for all n > 0 where {en}?n=0 is the canonical orthonormal basis for P2. It is straightforward to check that WOE can never be normal, and that W, is hyponormal if and only if an 0. Recall the Bram-Halmos criterion for subnormality, which states that an operator T E ?QJ(H) is subnormal if and only if S (T'x3, Tix) > 0 ij for all finite collections xoXIx ,Xk E R([1], [2, 111.1.9]). Using the Choleski algorithm for operator matrices, it is easy to see that this is equivalent to the positivity of the matrices (T*jTi TiT*i)i31 for k = 1, 2,. If we denote by [A, B] := AB BA the commutator of two operators A and B, and if we define T to be k-hyponormal whenever the k x k operator matrix Mk(T) := ([T*j, T'])i31 is positive, then the Bram-Halmos criterion can be rephrased as saying that T is subnormal if and only if T is k-hyponormal for every k > 1 ([7]). Recall ([3], [4]) that T E ?Q() is weakly k-hyponormal if Z=o siT' is hyponormal for every complex number si (0 quadratically hyponormal. In [3, Proposition 7], it is shown that there exists a quadratically hyponormal weighted shift which is not 2-hyponormal. Received by the editors October 4, 1998 and, in revised form, February 24, 1999. 1991 Mathematics Subject Classification. Primary 47B20, 47B37.

Let (X, L) be a polarized abelian variety defined over the complex number field. Then we classify (X, L) with h0(L) > 2 such that (k + 1)L is not k-jet ample nor k-very ample.

Let k k be a field, V V an r r -dimensional k k -vector space, and W W a finite subgroup of A u t k ( V ) \mathrm {Aut}_k(V ) . Let S = S [ V # ] S = S[V^{\#}] be the symmetric algebra on V # V^\# , the k k -dual of V V , and R = S W R = S^W the ring of invariants under the natural action of W W on S S . Define P ∗ P_* to be the quotient algebra S ⊗ R k S\otimes _R k . Steinberg has shown that R R is polynomial if k k is the field of complex numbers and the quotient algebra P ∗ = S ⊗ R k P_* = S\otimes _R k satisfies Poincaré duality. In this paper we use elementary methods to prove Steinberg’s result for fields of characteristic 0 0 or of characteristic prime to the order of W W . This gives a new proof even in the characteristic zero case. Theorem 0.1. If the characteristic of k k is zero or prime to the order of W W and P ∗ P_* satisfies Poincaré duality, then R R is isomorphic to a polynomial algebra on r r generators.

Let g ≠ 0, ±1 be a fixed integer. Given two sequences of complex numbers (ϕ m ) ∞ m=1 and (ψ n ) ∞ n=1 and two sufficiently large integers M and N, we estimate the exponential sums Σ Σϕ p ψ n e p (ag n ), a ∈ Z, p≤M 1≤n≤N gcd(ag,p)=1 where the outer summation is taken over all primes p ≤ M with gcd(ag, p) = 1.

The aim of this note is to give a short proof that 2-local derivations on M n , the n x n matrix algebra over the complex numbers are derivations and to give a shorter proof that 2-local *-automorphisms on M n are *-automorphisms.

In this short note, we determine the Kodaira dimension and some of the plurigenera of (a desingularization of) a symmetric power of a smooth projective variety. We use it to obtain bounds on the genus of curve passing through a fixed number of general points., 4 pages, latex

A Borel (or even analytic) subring of R either has Hausdorff dimension 0 or is all of R. Extensions of the method of proof yield (among other things) that any analytic subring of C having positive Hausdorff dimension is equal to either R or C.

We prove that the Kauffman bracket skein algebra of a cylinder over a surface with boundary, defined over complex numbers, is isomorphic to the observables of an appropriate lattice gauge field theory., 7 pages, 5 eps files

A unified and relatively simple proof is given for some well-known results involving finite unions of uniformly separated sequences.

