This article examines Gene x Environment (G x E) interactions in two comorbid developmental disorders--reading disability (RD) and attention-deficit/hyperactivity disorder (ADHD)--as a window on broader issues on G x E interactions in developmental psychology. The authors first briefly review types of G x E interactions, methods for detecting them, and challenges researchers confront in interpr...

Pick d+1 points uniformly at random on the unit sphere in Rd. What is expected value of angle sum simplex spanned by these points? Choose n d-dimensional ball. number faces their convex hull? We answer and some other, seemingly unrelated, questions stochastic geometry. To this end, we compute internal angles simplices whose vertices are independent sampled from one following distributions: (i) ...

This paper considers the approximate reconstruction of points, ~x ∈ RD, which are close to a given compact d-dimensional submanifold, M, of RD using a small number of linear measurements of ~x. In particular, it is shown that a number of measurements of ~x which is independent of the extrinsic dimension D suffices for highly accurate reconstruction of a given ~x with high probability. Furthermo...

s — A generalized process (XÁ; ¿ 2 E), where E is a nuclear space, is a random variable family such that the map ¿ 7! XÁ is linear and continuous. The Gaussian white noise process is a wellknown example. It is characterized by a Gaussian measure on the dual space E 0 ofE . Ordinary Gaussian processes can be constructed from the white noise process using the duality relation: (X Á = h_;¿ xi; x 2...

Let μ = (μ1, . . . , μd) be such that each μi is a signed measure on Rd belonging to the Kato class Kd,1. The existence and uniqueness of a continuous Markov process X on Rd, called a Brownian motion with drift μ, was recently established by Bass and Chen. In this paper we study the potential theory of X. We show that X has a continuous density qμ and that there exist positive constants ci, i =...

In the first part of the paper we establish the pointwise convergence as t → +∞ for convolution operators ∫ Rd tdK (ty)φ(x− y)dy under the assumptions that φ(y) has integrable derivatives up to an order α and that |K(y)| ≤ c (1 + |y|)−β with α+β > d. We also estimate the Hausdorff dimension of the set where divergence may occur. In particular, when the kernel is the Fourier transform of a bound...

We introduce a Fourier-based harmonic analysis for a class of discrete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space X comes with a finite-to-one endomorphism r : X → X which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in...

We prove that the L2([0, 1]d × SO(d))-norm of periodizations of a function from L1(Rd) is equivalent to the L2(Rd)-norm of the function itself in higher dimensions. We generalize the statement for functions from Lp(Rd) where 1 ≤ p < 2d d+2 in the spirit of the Stein-Tomas theorem. 0. Introduction. Let f be a function from L(R). Define a family of its periodizations with respect to a rotated int...

For a system of functional differential equations of an arbitrary order the conditions are established for the initial value problem to be solvable on an infinite interval, and the structure of the set of solutions to this problem is studied. Introduction For the p-th order functional differential system x(p)(t) = f(t, xt, . . . , x (p−1) t ), b ≤ t < +∞, (1) we consider the initial value probl...

Let (X,Z) be a continuous random vector in R × Rd, d ≥ 1. In this paper, we define the notion of a nonparametric residual of X on Z that is always independent of the predictor Z. We study its properties and show that the proposed notion of residual matches with the usual residual (error) in a multivariate normal regression model. Given a random vector (X,Y,Z) in R × R × Rd, we use this notion o...