Using a direct approach the return map near a focus of a planar vector field with nilpotent linear part is found as a convergent power series which is a perturbation of the identity and whose terms can be calculated iteratively. The first nontrivial coefficient is the value of an Abelian integral, and the following ones are explicitly given as iterated integrals.

Convolution type Calderón-Zygmund singular integral operators with rough kernels p.v. Ω(x)/|x| are studied. A condition on Ω implying that the corresponding singular integrals and maximal singular integrals map L → L for 1 < p < ∞ is obtained. This condition is shown to be different from the condition Ω ∈ H1(Sn−1).

Some Rational elliptic curves whose modular parametrization is given by an Eichler integral were considered. The points, other than cusps, that map to zero under modular parametrization were studied computationally. Surprisingly, these zeros appear to be CM-points. This paper is organized under the following section headings:

We analyze admissibility and exactness of observation operators arising in control theory for Volterra integral equations. We give a necessary and sufficient criterion for an unbounded observation operator to map a solution into L2. We then discuss the Hautus Lemma, giving a partial result and an example where it fails.

Consider Hill’s operator Q = − d2 dx2 + q(x) in which the potential q(x) is an almost surely continuous and rotation invariant Gaussian process on the circle x ∈ [0, 1). Viewing the classical Riccati map as a change of measure, we establish functional integral formulas for the probability density function of the ground state energy and also determine the density’s shape.

Spatial scaling is an integral aspect of many spatial tasks that involve symbol-toreferent correspondences (e.g., map reading, drawing). In the present study, we asked 3to 6year-old children and adults to locate objects in a two-dimensional spatial layout using information from a second spatial representation (map). We examined how scaling factor and reference features, such as the shape of the...

Convolution type Calderr on-Zygmund singular integral operators with rough kernels p.v. (x)=jxj n are studied. A condition on implying that the corresponding singular integrals and maximal singular integrals map L p ! L p for 1 < p < 1 is obtained. This condition is shown to be diierent from the condition 2 H 1 (S n?1).

We study the covering dimension of (positive ) solutions to varoius classes of nonlinear equations based on the nontriviality of the fixed point index of a certain condensing map. Applications to semilinear equations and to nonlinear perturbations of the Wiener-Hopf integral equations are given.

We prove that the classical integral cycle class map from algebraic cycles to étale cohomology factors through a quotient of `-adic étale cobordism over an algebraically closed field of positive characteristic. This shows that there is a strong topological obstruction for cohomology classes to be algebraic and that examples of Atiyah, Hirzebruch and Totaro also work in positive

We present here a canonical description for quantizing classical maps on a torus. We prove theorems analagous to classical theorems on mixing and ergodicity in terms of a quantum Koopman space L (A~, τ~) obtained as the completion of the algebra of observables A~ in the norm induced by the following inner product (A,B) = τ~ ( A†B ) , where τ~ is a linear functional on the algebra analogous to t...