We link Carey's exponential representation of the determining function of a perturbation pair with the moment problem. We prove that an operator sequence represents the moments of a phase operator if and only if there is another positively defined sequence of operators satisfying a boundedness condition. INTRODUCTION The L-moment problem consists in characterizing the moment sequence (1) ~~~~~An = |tnf (t) dt, n Ez N of a measurable function f (with prescribed support in R) which satisfies O < f < L a.e. This problem was formulated and completely solved by Achiezer and Krein in the 1 930s [ 1]. The problem may be formulated for operator-valued functions. On the other hand, R. W. Carey introduced in [2] a complete unitary invariant which occurs in the perturbation theory of self-adjoint operators. This invariant, called in [2] the "phase shift," is a direct generalization of the phase shift which has been encountered in perturbation situations of one-dimensional range. Carey proved in [2] that, for z a complex number with Im z :$ 0 and R = (A z) 1,the determining function q(z) = I + KRzK can be represented in the form: +(Z) = exp (fB({)/({z)d)d) where B is a summable operator function B: R -() with 0 < B({) < 1. In the same paper he also proved the converse result: suppose B () is a Received by the editors February 8, 1990 and, in revised form, June 29, 1990; presented at the International Colloquium on complex analysis and the VIth Romanian-Finnish Seminar, Bucharest, June 5-8, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 44A60; Secondary 47A55.

We show how for every integer n one can explicitly construct n distinct plane quartics and one hyperelliptic curve over the complex numbers all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When we say that the curves can be constructed ``explicitly'', we mean that the coefficients of the defining equations of the curves are simple rational expressions in algebraic numbers in R whose minimal polynomials over Q can be given exactly and whose decimal approximations can be given to as many places as is necessary to distinguish them from their conjugates. We also prove a simply-stated theorem that allows one to decide whether or not two plane quartics over the complex numbers, each with a pair of commuting involutions, are isomorphic to one another., Comment: 11 pages, AMS-LaTeX

We show that a recent result of Littlewood-Paley type, due to the author, is essentially best-possible. In a recent paper [W], the author proved Littlewood-Paley inequalities for certain finite linear sums f = Z1 AIq(I), defined on Rd. The summation here is indexed over D, the family of dyadic cubes in Rd; the A, are complex numbers. The functions 0(I), assumed to be smooth, belong to a family F that is "almost-orthogonal" and satisfies a mild decay condition. Precisely, we assume that there is an M > d/2 such that for all x E Rd and all I E X (1) kt(i)(X)| ? ?(I)IVq(I)(X)I ? III1/2(1 + IX-xiI/1(I))M. The notation is more or less standard: XI denotes I's center, ?(I) is its sidelength, and III is its Lebesgue measure (per tradition, we shall use JEJ to mean the Lebesgue measure of any measurable set E). We assume in addition that, for any finite linear sum Zi VI0(I), (2) I2 I ZVI0(I)1 dx d. For every a E A,, as above and 0 < p < oc, there is a constant C = C(M, d, p, a, b, p) such that, for all finite linear sums f = ZI AIq(I) from F, 2 ~~~~~~~p/2 If

In this paper we establish an upper bound for the Krull dimension of a module over a Weyl algebra in terms of a geometrical invariant of its char- acteristic variety, the involutive dimension. This is followed by some examples which show that this inequality may be strict. In this paper we show that the Krull dimension of a module over the Weyl algebra is related to geometrical invariants of the characteristic variety of that module. We begin with a survey of a few basic facts about modules over the Weyl algebra and their characteristic varieties. Let An denote the nth Weyl algebra over the field of complex numbers C. The generators of the algebra An will be denoted by xi and di = d/dxi for I gh} — g{f, h} + h{f, g}. It may be used to calculate the symbol of a bracket of two operators in A" as follows: If d £ A"(k) and d' £ An(m), then ak+m_2((d, d')) = {ak(d), am(d')}. For more details see (1) or (10). Let M be a finitely generated left ?("-module and F a good filtration of M. By that we mean a filtration of M for which %rF M is finitely generated over Sn . Let I(M) be the radical of the annihilator of grf M in S" . Note that I(M) is a homogeneous ideal of S" . The ideal I(M) is called the characteristic ideal of M and its variety char(AZ) in C2" , the characteristic variety of M. Both I(M) and char(Af ) are independent of the choice of the good filtration

For a compact Hausdorff space X X , let C ( X , H ) C(X,{\mathbf {H}}) denote the set of all quaternion-valued functions on X X . It is proved that if a real B ∗ {B^*} -algebra A A satisfies the following conditions: (i) the spectrum of every selfadjoint element is contained in the real line and (ii) every element in A A is normal, then A A is isometrically ∗ * -isomorphic to a closed ∗ * -subalgebra of C ( X , H ) C(X,{\mathbf {H}}) for some compact Hausdorff X X . In particular, a real C ∗ {C^*} -algebra in which every element is normal is isometrically ∗ * -isomorphic to a closed ∗ * -subalgebra of C ( X , H ) C(X,{\mathbf {H}}) .

We prove that if a weakly even sequence { c k : k = 0 , ± 1 , … } \{ {c_k}:k = 0, \pm 1, \ldots \} of complex numbers is such that for some p > 1 p > 1 we have \[ ∑ m = 1 ∞ 2 m / q ( ∑ k = 2 m − 1 2 m − 1 | Δ ( c k + c − k ) | p ) 1 / p > ∞ , 1 p + 1 q = 1 , \sum \limits _{m = 1}^\infty {{2^{m/q}}} {\left ( {\sum \limits _{k = {2^{m - 1}}}^{{2^m} - 1} {{{\left | {\Delta \left ( {{c_k} + {c_{ - k}}} \right )} \right |}^p}} } \right )^{1/p}} > \infty ,\frac {1}{p} + \frac {1}{q} = 1, \] then the symmetric partial sums of the trigonometric series ( ∗ ) ∑ k = − ∞ ∞ c k e i k x ( * )\sum \nolimits _{k = - \infty }^\infty {{c_k}{e^{ikx}}} converge pointwise, except possibly at x = 0 ( mod ⁡ 2 π ) x = 0(\operatorname {mod} 2\pi ) , to a Lebesgue integrable function, ( ∗ ) ( * ) is the Fourier series of its sum, and series ( ∗ ) ( * ) converges in L 1 ( − π , π ) {L^1}( - \pi ,\pi ) -norm if and only if lim | k | → ∞ c k ln ⁡ | k | = 0 {\lim _{|k| \to \infty }}{c_k}\ln |k| = 0 . In addition, we present new proofs of the theorems by J. Fournier and W. Self [6] and by ČC. V. Stanojević and V. B. Stanojević [10].

Differential equations of the form rp(z)q'(z) ― sp'(z)q(z) = γp(z) or rp(z)q'(z) ― sp'(z)q(z) = ―, where ― is a nonzero complex number and r, s are positive integers, have arisen in attempts to solve the two-variable Jacobian Conjecture. Solutions of such equations, in which p(z) and q(z) are monic complex polynomials of positive degrees r and s, give rise to extra-special pairs of polynomials in the sense of W. W. Stothers. Stothers showed that, modulo automorphisms of ℂ[z], there are only finitely many extra-special pairs of a given degree n. This implies that, modulo automorphisms of ℂ[z], there are only finitely many solutions of the above differential equations in which p(z) and q(z) are monic polynomials of given degrees r and s.

Let G be a subgroup of the additive group of real numbers and let C ⊆ G be infinite and convex in G. We show that G is definable in (R, +,.,C) and that Z is definable if G has finite rank. This has a number of consequences for expansions of certain o-minimal structures on the real field by multiplicative groups of complex numbers.

We use stratified Morse theory for a manifold with corners to give a new bound for the sum of the Betti numbers of a hypersurface in R^n_> defined by a polynomial with n+l+1 terms., 8 pages, 2 figures

A short proof is given of the elliptical range theorem concerning the numerical range of a 2× 2 complex matrix. Let A be a bounded linear operator acting on a Hilbert space. The numerical range (also known as the field of values) of A is the collection of complex numbers of the form (Ax, x) with x ranging through the unit vectors in the Hilbert space. As pointed out by many authors, the numerical range is very useful in studying operators and has applications to many subjects (e.g., see [AL], [H], [HJ], [I]). One of the basic properties, which is important in developing the theory, of numerical range is that the numerical range of an operator is always convex. There are many proofs of this interesting result, and many of them are done by reducing the problem to considering the numerical range of a 2×2 complex matrix, viewed as an operator acting on C. In this particular case, we have the following result, known as the Elliptical Range Theorem. Theorem. Let A be a 2×2 matrix with eigenvalues λ1 and λ2. The numerical range of A is an elliptical disk with λ1 and λ2 as foci, and {tr (A∗A) − |λ1| − |λ2|} as minor axis. Most proofs of this theorem are computational and are quite involved (e.g., see [D], [M], and [HJ, §1.3]). The purpose of this note is to present a simple proof of it. Proof of Theorem. Let W (A) be the numerical range of A. It is easy to verify that W (U∗AU) = W (A) for any unitary matrix U , and W (μA − ηI) = μW (A) − η for any μ, η ∈ C. These types of transformations will not change the conclusion of the theorem, and will be used in our proof. If A is normal, then A is unitarily similar to a diagonal matrix D = diag (λ1, λ2). One readily checks that W (A) = W (D) = {λ1|x1| + λ2|x2| : x1, x2 ∈ C, |x1| + |x2| = 1} is a line segment, which can be viewed as a degenerate ellipse with λ1 and λ2 as foci, and 0 as the length of the minor axis. Received by the editors January 17, 1995. 1991 Mathematics Subject Classification. Primary 15A60.

An elementary new and simple proof of Erdos-Lax theorem is given which in essence involves no analysis. Let P(z) be a polynomial of degree n with Maxlz I= P(z)I = 1, then Max IP'(z)I < n. (1) Inequality (1) is an immediate consequence of S. Bernstein's theorem on the derivative of a trigonometric polynomial (for reference see [5]). Inequality (1) can be sharpened if we restrict ourselves to the class of polynomials having no zeros in Izi < 1. In fact, P. Erdos conjectured and later P. D. Lax [3] proved the following THEOREM A. If P(z) is a polynomial of degree n with Maxlzl = P(z)I = 1 and P(z) has no zeros in the disk IzI < 1, then Max I P'(z) I < n /2. (2) JzJ=l The result is best possible and equality in (2) holds for P(z) = (a + /3z n)/2, where lal = I,I8 = 1. For other proofs of Theorem A see [1], [2], and [4]. In this paper we give an apparently new proof of Theorem A, which in essence involves no analysis. The proof depends on the following lemma which is also of independent interest. LEMMA 1. If P(z) is a polynomial of degree n and zl, Z2, ... , zn are the zeros of z n+ a, where a I -1 is any nonzero complex number, then for any complex number t, n 1+a Zk tP'(t) = 1 a P(t) + na i P(tZk) ( 2 (3) PROOF OF LEMMA 1. Let t be an arbitrary complex number. Consider the function FJ(z) defined by FJ(z) = (P(tz) P(t))/(z 1). Then FJ(z) is a polynomial of degree < n 1, and therefore, by using Lagrange's interpolation formula with zl, Z2, ... , zn as the basic points of interpolation we can write FJ(z) as t (Z) = tJ(Zk~) +a k=1 nzkn 1 (z zk) Received by the editors May 14, 1979 and, in revised form, August 27, 1979. AMS (MOS) subject classifications (1970). Primary 30A06, 30A40; Secondary 26A82, 26A84.

with sequence of approximants {Fp} has property (A) means there exists a complex number v such that each of the sequences { F2p-1I and { F2p} contains an infinite subsequence convergent to v. Here, c denotes the complex number sequence { cp }I P and d denotes the complex number sequence { dp } p0. When we speak of F(c, d) convergent absolutely at least in the wider sense we shall mean that there exists a positive integer n such that the continued fraction

We consider the degenerate parabolic operator L [ u ] = γ B [ u ] − u t L[u] = \gamma B[u] - {u_t} on a domain D = Ω × ( 0 , T ] D = \Omega \times (0,T] where B [ u ] = Σ i , j = 1 n ( a i j ( x ) u x j ) x i B[u] = \Sigma _{i,j = 1}^n{({a_{ij}}(x){u_{{x_j}}})_{{x_i}}} and γ \gamma is an arbitrary complex number. Classically, γ = 1 \gamma = 1 and the real-valued matrix ( a i j ) ({a_{ij}}) is positive definite. We assume ( a i j ) ({a_{ij}}) is a real-valued symmetric matrix but not necessarily definite. We prove that any complex-valued function u which satisfies the inequality | L [ u ] | ⩽ c | u | |L[u]| \leqslant c|u| for some nonnegative constant c and vanishes initially as well as on the boundary of Ω \Omega must vanish on all of D. The theorem is particularly useful in studying uniqueness for many systems which are not parabolic.

Let M M be a ternary matroid, t ( M , x , y ) t\left ( {M,x,y} \right ) be its Tutte polynomial and d ( M ) d\left ( M \right ) be the dimension of the bicycle space of any representation of M M over GF ( 3 ) {\text {GF}}\left ( 3 \right ) . We show that, for j = e 2 i π / 3 j = {e^{2i\pi /3}} , the modulus of the complex number t ( M , j , j 2 ) t\left ( {M,j,{j^2}} \right ) is equal to ( 3 ) d ( M ) {\left ( {\sqrt 3 } \right )^{d\left ( M \right )}} . The proof relies on the study of the weight enumerator W C ( y ) {W_\mathcal {C}}\left ( y \right ) of the cycle space C \mathcal {C} of a representation of M M over GF ( 3 ) {\text {GF}}\left ( 3 \right ) evaluated at y = j y = j . The main tool is the concept of principal quadripartition of C \mathcal {C} which allows a precise analysis of the evolution of the relevant invariants under deletion and contraction of elements. Soit M M un matroïde ternaire, t ( M , x , y ) t\left ( {M,x,y} \right ) son polynôme de Tutte et d ( M ) d\left ( M \right ) la dimension de l’espace des bicycles d’une représentation quelconque de M M sur GF ( 3 ) {\text {GF}}\left ( 3 \right ) . Nous montrons que, pour j = e 2 i π / 3 j = {e^{2i\pi /3}} , le module du nombre complexe t ( M , j , j 2 ) t\left ( {M,j,{j^2}} \right ) est égal à ( 3 ) d ( M ) {\left ( {\sqrt 3 } \right )^{d\left ( M \right )}} . La preuve s’appuie sur l’étude de l’énumérateur de poids W C ( y ) {W_\mathcal {C}}\left ( y \right ) de l’espace des cycles C \mathcal {C} d’une représentation de M M sur GF ( 3 ) {\text {GF}}\left ( 3 \right ) pour la valeur y = j y = j . L’outil essentiel est le concept de quadripartition principale de C \mathcal {C} qui permet une analyse précise de l’évolution des invariants concernés relativement à la suppression ou contraction d’éléments.

We prove that the hairs of the exponential-like maps f (z) = Aez are smooth curves. This answers affirmatively a question of Devaney and Krych. The proof is constructive in the sense that a dynamically defined CO' parametrization is presented. 0. Introduction. The study of the dynamical behaviour of the complex exponential map was begun by Fatou in 1926, following the work of Julia and Fatou himself concerning the dynamics of the rational maps on the sphere. Recently Devaney and Krych [DK] were able to give a symbolic description of that behaviour, by introducing the idea of "itinerary of a complex number under the action of exp" (see ?1 or [DK] for the definition). This led them to define "hair of exp associated to a given itinerary" as, roughly, the set of complex numbers sharing that itinerary and having fastly growing iterates. The hairs turn out to be quite simple sets (in particular they are curves) and have very simple dynamical properties. In fact these properties may be used to define "hair of an entire transcendental function" and it turns out that hairs really exist for a very large class of such functions (see [DT]). This, naturally, makes them an important tool for understanding the dynamics of these functions and increases the interest in their study. In this note we complement the description of the hairs of exp given in [DK], proving that they are C??-differentiable curves. To do this we construct in ?2 a parametrization / of a hair as the uniform limit of a sequence (/3n,O)n and, in ?3, we prove / to be COO by showing that all the sequences of derivatives ( (k) are also uniformly convergent. As far as we know it is still an open question whether the hairs are or are not analytic. We do not even know if / is an analytical parametrization. I would like to thank A. Douady for suggesting me this problem and for helpful conversations on the subject. 1. Definitions and results. Let A E C {O} and take A = peio, with p > 0 and -7r < 0 < 7r. Let f(z) = Aez, z E C and g(x) = pex, x E R. Take, for s E Z, B(s) = {z (2s 1)7r < Imz + 0 < (2s + 1)7r}. Since fIB(8) is a diffeomorphism, denote fs = (fIB(s))1 Received by the editors March 29, 1987 and, in revised form, July 8, 1987. A short version of this paper (in Portuguese) was presented to the VII Congress of the Group of Mathematicians of Latin Expression, held at the Universidade de Coimbra/Portugal in September 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 30D05, 58F15.

The closure, W, of the set of values (Cx, x) when ||x|| = 1 is a closed convex set (Hausdorff, cf. [8, p. 34]). A complex number z will be said to belong to the interior of W if z is in W and if one of the following conditions holds: (i) If W is two-dimensional, then z does not lie on the boundary of W; (ii) If W is a line segment, then z is not an end point; (iii) W consists of z alone. It was shown in [4] that if A (or B) is normal, or even semi-normal, so that AA* -A *A is semi-definite, then 0 belongs to W, but is not necessarily in the interior of W. (That, for arbitrary A and B, in general 0 need not even belong to W was shown in [2].) In fact, if A =(aij) is defined by aj,+=1,aij=0 ifj$i+1, then C=AA*-A*A = (cij) is the self-adjoint matrix all elements of which are zero except cl =1. Consequently, C>0 with a spectrum consisting of X=0, 1; hence W is the segment 0

THEOREM. Suppose that A is a complex sequence, t> 0, m is a nonnegative integer, API t(p + 1 )m for each nonnegative integer p, and f is a function such that, for each complex number z with IzI

The determination of the coefficients is quite complicated (and, for the coefficient of n-4, incorrect). In the present paper we obtain Copson's series by a simpler method which yields an asymptotic expansion for Sn(z) valid for every complex number z except z= 1. For our principle result we require the following three lemmas concerning the function Sn(z) defined by (1) and the function Tn(z) defined by

The parameter p in (1) is required to be a positive odd integer and z is any complex number with 9T(z) >0. The parameters ae and f3 are real numbers restricted as follows: When p = 1 we must have 0 ? ce ? 1 and 0 1, Equation (1) is valid for 0 1, it yields a transformation formula for the Lambert series

in which a and b are complex numbers, transforms the formal development F= Ei,j aijxiyi into a power series Fa,b(t) with a nonvanishing radius of convergence, the series F= Ei I aijxiyil converges for sufficiently small I xj and I yI. The conclusion is, in other terms: F is the Taylor series of a function of two complex variables x, y which is analytic at the point (x=O, y=O). M. A. Zorn suggested, and Rimhak Ree [2] proved recently, that it is sufficient to consider, in the hypothesis, the substitutions Sab with a and b real. This result is however incomplete: we are here resolving the following problem: what are the minimal conditions which must be imposed on a class [Sab] of substitutions (1.1) transforming F into a convergent power series Fa,b(t), in order that the preceding conclusion hold for any given formal development F? It is convenient, for our purpose, to introduce the following definition. DEFINITION. The class [Sa,b] of substitutions (1.1) is called normal if the convergence of the power series Fa,b(t) for every SabE [Sa,b] implies for any given formal development F to be the Taylor series of an analytic function of x, y. The following remarks will be useful: if X is a complex number different from zero, and a ==Xa, j3 =Xb, then the series Fao(t) and Fab(t) are simultaneously convergent or divergent; the substitution S00 is degenerate and we may suppose that it is excluded from [Sa,b]. We consider therefore the correspondences

Introduction. The title refers to analytic functions s(x) which behave like nonconstant logarithmic monomials M(x) = cxmo(log x)ml ... (log, x)mr (where c is a complex number 0, the mi are real, and logN is the N-fold iterate of the principal determination of log) in the sense that s(x)/M(x)-*1 as x-* oo in the complex plane. DEFINITION. An analytic function E is said to -3O rapidly enough for M if E- O and (M/M')E'- O as x- oo. Theorem 2 states that if s = M(1 +E) where E-*O rapidly enough for M, then s is 1-1 in some neighborhood of infinity. The neighborhood bases for oo with which we shall be concerned are families F(a, ,B) whose elements are sector-like regions V(ax, ,B, t) defined as follows: Let -r?a<

The closed invariant subspaces of multiplication by z in H2 were determined by Beurling [1, Theorem IV, p. 253]. Vector generalizations of this theorem are known (Halmos [3] and the author [6]), but they involve an unnecessary use of analysis. We can now prove the theorem of [6] by purely algebraic and geometric methods. To emphasize these methods, we work with sequences, which we write as formal power series, rather than functions analytic in the unit disk. Let e be a Hilbert space with elements denoted by a, b, c, and with norm j * |. If b is a vector in C, then b is the linear functional on C such that ba = (a, b) for every a in C. A formal power series is a sequence (ao, a,, a2, * * ) written f(z) = Janzn with an indeterminate z. Let f(z) = 2anZn and g(z) = EbnZn be formal power series with coefficients an and bn in C; let B(z) = EBnzn be a formal power series whose coefficients Bn are (bounded) operators in C; let a be a complex number, and let c be a vector in C. Then f(z) +g(z), af(z), zf(z), and B(z)f(z) are the formal power series Z(an +bn)zn, (aan)Zn, (jan)zn, and ( X_0 Bkan-k)Zn, respectively. A sequence (fk(z)) of formal power series with coefficients in C is said to be formally convergent if, for each n = 0, 1, 2, * * * , the corresponding sequence of nth coefficients is convergent. Let C(z) be the Hilbert space of formal power series f(z) = anZn with coefficients an in C, such that

for'} =0,1, 2,) This paper is concerned with the determination of those generalized Hausdorff matrices which sum all absolutely convergent sequences. Theorem 2 gives a sufficiency condition stated in terms of the behavior of the points of continuity of the graph of g and Theorems 3 and 4 provide examples which serve to further describe functions which generate such matrices. Theorem 5 gives a necessary and sufficient condition in terms of moment sequences. Theorem 1 gives conditions which are necessary and sufficient for a semi-infinite complex number matrix to sum all absolutely convergent sequences and although the theorem is known [31, a brief proof is included here for completeness since the author does not know of one in the literature.

Must a sequence { z k } \{ {z_k}\} of complex numbers be identically zero if ∑ f ( z k ) = 0 \sum {f({z_k}) = 0} for every entire function f f vanishing at the origin? Lenard’s example of a nonzero sequence of complex numbers whose power sums ("moments") all vanish is shown to give a negative answer to this question and to lead to a novel representation theorem for entire functions. On the positive side it is proved that if { z k } \{ {z_k}\} is in l p {l^p} where p > ∞ p > \infty , then vanishing moments imply { z k } \{ {z_k}\} is identically zero. Virtually the same proof shows that, on a Hubert space, two compact normal operators A A and B B with trivial kernels are unitarily equivalent if some power of each belongs to the trace class and tr ⁡ ( A n ) = tr ⁡ ( B n ) \operatorname {tr}({A^n}) = \operatorname {tr}(B^n) for all n n in a set of positive integers with asymptotic density one.

The sequence is said to be H(ck) evaluable to L if yn-*L as n-oo. The transformation H(Q) is regular (i.e., lim xn=L implies lim yn=L) if and only if c(0+)=0 and c(1)=1. If x is a complex number sequence, then C(x) denotes the set of limit points of x. Among the regular Hausdorff transformations are the methods C, (Re r>0) of Cesaro, Hr (Re r>0) of H 6lder and Er (0

The purpose of this note is to give a simplified proof of a theorem of Gelfand and Silov in the theory of normed, commutative rings. Let K be a complex Banach space which is also a commutative ring with unit element e, the norm being subject to the conditions ||ell = 1 and I I xy| I < I I X| I I YI I for all x and y in K. Let 9M be the space of maximal ideals of K. Then, for each element xeK, and each maximal ideal MGC_, there is a unique complex number x(M) defined by

Let f ( z ) f(z) be a normalized function, analytic and univalent in the open unit disc. It is shown that if g ( z ) = ∫ 0 z ( f ( t ) / t ) α d t g(z) = \int _0^z {{{(f(t)/t)}^\alpha }dt} , then g g is univalent in the open unit disc if α \alpha is a complex number satisfying 0 ≦ | α | ≦ ( √ 2 − 1 ) / 4 0 \leqq |\alpha | \leqq (\surd 2 - 1)/4 .

The singular integral characterization of H' on simple martingales was given by S. Janson. We show that his result cannot be extended to lIP if p ( > 0) is very small. Let Q = (0, 1]. Let F be the a-field of all Borel sets in Q. Let dx be the Lebesgue measure. Then (Q, F, dx) is a probability space. Let d > 3 be an integer. For each integer n > 0, let F,, be the sub-a-field of F generated by ((k )d-n, kd-n], k = 1,...,d'. Set I(kl1..k,k) ((k, -)d-' + +(kn1I)d-1 +(k, -I)d(k I)d-' + +(knI)d'+ knd-n] foreachk . k,iE{1. d). A martingale is a sequence of complex-valued integrable functions {fn})3?0 such that E[fln+ 1 F,1] = , where E I F,,] denotes the conditional expectation with respect to the sub-a-field F,. We writef for {f n)}0. Iff is generated from a function f(X) E L'(Q) by (1I) fn = E[f I Fn] we identify f and f. For a martingalef we define f*(x) = sup Ifn(x). n O For p E (0, oo), f is said to belong to HP if f* 11 p + x, where IIf IIP f*(X) dx It is well known that if p > 1, then HP and LP(Q) can be identified. That is, f* E LP(Q) if and only if there exists a function f(x) E LP(Q) such that (1) and CPjIf*jj IIP1SpII 0, hold. It is also known that if f* E L'(Q), thenf is generated from an L'-function but that the converse is not true. Received by the editors September 9, 1982 and, in revised form, November 30, 1982. 1980 Mathematics Subject Classification. Primary 42B30; Secondary 46J 15. 1 Research partly supported by NSF MCS-82033 19 and Science Research Foundation of Japan. (C)1983 American Mathematical Society 0002-9939/83/0000-1 255/$02.25 617 This content downloaded from 157.55.39.138 on Sun, 26 Jun 2016 05:32:58 UTC All use subject to http://about.jstor.org/terms 618 AKIHITO UCHIYAMA For n > 1, set A fn = f, f, _. Since Afn is F11-measurable, the notation f\n(I(kI, * . .,k,,)) makes sense. Let Z ={ T(X = )kI E~ C : E X k = 0, where (xk )k=d denotes a d-dimensional column vector and C is the set of all complex numbers. Note that (L\f,(I(k ,...,k,_1, k)))d= E V. Let A be a linear operator from V to V. Set (Agn( I(k 1 * ,kn k )))d kI= = A(AfjslI(kj,. . .,k*_ k )))dk1